Design Explorations2.bitdownload.ir/Engineering/ALL.ANSYS/ANSYS 12... · To view the ANSYS...

Design Exploration

Release 12.0ANSYS, Inc.April 2009Southpointe

275 Technology Drive ANSYS, Inc. iscertified to ISO9001:2008.

Canonsburg, PA [email protected]://www.ansys.com(T) 724-746-3304(F) 724-514-9494

Copyright and Trademark Information

© 2009 SAS IP, Inc. All rights reserved. Unauthorized use, distribution or duplication is prohibited.

ANSYS, ANSYS Workbench, Ansoft, AUTODYN, EKM, Engineering Knowledge Manager, CFX, FLUENT, HFSS and any andall ANSYS, Inc. brand, product, service and feature names, logos and slogans are registered trademarks or trademarksof ANSYS, Inc. or its subsidiaries in the United States or other countries. ICEM CFD is a trademark used by ANSYS, Inc.under license. CFX is a trademark of Sony Corporation in Japan. All other brand, product, service and feature namesor trademarks are the property of their respective owners.

Disclaimer Notice

THIS ANSYS SOFTWARE PRODUCT AND PROGRAM DOCUMENTATION INCLUDE TRADE SECRETS AND ARE CONFIDENTIALAND PROPRIETARY PRODUCTS OF ANSYS, INC., ITS SUBSIDIARIES, OR LICENSORS. The software products and document-ation are furnished by ANSYS, Inc., its subsidiaries, or affiliates under a software license agreement that contains pro-visions concerning non-disclosure, copying, length and nature of use, compliance with exporting laws, warranties,disclaimers, limitations of liability, and remedies, and other provisions. The software products and documentation maybe used, disclosed, transferred, or copied only in accordance with the terms and conditions of that software licenseagreement.

ANSYS, Inc. is certified to ISO 9001:2008.

U.S. Government Rights

For U.S. Government users, except as specifically granted by the ANSYS, Inc. software license agreement, the use, du-plication, or disclosure by the United States Government is subject to restrictions stated in the ANSYS, Inc. softwarelicense agreement and FAR 12.212 (for non-DOD licenses).

Third-Party Software

See the legal information in the product help files for the complete Legal Notice for ANSYS proprietary software andthird-party software. If you are unable to access the Legal Notice, please contact ANSYS, Inc.

Published in the U.S.A.

Table of Contents

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Introduction to Design Exploration .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Available Tools ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ANSYS Design Exploration: a Typical Session, What to Look For? .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Initial Simulation: Analysis of the Product Under all Operating Conditions .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Identify Design Candidates .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Assess the Robustness of the Design Candidates .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Determine the Number of Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Increase the Response Surface Accuracy .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Systems Support ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The User Interface .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Design Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Response Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Workflow .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Adding Design Exploration Templates to the Schematic ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Duplicating Existing Design Exploration Systems .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Running Design Exploration Analyses .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Monitoring Design Exploration Analyses .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Design Exploration Options .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Using Design Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

What is Design Exploration? .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Design of Experiments .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Parameters Parallel Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Design Points vs Parameter Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Parameters Correlation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Correlation Scatter Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Correlation Matrix ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Sensitivities Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Response Surface .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Response Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Local Sensitivity Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Spider Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Goal Driven Optimization .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Tradeoff Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Samples Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Sensitivities Chart (GDO) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Six Sigma Analysis ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Design of Experiments (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Six Sigma Analysis ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Sensitivities Chart (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Working with Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Input Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Setting Up Design Variables .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Setting Up Uncertainty Variables .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Output Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Design Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33The Design of Experiments Solution .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Design Criteria Considered in DOE .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34DOE Using a Central Composite Design .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

iiiRelease 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information

of ANSYS, Inc. and its subsidiaries and affiliates.

Upper and Lower Locations of DOE Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38DOE Matrix Generation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Using Response Surfaces .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Meta Model Types .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Standard Response Surface - Full 2nd-Order Polynomials ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Kriging .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Non-Parametric Regression .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Suggested Best Practices .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Neural Network .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Meta Model Refinement .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Change Meta Model Types .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Refine the Model Using the Kriging Meta Model ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Goodness of Fit ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Response Surface Charts ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Using the Response Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Using the Spider Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Using the Local Sensitivity Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Global Sensitivities ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Adding Design Points and Response Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Using Parameters Correlation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Running a Parameters Correlation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Purpose of Parameters Correlation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Sample Generation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Determining Significance .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Viewing Significance and Correlation Values .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Using Goal Driven Optimization .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Optimization Options .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Screening .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56MOGA ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57NLPQL .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Defining Design Objectives .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Creating New Design Points and Response Points from Candidates .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Goal Driven Optimization Charts ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Using the Sensitivities Chart (GDO) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Using the Tradeoff Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Using the Samples Chart ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Using Six Sigma Analysis ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Performing a Six Sigma Analysis ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Using Statistical Postprocessing .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Tables (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Using Parameter Charts (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Using the Sensitivities Chart (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Statistical Measures .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Understanding Goal Driven Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Principles (GDO) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Guidelines and Best Practices (GDO) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Postprocessing GDO Results ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Postprocessing Overview .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Understanding Sensitivities Charts ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Understanding Tradeoff Charts ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Understanding Samples Charts ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Goal Driven Optimization Theory .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential informationof ANSYS, Inc. and its subsidiaries and affiliates.iv

Design Exploration

Shifted Hammersley Sampling Method .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Pareto Dominance in Multi-Objective Optimization .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72MOGA (Multi-Objective Genetic Algorithm) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Decision Support Process .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Non-linear Programming by Quadratic Lagrangian (NLPQL) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Understanding Six Sigma Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Principles (SSA) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Guidelines for Selecting SSA Variables .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Choosing and Defining Uncertainty Variables .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Uncertainty Variables for Response Surface Analyses .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Choosing a Distribution for a Random Variable .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Measured Data .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Mean Values, Standard Deviation, Exceedance Values .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90No Data .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Distribution Functions .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Sample Generation .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Weighted Latin Hypercube Sampling .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Postprocessing SSA Results ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Histogram ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Cumulative Distribution Function .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Probability Table .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Statistical Sensitivities in a Six Sigma Analysis ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Six Sigma Analysis Theory .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Glossary of General Terms .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Index .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vRelease 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information


Design Exploration

Overview

The following links provide quick access to information concerning Design Exploration and its use:

• Using Design Exploration Help (p. 1)

• Systems Support (p. 4)

• What is Design Exploration? (p. 17)

• The User Interface (p. 5)

• Parameters (p. 7)

• Design Points (p. 8)

• Workflow (p. 9)

• Design Exploration Options (p. 11)

• "Using Design Exploration" (p. 17)

Using Design Exploration Help

Design Exploration Help is included as part of the ANSYS Workbench Help, where you can also find an ex-planation of general Workbench features. To view the ANSYS Workbench Help, see Getting Started in ANSYSWorkbench.

Introduction to Design Exploration

Because a good design point is often the result of a trade-off between various objectives, the explorationof a given design cannot be performed by using only direct optimization algorithms leading to a singledesign point. It is important to gather enough information about the current design so as to be able to answerthe so-called “what-if” questions – quantifying the influence of design variables on the performance of theproduct in an exhaustive manner. By doing so, the right decisions can be made based on accurate information- even in the event of an unexpected change in the design constraints.

ANSYS Design Exploration describes the relationship between the design variables and the performance ofthe product by using Design of Experiments (DOE), combined with Response Surfaces. DOE and ResponseSurfaces provide all of the information required to achieve Simulation Driven Product Development. Oncethe variation of the performance with respect to the design variables is known, it becomes easy to understandand identify all changes required to meet the requirements for the product. Once the Response Surfacesare created, the information can be shared in easily understandable terms: curves, surfaces, sensitivities, etc.They can be used at any time during the development of the product without requiring additional simulationsto test a new configuration.

Available Tools

ANSYS Design Exploration software has a powerful suite of DOE schemes: Central Composite Design (CCD)or Optimal Space-Filling. CCD provides a traditional DOE sampling set, while Optimal Space-Filling’s objectiveis to gain the maximum insight with the fewest number of points. This feature is very useful when thecomputation time available to the user is limited.

1Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information


After sampling, ANSYS Design Exploration software provides four different meta models to represent thesimulation’s responses: Full Second-Order Polynomial, Kriging, Non-Parametric Regression and Neural Network.These meta-models can accurately represent highly nonlinear responses such as those found in high frequencyelectromagnetics.

Once the simulation’s responses are characterized, ANSYS Design Exploration supplies three different typesof optimization algorithms: Screening (Shifted Hammersley), Multi-Objective Genetic Algorithm (MOGA) andNon-linear Programming by Quadratic Lagrangian (NLPQL).

Several graphical tools are available to investigate a design: sensitivity plots, correlation matrices, curves,surfaces, trade-off plots and parallel charts with Pareto Front display, and spider charts.

Correlation matrix techniques are also provided to help the user identify the key parameters of a designbefore creating response surfaces.

ANSYS Design Exploration: a Typical Session, What to Look For?

The main purpose of ANSYS Design Exploration is to identify the relationship between the performance ofthe product (maximum stress, mass, fluid flow, velocities, etc.) and the design variables (dimensions, loads,material properties, etc.). Based on these results, the analyst will be able to influence the design so as tomeet the product’s requirements. He will be able to identify the key parameters of the design and how theyinfluence the performance.

ANSYS Design Exploration does provide tools to analyze a parametric design with a reasonable number ofparameters. The response surface methods described here are suitable for problems using about 10 to 15input parameters.

Initial Simulation: Analysis of the Product Under all Operating Conditions

The first step of any design simulation is to create the simulation model. The simulation model can useanything from a single physics up to a complex multiphysics simulation involving multiple conditions andphysics coupling.

In addition to performing the standard simulation, this step is also used to define the parameters to be in-vestigated. The input parameters (also called design variables) are identified, and may include CAD parameters,loading conditions, material properties, etc.

The output parameters (also called performance indicators) are chosen from the simulation results and mayinclude maximum stresses, fluid pressure, velocities, temperatures, or masses, and can also be custom defined.Product cost could be a custom defined parameter based on masses, manufacturing constraints, etc.

CAD parameters are defined from a CAD package or from ANSYS DesignModeler. Material properties arefound in the Engineering Data section of the project, while other parameters will have their origin in thesimulation model itself. Output parameters will be defined from the various simulation environments(mechanics, CFD, etc.). Custom parameters are defined directly in the Parameter Set (the bus bar that collectsall parameters) available from the project schematic.

Identify Design Candidates

Once the initial model has been created and parameters defined, the next step in the session is to create aresponse surface. After inserting a Response Surface system in the project, you need to define the designspace by giving the minimum and maximum values to be considered for each of the input variables. Basedon this information, the Design of Experiment (DOE) part of the Response Surface system will create the

Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential informationof ANSYS, Inc. and its subsidiaries and affiliates.2

Overview

design space sampling. Note that this sampling depends upon the choice made for the DOE scheme –usually, the default CCD scheme will provide good accuracy for the final approximation. Then, the DOE needsto be computed. All points will be sequentially computed.

Once the DOE has been updated, a response surface is created for each output parameter. A response surfaceis an approximation of the response of the system. Its accuracy depends on several factors: complexity ofthe variations of the output parameters, number of points in the original DOE, and choice of the responsesurface type. Four main types of response surfaces are available in ANSYS Design Exploration. As a startingpoint, the Standard Response Surface (based on a modified quadratic formulation) will provide satisfyingresults when the variations of the output parameters is mild, while the Kriging scheme will be used forstronger variations.

After the response surfaces have been computed, the design can be thoroughly investigated using a varietyof graphical and numerical tools, and valid design points identified by optimization techniques.

Usually, the investigation will start with the sensitivity graphs. This bar or pie chart will graphically showhow much the output parameters are locally influenced by the input parameters around a given responsepoint. Note that varying the location of the response point may provide totally different graphs. Thinkingof the hill/valley analogy, if the response point is in a flat valley, the influence of the input parameters willbe small. If the point is at the top of a steep hill, the influence of the parameters will be strong. The sensit-ivity graphs provide the first indication about the relative influence of the input parameters.

The response surfaces will provide curves or surfaces that show the variation of one output parameter withrespect to one or two input parameters at a time. These curves/surfaces also are dependent on the responsepoint.

Both sensitivity charts and response surfaces are key tools for the analyst to be able to answer the “What-if” questions (“What parameter should we change if we want to reduce the cost?”).

ANSYS Design Exploration provides additional tools to identify design candidates. While they could be de-termined by a thorough investigation of the curves, it might be convenient to be guided automatically tosome interesting candidates. Access to optimization techniques that will find design candidates from theresponse surfaces is provided by the Goal Driven Optimization (GDO) systems. These systems can be draggedand dropped over an existing response surface system so as to share this portion of the data. Several GDOsystems can be inserted in the project, which is useful if several hypothesis are to be analyzed.

Once a GDO system has been introduced, the optimization cell needs to be defined, which includes choosingthe optimization method and setting the objectives. Then the optimization problem can be solved. In manycases, there will not be a unique solution to the optimization problem, and several candidates will be iden-tified. The results of the optimization process is also very likely to provide candidates that cannot be manu-factured (a radius of 3.14523 mm is probably hard to achieve!). But since all information about the variabilityof the output parameters is provided by the response surface, it becomes easy to find a design candidateclose to the one indicated by the optimization process that will be acceptable.

As a good practice, it is also recommended to check the accuracy of the response surface for the designcandidates. To do so, the candidate should be converted to a design point and a full simulation performedfor that point, so as to check the validity of the output parameters.

Assess the Robustness of the Design Candidates

Once one or several design points have been identified, the probabilistic analysis will help quantify the reli-ability or quality of the product by means of a statistical analysis. Probabilistic analysis typically involves fourareas of statistical variability: geometric shape, material properties, loading, and boundary conditions. Forexample, the statistical variability of the geometry of a product would try to capture product-to-product



Assess the Robustness of the Design Candidates

differences due to manufacturing imperfections quantified by manufacturing tolerances. Probabilistic char-acterization provides a probability of success or failure and not just a simple yes/no evaluation. For instance,a probabilistic analysis could determine that one part in 1 million would fail, or the probability of a productsurviving its expected useful life.

Inserting a Six-Sigma system in the schematic will provide the robustness analysis. The process here is verysimilar to the response surface creation. The main difference is in the definition of the parameters: insteadof giving minimum and maximum values defining the parameter's ranges, the parameters will be describedin terms of statistical curves and their associated parameters. Once the parameters are defined, a new DOEwill be computed, as well as the corresponding response surfaces. Additional results in the form of probab-ilities and statistical distributions of the output parameters are provided, along with the sensitivity chartsand the response surfaces.

Determine the Number of Parameters

To get accurate response surfaces within a reasonable amount of time, the number of input parametersshould be limited to 10 to 15. If more parameters need to be investigated, the correlation matrix will providea way to identify some key parameters before creating the response surface. The Parameters Correlationsystems should be used prior to the Response Surface systems, to reduce the number of parameters to theabove mentioned limit. The parameters correlation method will perform simulations based on a randomsampling of the design space, so as to identify the correlation between all parameters. The number of sim-ulations will depend upon the number of parameters, as well as the convergence criteria for the means andstandard deviations of the parameters. The user can provide a hard limit for the number of points to becomputed—it should be noted that the accuracy of the correlation matrix might be affected if not enoughpoints are computed.

Increase the Response Surface Accuracy

Usually, a richer set of points for the DOE will provide a more accurate response surface. However, it maynot be easy to know before the response surface is created how many additional points are required. TheKriging response surface will allow ANSYS Design Exploration to determine the accuracy of the responsesurface as well as the points that would be required to increase this accuracy. To use this feature, the responsesurface type should be set to Kriging. The refinement options will then be available to the user. The refinementcan be automated or user-defined. This last option should be used to control the number of points to becomputed.

Systems Support

Design Exploration is only supported on the Windows platform. A complete listing of operating system re-quirements is listed in the ANSYS, Inc. Installation Guide for Windows.

Limitations

• The Design Exploration environment is not localized.

• Suppressed properties are not available for the Design of Experiments (DOE) method.

• Databases from the previous releases cannot be resumed in release 12.


Overview

Note

You cannot resume a DesignXplorer database that was created in a previous release of Workbench.You can resume a project that was created in an earlier version of Workbench, but any associatedDesignXplorer database will be ignored. However, all parameters originally defined by the projectwill still be available to Design Exploration, and the Design Exploration components can be usedto rebuild and re-solve the DesignXplorer analyses.

The User Interface

The Workbench User Interface allows you to easily build your project by adding systems from a Toolboxarea on the left of the window to the Project Schematic area on the right. Design Exploration systems areavailable in the Toolbox to allow you to perform the different types of parametric analyses on your projectas described in the previous section. Once you add the Design Exploration systems to your Project Schem-atic, you edit the cells in the systems to open the workspaces for those cells.

Project Schematic

You will initially create your project containing analysis systems, models, and other systems in the ProjectSchematic. Then you can choose which of the parameters in those systems will be exposed to the DesignExploration systems through the Parameter Set bar. Once the analysis systems and Parameter Set are defined,you can add the Design Exploration systems that you need. A project schematic containing Design Explorationsystems would look something like the following:

Toolbox

When you are viewing the Project Schematic, the following system templates are available from the DesignExploration Toolbox.



Toolbox

To perform a particular type of analysis, drag the template from the Toolbox onto your Project Schematicbelow the Parameter Set bar.

When you are viewing a workspace, the Toolbox will display items that can be added to the Outline basedon the currently selected cell. For example, if you are in a Response Surface workspace and you select aResponse Point cell in the outline, the Toolbox will contain chart templates that you can add to that ResponsePoint. Double click on template or drag it onto the selected cell in the Outline view to add it to the Outline.

Workspace

Once a Design Exploration system is added to the schematic, you can edit the cells to open the Workspace,which allows you to set up the analysis options, run the analysis, and view the results. For example, if youright click on the Design of Experiments cell and select Edit ..., your workspace will show the followingconfiguration:

You will see this window configuration for any Design Exploration system cell that you edit. The four panelsvisible are:

• Outline: provides a summary of all of the item that make up the cell you are editing.

• Table: provides a list of either the design points or response points associated with that cell.

• Properties: provides access to the properties of the various cells in the currently displayed Outline. Forexample, you can set parameter limits, analysis types, and chart options from the properties panel.

• Chart: displays the various charts available for the different Design Exploration system cells.


Overview

In general, you select a cell in the Outline view and either set up its Properties or review the Chart and/orTable associated with that cell.

Toolbar

When you are editing a Design Exploration cell, additional buttons are added to the Workbench toolbar:

• a button that allows you to update the particular object you are editing (for example, Update Design

of Experiments)

• for some objects, a button that allows you to preview the results of the update (for example, Preview

Design of Experiments)

Parameters

The following types of parameters are used in Design Exploration:

• Input Parameters (p. 7)

• Output Parameters (p. 8)

See Working with Parameters and Design Points for more information about parameters.

Input Parameters

Input parameters are those parameters that define the inputs to the analysis for the model under investigation.Input parameters have predefined ranges which can be changed. These include (and are not limited to) CADparameters, Analysis parameters, and DesignModeler parameters. CAD and DesignModeler input parametersmay include length, radius, etc.; Analysis input parameters may include pressure, material properties, mater-ials, sheet thickness, etc.

When you start a Design of Experiments analysis in Design Exploration, a default of +/- 10% of the currentvalue of each input parameter is used to initialize the parameter range. If any parameter has a current valueof 0.0, then the initial parameter range will be computed as 0.0 → 10.0. Since Design Exploration is not awareof the physical limits of parameters, you will need to check that the assigned range is compatible with thephysical limits of the parameter. Initially, the current value of the parameter within Design Exploration willbe the midpoint between the upper bound and lower bound.

Input parameters can be discrete or continuous, and each of these have specific forms.

ExampleDescriptionParameter Type

Number of holes, number ofweld points

Valid only at integer valuesDiscrete Parameters

Thickness, force, temperatureContinuous non-interruptedrange of values

Continuous Parameters

Drill bit sizes, readily availablebolts, screws

The theoretical values arecontinuous, but manufactur-

Continuous Parameters withUsability Constraints

ing or real-world constraintslimit actual values

Continuous parameters physically vary in a continuous manner between a lower and an upper bound definedby the user. Examples are CAD dimension or load magnitude.



Input Parameters

Discrete parameters physically represent different configurations or states of the model. An example is thenumber of holes in a geometry. You can set a parameter to be discrete by selecting Discrete as the Classi-

fication for the parameter in the Properties view. You would then enter the values for the Levels that youwant the parameter to have in the Table view for the parameter. Levels are limited to integer values andwill not be sorted as you enter them.

Usability parameters are continuous parameters for which a post-processing filter is defined. You can entera restricted list of levels that are the only values authorized for post-processing. You can change a continuousparameter to a usability parameter by selecting Usability as the Classification for the parameter in theProperties view. You would then enter the values for the Levels that you want the parameter to have in theTable view for the parameter. Levels are sorted numerically as you enter them. You are not restricted to onlyentering integers for Usability Levels. Once you have the levels defined, all of the analysis done on thesample set, from charts to optimization, will only use the usability levels. For instance, optimization won'tsuggest a design where the usability parameter has a value not contained in the list of levels.

Output Parameters

Output parameters are those parameters that result from the geometry or are the response outputs fromthe analysis. These include (and are not limited to) volume, mass, frequency, stress, heat flux, and so forth.

Derived Parameters

Derived parameters are specific output parameters defined as an analytical combination of output or inputparameters. As the definition suggests, derived parameters are calculated from other parameters by usingequations that you provide. They are created in the analysis system and passed into Design Exploration asoutput parameters. See Working with Parameters and Design Points for more information.

Design Points

A Design Point is defined by a snapshot of parameter values where output parameter values were calculateddirectly by a project update. Design Points are created by Design Exploration; for instance, when processinga Design of Experiments or a Correlation Matrix, or refining a Response Surface.

It is also possible to insert a Design Point at the project level from an optimization Candidate design, in orderto perform a validation update. Note that the output parameter values are not copied to the created DesignPoint since they were calculated by Design Exploration and are, by definition, approximated. Actual outputparameters will be calculated from the Design Point input parameters when a project update is done. SeeAdding Design Points and Response Points (p. 54) for more information on adding Design Points to the project.

Response Points

A Response Point is defined by a snapshot of parameter values where output parameter values were calculatedin Design Exploration from a Response Surface. As such, the parameter values are approximate and calculatedfrom response surfaces, and the most promising designs should be verified by a solve in the analysis systemusing the same parameter values.

Using Response Points

New Response Points can be created when editing a Response Surface cell from the Outline, or from theTable view. Response Points as well as Design Points can also be inserted from the Table view or the charts,using the appropriate right mouse button menu option when selecting a table row or a point on a chart.For instance, it is possible to insert a new Response Point from a point selected with the mouse on a Response


Overview

Chart. See Adding Design Points and Response Points (p. 54) for more information on creating custom ResponsePoints.

Workflow

To run a parametric analysis in Design Exploration, you will need to:

• create your model (either in DesignModeler or other CAD software) and load it into one of the Analysissystems available in Workbench

• select the parameters you are interested in

• add to your project the Design Exploration features that you want to use

• set the parameter limits and update the analysis

• view the results of the Design Exploration analysis

Adding Design Exploration Templates to the Schematic

You can use as many of the Design Exploration features as you like and rerun them with different parametersand limits as many times as you need to refine your design. The following steps outline the process forputting Design Exploration elements into your Project Schematic:

1. Create parts and assemblies in either the DesignModeler solid modeler or any of the supported CADsystems. Features in the geometry that will be important to the analysis should be exposed as para-meters. Such parameters can be passed to Design Exploration.

2. Drag an analysis system into the Project Schematic , connecting it to the DesignModeler or CAD file.

3. Click on the Parameter Set bar

4. In the Outline view, click on the various input parameters that you need to modify. Set the limits ofthe selected parameter in the Properties view.

5. Drag the Design Exploration template that you would like to use into the Project schematic below theParameter Set bar.

Duplicating Existing Design Exploration Systems

You can duplicate any Design Exploration systems that you have created on your schematic. The mannerin which you duplicate them dictates the data that is shared between the duplicated systems.

• For any Design Exploration system, right click on the B1 cell of the system and select Duplicate. A newsystem of that type will be added to the schematic under the Parameter Set bar. No data is shared withthe original system.

• For a Design Exploration system that has a DOE cell, click on the DOE cell and select Duplicate. A newsystem of that type will be added to the schematic under the Parameter Set bar. No data is shared withthe original system.

• For a Design Exploration system that has a Response Surface cell, click on the Response Surface celland select Duplicate. A new system of that type will be added to the schematic under the ParameterSet bar. The DOE data will be shared from the original system.

• For a GDO or SSA Design Exploration system, if you click on those cells and select Duplicate, a newsystem of that type will be added to the schematic under the Parameter Set bar. The DOE and ResponseSurface data will be shared from the original system.



Duplicating Existing Design Exploration Systems

Any cell in the duplicated Design Exploration system that contains data that is not shared from the originalsystem will be marked as Update Required.

Running Design Exploration Analyses

After you have placed all of the Design Exploration templates on your schematic you should set up the in-dividual Design Exploration systems and then run the analysis for each cell.

1. Right click on each cell in each of the Design Exploration systems to open the workspace for that cell.

2. In the workspace, set up any analysis options that are needed (parameter limits, optimization objectives,optimization type, six sigma parameter distributions, etc.).

3. To run the analysis for a cell from its workspace, click on the Update button in the toolbar for that celltype (Update Optimization, Update Design of Experiments, etc.). You can also click on the Update

Project button to update the entire project.

4. Make sure that you set up and solve each cell in a Design Exploration system to complete the analysisfor that system.

5. View the results of each analysis from the workspaces of the various cells in the Design Explorationsystems (charts, tables, statistics, etc.).

When you are in the Project Schematic view, cells in the various systems contain icons to indicate theirstate. If they are out of date and need an update, you can right click on the cell and select Update fromthe menu.

Monitoring Design Exploration Analyses

After starting a Design Exploration analysis Update, there are two different views available to help youmonitor your analysis.

Progress View

There is a Progress view that you can open from the View menu or from Show Progress button at thelower right corner of the window. During execution, a progress execution status bar appears in the Progresscell. The name of the task that is executing appears in the Status cell and information about the executionappears in the Details cell. This view continuously reports the status of the execution, until it is complete.You can stop an execution by clicking the Stop button to the right of the progress bar and you can restartit at a later time by using any of the Update methods that you would use to start a new Update for the cellor workspace.

Messages View

If solution errors exist and the Messages view is not open, the Show Messages button in the lower rightof the window will flash orange and indicate the number of messages. Click this button to examine solutionerror information. You can also open the Messages view from the View menu in the toolbar.


Overview

Design Exploration Options

You can set the default behavior of Design Exploration features through the Tools>Options dialog box.These options will be used as the default values for the analyses that you set up in your project workspace.In the workspace, each system has Properties panels that allow you to change the default values for anyspecific cell in an analysis. Changing the options in a cell's Properties view does not affect the global defaultoptions described here in the Options dialog box. To access Design Exploration default options:

1. From the main menu, choose Tools>Options. An Options dialog box appears. Expand the DesignExploration item in the tree.

2. Click on Design Exploration or one of its sub-options.

3. Change any of the option settings by clicking directly in the option field on the right. You may needto scroll down to see all of the fields. You will see a visual indication for the kind of interaction requiredin the field (examples are drop-down menus, and direct text entries). Some sections of the screensmay be grayed out unless a particular option above them is selected.

4. Click OK when you are finished changing the default settings.

The following options appear in the Options dialog box:

Design Exploration

Design of Experiments

Response Surface

Sampling and Optimization

There are many general Workbench options in the Options dialog box. You can find more information aboutthem in ANSYS Workbench Options.



Design Exploration Options

Design Exploration

The Design Points category includes:

• Preserve Design Points After DX Run: If checked, after the solution of a Design Exploration cell com-pletes, save the design points created for the solution to the project Table of Design Points. If thisoption is selected, the following option is available:

– Retain Files for Preserved Design Points: If checked, in addition to saving the design points tothe project Table of Design Points, exports each of the Design Exploration design points as a sep-arate project.

The Graph category includes:

• Number of Gridlines: Changes the number of points used by the individual continuous input parameteraxes in the 2D/3D response surface charts. This enhances the viewing resolution of these charts. Therange is from 2 to 100. The default is 10.

• Number of Colors: Controls the number of colors used by 2D and 3D Tradeoff charts. Choosing morecolors provides a better color quality display at the expense of slower chart rendering. Choosing lesscolors provides faster chart rendering at the expense of lower color quality. The following choices areavailable:

– [8]

– [16]

– [32] (default)

– [64]

– Max (may be slow)

The Sensitivity category includes:

• Significance Level: Controls the relative importance or significance of input variables. The allowablerange is from 0.0, meaning all input variables are "assumed" to be insignificant, to 1.0, meaning all inputvariables are "assumed" to be significant. The default is 0.025.

• Correlation Coefficient Calculation Type: Specifies the calculation method for determining sensitivitycorrelation coefficients. The following choices are available:

– Rank Order (Spearman) (default): Correlation coefficients are evaluated based on the rank ofsamples.

– Linear (Pearson): Correlation coefficients are evaluated based on the sample values.

The Parameter Options category includes:

• Display Parameter Full Name: If checked (default), displays the full parameter everywhere in the DesignExploration user interface. If unchecked, displays the short parameter name everywhere in the DesignExploration user interface, except for the Outline view which always displays the full parameter name.

• Parameter Naming Convention: Sets the naming style of input parameters within Design Exploration.The following choices are available:

– Taguchi Style: Names the parameters as Continuous Variables and Noise Variables.

– Uncertainty Management Style (default): Names the parameters as Design Variables and UncertaintyVariables.


Overview

– Reliability Based Optimization Style: Names the parameters as Design Variables and RandomVariables.


The Design of Experiments Type category includes the following settings. Descriptions of each setting areincluded in the Design Points (p. 33) section under "Using Design Exploration" (p. 17).

• Central Composite Design (default): Note that changing the CCD type in the Options dialog box willgenerate new design points provided the study has not yet been solved.

• Optimal Space Filling Design

The Central Composite Design Options category includes:

• Design Type: The following choices are available:

– Auto Defined (default)

– G-Optimality

– VIF-Optimality

– Rotatable

– Face Centered

• Enhanced Template: Select this option to enable the enhanced template for the Rotatable and Face

Centered design types. Selecting it with other design types will have no effect.

The Optimal Space Filling Options category includes:

• Design Type: The following choices are available:

– Max-Min Distance (default)

– Centered L2

– Maximum Entropy

• Max Number of Cycles: Indicates the maximum number of iterations which the base DOE undergoesfor the final sample locations to conform to the chosen DOE type.

• Sample Type: : This indicates the specific method chosen to determine the number of samples. Thefollowing choices are available:

– CCD Samples (default): Number of samples is the same as that of a corresponding CCD design.

– Linear Model Samples: Number of samples is the same as that of a design of linear resolution.

– Pure Quadratic Model Samples: Number of samples is the same as that of a design of pure quad-ratic (constant and quadratic terms) resolution.

– Full Quadratic Model Samples: Number of samples is the same as that of a design of full quadratic(all constant, quadratic and linear terms) resolution.

– User Defined Samples: User determines the number of DOE samples to be generated.

• Number of Samples: For a Sample Type of User Defined Samples, specifies the default number ofsamples.

Response Surface

The Type category includes:



Response Surface

• Standard Response Surface - Full 2nd Order Polynomials (default):

• Kriging: Accurate interpolation method. Kriging Options include:

– Kernel Variation Type, which can be set to:

→ Variable Kernel Variation (default): Pure Kriging mode.

→ Constant Kernel Variation: Radial Basis Function mode.

• Non-Parametric Regression: Provides improved response quality. Initialized with one of the availableDOE types.

• Neural Network: Non-linear statistical approximation method inspired from biological neural processing.Neural Network Options include:

– Number of Cells: The number of cells controls the quality of the meta model. The higher it is, thebetter the network can capture parameters interactions. The recommended range is from 1 to 10.The default is 3.

Once a meta model is solved, it is possible to switch to another meta model type or change the options forthe current meta model in the Properties view for that Response Surface workspace in the analysis system.After changing the options in the Properties view, you must Update the Response Surface to obtain the newfitting.


The Random Number Generation category includes:

• Repeatability: When checked, seeds the random number generator to the same value each time yougenerate uncertainty analysis samples. With Repeatability unchecked , the random number generatoris seeded with the system time every time you generate samples. This applies to all methods in DesignExploration where random numbers are needed, as in Six Sigma Analysis or Goal Driven Optimization.The default is checked.

The Weighted Latin Hypercube category includes:

• Sampling Magnification: To indicate number of times used to reduce regular Latin Hypercube sampleswhile achieving a certain probability of failure (Pf ). For example, the lowest probability of failure for1000 Latin Hypercube samples is approximately 1/1000; magnification of 5 is meant to use 200weighted/biased Latin Hypercube samples to approach the lowest probability of 1/1000. It is recommen-ded not to use magnification greater than 5. A greater magnification may result in significant Pf errordue to highly biased samples. The default is 5.

The Optimization category includes:

• Constraint Handling (GDS)

This option can be used for any optimization application and is best thought of as a "constraint satis-faction" filter on samples generated from the Screening, MOGA, or NLPQL runs. This is especially usefulfor Screening samples to detect the edges of solution feasibility for highly constrained nonlinear optim-ization problems. The following choices are available:

– As Goals (default): Implies that the upper, lower and equality constrained objectives of the candidatedesigns shown in the Table of Optimization are treated as goals; thus any violation of the objectivesis still considered feasible.

– As Hard Constraints: When chosen, the upper, lower and equality constrained objectives are treatedas hard constraints; that is, if any of them are violated then the candidate is no longer displayed.


Overview

So, in some cases no candidate designs may be displayed depending on the extent of constraintviolation.




Using Design Exploration

The topics in this section cover the basics of using the Design Exploration templates.What is Design Exploration?Design of ExperimentsParameters CorrelationResponse SurfaceGoal Driven OptimizationSix Sigma AnalysisWorking with ParametersDesign PointsThe Design of Experiments SolutionUsing Response SurfacesGlobal SensitivitiesAdding Design Points and Response PointsUsing Parameters CorrelationUsing Goal Driven OptimizationUsing Six Sigma Analysis

What is Design Exploration?

Design Exploration is a powerful tool for designing and understanding the analysis response of parts andassemblies. It uses a deterministic method based on Design of Experiments (DOE) and various optimizationmethods, with parameters as its fundamental components. These parameters can come from any supportedanalysis system, DesignModeler, and various CAD systems. Responses can be studied, quantified, and graphed.Using a Goal Driven Optimization method, the deterministic method can obtain a multiplicity of designpoints. You can also explore the calculated Response Surface and generate design points directly from thesurface.

After setting up your analysis, you can pick one of the system templates from the Design Exploration toolboxto:

• parameterize your solution and view an interpolated response surface for the parameter ranges

• view the parameters associated with the minimum and maximum values of your outputs

• create a correlation matrix that shows you the sensitivity of outputs to changes in your input parameters

• set output objectives and see what input parameters will meet those objectives

• perform a Six Sigma analysis on your model


Design of Experiments (p. 18) is a technique used to determine the location of sampling points and is includedas part of the Response Surface, Goal Driven Optimization, and Six Sigma systems. There are several versionsof design of experiments available in engineering literature. These techniques all have one common charac-teristic: they try to locate the sampling points such that the space of random input parameters is exploredin the most efficient way, or obtain the required information with a minimum of sampling points. Sample



points in efficient locations will not only reduce the required number of sampling points, but also increasethe accuracy of the response surface that is derived from the results of the sampling points. By default thedeterministic method uses a central composite design, which combines one center point, points along theaxis of the input parameters, and the points determined by a fractional factorial design.

Parameters Correlation

Probabilistic sensitivities are based on a statistical correlation analysis between the individual probabilisticdesign variables. You can review the Parameters Correlation (p. 20) data that has been used to derive sens-itivities and decide if individual sensitivity values are significant or not. This information is collected in thecorrelation matrix of the random output parameters versus the random input variables. You can also reviewthe correlations that have been sampled between random input variables. The correlations between randomoutput parameters are important if you want to use the probabilistic results of your probabilistic analysis asinput for another probabilistic analysis.

Goal Driven Optimization

You can use "Understanding Goal Driven Optimization" (p. 65) to state a series of design goals, which will beused to generate optimized designs.

The optimization approach used in Design Exploration departs in many ways from traditional optimizationtechniques. Many goals are allowed and can be assigned to each output parameter. Goals may be weightedin terms of importance.

Response Surface

There is one Response Surface (p. 22) or Curve for every output parameter. Output parameters are representedin terms of the input parameters, which are treated as independent variables.

For the deterministic method, response surfaces for all output parameters are generated in two steps:

• Solving the output parameters for all design points as defined by a design of experiments

• Fitting the output parameters as a function of the input parameters using regression analysis techniques

Six Sigma Analysis

Six Sigma Analysis (SSA) is an analysis technique for assessing the effect of uncertain input parameters andassumptions on your model.

A Six Sigma Analysis allows you to determine the extent to which uncertainties in the model affect the resultsof an analysis. An "uncertainty" (or random quantity) is a parameter whose value is impossible to determineat a given point in time (if it is time-dependent) or at a given location (if it is location-dependent). An exampleis ambient temperature: you cannot know precisely what the temperature will be one week from now in agiven city.


A Design of Experiments cell is part of the Response Surface, Goal Driven Optimization, and Six Sigma systems(although the DOE for a six sigma system treats the inputs and outputs differently and cannot share datawith the other types of DOEs). The DOE workspace allows you to preview (generates the points but doesnot solve them) or generate and solve a DOE matrix. Within the DOE workspace, you can set the inputparameter limits, set the properties of the DOE solve, view the design points table, and view several para-



meter charts. The following views are available in the DOE workspace to allow you to customize your DOEand view the updated results:

Outline: Allows you to:

• Select the DOE cell and change its properties.

• Select the input parameters and change their limits.

• Select the output parameters and view their minimum and maximum values.

• Select one of the available charts for display. You can insert directly into the Outline as many new chartsfrom the chart toolbox as you want. When a chart is selected, you can change the data properties ofthe chart (chart type and parameters displayed).

Properties: (See Design Criteria Considered in DOE (p. 34) for complete descriptions of DOE options.)

• Preserve Design Points After DX Run – check this box if you want to retain design points at the projectlevel from each update of this DOE you perform.

• DOE Type

– Central Composite Design

– Optimal Space-Filling Design

– Custom

• Settings for selected DOE Type (not for Custom)

Table: Displays data grid of the design points that populates automatically during the solving of the points.Displays input and output parameters for each design point. You can add points manually if the DOE Type

is set to Custom.

Chart: Displays the available charts described below.

Related Topics:

• The Design of Experiments Solution (p. 34)


• What is Design Exploration? (p. 17)

• Design Exploration Options (p. 11)

Parameters Parallel Chart

Generates a graphical display of the DOE matrix using parallel Y axes to represent all of the inputs andoutputs. Select the chart from the Outline to display it in the Chart view. Use the Properties view as follows:

Properties:

• Parameter Axes: The data properties for the chart allow you to set which parameter axes to displayon the chart. Simply check the box next to the parameters you want to display.

• Various generic chart properties can be changed for this chart.

You can also look at this chart data a different way. Right click on the chart background and select Edit

Properties, then change Chart Type to Spider Chart, which will show all input and output parameters ar-ranged in a set of radial axes spaced equally. Each DOE point is represented by a corresponding envelopedefined in these radial axes.



Parameters Parallel Chart

Design Points vs Parameter Chart

Generates a graphical display for plotting design points vs. any input or output parameter. Select the chartcell in the Outline to display it in the Chart view. Use the Properties view as follows:

Properties:

• X-Axis (top,bottom), Y-Axis (right,left): Design points (X top and bottom axes) can be plotted againstany of the input and output parameters (any of the axes).


Parameters Correlation

Probabilistic sensitivities are based on a statistical correlation analysis between the individual probabilisticdesign variables. You can review the correlation data that has been used to derive sensitivities and decideif individual sensitivity values are significant or not. The following views are available in the ParametersCorrelation workspace to allow you to customize your search and view the results:

Outline: allows you to:

• Select the Parameters Correlation cell and change its properties and view the number of samples gen-erated for this correlation.

• Select the input parameters and change their limits.

• Select the output parameters and view their minimum and maximum values.

• Select one of the available charts for display. When a chart is selected, you can change the data prop-erties of the chart (chart type and parameters displayed).

Properties:

• Preserve Design Points After DX Run – check this box if you want to retain design points at the projectlevel from this parameters correlation.

Note

There is no way to view the design points created for a parameters correlation other thanpreserving them at the project level.

• Reuse the samples already generated – check this box if you want to reuse the samples generatedin a previous correlation.

• Choose Correlation Type algorithm:

– Spearman

– Pearson

• Number of Samples – maximum number of samples to generate for this correlation.

• Choose Auto Stop Type:

– Enable Auto Stop

– Execute All Simulations

• Mean Value Accuracy – (if Auto Stop is enabled) set the desired accuracy for the mean value of thesample set.



• Standard Deviation Accuracy – (if Auto Stop is enabled) set the desired accuracy for the standarddeviation of the sample set.

• Convergence Check Frequency – (if Auto Stop is enabled) number of simulations to execute beforechecking for convergence.

Table: displays the input and output parameters in a correlation matrix. The table can be sorted by an indi-vidual parameter if desired.

Chart: displays the available charts described below.

Related Topics:

• Using Parameters Correlation (p. 54)


• Correlation Coefficient Theory

Correlation Scatter Chart

Allows you to graphically view the sample values of one parameter vs. another. A trend line is drawnshowing the correlation of the sample values. Select the Correlation Scatter chart cell in the Outline todisplay the Correlation Scatter chart in the Chart view. Use the Properties view as follows:

Properties:

• Choose the parameters to display on the X Axis and Y Axis.


Correlation Matrix

Allows you visualize how closely the various input and output parameters are coupled. The strength ofcorrelation is indicated by color in the matrix. Placing your cursor over a particular square will show you thecorrelation value for the two parameters associated with that square. Select the Correlation Matrix chartcell in the Outline to display the chart in the Chart view. Use the Properties view as follows:

Properties:

• Enable or disable the parameters that are displayed on the chart.


Related Topics:

• Viewing Significance and Correlation Values (p. 55)

Sensitivities Chart

Allows you to graphically view the global sensitivities of each output parameter with respect to the inputparameters. Select the Sensitivities chart cell under Charts in the Outline to display the chart in the Chartview. Use the Properties view as follows:

Properties:

• Chart Mode – set to Bar or Pie.



Sensitivities Chart



Related Topics:

• Global Sensitivities (p. 53)

• Viewing Significance and Correlation Values (p. 55)

Response Surface

Builds a response surface from the DOE Design Points input and output values based on the chosen Response

Surface Type. Within the Response Surface workspace, you can view the input parameter limits and initialvalues, set the properties of the response surface algorithm, view the response points table, and view severaltypes of response charts. The following views are available in the Response Surface workspace to allow youto customize your Response Surface and view the results:


• Select the Response Surface cell and change its properties.

• Select the output parameters and view their minimum and maximum values, set their transformationtype, and view their goodness of fit.

• Select a response point and view its input and output parameter properties. You can change the inputvalues and see the corresponding output value.

• Select one of the available charts for display. When a chart is selected, you can change the data prop-erties of the chart (chart type and parameters displayed).

Properties:

• Preserve Design Points After DX Run – check this box if you want to retain at the project level designpoints that are created when refinements are run for this Response Surface

Note

Checking this box will not preserve any design points unless you run a Kriging refinement,since the Response Surface uses the design points generated by the DOE. If the DOE of theResponse Surface does not preserve design points, when you perform a refinement, the re-finement design points will be preserved at the project level. If the DOE is set to preservedesign points, and the Response Surface is set to preserve design points, when you performa refinement the project will contain the DOE design points and the refinement design points.

• Choice of Response Surface Type algorithm

– Standard Response Surface – Full 2nd Order Polynomials

– Kriging

– Non-Parametric Regression

– Neural Network

• Settings for selected algorithm, as applicable

Table: Displays data grid of the Response Points for that response surface. Displays input and output para-meters for each response point. You can add response points by right-clicking on the response surface and



selecting Insert as Response Points. You can also create response points by directly entering values into theinput parameter cells of the row designated with an asterisk (*).

Chart: Displays the available charts described below for each response point.

Related Topics:


• Using Response Surfaces (p. 39)

• Goodness of Fit (p. 49)

• Standard Response Surface - Full 2nd-Order Polynomials (p. 39)

• Kriging (p. 41)

• Non-Parametric Regression (p. 42)

• Suggested Best Practices (p. 45)

• Neural Network (p. 46)

Response Chart

Allows you to graphically view the impact that changing each input parameter has on the displayed outputparameter. Select the Response chart cell in the Outline to display the response chart in the Chart view.You can add response points to the Table of Response Surface by right-clicking on the response chart andselecting Insert as Response Points. You can add design points to the Project by right-clicking on the re-sponse chart and selecting Insert as Design Points. Use the Properties view as follows:

Properties:

• Chart Mode – set to 2D or 3D.

• Number of Points on X (and Y, in 3D) – set the number of points that you want to appear on the re-sponse curve.

• Choose the input parameter(s) to display on the X Axis and Y Axis (3D).

• Choose the output parameter to display on the Z Axis (3D) or Y Axis (2D).

• Use the sliders to change the values of the input parameters that are not displayed on the curve to seehow they effect the values for the parameters that are displayed on the curve. You may also enterspecific values in the boxes above the sliders.

• View the interpolated output parameter values for the selected set of input parameter values.


You can view Goodness of Fit information for the output parameters on the generated response curve. Selectone of the output parameter cells in the Outline to display its properties. For each output, you can alsochange the transformation function to improve the quality of the approximation.

Related Topics:

• Response Surface Charts (p. 49)

• Adding Design Points and Response Points (p. 54)



Related Topics:

Local Sensitivity Chart

Allows you to graphically view the impact that changing each input parameter has on the output parameters.Select the Local Sensitivity chart cell under Response Point in the Outline to display the chart in the Chartview. Use the Properties view as follows:

Properties:


• Use the sliders to change the values of the input parameters to see how the sensitivity is changed foreach output.



Related Topics:


• Using the Local Sensitivity Chart (p. 52)

Spider Chart

Allows you visualize the impact that changing the input parameters has on all of the output parameterssimultaneously. Select the Spider chart cell in the Outline to display the chart in the Chart view. Use theProperties view as follows:

Properties:

• Use the sliders to change the values of the input parameters to see how they affect the output paramet-ers.



You can also look at this chart data a different way. Right click on the chart background and select Edit

Properties, and change Chart Type to Parallel Coordinate. Then click on the Spider cell under ResponsePoint in the Outline view and use the input parameter sliders in Properties view to see how specific settingsof various input parameters change the output parameters, which are arranged in parallel 'Y' axes in thechart.

Related Topics:


• Using the Spider Chart (p. 51)

Goal Driven Optimization

Goal Driven Optimization (GDO) is a constrained, multi-objective optimization technique in which the “best”possible designs are obtained from a sample set given the objectives you set for parameters. The DOE andResponse Surface Cells are used in the same way as described previously in this section. The following viewsare available in the Optimization cell workspace to allow you to customize your GDO and view the results:




• Select the Optimization cell and change its properties, and view the size of the generated sample set.

• Select the inputs and outputs to view the input values used in the analysis and the calculated maximumand minimum values of each of the outputs.

• Select one of the available charts for display. When a chart is selected, you can change the data prop-erties of the chart (x- and y-axis parameters, parameters displayed on bar chart, etc.).

Properties: (See Optimization Options (p. 56) for complete descriptions of GDO options.)

• Optimization Method

– MOGA

– NLPQL

– Screening

• Settings for selected Optimization Method

• Constraint Handling (GDS)

– As Goal (default)

– As Hard Constraints

Table: Before the update, provides fields to allow you to set your objective for each input and output, orset a target value, and indicate the relative importance of that objective. (See Defining Design Objectives (p. 59)for complete descriptions of GDO options.) After the update, displays up to three best candidate points forthe stated objectives. A number of gold stars or red crosses are displayed next to each objective-drivenparameter to indicate how well it meets the stated objective, from three red crosses (the worst) to threegold stars (the best).

Chart: Displays the available charts described below.

Related Topics:

• Using Goal Driven Optimization (p. 56)

• "Understanding Goal Driven Optimization" (p. 65)

• Principles (GDO) (p. 65)

• Guidelines and Best Practices (GDO) (p. 66)

• Postprocessing GDO Results (p. 68)

• Goal Driven Optimization Theory (p. 71)

Tradeoff Chart

Allows you to view the Pareto fronts created from the samples generated in the GDO. Select the Tradeoff

chart cell under Charts in the Outline to display the chart in the Chart view. Use the Properties view as follows:

Properties:

• Chart Mode – set to 2D or 3D.

• Number of Pareto Fronts to Show – Set the number of Pareto fronts that are displayed on the chart.

• Show infeasible points – Enable or disable the display of infeasible points. (Available when some ofthe objectives are constraints.)

• Set the parameters to be displayed on the X- and Y-axis.



Tradeoff Chart


Related Topics:

• Using the Tradeoff Chart (p. 62)

• Understanding Tradeoff Charts (p. 69)

Samples Chart

Allows you to visually explore a sample set given defined objectives. Select the Samples chart cell underCharts in the Outline to display the chart in the Chart view. Use the Properties view as follows:

Properties:

• Chart Mode – set to Candidates or Pareto Fronts. For Pareto Fronts, the following options can be set:

– Number of Pareto Fronts to Show – either enter the value or use the slider to select the numberof Pareto Fronts displayed.

– Coloring Method – can be set to per Samples or per Pareto Front.

• Show infeasible points – Enable or disable the display of infeasible points. (Available when some ofthe objectives are constraints.)



Related Topics:

• Understanding Samples Charts (p. 70)

Sensitivities Chart (GDO)

Allows you to graphically view the global sensitivities of each output parameter with respect to the inputparameters. Select the Sensitivities chart cell under Charts in the Outline to display the chart in the Chartview. Use the Properties view as follows:

Properties:




Related Topics:


• Using the Sensitivities Chart (GDO) (p. 61)

Six Sigma Analysis

The Six Sigma Analsysis (SSA) system consists of a DOE cell, a Response Surface cell, and a Six Sigma Analysiscell. The DOE cell allows you to set up the input parameters and generate the samples for the analysis. TheResponse Surface that results will be the same as the Response Surface from a standard Response Surfaceanalysis. The SSA cell allows you to set up the analysis type and view the results of the analysis.



By default, the Six Sigma Analysis DOE includes all of the input parameters and sets them to be uncertaintyparameters. When you run a Six Sigma Analysis, you will need to set up the uncertainty parameters in theSSA DOE. If you want any input parameters to be treated as deterministic parameters, uncheck the box nextto those parameters in the SSA DOE Outline view.

Design of Experiments (SSA)

The Design of Experiments part of the Six Sigma templates is the same as the DOE in other templates, withthe exception of the input parameters being designated as uncertainty parameters. You will need to set upthe input parameter options before solving the DOE. The following views in the DOE workspace containcomponents that are unique to the Six Sigma DOE:


• Select the input parameters and change their distribution properties.

• View the Skewness and Kurtosis properties for each input parameter distribution.

• View the calculated Mean and Standard Deviation for all Distribution Types except Normal and TruncatedNormal where you can set those values.

• View the Initial Value for each input parameter.

Properties: (The properties for the uncertainty input parameters are described here.)

• Distribution Type – Type of distribution associated with the input parameter

– Uniform

– Triangular

– Normal

– Truncated Normal

– Lognormal

– Exponential

– Beta

– Weibull

• Maximum Likely Value (Triangular Only)

• Log Mean (Lognormal only)

• Log Standard Deviation (Lognormal only)

• Exponential Decay Parameter (Exponential only)

• Beta Shape R (Beta only)

• Beta Shape T (Beta only)

• Weibull Exponent (Weibull only)

• Weibull Characteristic Value (Weibull only)

• Non-Truncated Normal Mean (Truncated Normal only)

• Non-Truncated Normal Standard Deviation (Truncated Normal only)

• Mean (Normal only)

• Standard Deviation (Normal only)



Design of Experiments (SSA)

• Lower Bound (Settable for all but Normal and Lognormal)

• Upper Bound (Uniform, Triangular, Truncated Normal, and Beta only)

Table: When output parameters or charts are selected in the Outline, displays the normal DOE data grid ofthe design points that populates automatically during the solving of the points. When an input parameteris selected in the Outline, displays a table that shows for each of the samples in the set:

• Quantile: the input parameter value point for the given PDF and CDF values.

• PDF: (Probability Density Function) density of the input parameter along the X-axis.

• CDF: (Cumulative Distribution Function) is the integration of PDF along the X-axis.

Chart: When an input parameter is selected in the Outline view, displays the Probability Density Functionand the Cumulative Distribution Function for the Distribution Type chosen for the input parameter. Thereare also Parameters Parallel and Design Points vs Parameter charts as you would have in a standard DOE.

Related Topics:

• Design of Experiments (p. 18)

Six Sigma Analysis

Once you have assigned the distribution functions to the input parameters in the DOE, you can update theproject to perform the Six Sigma Analysis. Once the analysis is finished, edit the Six Sigma Analysis cell inthe project to see the results. The following views are available in the Six Sigma Analysis workspace to allowyou to customize your analysis and view the results:


• Select each input parameter and view its distribution properties, statistics, upper and lower bounds,initial value, and distribution chart.

• Select each output parameter and view its calculated maximum and minimum values, statistics anddistribution chart.

• Set the table display format for each parameter to Quantile-Percentile or Percentile-Quantile .

Properties:

For the Six Sigma Analysis, the following properties can be set:

• Sampling Type – Type of sampling used for the Six Sigma Analysis

– LHS

– WLHS

• Number of Samples

For the parameters, the following properties can be set:

• Probability Table – Manner in which the analysis information is displayed in the Table view.

– Quantile-Percentile

– Percentile-Quantile

Table: Displays a table that shows for each of the samples in the set:



• <Parameter Name>: a valid value in the parameter's range.

• Probability: Probability that the parameter will be less than or equal to the parameter value shown.

• Sigma Level: The approximate number of standard deviations away from the sample mean for thegiven sample value.

If Probability Table is set to Quantile-Percentile in the Statistics section of the Properties view, you can edita parameter value and see the corresponding Probability and Sigma Level values. If it is set to Percentile-Quantile, the columns are reversed and you can enter a Probability or Sigma Level value and see the corres-ponding changes in the other columns.

Chart: When a parameter is selected in the Outline view, displays its Probability Density Function and theCumulative Distribution Function. A global Sensitivities chart is available in the outline.

Related Topics:

• Using Six Sigma Analysis (p. 62)

• Statistical Measures (p. 64)

Sensitivities Chart (SSA)

Allows you to graphically view the global sensitivities of each output parameter with respect to the inputparameters in the Six Sigma Analysis. Select the Sensitivities chart cell under Charts in the Outline to displaythe chart in the Chart view. Use the Properties view as follows:

Properties:




Related Topics:


• Statistical Sensitivities in a Six Sigma Analysis (p. 99)

Working with Parameters

General Information

Parameters are exposed from the individual analysis applications. Edit the cells (DOE, Response Surface,Parameters Correlation, etc.) of a Design Exploration template to see the available input and output para-meters in the Outline view. You can set options for those parameters (such as upper and lower limits) byclicking on the parameter in the Outline and then setting the options for that parameter in the Propertiesview. The Properties view will also give you additional information about the parameters.



General Information

Note

If you modify your analysis database after a Design Exploration solution (causing the parametersto change), your Design Exploration studies may not reflect your analysis anymore. If you modifyyour analysis, the Update Required symbol will appear on the Design Exploration cells to indicatethat they are out of date.

Parameters as Design Variables for Optimization

When you create a Design Exploration design study, each input parameter's range will default to +/- 10%of the parameter's initial value. These default values may not be valid in certain situations, so be sure toverify that the ranges are valid with respect to the input geometry and Analysis System scenario.

Effects of a Parameter Change on Design Exploration Systems

When a Parameter is added, deleted or modified in the project, either the existing Design Exploration systemstake the change into account immediately or they are set to a Refresh Required state. Depending on thenature of the change, a Refresh or an Update operation is required to take the changes in account.

When non-parametric changes occur in the model (e.g. direction of a non-parameterized force, shape of thegeometry, non-parameterized dimension of the model, etc), the Design Exploration systems are set to aRefresh Required state and an Update operation is required to rebuild the results. In some situations, non-parametric changes may not be caught and it is necessary to perform a Clear Generated Data operationfollowed by an Update operation in order to synchronize the Design Exploration systems with the new stateof the project. For example, editing the input file of a Mechanical APDL system outside of Workbench is anon-parametric change that is not caught. Inserting, Deleting, Duplicating or Replacing systems in a projectmay not be caught as well in very rare situations.

Note

You can change the way that units that are displayed in your Design Exploration systems fromthe Units menu. Changing the units display in this manner simply causes the existing data ineach system to be displayed in the new units system, and does not require an update of theDesign Exploration systems. If the user changes a parameter's unit from the Parameters cell onthe schematic, all Design Exploration systems become out of date, and will be marked as UpdateRequired.

Input Parameters

In the Outline view of your Design Exploration template, select the parameter that you want to edit. Mostchanges will be made using the Properties view for that parameter. Changes made for parameters will onlyaffect the parameter information for the current template and any templates that share a project cell (DOE,Response Surface) from the template.

• An input parameter may be selected or unselected in a template by checking the box to the right of itin the Outline view.

• In the Properties view, specify further attributes for the selected input parameter, such as a range (upperand lower bounds) of values for a continuous input parameter or values for a discrete input parameter.



Caution

Using the DOE Method, the number of generated design points in the DOE Matrix is directly re-lated to the number of selected input parameters. The design and analysis Workflow shows thatspecifying many input parameters will make heavy demands on computer time and resources,including system analysis, DesignModeler geometry generation, and/or CAD system generation.Also, large input parameter ranges may lead to inaccurate results.

Setting Up Design Variables

A range (upper and lower bounds) of values is required for all continuous input parameters, and values arerequired for discrete and usability input parameters.

Setting Up Uncertainty Variables

For uncertainty variables (used in a Six Sigma analysis), you must specify the type of statistical distributionfunction used to describe its randomness as well as the parameters of the distribution function. For thedistribution type, you can select one of the following:

• Uniform

• Triangular

• Normal

• Truncated Normal

• Lognormal

• Exponential

• Beta

• Weibull

For more information on the distribution types, see Distribution Functions (p. 92).

In the example below of a beam supporting a roof with a snow load, you could measure the snow heighton both ends of the beam 30 different times. Suppose the histograms from these measurements look likethe figures given below.



Setting Up Uncertainty Variables

Figure: Histograms for the Snow Height H1 and H2

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

5.0E-01

6.0E-01

4.07E+01

1.22E+02

2.03E+02

2.85E+02

3.66E+02

Snow Height H1

Rela

tive

Freq

uenc

y

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

5.0E-01

6.0E-01

7.0E-01

8.0E-01

9.85E+01

2.95E+02

4.92E+02

6.89E+02

8.86E+02

1.08E+03

Snow Height H2

Rela

tive

Freq

uenc

y

From these histograms, you can conclude that an exponential distribution is suitable to describe the scatterof the snow height data for H1 and H2. Suppose from the measured data we can evaluate that the averagesnow height of H1 is 100 mm and the average snow height of H2 is 200 mm. The parameter λ can be directlyderived by 1.0 divided by the mean value, which leads to λ1 = 1/100 = 0.01 for H1, and λ1 = 1/200 = 0.005for H2.

Output Parameters

Each output parameter corresponds to a response surface which is expressed as a function of the inputparameters. Some typical output parameters are: Equivalent Stress, Displacement, Maximum Shear Stress,etc.



Design Points

When you solve a Design Exploration system, design points are generated based on the number of inputparameters in the system. Before solving a Design of Experiments or Response Surface (if performing a re-finement) cell for a Six Sigma or non-Six Sigma system, you can preview the generated design points byclicking the corresponding Preview button in the toolbar. You are not able to preview the generated designpoints for a Parameters Correlation solution.

You can solve for the output parameter values for the generated design points in several ways:

• From the workspace, click the corresponding Update button in the toolbar to solve for all of the gen-erated design points.

• From the Project Schematic, right click the cell and select Update from the menu.

• From the Project Schematic, click the Update Project button in the toolbar to update all of the systemsin the project.

For the deterministic method, the locations of the generated design points are determined according to adesign of experiments method that is by default a central composite design with a fractional factorial design.The number of generated design points is calculated using the number of input parameters selected in theparameters view. Response Surfaces will be generated from the associated DOE solution.

Saving Generated Design Points to the Project

The generated design points are temporarily written to the Table of Design Points in the project's ParameterSet Bar during the solution of a Design Exploration cell. These generated design points will be removedfrom the Table of Design Points after the solution unless:

• The user has selected Preserve Design Points After DX Run in the Design Exploration cell's Propertiesview

• The user has selected Preserve Design Points After DX Run in the Tools>Options>Design Exploration

dialog

You may want to preserve generated design points for the following reasons:

• Preserving design points has the advantage of letting the user see the exact design points used by aDesign Exploration cell. In a Parameters Correlation analysis for instance, it is the only way to see theactual design points used to calculate the correlation since there is no preview for that cell.

• If you preserve the design points, a subsequent update of that Design Exploration system can reusethe existing Design Points, which will reduce the CPU cost.

Note

When a Design Exploration design point fails to update, it is retained automatically in the project,even if the preserve option is not checked.

In addition to preserving the Design Points in the project, you can also export the preserved Design Pointsfrom the project. If you check the Retain Files for Preserved Design Points option in the Properties underthe Preserve Design Points After DX Run property, in addition to saving the design points to the projectTable of Design Points, each of the Design Exploration Design Points is exported as a separate project.



Saving Generated Design Points to the Project

Note

If you run a Design Exploration analysis and export Design Points, and then you uncheck theoption to export them, any Design Points that were marked for export and are reused in ensuinganalyses will still be exported, although newly created Design Points will not be exported.

The Design of Experiments Solution

One you have set up your input parameters, you can update the DOE, which submits the generated designpoints to the analysis system for solution. After the solution is complete, you can update the ResponseSurface cell, which generates response surfaces for each output parameter based on the data in the generateddesign points.

You should set up your DOE Properties before generating your DOE design point matrix.Design Criteria Considered in DOEDOE Using a Central Composite DesignUpper and Lower Locations of DOE PointsDOE Matrix Generation

Design Criteria Considered in DOE

When you are editing a DOE cell, the Properties view allows you to set up the analysis options as describedbelow. You select the choice from the Design of Experiments Type drop down menu located in theProperties view of the Design of Experiments cell. You can also set defaults for most of these settingsfrom the Design Exploration > Design of Experiments section of the Options dialog box, which is accessiblefrom the Tools menu. The DOE types available are:

• Central Composite Design (default): Provides a screening set to determine the overall trends of theMeta Model to better guide the choice of options in the Optimal Space-Filling Design.

A Design Type Specifies a Central Composite Design (CCD) type to help improve the response surfacefit for DOE studies. For each CCD type, the alpha value is defined as the location of the sampling pointthat accounts for all quadratic main effects. The following choices are available:

– Auto Defined (default): Design Exploration automatically selects the Design Type based on thenumber of input variables. Use of this option is recommended for most cases as it automaticallyswitches between the G-Optimal (if the number of input variables is 5) or VIF-optimal otherwise.

However, the Rotatable design may be used if the default option does not provide good values forthe goodness of fits from the response surface plots. Also, the Enhanced template may be used ifthe default Standard template does not fit the response surfaces too well.

– Face Centered: A three-level design with no rotatability. The alpha value equals 1.0. A Template

Type setting automatically appears, with Standard and Enhanced options. Choose Enhanced fora possible better fit for the response surfaces.

– Rotatable: A five-level design that includes rotatability. The alpha value is calculated based on thenumber of input variables and a fraction of the factorial part. A design with rotatability has the samevariance of the fitted value regardless of the direction from the center point. A Template Type

setting automatically appears, with Standard and Enhanced options. Choose Enhanced for a possiblebetter fit for the response surfaces.

– VIF-Optimality: A five-level design in which the alpha value is calculated by minimizing a measureof non-orthogonality known as the Variance Inflation Factor (VIF). The more highly correlated theinput variable with one or more terms in a regression model, the higher the Variance Inflation Factor.



– G-Optimality: Minimizes a measure of the expected error in a prediction and minimizes the largestexpected variance of prediction over the region of interest.

See DOE Using a Central Composite Design (p. 36) for more information.

• Optimal Space-Filling Design: Creates optimal space filling design of experiments (DOE) plans accordingto some specified criteria. The objective is to fill the design space efficiently with as small number ofpoints as possible while keeping the discrepancy as low as possible. This is a more efficient DOE schemewhen a more complex Meta Modeling technique such as Kriging, Non Parametric Regression or NeuralNetworks is desired. In these situations, traditional CCD based DOEs are often not suitable as they havepoor space filling capabilities.

Note

If you have several parameters and one is discrete (having 4 values, for example) and youask for an Optimal Space Filling analysis, User-Defined with 20pts, you will actually have20x4=80 pts. This is by design; a different DOE must be created for each combination of thediscrete parameters, which may quickly result in a lot of design points being generated.

The following options are available with Optimal Space-Filling Design:

– Design Type: The following choices are available:

→ Max-Min Distance (default): Computationally faster than Maximum Entropy.

→ Centered L2: Computationally faster than Maximum Entropy.

→ Maximum Entropy: This option often provides better results for highly correlated design spaces.

– Maximum Number of Cycles: Determines the number of optimization loops the algorithm needs,which in turns determines the discrepancy of the DOE. The optimization is essentially combinatorial,so a large number of cycles will slow down the process; however, this will make the discrepancy ofthe DOE smaller. For practical purposes, 10 cycles usually is good for up to 20 variables. Must begreater than 0. The default is 10.

– Samples Type: Determines the number of DOE points the algorithm should generate. This optionis suggested if you have some advance knowledge about the nature of the Meta Model. The followingchoices are available:

→ CCD Samples (default): Generates the same number of samples a CCD DOE would generate forthe same number of inputs. You may want to use this to generate a space filling design whichhas the same cost as a corresponding CCD design.

→ Linear Model Samples: Generates the number of samples as needed for a linear Meta Model.

→ Pure Quadratic Model Samples: Generates the number of samples as needed for a pure quad-ratic Meta Model (no cross terms).

→ Full Quadratic Samples: Generates the number of samples needed to generate a full quadraticmodel.

→ User-Defined Samples: Expects the user to enter the desired number of samples. When selectedwith any design type, a Number of Samples option is available below the Samples Type entry.You must enter a positive number. The default is 10.

• Custom: Allows definition of a custom DOE matrix. You can manually add new design points, enteringthe input parameter values directly into the table. If you previously solved the DOE using one of theother algorithms, those design points will be retained and you can add new design points to that table.



Design Criteria Considered in DOE

If you have not previously solved the DOE, an initial design point with the initial values of the inputparameters is entered in the table.

Once the matrix is defined, choose the Update button as usual to solve all of the unsolved designpoints. There are no specific controls that you can perform on the validity of the customized matrix.

Note

You can generate a DOE matrix using one of the other design options provided, then switchto Custom to modify the matrix. The opposite however is not possible; that is, if you definea Custom matrix first, then change to one of the other design options provided, the matrixis cleared.

It is expected that the Custom option will be used to enter DOE plans built externally. If youuse this feature to enter all of the points in the matrix manually, you must make sure to enterenough points so that a good fitting can be created for the response surface. This is an ad-vanced feature that should be used with caution; always verify your results with a directsolve.

DOE Using a Central Composite Design

In Central Composite Design (CCD), a Rotatable (spherical) design is preferred since the prediction varianceis the same for any two locations that are the same distance from the design center. However, there areother criteria to consider for an optimal design setup. Among these criteria, there are two commonly con-sidered in setting up an optimal design using the design matrix.

1. The degree of non-orthogonality of regression terms can inflate the variance of model coefficients.

2. The position of sample points in the design can be influential based on its position with respect toothers of the input variables in a subset of the entire set of observations.

An optimal CCD design should minimize both the degree of non-orthogonality of term coefficients and theopportunity of sample points having abnormal influence. In minimizing the degree of non-orthogonality,the Variation Inflation Factor (VIF) of regression terms is used. For a VIF-Optimality design, the maximumVIF of the regression terms is to be minimized, and the minimum value is 1.0. In minimizing the opportunityof influential sample points, the leverage value of each sample points is used. Leverages are the diagonalelements of the Hat matrix, which is a function of the design matrix. For a G-Optimality design, the maximumleverage value of sample points is to be minimized.

For a VIF-Optimality design, the alpha value/level is selected such that the maximum VIF is minimum.Likewise, for a G-Optimality design, the alpha value/level is selected such that the maximum leverage isminimum. The rotatable design is found to be a poor design in terms of VIF- and G-Efficiencies.

For an optimal CCD, the alpha value/level is selected such that both the maximum VIF and the maximumleverage are the minimum possible. For the Auto Defined design, the alpha value is selected from eitherthe VIF- or G-Optimality design that meets the criteria. Since it is a multi-objective optimization problem, inmany cases, there is no unique alpha value such that both criteria reach their minimum. However, the alphavalue is evaluated such that one criterion reaches minimum while another approaches minimum.

For the current Auto Defined setup (except for a problem with five variables that uses G-Optimality design)all other multi-variable problems use VIF-Optimality. In some cases, despite the fact that Auto Defined

provides an optimal alpha meeting the criteria, an Auto Defined design might not give as good of a response



surface as anticipated due to the nature of the physical data used for fitting in the regression process. Inthat case, you should try other design types that might give a better response surface approximation.

Note

Default values for CCD Types can be set in the Design Exploration > Design of Experiments

section of the Options dialog box, which is accessible from the Tools menu.

It is good practice to always verify some selected points on the response surface with an actual simulationevaluation to determine its validity of use for further analyses in Design for Six Sigma. In some cases, a goodresponse surface does not mean a good representation of an underlying physics problem since the responsesurface is generated according to the predetermined sampling points in the design space, which sometimesmisses capturing an unexpected change in some regions of the design space. In that case, you should tryExtended DOE. In Extended DOE, a mini CCD is appended to a standard CCD design, where a second alphavalue is added, and is set to half the alpha value of the Standard CCD. The mini CCD is set up in a way thatthe essences of CCD design (rotatability and symmetry) are still maintained. The appended mini CCD servestwo purposes:

1. to capture a drastic change within the design space, if any.

2. to provide a better response surface fit.

The two purposes seem to be conflicting in some cases where the response surface might not be as goodas that of the Standard DOE due to the limitation of a quadratic response surface in capturing a drasticchange within the design space.

The location of the generated design points for the deterministic method is based on a central compositedesign. If N is the number of input parameters, then a central composite design consists of:

• One center point.

• 2*N axis point located at the -α and +α position on each axis of the selected input parameters.

• 2(N-f) factorial points located at the -1 and +1 positions along the diagonals of the input parameterspace.

The fraction f of the factorial design and the resulting number of design points are given in the followingtable:

Table 1 Number of Generated Design Points as a Function of the Number of Input Parameters

Number of

design points

Factorial num-

ber fNumber of in-

put parameters

501

902

1503

2504

2715

4516

7917

8128

14729



DOE Using a Central Composite Design

Number of

design points

Factorial num-

ber fNumber of in-

put parameters

149310

151411

281412

283513

285614

287715

289816

291917

549918

5511019

5531120

Upper and Lower Locations of DOE Points

The upper and lower levels of the DOE points depend on whether the input variable is a design variable(optimization) or an uncertainty variable (Six Sigma Analysis). Generally, a response surface will be moreaccurate when closer to the DOE points. Therefore, the points should be close to the areas of the inputspace that are critical to the effects being examined.

For example, for Goal Driven Optimization, the DOE points should be located close to where the optimumdesign is determined to be. For a Six Sigma Analysis, the DOE points should be close to the area wherefailure is most likely to occur. In both cases, the location of the DOE points depends upon the outcome ofthe analysis. Not having that knowledge at the start of the analysis, you can determine the location of thepoints as follows:

• For a design variable, the upper and lower levels of the DOE range coincide with the bounds specifiedfor the input parameter.

It often happens in optimization that the optimum point is at one end of the range specified for oneor more input parameters.

• For an uncertainty variable, the upper and lower levels of the DOE range are the quantile values corres-ponding to a probability of 0.1% and 99.9%, respectively.

This is the standard procedure whether the input parameter follows a bounded (e.g., uniform) or un-bounded (e.g., Normal) distribution, because the probability that the input variable value will exactlycoincide with the upper or lower bound (for a bounded distribution) is exactly zero. That is, failure cannever occur when the value of the input variable is equal to the upper or lower bound. Failure typicallyoccurs in the tails of a distribution, so the DOE points should be located there, but not at the very endof the distribution.

DOE Matrix Generation

The DOE matrix is generated after you Update your DOE. This matrix consists of generated design pointsthat are submitted to the parent analysis system for solution. You can monitor the progress of the designgeneration by clicking the Show Progress button in the lower right corner of the window when the DOEmatrix generation is prolonged; for example, Optimal Space Filling or certain VIF or G-optimal designs.



To clear the design points generated for the DOE matrix, return to the Project Schematic, right click on theDOE cell, and select Clear Generated Data. You can clear data from any Design Exploration cell in the ProjectSchematic in this way, and regenerate your solution for that cell with changes to the parameters if you sodesire.

Using Response Surfaces

There is one Response Surface or Curve for every output parameter. Output parameters are represented interms of the input parameters, which are treated as independent variables.

For the deterministic method, response surfaces for all output parameters are generated in two steps:

• Solving the output parameters for all design points as defined by a design of experiments.

• Fitting the output parameters as a function of the input parameters using regression analysis techniques.

As many charts as needed can be generated and inserted in the Charts section of the Outline view for theResponse Surface. You can also save individual charts as graphic files.

Meta Model Types

Several different modeling algorithms are available to create the response surface:

• Standard Response Surface-Full 2nd-Order Polynomials (Default)

• Kriging

• Non-Parametric Regression

• Neural Network

Standard Response Surface - Full 2nd-Order Polynomials

This is the default.

Regression analysis is a statistical methodology that utilizes the relationship between two or more quantit-ative variables so that one dependent variable can be estimated from the other or others.

A regression analysis assumes that there are a total of n sampling points and for each sampling point thecorresponding values of the output parameters are known. Then the regression analysis determines the re-lationship between the input parameters and the output parameter based on these sample points. This re-lationship also depends on the chosen regression model. Typically for the regression model, a second-orderpolynomial is preferred. In general, this regression model is an approximation of the true input-to-outputrelationship and only in special cases does it yield a true and exact relationship. Once this relationship isdetermined, the resulting approximation of the output parameter as a function of the input variables iscalled the response surface.

General Definitions

The error sum of squares SSE is:

(1)SSE y y y y y yi ii

nT= − = − −

=∑ ( ) ({ } { }) ({ } { })^ ^ ^2

1

where:



General Definitions

yi = value of the output parameter at the ith sampling point

yi^

= value of the regression model at the ith sampling point

The regression sum of squares SSR is:

(2)SSR y yii

n

= −=∑ ( )^ 2

1

where:

yn

yii

n

==∑1

1

The total sum of squares SST is:

(3)SST y yii

n

= −=∑( )2

1

For linear regression analysis the relationship between these sums of squares is:

(4)SST SSR SSE= +

Linear Regression Analysis

For a linear regression analysis, the regression model at any sampled location {x}i, with i = 1, ... , n in the m-dimensional space of the input variables can be written as:

(5)y t ci i= +{ } ε

where:

ti = row vector of regression terms of the response surface model at the ith sampled location

{c} = c c cp

T

1 2 ... = vector of the regression parameters of the regression modelp = total number of regression parameters. For linear regression analysis, the number of regressionparameters is identical to the number of regression terms.

The regression coefficients {c} can be calculated from:



(6){ } ([ ] [ ]) [ ] { }c d d d yT T= −1

Forward-Stepwise-Regression

In the forward-stepwise-regression, the individual regression terms are iteratively added to the regressionmodel if they are found to cause a significant improvement of the regression results. For DOE, a partial Ftest is used to determine the significance of the individual regression terms.

Improving the regression model with transformation functions

Only in special cases can output parameters of a finite element analysis, such as displacements or stresses,be exactly described by a second-order polynomial as a function of the input parameters. Usually a secondorder polynomial provides only an approximation. The quality of the approximation can be significantlyimproved by applying a transformation function input parameter as well as the output parameters. For DOE,the Yeo-Johnson transformation is used. The parameter λ of the Yeo-Johnson transformation for the indi-vidual input parameters and output parameters is determined using a nonlinear optimization algorithm.

Transformation Families

You can select between Box-Cox, Yeo-Johnson and None for the transformation used in standard responsesurface regression. None means that we compute the standard response surface regression without anytransformations. It is an option, on a per output basis, used to transform output parameter samples. Thedefault is Yeo-Johnson transformation since it is more numerically stable in its back-transformation. On thecontrary, Box-Cox transformation is more numerically unstable in its back-transformation, but it gives betterfit in some cases. It is recommended to try Box-Cox transformation if the goodness of fit of the responsesurface is not as good. All applications use the response surface to generate samples; for example, GoalDriven Optimization (GDO) and Design for Six Sigma (DFSS) samples. To access, click on an output parameterin the Outline view of a response surface. In the Properties view under Output Settings, select the Trans-

formation Type you want to apply and click Update to re-solve for a new response surface.

Kriging

Use the Kriging option to provide a comprehensive goodness of fit report of a standard response surfaceto determine the adequacy to be applied in Goal Driven Optimization (GDO) and Design for Six Sigma (DFSS)samples. To view the goodness of fit information, click on one of the output parameters in the Outline viewof your Repsonse Surface. A Goodness-of-Fit Information section will be shown in the Properties view forthe parameter. Kriging is an accurate interpolation method.

Kriging postulates a combination of a polynomial model plus departures of the form given by Equa-

tion 7 (p. 41):

(7)y x f x Z x( ) ( ) ( )= +

where y(x) is the unknown function of interest, f(x) is a polynomial function of x, and Z(x) is the realizationof a normally distributed Gaussian random process with mean zero, variance σ2, and non-zero covariance.The f(x) term in Equation 7 (p. 41) is similar to the polynomial model in a response surface and provides a“global” model of the design space.

While f(x) “globally” approximates the design space, Z(x) creates “localized” deviations so that the krigingmodel interpolates the N sample data points. The covariance matrix of Z(x) is given by Equation 8 (p. 42).




(8)Cov Z x Z x R r x xi j i j( ) ( ) ([ ( )]), ,

= σ2

In Equation 8 (p. 42) R is the correlation matrix, and r(xi, xj) is the spatial correlation of the function betweenany two of the N sample points xi and xj. R is an N*N symmetric, positive definite matrix with ones alongthe diagonal. The correlation function r(xi, xj) is Gaussian correlation function:

(9)r x x x xi j k ki

kj

k

M( ) exp, = − −

=

∑ θ1

2

The θk in Equation 9 (p. 42) are the unknown parameters used to fit the model, M is the number of design

variables, and xki

andxk

j

are the kth components of sample points xi and xj. In some cases, using a singlecorrelation parameter gives sufficiently good results; the user can specify the use of a single correlationparameter, or one correlation parameter for each design variable (Tools>Options>Design Exploration>ResponseSurface>Kriging Options>Kernel Variation Type: Variable or Constant).

Z(x) can be wrote :

(10)Z x r x xii

i

N( ) ( )=

=∑ λ ,

1

Non-Parametric Regression

Non-parametric regression provides improved response quality and is initialized with one of the availableDOE types.

The non-parametric regression (NPR) algorithm is implemented in the ANSYS Workbench Design Explorationas a metamodeling technique prescribed for predictably high nonlinear behavior of the outputs with respectto the inputs. The NPR feature is accessible via Tools>Options>Design Exploration>Response Surface>Type:Non-Parametric Regression.

NPR belongs to a general class of Support Vector Method (SVM) type techniques. These are data classificationmethods which use hyperplanes to separate data groups. The regression method works almost similarly,the main difference being that the hyperplane is used to categorize a subset of the input sample vectorswhich are deemed sufficient to represent the output in question. This subset is called the “support vector”set. The objective is to find a minimal, sparse support vector set from the input samples provided. Supportvector methods are preferred over other non-parametric regression methods because of two inherent ad-vantages. First, SVM-type approaches map the input space to a feature space by a kernel map before doingthe fitting; this ensures that the “curse of dimensionality” as suffered by high dimensional input spaces nolonger exist. Second, the training objective function is quadratic in the feature space, thus, QP based methodsmay be used easily to determine the support vectors without the risk of the solution being trapped in alocal minimum.

Let the input sample (as generated from a DOE method) be X = {x1 , x2, x3, …, xM}, where each xi is an N-dimensional vector and represents an input variable. The objective is to determine the equation of the form:



(11)Y W X b= < > +,

Where, W is a weighting vector. In case of generic non-parametric case, the Equation 11 (p. 43) is rewrittenas the following:

(12)Y A A K X X bi i ii

N= − +

=∑ ( ) ( , )*

r r

1

Where, K X Xi( , )r r

is the kernel map and the quantities Ai and Ai*

are Lagrange multipliers whose derivationwill be shown in later sections.

In order to determine the Lagrange multipliers, we start with the assumption that the weight vector W mustbe minimized such that all (or most) of the sample points lie within an error zone around the fitted surface.For a simple demonstration of this concept, please see Figure : Fitting a regression line for a group of sample

points with a tolerance of ε which is characterized by slack variables ξ* and ξ (p. 43) .

Figure: Fitting a regression line for a group of sample points with a tolerance of ε which is

characterized by slack variables ξ* and ξ

ε

ε

ε

ξ

ξ ∗

Thus, we write a primal optimization formulation as the following:




(13)

Minimize L

S.T.

= + +

− < > + ≤ +

<

=∑0 5

2

1. ( )

( , )

( ,

*WC

N

y W X b

W

i ii

N

i i i

ξ ξ

ε ξ

XXi b yi i> + ≤ +) - *ε ξ

Where, C is an arbitrary (> 0) constant. In order to characterize the tolerance properly, we define a lossfunction in the interval (-ε, ε) which defacto becomes the residual of the solution. In the present implement-ation, we use the ε-insensitive loss function which is given by the equation:

(14)L f x y

L f x y

ε

ε

ε

ε

= ∀ − <

= − −

0 ( ) )

( ) )

The primal problem in Equation 13 (p. 44) can thus be rewritten as the following using generalized lossfunctions as:

(15)

L W C l l

W X b y

y

i

N

i i

i i

i i i

i

= + +

≥< > + − ≤ +

=∑0 5

0

2

1. ( ( ) ( ))

,

,

*

*

ξ ξ

ξ ξε ξ

S.T.

−− < > − ≤ +W X bi i, *ε ξ

In order to solve this efficiently, a Lagrangian dual formulation is done on this to yield the following expression:

(16)

L W C l l A W X b y

A

i i ii

N

i

N

i i i= + + + < > + − +

+

==∑∑0 5

2

11. ( ( ) ( )) [ , ( )]* *ξ ξ ε ξ

iii

N

i i i

i i i ii

N

y W X b* *

* *

[ , ( )]

( )

=

=

∑

∑

− < > − − +

+ +

1

1

ε ξ

η ξ η ξ

After some simplification, the dual Lagrangian can be written as the following Equation 17 (p. 45):



(17)

min L = − − < >

+ − −

==

=

∑∑0 511

. ( )( ) ,

[ ( )

* *

*

A Ai A A X X

A A y

ij

N

i

N

j j i j

i i ii

ε11

1

N

i i

ii

ii

i

N

ii

i

A A

Cl

l Cl

∑

∑

−

+∂∂

−

+

∂

∂=

( )]

( )( )

( )

*

**

ξξξ

ξ ξξ

ξ***

(*)

(*)(

( )

( )

(

−

− =

≤ ≤∂

=

=

∑

∑

l

A A

A Cl

ii

N

i

N

i i

ii

ξ

ξ

1

10

0

S.T.

**)

(*)

(*)(*)

(*)

(*)

)

inf( )

∂

=∂

∂≥

ξ

ξ ξξ

ξ

i

i

i

Cl

A

Equation 17 (p. 45) is a quadratic constrained optimization problem, and the design variables are the vectorA. Once this is computed, the constant b in Equation 11 (p. 43) can be obtained by the application of Karush-Kuhn-Tucker (KKT) conditions. Once the ε-insensitive loss functions are used instead of the generic lossfunctions l(ξ) in Equation 17 (p. 45), this can be rewritten in a much simpler form as the following:

(18)

MIN L A A A A K X X

A

A A ij

N

i

N

i j j i j

ii

N

,

* *

*

* . ( )( ) ( , )

[ (

= − −

+ ∑

==

=

∑∑0 511

1ε ++ − −

≤ ≤

∑ − ==

A yi A A

A C

A A

i i i

i

ii

N

i

) ( )]

( )

*

(*)

*

S.T.

0

01

Which is be solved by a QP optimizer to yield the Lagrange multipliers Ai

(*)

and the constant b is obtainedby applying the KKT conditions.

Suggested Best Practices

The choice of the most optimal Non-Parametric Regression meta model is done in terms of the selection ofthe optimal hyperparameters, i.e. the tube diameter ε and the Gaussian kernel density "k" of the kernelfunction chosen. The sensitivity of the meta model to the hyperparameter "C" is not very prominent. However,in the current version, the hyperparameters are fixed to constant values and are not optimized. The valuesare determined from a series of benchmark tests and strike a compromise between the meta model accuracyand computational speed. For a large family of problems, the current settings of the hyperparameters willprovide good results; however, for some problem types (like ones dominated by flat surfaces or lower orderpolynomials), some oscillations may be noticed between the DOE points. The choice of Gaussian radial basisfunctions as kernel functions also contributes to these oscillations.




In order to circumvent this, you can use a larger number of DOE points or, depending on the fitness landscapeof the problem, use one of the several optimal space filling DOEs provided. In general, it is suggested thatthe problems first be fitted with a quadratic response surface, and the NPR fitting adopted only when thegoodness of fit metrics from the quadratic response surface model is unsatisfactory. This will ensure thatthe NPR is only used for problems where low order polynomials do not dominate.

Neural Network

This mathematical technique is based on the natural neural network in the human brain.

In order to interpolate a function, we build a network with three levels (Inp, Hidden, Out) where the connec-tions between them are weighed, like this:

Each arrow is associated with a weight (w) and each ring is called a cell (like a neural).

If the inputs are xi, the hidden level contains function gj (xi) and the output solution is also:

f x K w g xk i jk j i( ) ( ( ))= ∑

where K is a predefined function, such as the hyperbolic tangent or an exponential based function, in orderto obtain something similar to the binary behavior of the electrical brain signal (like a step function). Thefunction is continuous and differentiable.

The weight functions (wjk) are issued from an algorithm which minimizes (as the least squares method) thedistance between the interpolation and the known values (design points). This is called the learning. Theerror is checked at each algorithm iteration with the design points which are not used for the learning. Weneed to separate learning design points and error checking design points.

The error decreases and then increases when the interpolation order is too high. The minimization algorithmis stopped when the error is the lowest.

This method uses a limited number of design points to build the approximation. It works better when thenumber of design points and the number of intermediate cells are high. And it can give interesting resultswith several parameters.



Meta Model Refinement

Once a meta model is solved, you can attempt to improve it by using one of the following refinement pro-cedures. Review the goodness of fit information in the Properties view for each output parameter to evaluatewhether the refinement was successful.

Change Meta Model TypesRefine the Model Using the Kriging Meta Model

Change Meta Model Types

Changing meta model types or changing the available options for the current meta model can improve thegoodness of fit. To change the meta model type, select the Response Surface cell in the Outline view.Change the Response Surface Type in the Properties view. Click Update Response Surface in the toolbarto run an analysis using the new meta model. Look at the goodness of fit information for the output para-meters to see how the refinement improved the fitting.

Refine the Model Using the Kriging Meta Model

You can use the Kriging Meta Model to perform several different types of refinement for the Response Surface.Interactive and Auto Refinement are not supported for discrete parameters. In order to run a refinement,you must first run an analysis of the response surface using the Kriging Meta Model with No Refinement

selected for Refinement Type. After the initial analysis, use the different refinement types as described inthe following sections.

Interactive Refinement

The following Properties can be set for Kriging Interactive refinement.

• Maximum Number of Points: Maximum number of design points that will be generated during therefinement.

• Maximum Relative Error: Can be set between 1% - 99%. If the user sets 5% as the maximum relativeerror, it means that he will accept a Response Surface with an error less than 5%.

• Output Parameter: Select an output parameter. Interactive refinement can be associated with only oneoutput parameter. If the user needs to refine the response surface for several output parameters, heneeds to do several interactive refinements.

The interactive refinement can be done in one step (Update Response Surface) or in several steps:

1. Click the Preview Response Surface button in the toolbar to see the new design points that are goingto be inserted (only input parameter values).

2. If the user wants to delete or modify certain points before computing them, he can edit the table. Todo so, change the refinement type from Interactive to Manual, which makes the table editable.

3. Click Update Response Surface in the toolbar to complete the refinement.

The design points generated for the refinement will appear in the Response Surface Table view under Design

Set of Response Surface. If the response surface is accurate enough, interactive refinement does not providenew points to improve the response surface.

Auto Refinement

Auto refinement enables the user to refine the Kriging response surface automatically by choosing certainrefinement criteria and allowing an automated optimization-type procedure to add more DOE points to the



Auto Refinement

design domain where they are most needed. The number and the spread of the DOE points are determinedby the following Response Surface Properties.

• Maximum Number of Points: Maximum number of Design Points that will be generated during therefinement.

• Error Minimization Criteria:

– Maximum Output - the maximum mean square error of all output parameters is minimized.

– All Outputs - the maximum product of the output and the mean square error is minimized.

– Sum of Outputs - the maximum sum of the output and the mean square error is minimized.

• Output Variable Combination:

– Mean Square Error - only the output with the largest mean square error is considered.

– Sum Of Outputs - mean square error of all outputs are considered, thus multiple DOE points aregenerated in each iteration depending on the number of outputs.

– Product Of Outputs - combined mean square error of all outputs is considered.

• Crowding Distance Separation Percentage: Minimum distance between 2 new DOE points in the inputspace, implemented as a constraint in the search for the DOE points.

When the refinement type is Auto, the output parameters will have an additional option, Stop Criterion,added to their properties. If the Stop Criterion box is checked for an output, a Relative Error property canbe set that corresponds to the maximum relative error that the user will accept for that parameter (0.01 to0.99).

The Relative Error works in the following manner. At each step of the automatic refinement:

1. The refinement finds new Design Points to improve the response surface; the output parameter valuesare evaluated with the previous response surface.

2. The real values of the output parameters are computed.

3. The Response Surface can be updated with these new values.

4. For direct output parameters with Stop Criterion enabled, the values (1) are compared with the realvalues (2). If the relative error is greater than the Relative Error specified, then the auto refinement iscontinued (need to add new a Design Point -> go to step 1).

Once you have entered all of the properties, click Update Response Surface in the toolbar to complete therefinement. The design points generated for the refinement will appear in the Response Surface Table viewunder Design Set of Response Surface.

The automatic refinement stops when the maximum number of Design Points is reached or if the responsesurface is accurate enough for direct output parameters with the Stop Criterion enabled. If there is no directoutput parameter with Stop Criterion enabled, automatic refinement executes all Design Points.

Manual Refinement

If you set the refinement type to Manual you can add, delete, or modify the additional design points usedto generate the response surface. If you choose manual after doing an auto or interactive refinement, youcan edit the points that were generated from those refinements. If you do a manual refinement as yourinitial refinement, you can enter new as many new design points as you would like into the Table view.



Goodness of Fit

You can view goodness of fit information for any of the output parameters in a response surface. To do so,edit the Response Surface cell from the Project Schematic. Then click on one of the output parameter cellsin the Outline view. The Goodness-of-Fit Information section will be displayed in the Properties view.The following Goodness of Fit information is displayed:

• Coefficient of Determination: The percent of the variation of the output parameter that can be explainedby the response surface regression equation. That is, coefficient of determination is the ratio of the ex-plained variation to the total variation.

The design points used to create the response surface are likely to contain variation for each outputparameter (unless all the output values are the same, which will result in a flat response surface). Thisvariation is explained by the response surface that the DOE generates. If the response surface were topass directly through each design point, the coefficient of determination would be 1 or 100%, meaningthat all variation is explained.

• Adjusted Coefficient of Determination: The Adjusted Coefficient of Determination takes the samplesize into consideration when computing the Coefficient of Determination. Usually this is more reliablethan the usual coefficient of determination when the number of samples is small ( < 30).

• Maximum Relative Residual: The maximum distance (relatively speaking) out of all of the generateddesign points from the calculated response surface to each generated design point.

• Root mean square error: This is the square root of the average square of the residuals at the DOEpoints for regression methods. For Kriging based methods, this is done on a set of testing points andthe prediction error is used instead.

• Relative Root Mean Square Error: This is the square root of the average square of the residuals scaledby the actual output values at the DOE points for regression methods. For Kriging based methods, thisis done on a set of testing points and the relative prediction error is used instead.

• Relative Maximum Absolute Error: This is the absolute maximum residual value relative to thestandard deviation of the actual outputs.

• Relative Average Absolute Error: This is the absolute maximum residual value relative to the standarddeviation of the actual output data, modified by the number of samples. This is useful when the numberof samples is low ( < 30).

Response Surface Charts

The various Response Surface charts allow you to graphically view the impact that parameters have on oneanother. There are currently three types of charts available: response charts, spider charts, and local sensit-ivity charts. When you initially edit a Response Surface, you will see one of each of the chart types. For eachchart type, the parameter values for the current response point are shown in the Properties view of thechart, initially.

When you click on a response point cell in the Outline view, the toolbox will display all available chart typesfor the cell. You can drag a chart template out of the toolbox and drop it on the response point to add an-other chart of that type. You can have as many charts as you like under a response point. You can changethe name of a chart cell by clicking on it twice and entering the new name. (This will not affect the title ofthe chart, which is set as part of the chart properties.) Charts can also be saved as graphics by right clickingon them and selecting Save Image As.... See Saving a Chart for more information.

Each chart provides the ability to visually explore the parameter space by using the input parameter slidersavailable in the chart's Properties view. These sliders allow you to modify an input parameter value and



Response Surface Charts

view its effect on the displayed output parameter. Spider charts for Response Points are available from theOutline view. Spider charts are useful for visualizing all output parameter values simultaneously.

You can change the parameter values to explore other designs by moving the slider bars, or entering spe-cific values in the accessible value boxes.

The triad control at the bottom left in the 3D Response chart window allows you to rotate the chart infreehand mode or quickly view the chart from a particular plane. To zoom any part of the chart, use thenormal zooming controls (Shift-middle mouse button, or scroll wheel). See Using the Triad for more inform-ation on chart manipulation.

Using the Response Chart

The Response chart can be viewed as a 2-D or 3-D contour graph. In the Properties view for the chart, setthe Chart Mode to be 2D or 3D, and choose the parameters to be shown on the X, Y, and Z (for 3D) axes.Use the sliders to explore how changing the values of the parameters that are not displayed on the chartaffect the shape and position of the curve.

Also indicated in the Properties view are the values of the remaining output parameters. Thus, instantaneousdesign and analysis is possible, leading to the generation of additional design points.

Viewing a 2-D Contour Graph

A 2-D contour graph is a flat 3-D response chart. To access this feature, set the Chart Mode to be 2D, andchoose the parameters to be shown on the X and Y axes.

An example of a 2-D response chart is shown below.




Set the Chart > Mode to be 3D, and choose the parameters to be shown on the X, Y, and Z axes. A smooth3-D contour of Z versus X and Y is presented. In the Chart view, the 3-D contour can be rotated by clickingand dragging the mouse. Moving the cursor over the response surface shows the values of Z, X, and Y. Thevalues of other input parameters can also be adjusted in the Properties view using the sliders, showing dif-ferent contours of Z (and the corresponding values of other output parameters can be seen in the Propertiesview). Thus, instantaneous design and analysis is possible, leading to the generation of additional designpoints.

The actual appearance of the graph will vary depending on the input parameter pair you select — discretevs. discrete, continuous vs. continuous, or discrete vs. continuous. Two examples of this are shown below.

Using the Spider Chart

Spider charts allow you to visualize the impact that changing the input parameter(s) has on all of the outputparameters simultaneously. When you solve a Response Surface, a Spider chart will appear in the Outline

view for the default Response Point.




You can use the slider bars in the chart's Properties view to adjust the value for the input parameter(s) tovisualize different designs. You can also enter specific values in the value boxes. The parameter legend boxat the top left in the Chart view allows you to select the parameter that is in the primary (top) position. Onlythe axis of the primary parameter will be labeled with values.

Using the Local Sensitivity Chart

Local Sensitivity charts allow you to see the impact of the input parameters on the output parameters. Whenyou solve a Response Surface, a Local Sensitivity chart will appear in the Outline view for the default ResponsePoint. The Local Sensitivity chart can be displayed as a bar chart or a pie chart. At this level, the sensitivitycharts are “Single Parameter Sensitivities.” This means that Design Exploration calculates the change of theoutput based on the change of each input independently at the current value of each input parameter. Thelarger the change of the output, the more significant is the input parameter that was varied. As such, singleparameter sensitivities are local sensitivities. Changing the input parameter values using the sliders in thechart's Properties view will update the sensitivities.



For discrete parameters, you can change the parameter values to explore other designs by moving the sliderbars, or entering specific values in the value boxes.

Local Sensitivity Chart Limitations

• Sensitivity charts are not available if ALL input parameters are discrete.

• Sensitivities are calculated for continuous parameters only.

• Changes in sensitivities due to different discrete parameter combinations are supported.

Global Sensitivities

The sensitivity plots available in Design Exploration allow you to efficiently improve your design toward amore reliable and better quality design, or to save money in the manufacturing process while maintainingthe reliability and quality of your product. You can request a sensitivity plot for any output parameter inyour model.

The sensitivities available under the Six Sigma Analysis and the Goal Driven Optimization views are statisticalsensitivities. Statistical sensitivities are global sensitivities, whereas the single parameter sensitivities availableunder the Responses view are local sensitivities. The global, statistical sensitivities are based on a correlationanalysis using the generated sample points, which are located throughout the entire space of input para-meters. The local single parameter sensitivities are based on the difference between the minimum andmaximum value obtained by varying one input parameter while holding all other input parameters constant.As such, the values obtained for single parameter sensitivities depends on the values of the input parametersthat are held constant. Global, statistical sensitivities do not depend on the settings for the input parameters,because all possible values for the input parameters are already taken into account when determining thesensitivities.

Global Sensitivity Chart Limitations

• Sensitivity charts are not available if ALL input parameters are discrete.

• Sensitivities are calculated for continuous parameters only.

Related Topics:

Using the Local Sensitivity Chart (p. 52)Using the Sensitivities Chart (GDO) (p. 61)Statistical Sensitivities in a Six Sigma Analysis (p. 99)



Related Topics:

Adding Design Points and Response Points

Design Points can be added to the project and Response Points can be added to a response surface tablefrom a number of results views. You can either Insert as Response Point(s), Insert as Design Point(s) or evencopy values to the current Response Point. These options are available from the right mouse button menuon the rows of the Table view, or from various charts. The same mechanism is used, except that availabilityof the three operations depends on the context. Some of the different charts and tables that allow this op-eration are: Optimization candidate table, Samples chart, Tradeoff chart, and Response chart.

Selected points can be stored as Response Points or Design Points. Right click on a table row or chart pointand select either Insert as Response Point(s), Copy Values to Response Point, or Insert as Design Point(s)

from the menu.

• Insert as Response Point(s) inserts new Response Points into the Table of Response Surface bycopying the input and output parameter values of the selected Candidate Design(s).

• Copy Values to Response Point copies the input and output parameter values of the selected CandidateDesign(s) to the current Response Point.

• Insert as Design Point(s) creates new Design Points at the project level (edit the Parameter Set in theSchematic view to list the existing Design Points) by copying the input parameter values of the selectedCandidate Design(s). The output parameter values are not copied because they are approximated valuesprovided by the Response Surface.

The above insertion operations are available from Optimization charts, depending on the context. For instance,it is possible to right click on a point in a Tradeoff chart in order to insert the corresponding sample as aResponse Point or a Design Point. The same operation is available from a Samples chart, unless it is beingdisplayed as a spider chart.

Using Parameters Correlation

This section contains information about using the Parameters Correlation analysis.Running a Parameters CorrelationPurpose of Parameters CorrelationSample GenerationDetermining SignificanceViewing Significance and Correlation Values

Running a Parameters Correlation

To create a Parameters Correlation system on your project schematic, do the following:

1. With the Project Schematic displayed, drag a Parameters Correlation template from the Design Ex-ploration area of the toolbox directly under the Parameter Set bar of the schematic.

2. After placing the Parameters Correlation system on the Schematic, double click on the ParametersCorrelation cell to open the workspace and set up the correlation options that are described in Para-

meters Correlation (p. 20).

3. Click the Update Parameters Correlation button on the toolbar. When the update is finished, examinethe results using the various charts shown in the Outline view. See Parameters Correlation (p. 20) formore information about the available charts.



Purpose of Parameters Correlation

The application of Goal Driven Optimization (GDO) and Six Sigma Analysis (SSA) in a finite element basedanalysis framework is always a challenge in terms of solving time, especially when the finite element modelis large. For example, hundreds or thousands of finite element simulation runs in SSA is not uncommon. Ifone simulation run takes hours to complete, it is almost impractical to perform SSA at all with thousands oreven hundreds of simulations.

To perform GDO or SSA in finite element based analysis framework, it is always recommended to performa Design of Experiments (DOE) study. From the DOE study, a response surface is built within the designspace of interest. After a response surface is created, all simulation runs of GDO or SSA will then becomefunction evaluations.

In the DOE study, however, the sampling points increase dramatically as number of input parameters increases.For example, a total of 149 sampling points (finite element evaluations) are needed for 10 input variablesusing Central Composite Design with fractional factorial scheme. As the number of input variables increases,the analysis becomes more and more intractable. In this case, one would like to exclude unimportant inputparameters from the DOE sampling in order to reduce unnecessary sampling points. A correlation matrix isa tool to help users identify input parameters deemed to be unimportant, and thus treated as deterministicparameters in SSA.

Sample Generation

The importance of an input to an output is determined from their correlation. The samples used for correlationcalculation are generated with Latin Hypercube Sampling method. The Latin Hypercube samples are generatedin such a way that the correlations among the input parameters are less than or equal to 5%. Also, the LatinHypercube samples are generated such a way that each sample has equal probability of pick (uniformlydistributed). Two correlation calculation methods are provided for the user to select from; Spearman’s rankcorrelation and Pearson’s linear correlation. Pearson’s uses actual data for correlation evaluation whileSpearman’s uses ranks of data, which is deemed to be more accurate, and therefore is recommended.

If Auto Stop Type is set to Enable Auto Stop for the correlation, the number of samples required to calculatecorrelation is determined according to convergences of means and standard deviations of output parameters.At each iteration of simulation runs (finite element runs in this case), convergences of means and standarddeviations are checked against accuracy provided by the user. The iteration stops when accuracy is met, ornumber of samples exceeds the maximum value set by the user, whichever comes first.

Determining Significance

Significance of an input parameter to an output parameter is determined using a statistical hypothesis test.In the hypothesis test, a NULL hypothesis of insignificance is assumed, and is tested if it is statistically trueaccording to significance level (or acceptable risk) set by the user. From the hypothesis test, a p-Value(probability value) is calculated, and is compared with the significance level. If the p-Value is greater thanthe significance level, it is concluded that the NULL hypothesis is true, and that the input parameter is insig-nificant to the output parameter, and vice versa. See the Six Sigma Theory section for more information.

Viewing Significance and Correlation Values

If you select Sensitivities from the Outline view in the Six Sigma workspace, you can review the sensitivitiesderived from the samples generated for the Parameters Correlation. The Parameters Correlation sensitivitiesare global sensitivities. In the Properties view for the Sensitivities chart, you can choose the output para-meters for which you want to review sensitivities, and the input parameters that you would like to evaluatefor the output parameters.



Viewing Significance and Correlation Values

The default setting for the Significance Level (found in the Design Exploration section of the Tools >

Options dialog) is 0.025. Parameters with a sensitivity above this significance will be shown with a flat lineon the Sensitivity chart, and the value displayed for those parameters when you mouse over them on thechart will be 0. In order to view the actual correlation value of the insignificant parameter pair, you caneither select the Correlation Matrix from the Outline and mouse over the square for that pair in the matrix,or you can set the Significance Level to 1, which bypasses the significance test and displays all inputparameters on the Sensitivities chart with their actual correlation values.

Using Goal Driven Optimization

This section contains information about running a Goal Driven Optimization analysis. For more information,see "Understanding Goal Driven Optimization" (p. 65).

Optimization OptionsDefining Design ObjectivesCreating New Design Points and Response Points from CandidatesGoal Driven Optimization Charts

Optimization Options

To create a GDO system on your project schematic, do the following:

1. With the Project Schematic displayed, drag a Goal Driven Optimization template from the DesignExploration area of the toolbox to the schematic. You may drag it:

• directly under the Parameter Set bar, in which case it will not share any data with any other systemson the schematic.

• onto the Design of Experiments cell of a Response Surface, in which case it will share all of thedata generated by the DOE.

• onto the Response Surface cell of a Response Surface, in which case it will share all of the datagenerated for the DOE and Response Surface cells.

2. After placing the GDO system on the Schematic, if you are not sharing the DOE and Response Surfacecells, edit the DOE and set it up as described previously in Design of Experiments (p. 18) and solve theDOE and Response Surface cells.

3. Edit the Optimization cell. Make sure that cell A2 is selected in the Outline view, and set up the optionsfor your optimization in the Properties view.

The following optimization options are available.ScreeningMOGANLPQL

Screening

The Screening option allows you to generate a new sample set and sort its samples based on objectives. Itis available for all types of input parameters. Usually the Screening approach is used for preliminary design,which may lead you to apply the MOGA or NLPQL options for more refined optimization results. See thePrinciples section under Goal Driven Design for more information.

To perform a Screening optimization, in the Properties view of the Optimization:

1. Enter the following settings in the Properties view:

• Select Screening for the Optimization Method.



• Enter the Number of Initial Samples to generate for the optimization.

• Specify the Constraint Handling (GDO). This option can be used for any optimization applicationand is best thought of as a "constraint satisfaction" filter on samples generated from the Screening,MOGA, or NLPQL runs. This is especially useful for Screening samples to detect the edges of solutionfeasibility for highly constrained nonlinear optimization problems. The following choices are available:

– As Goals (default): Implies that the upper, lower, and equality constrained objectives of thecandidate designs shown in the Table of Optimization are treated as goals; thus any violationof the objectives is still considered feasible.

– As Hard Constraints: When chosen, the upper, lower and equality constrained objectives aretreated as hard constraints; that is, if any of them are violated then the candidate is no longerdisplayed. So, in some cases no candidate designs may be displayed depending on the extentof constraint violation.

2. Set input and output parameter Objectives in the Table of Optimization view. See Defining Design

Objectives (p. 59) for more information.

3. Click the Update Optimization toolbar button.

The result is a group of points or “sample set.” The points that are most in alignment with the objectivesare displayed in the table as the Candidate Designs for the optimization. The result also displays theread-only field Size of Generated Sample Set in the Properties view.

Note

The Tradeoff chart for a screening optimization where some objectives are constraints willshow all of the feasible points (that meet the constraints) as the first Pareto front. Pointsfrom the sample set that do not meet the constraints will be shown as infeasible points.The Samples chart will also indicate the infeasible points by color.

MOGA

The MOGA (Multi-Objective Genetic Algorithm) option allows you to generate a new sample set or use anexisting set for providing a more refined approach than the Screening method. It is available for continuousinput parameters only and can handle multiple goals. See the MOGA (Multi-objective Genetic Algorithm)section for more information on this algorithm.

To perform a MOGA optimization:


• Optimization Method: Set to MOGA.

• Number of Initial Samples: Complete this field if you wish to start from a new set. This numbershould be at least 10 times the number of continuous input parameters, but not more than 300.If you want to use an existing Screening sample set, you cannot edit the Number of Initial Samples

field.

• Number of Samples Per Iteration: The number of samples that are iterated and updated at eachiteration. This setting must be less than or equal to the number of initial samples.

• Maximum Allowable Pareto Percentage: The approximate maximum percentage of first Paretofront points (computed with respect to the Number of Samples Per Iteration) that you wish tocompute. For example, if the Maximum Allowable Pareto Percentage is set to 70 and the Number

of Samples Per Iteration is set to 200, then you can expect to have, at most, approximately 140



Optimization Options

first Pareto front points. If the Maximum Allowable Pareto Percentage is too low (below 30), thenthere is a chance that the iterations will converge slowly. If the value is too high (above 80), thenthe problem may converge prematurely. The value of the Maximum Allowable Pareto Percentage

depends on the number of output and input parameters of the problem, in addition to the natureof the design space itself. Using a value between 55 and 75 works best for most problems.

• Maximum Number of Iterations: The maximum possible number of iterations the algorithm ex-ecutes. If convergence happens before this number is reached, the iterations will cease. This alsoprovides an idea of the maximum possible number of function evaluations that are needed for thefull cycle, as well as the maximum possible time it may take to run the optimization. For example,the absolute maximum number of evaluations is given by: Number of Initial Samples + Number ofSamples Per Iteration * (Maximum Number of Iterations - 1).

• Initial Samples drop down menu allows you to either start a completely new sample generation(Generate Initial Samples) or select from a list of existing Screening samples.

• Constraint Handling (GDO): This option can be used for any optimization application and is bestthought of as a "constraint satisfaction" filter on samples generated from the Screening, MOGA, orNLPQL runs. This is especially useful for Screening samples to detect the edges of solution feasibilityfor highly constrained nonlinear optimization problems. The following choices are available:

– As Goals (default): Implies that the upper, lower, and equality constrained objectives of thecandidate designs shown in the Table of Optimization are treated as goals; thus any violationof the objectives is still considered feasible.

– As Hard Constraints: When chosen, the upper, lower, and equality constrained objectives aretreated as hard constraints; that is, if any of them are violated then the candidate is no longerdisplayed. So, in some cases no candidate designs may be displayed depending on the extentof constraint violation.

2. Set input and output parameter objectives in the Table of Optimization view. At least one of theoutput parameter objectives must be a goal. Multiple output parameter goals can be set. See Defining

Design Objectives (p. 59) for more information.


The result is a group of points or “sample set”. The points that are most in alignment with the objectivesare displayed in the table as the Candidate Designs for the optimization. The result also displays theread-only field Size of Generated Sample Set in the Properties view.

NLPQL

The NLPQL (Non-Linear Programming by Quadratic Lagrangian) option allows you to generate a new sampleset or use an existing set to provide a more refined approach than the Screening method. It is available forcontinuous input parameters only and can only handle one output parameter goal (other output parameterscan be defined as constraints). See the Non-Linear Programming by Quadratic Lagrangian section for moreinformation on this algorithm.

To generate samples and perform a NLPQL optimization:


• Optimization Method: Set to NLPQL.

• Allowable Convergence Percentage: The tolerance to which the Karush-Kuhn-Tucker (KKT) optim-ality criterion is generated during the NLPQL process. (See the NLPQL theory section for more in-formation.) A smaller value indicates more convergence iterations and a more accurate (but slower)solution. A larger value indicates less convergence iterations and a less accurate (but faster) solution.



The typical default is 1.0e-06, which is consistent across all problem types since the inputs, outputs,and the gradients are scaled during the NLPQL solution.

• Maximum Number of Iterations: The maximum possible number of iterations the algorithm ex-ecutes. If convergence happens before this number is reached, the iterations will cease. This alsoprovides an idea of the maximum possible number of function evaluations that are needed for thefull cycle, as well as the maximum possible time it may take to run the optimization. For example,the absolute maximum number of evaluations is given by: Number of Initial Samples + Number ofSamples Per Iteration * (Maximum Number of Iterations - 1).

• Constraint Handling (GDO): This option can be used for any optimization application and is bestthought of as a "constraint satisfaction" filter on samples generated from the Screening, MOGA, orNLPQL runs. This is especially useful for Screening samples to detect the edges of solution feasibilityfor highly constrained nonlinear optimization problems. The following choices are available:

– As Goals (default): Implies that the upper, lower, and equality constrained objective of thecandidate designs shown in the Table of Optimization are treated as goals; thus any violationof the objectives is still considered feasible.

– As Hard Constraints: When chosen, the upper, lower, and equality constrained objectives aretreated as hard constraints; that is, if any of them are violated then the candidate is no longerdisplayed. So, in some cases no candidate designs may be displayed depending on the extentof constraint violation.

2. Set input and output parameter objectives in the Table of Optimization view. See Defining Design

Objectives (p. 59) for more information. Remember, you can only have one output parameter goal forNLPQL.

3. Because NLPQL is a gradient based method, the candidates found depend on the starting point in theparameter space. NLPQL could find a local optimum which is not the global optimum. To give controlto the user, the Optimization table has an additional row titled Initial Value when using NLPQL. Thisallows the user to determine the starting point of the optimization by entering the value of each inputparameter in the Initial Value row.


The result is a group of points or “sample set.” The points that are most in alignment with the objectivesare displayed in the table as the Candidate Designs for the optimization. The result also displays theread-only field Size of Generated Sample Set in the Properties view. This value is always equal to 1for an NLPQL result.

Defining Design Objectives

When you edit the Optimization cell of a Goal Driven Optimization that you've placed on your schematic,you can state a series of design objectives, which will be used to generate optimized designs. You will seea Table of Optimization view which will allow you to set objectives for all input and output parameters asdescribed below.

The optimization approach used in the Design Exploration environment departs in many ways from tradi-tional optimization techniques. Many objectives are allowed and can be assigned to each output parameter.Objectives may be weighted in terms of importance.

This departure from traditional optimization also allows objectives to be set for input parameters, which areindependent variables, allowing flexibility in obtaining the desired design configuration.

For both the input and output parameters, you can also specify an importance for each parameter, withpossible values of Default, Higher, or Lower.



Defining Design Objectives

The following tables show the objectives that may be applied to input and output parameters.

Table 2 Objectives and Their Meanings for Continuous Input Parameters

Mathematical MeaningMeaningGoal

XLower <= X <= XUpperKeep the input parameter within its upperand lower bounds

No Objective

X → XLowerMinimize the input parameter within itsrange

Minimize

X → ½ (XLower + XUpper)Attain the average value of the inputparameter within its range

Seek Midpoint

X → XUpperMaximize the input parameter within itsrange

Maximize

(X = an input parameter, XLower = lower limit, XUpper = Upper Limit)

Table 3 Objectives and Their Meanings for Discrete or Usability Input Parameters

Mathematical MeaningMeaningGoal

XLower <= X <= XUpperUses the full discrete range allowed for theparameter

No Objective

X → XTargetSet the objective to in-range values closeto the target value

Values = Target

X <= XTargetSet the objective to in-range values belowthe target value

Values <= Target

X >= XTargetSet the objective to in-range values abovethe target value

Values >= Target

(X = an input parameter, XLower = lower limit, XUpper = Upper Limit, XTarget = Target Value)

Table 4 Objectives and Their Meanings for Output Parameters

GDO TreatmentMathematical MeaningMeaningGoal

N/AN/ANo objective is specifiedNo Objective

GoalMinimize YAchieve the lowest possiblevalue for the output parameter

Minimize

GoalMaximize YAchieve the highest possiblevalue for the output parameter

Maximize

GoalY → YTargetIf a Target Value is specified,then achieve an output para-

Seek Target

meter value that is close to theTarget Value

Inequality constraintY <= YTargetIf a Target Value is specified,then achieve an output para-

Values <= Tar-get

meter value that is less thanthe Target Value

Inequality constraintY > =YTargetIf a Target Value is specified,then achieve an output para-

Values >= Tar-get

(Y = output parameter, YTarget = Target Value)



GDO TreatmentMathematical MeaningMeaningGoal

meter value that is greaterthan the Target Value

ConstraintY → YTargetIf a Target Value is specified,then achieve an output para-

Values = Tar-get

meter value that is close to theTarget Value

(Y = output parameter, YTarget = Target Value)

When the MOGA or NLPQL options are used for Goal Driven Optimization, the search algorithm uses theObjective and Target Value properties of the output parameters and ignores the similar properties of theinput parameters. However, when the Candidate Designs are reported, the first Pareto fronts (in case ofMOGA) or the best solution set (in case of NLPQL) are filtered through the Decision Support Process, whichapplies the input and output parameter properties like Importance, Target, and Objective, and reports the3 best candidate designs. This implies that the candidate designs may not necessarily correspond to thebest input targets, but definitely reports the optimal designs with respect to the output properties; this isconsistent with the mechanism of optimization.

The Screening option is not strictly an optimization approach; it uses the properties of the input as well theoutput parameters, and uses the Decision Support Process on a sample set generated by the Shifted Ham-mersley technique to rank and report the candidate designs. Thus, the candidate designs in this case corres-pond to both the input and output objectives specified by the user, although the Candidate Designs willnot correspond to the best optimal solutions.

Creating New Design Points and Response Points from Candidates

Once the objectives are stated and Candidate Designs are generated, Candidate Designs are listed in theOptimization table view. They can be stored as Response Points or Design Points. Right click on a CandidateDesign row and select either Insert as Response Point(s) or Insert as Design Point(s) from the menu.

• Insert as Response Point(s) inserts new Response Points into the Table of Response Surface bycopying the input and output parameter values of the selected Candidate Design(s).

• Insert as Design Point(s) creates new Design Points at the project level (edit the Parameter Set in theSchematic view to list the existing Design Points) by copying the input parameter values of the selectedCandidate Design(s). The output parameter values are not copied because they are approximated valuesprovided by the Response Surface.

Both of the above insertion operations are also available from Optimization charts, depending on the context.For instance, it is possible to right click on a point in a Tradeoff chart in order to insert the correspondingsample as a Response Point or a Design Point. The same operation is available from a Samples chart.

Goal Driven Optimization Charts

After solving an Optimization cell, several charts are created by default in the Charts section of the Outlinetree.

Using the Sensitivities Chart (GDO)

The Sensitivities chart shows the global sensitivities of the output parameters with respect to the inputparameters. The chart can be displayed as a bar chart or a pie chart by changing the Mode in the chart's



Goal Driven Optimization Charts

Properties view. You can also select which parameters you want to display on the chart from the Propertiespanel by selecting or deselecting the checkbox next to the parameter in the Properties view.

Various generic chart properties can be changed for this chart.

Note

If the p-Value calculated for a particular input parameter is above the Significance Level specifiedin the Design Exploration section of the Tools > Options dialog, the bar for that parameter willbe shown as a flat line on the chart. See Determining Significance (p. 55) for more information.

Using the Tradeoff Chart

After solving an Optimization cell, several charts are created by default in the Charts folder of the Outlinetree. The Tradeoff chart is a 2D or 3D scatter chart representing the generated samples. The colors appliedto the points represents the Pareto front they belongs to, from red (the worst) to blue (the best points).

The chart is controlled by changing its properties in the Property view.

• The chart can be displayed as in 2D or 3D by changing the Mode in the chart's Properties view.

• You can also select which parameter to display on each axis of the chart by selecting the parameterfrom the drop down menu next to the axis name in the Properties view.

• Limit the pareto fronts shown by moving the slider or entering a value in the field above the slider inthe chart's Properties view.


If there are several equivalent solutions for the defined Optimization Study, the Tradeoff chart can be usedto visualize them. They belong to the first Pareto front which is represented in blue. By exploring the samplesfor different parameters, you can determine the best compromise by varying secondary parameters.

For further information on results, see Postprocessing GDO Results (p. 68).

Using the Samples Chart

The Samples chart has two modes, to display the candidates with the full sample set, or to display thesamples Pareto front by Pareto front (same as the Tradeoff chart). Compared to the Tradeoff chart, theSamples chart has the advantage of showing all the parameters at once, whereas the Tradeoff chart canonly show 3 parameters at a time. The Samples chart is better for exploring the parameter space. With theSamples chart axis sliders, you can filter for each parameter very easily, providing an intuitive way to explorethe alternative designs.


Using Six Sigma Analysis

This section contains information about running a Six Sigma Analysis. In this section you will find informationon the following:

• Performing a Six Sigma Analysis (p. 63)

• Using Statistical Postprocessing (p. 63)

For more information, see Six Sigma Analysis (p. 26) and "Understanding Six Sigma Analysis" (p. 87).



Performing a Six Sigma Analysis

To perform a six sigma analysis, you must first add a Six Sigma Analysis template to your schematic fromthe Design Exploration section of the toolbox. You then edit the Design of Experiments (SSA) cell of theSix Sigma Analysis and specify which input parameters to evaluate, assign distribution functions to themas described in Setting Up Uncertainty Variables (p. 31), and specify the distribution attributes (this will gen-erate your samples for each input variable). Once this is completed, the procedure for a Six Sigma Analysisis:

Solve the Six Sigma Analysis by:

• updating each individual cell in the analysis either from the right menu in the Project Schematic orthe Update button in the workspace toolbar while editing the cell.

• right clicking on the Six Sigma Analysis cell in the project schematic and selecting Update to updatethe entire analysis at once.

• clicking the Update Project button in the toolbar to update the entire project.

Using Statistical Postprocessing

From the Six Sigma Analysis workspace you can see the statistics in the Properties view, the probabilitytables in the Table view (Tables (SSA) (p. 63)), and statistics charts in the Chart view (Using Parameter Charts

(SSA) (p. 63)) associated with each parameter by selecting a parameter in the Outline. You can open theSensitivities chart by selecting it from the Charts section of the Outline (see Using the Sensitivities Chart

(SSA) (p. 63)).

Tables (SSA)

In the Six Sigma Analysis workspace, you are able to view the probability tables of any input or outputvariable. To select which table view is displayed, select the desired value (Quantile-Percentile or Percentile-

Quantile) for the Probability Table field in the parameter's Properties view.

You can modify both types of tables by adding or deleting values. To add a value to the Quantile-Percentiletable, type the desired value into the New Parameter Value cell at the end of the table. A row with thevalue you entered will be added to the table in the appropriate location. To add a new value to the Percentile-Quantile table, type the desired value into the appropriate cell (New Probability Value or New Sigma Level)at the end of the table. To delete a row from either table, right click on the row and select Remove Level

from the menu. The row will be deleted from the table. You can also overwrite any value in an editablecolumn and corresponding values will be displayed in the other columns in that row.

Using Parameter Charts (SSA)

You can review the statistical results of the analysis by selecting an input or output parameter to view itsChart. The results of a Six Sigma Analysis are visualized using histogram plots and cumulative distributionfunction plots.


Using the Sensitivities Chart (SSA)

If you select Sensitivities from the Outline view in the Six Sigma workspace, you can review the sensitivitiesderived from the samples generated for the Six Sigma Analysis. The Six Sigma Analysis sensitivities areGlobal Sensitivities (p. 53), not Using the Local Sensitivity Chart (p. 52). In the Properties view for the Sensitiv-



Using Statistical Postprocessing

ities chart, you can choose the output parameters for which you want to review sensitivities, and the inputparameters that you would like to evaluate for the output parameters.


Note

If the p-Value calculated for a particular input parameter is above the Significance Level specifiedin the Design Exploration section of the Tools > Options dialog, the bar for that parameter willbe shown as a flat line on the chart. See Determining Significance (p. 55) for more information.

Statistical Measures

When the Six Sigma Analysis is updated, the following statistical measures are displayed in the Properties

view for each parameter

• Mean

• Standard Deviation

• Skewness

• Kurtosis

• Shannon Entropy (Complexity)

• Signal-Noise Ratio (Smaller is Better)

• Signal-Noise Ratio (Nominal is Best)

• Signal-Noise Ratio (Larger is Better)

• Sigma Minimum

• Sigma Maximum



Understanding Goal Driven Optimization

Goal Driven Optimization (GDO) is a set of constrained, multi-objective optimization (MOO) techniques inwhich the "best" possible designs are obtained from a sample set given the goals you set for parameters.There are three available optimization methods: Screening, MOGA, and NLPQL. MOGA and NLPQL can onlybe used when all input parameters are continuous.

The GDO process allows you to determine the effect on input parameters with certain objectives appliedfor the output parameters. For example, in a structural engineering design problem, you may want to de-termine which set of designs (in terms of geometric problem dimensions and material types) best satisfyminimum mass, maximum natural frequency, maximum buckling and shear strengths, and minimum cost,with maximum value constraints on the von Mises stress and maximum displacement.

This section describes the GDO system and its use in performing MOO.

Principles (GDO)

GDO can be used for design optimization in three ways: the Screening approach, the MOGA approach, orthe NLPQL approach. The Screening approach is a non-iterative direct sampling method by a quasi-randomnumber generator based on the Hammersley algorithm. The MOGA approach is an iterative Multi-ObjectiveGenetic Algorithm, which can optimize problems with continuous input parameters. NLPQL is a gradientbased single objective optimizer which is based on quasi-Newton methods.

MOGA is better for calculating the global optima while NLPQL is a gradient-based algorithm ideally suitedfor local optimization. So you can start with Screening or MOGA to locate the multiple tentative optima andthen refine with NLPQL to zoom in on the individual local maximum or minimum value. Problems withmixed parameter types (i.e., usability, discrete, or scenario parameters with continuous parameters) or discreteproblems cannot currently be handled by the MOGA or NLPQL techniques, and in these cases you will onlybe able to use the Screening technique.

The GDO framework uses a Decision Support Process (DSP) based on satisficing criteria as applied to theparameter attributes using a weighted aggregate method. In effect, the DSP can be viewed as a postprocessingaction on the Pareto fronts as generated from the results of the MOGA, NLPQL, or Screening process.

Usually the Screening approach is used for preliminary design, which may lead you to apply the MOGA orNLPQL approaches for more refined optimization results.

In either approach, the TradeOff chart, as applied to the resulting sample set, shows the Pareto-dominantsolutions. However, in the MOGA approach, the Pareto fronts are better articulated and most of the feasiblesolutions lie on the first front, as opposed to the usual results of the Screening approach where the solutionsare distributed across all the Pareto fronts. This is illustrated in the following two figures. Figure : 6,000 Sample

Points Generated by Screening Method (p. 66) shows the sample as generated by the Screening method,and Figure : Final Sample Set After 5,100 Evaluations by MOGA Method (p. 66) shows a sample set generatedby the MOGA method for the same problem.



Figure: 6,000 Sample Points Generated by Screening Method

Figure: Final Sample Set After 5,100 Evaluations by MOGA Method

Figure : Pareto Optimal Front Showing Two Non-dominated Solutions (p. 66) shows the necessity of generatingPareto fronts. The two axes represent two output parameters that are conflicting by nature. Two optimalsolutions, marked "1" and "2," are shown. Both of these solutions are "non-dominated." That is, they areequally good in terms of Pareto optimality. Point 1 corresponds to a better solution in terms of the outputmarked "v." Point 2 corresponds to a better solution in terms of the output marked "buck."

Figure: Pareto Optimal Front Showing Two Non-dominated Solutions

Guidelines and Best Practices (GDO)

Typically, the use of NLPQL is suggested for continuous problems when there is only one objective function.The problem may or may not be constrained and must be analytic (i.e. must be defined only by continuous



input parameters, and the objective functions and constraints should not exhibit sudden 'jumps' in theirdomain.) The main difference from MOGA lies in the fact that MOGA is designed to work with multiple ob-jectives and does not require full continuity of the output parameters. However, for continuous single ob-jective problems, the use of NLPQL gives greater accuracy of the solution as gradient information and linesearch methods are used in the optimization iterations. MOGA is a global optimizer designed to avoid localoptima traps and NLPQL is a local optimizer designed for accuracy.

The default convergence rate for the NLPQL algorithm is set to 1.0E-06. This is computed based on the(normalized) Karush-Kuhn-Tucker (KKT) condition. This implies that the fastest convergence rate of the de-rivatives or the functions (objective function and constraint) determine the termination of the algorithm.The advantage of this approach is that for large problems, it is possible to get a near-optimal feasible solutionquickly without being trapped into a series of iterations involving small solution steps near the optima.However, if the user prefers a more numerically accurate solution which may involve these iterations, thenthe convergence rate can be set to as low as 1.0E-12. The range of (1.0E-06 - 1.0E-12) covers a vast majorityof all optimization problems and the choice of the convergence rate is based on the user's preference.

At least one of the output parameters should have an Objective of Maximize, Minimize, or Seek Target

in order to do optimization with the MOGA or NLPQL methods (only one output can have an objective forthe NLPQL method).

If this is not done, then the optimization problem is either undefined (Objective for each parameter set toNo Objective) or is merely a constraint satisfaction problem (Objective set to >= Target or <= Target or= Target . When the problem is not defined, as in Figure : Case Where the Optimization Cannot be Run (p. 67),the MOGA or NLPQL analysis cannot be run. If the problem defined is only one of constraint satisfaction,as in Figure : Case Where GDO Solves a Constraint Satisfaction Problem (p. 68), the single Pareto fronts displayed(from the Tradeoff chart) represents the feasible points that meet the constraints. The infeasible points thatdo not satisfy the constraints can also be displayed on the chart. This is a good way to demonstrate thefeasibility boundaries of the problem in the output space (i.e., to determine the boundaries of the designenvelope that reflects your preferences). Infeasible points can also be displayed on any chart where theoptimization contains at least one constraint objective.

As shown in Figure : Case Where the Optimization Cannot be Run (p. 67), if the entire Objective column is setto "No Objective," the optimization process cannot be run, as no outputs have goals specified which coulddrive the algorithm. The fact that user input is required before the algorithm can be run is reflected bothby the state icon and the Quick Help of the Optimization cell on the Schematic view. If you try to updatethe optimization with no goals set, you will also get an error message in the Message view.

Figure: Case Where the Optimization Cannot be Run

As shown in Figure : Case Where GDO Solves a Constraint Satisfaction Problem (p. 68), some combinations ofObjective and Target Value settings specify that the problem is one of constraint satisfaction. The resultsobtained from these settings indicate only the boundaries of the feasible region in the design space. Currentlythe MOGA and NLPQL options cannot be used to solve a pure constraint satisfaction problem.



Guidelines and Best Practices (GDO)

Figure: Case Where GDO Solves a Constraint Satisfaction Problem

The Target and Objective settings of the input parameters do not affect the sample generation part of theMOGA and NLPQL methods. However, these settings will affect the ranking of the samples which generatethe three candidate designs, if the Optimization is updated.

Sample generation using the Screening method is not affected by any parameter setting. However, theparameter settings do still affect the generation of candidate designs.

Setting the Objective to Maximize, Minimize , or Seek Target, will always treat the objective as a goal.Setting the Objective to = Target, >= Target or <= Target treats the objectives as constraints if Constraint

Handling (GDO) in the Optimization Properties view is set to As Hard Constraints. If Constraint Handling

(GDO) is set to As Goals, those objectives are treated as goals and not constraints. If all objectives are con-straints, Screening is your only optimization option. If at least one of the objectives is a goal, MOGA andNLPQL are optimization options.

Typically, the Screening method is best suited for conducting a preliminary design study, because it is alow-resolution, fast, and exhaustive study that can be useful in locating approximate solutions quickly. Thesolutions may then be used as a starting sample for the MOGA and NLPQL methods. Since the Screening

method does not depend on any parameter objectives, you can change the objectives after performing theScreening analysis to view the candidates that meet different objective sets, allowing you to quickly performpreliminary design studies. It's easy to keep changing the goals and constraints to view the different corres-ponding candidates, which are drawn from the original sample set.

For example, you may run the Screening process for 3,000 samples, then use the Tradeoff or Samples

charts to view the Pareto fronts. The solutions slider can be used in the chart Properties to display only theprominent points you may be interested in (usually the first few fronts). When you do a MOGA optimization,you can limit the number of Pareto fronts that are computed in the analysis.

Once all of your objectives are defined, click Update Optimization to generate up to three candidate designs.You may save any of the candidates by right-clicking and selecting Insert as Response Point(s) or Insert

as Design Point(s) . See Adding Design Points and Response Points (p. 54) for more information.

Because the GDO algorithm works off of response surfaces, you should validate the best obtained candidateresults by saving the corresponding Design Points, solving them at the Project level, and comparing theresults.

Postprocessing GDO Results

The following topics are discussed in this section:Postprocessing OverviewUnderstanding Sensitivities ChartsUnderstanding Tradeoff ChartsUnderstanding Samples Charts



Postprocessing Overview

Postprocessing of GDO optimization results is explained below. The postprocessing phase follows the gen-eration of samples. Once you have Updated your optimization, candidates are displayed in the table andone instance of each available chart is created for you in the outline (you can create more charts by double-clicking in the toolbox). Select a chart and view the results, changing the chart's options to see differentviews of the parameter space.

Understanding Sensitivities Charts

The sensitivities charts available for Goal Driven Optimization display statistical, global sensitivities. For moreinformation about the Sensitivities charts, see:


• Using the Sensitivities Chart (GDO) (p. 61)

• Sensitivities Chart (GDO) (p. 26)

Understanding Tradeoff Charts

In theTradeoff chart produced after the optimization analysis, the samples are ranked by non-dominatedPareto fronts and the tradeoffs which concern you may be viewed. To make sense of the true nature of thetradeoffs, the plots must be viewed with the output parameters as the axes. This shows which output goalscan be achieved and whether this entails sacrificing the goal attainment of other outputs. Typically, a tradeoffchart shows you a choice of possible, non-dominated solutions from which to choose.

When an optimization is updated, you can view the best candidates (up to three) from the sample set basedon the stated objectives. However, these results are not truly representative of the solution set, as this ap-proach obtains results by ranking the solution by an aggregated weighted method. Schematically, this rep-resents only a section of the available Pareto fronts. Changing the weights (using the Importance optionin the Optimization Table) will display different such sections. This postprocessing step helps in selectingsolutions if you are sure of your preferences for each parameter.

Figure : Typical TradeOff Study of Sample Showing First Pareto Front (p. 70) shows the results of the tradeoffstudy performed on a MOGA sample set. The first Pareto front (non-dominated solutions) is shown as bluepoints on the output-axis plot. The slider in the Properties view can be moved to the right to add morefronts, effectively adding more points to the tradeoff chart. The additional points added this way are inferiorto the points in the first Pareto front in terms of the goals you specified, but in some cases, where there arenot enough first Pareto front points, these may be necessary to obtain the final design. You can right-clickon individual points and save them as Design Points or Response Points.



Understanding Tradeoff Charts

Figure: Typical TradeOff Study of Sample Showing First Pareto Front

It is also important to discuss the representation of feasible and infeasible points (infeasible points areavailable if any of the objectives are defined as constraints). The MOGA algorithm always ensures thatfeasible points are shown as being of better quality than the infeasible points, and different markers areused to indicate them in the chart. In the 2-D and 3-D tradeoff chart, rectangles denote feasible points andcirlcles denote infeasible ones. You can enable or disable the display of infeasible points in the Chart Prop-erties view.

Also, in both types of charts, the best Pareto front is blue and the fronts gradually transition to red (worstPareto front). Figure : Two-dimensional TradeOff Plot Showing Feasible and Infeasible Points (p. 70) is a typical2-D tradeoff chart with feasible and infeasible points.

Figure: Two-dimensional TradeOff Plot Showing Feasible and Infeasible Points

Understanding Samples Charts

The Samples chart is a postprocessing feature that allows you to explore a sample set given defined goals.After solving an Optimization cell, a Samples chart will appear in the Outline under Charts.

The aim of this chart is to provide a multidimensional graphical representation of the parameter space youare studying. The chart uses the parallel Y axes to represent all of the inputs and outputs. Each sample isdisplayed as a group of line curves where each point is the value of one input or output parameter. Thecolor of the curve identifies the Pareto front that the sample belongs to, or the chart can be set so that thecurves display the best candidates and all other samples.

The Samples chart is a powerful exploration tool because of its interactivity. It allows you to first pick a Yaxis with the mouse then slide the yellow arrows that appear on the bottom and the top of each axis up ordown in order to increase or decrease the axis bounds. Samples are dynamically hidden if they fall outside



of the bounds. Repeating the same operation with each axis allows you to manually explore and find trade-offs.

Goal Driven Optimization Theory

In this section, theoretical aspects of goal driven optimization are briefly discussed.

• Shifted Hammersley Sampling Method (p. 71)

• Pareto Dominance in Multi-Objective Optimization (p. 72)

• MOGA (Multi-Objective Genetic Algorithm) (p. 73)

• Decision Support Process (p. 73)

• Non-linear Programming by Quadratic Lagrangian (NLPQL) (p. 77)

Shifted Hammersley Sampling Method

The Shifted Hammersley method is the sampling strategy used for all sample generation, except for theNLPQL method. The conventional Hammersley sampling algorithm is a quasi-random number generatorwhich has very low discrepancy and is used for quasi-Monte Carlo simulations. A low-discrepancy sequenceis defined as a sequence of points that approximate the equidistribution in a multi-dimensional cube in anoptimal way. In other words, the design space is populated almost uniformly by these sequences and, dueto the inherent properties of Monte Carlo sampling, dimensionality is not a problem (i.e., the number ofpoints does not increase exponentially with an increase in the number of input parameters). The conventionalHammersley algorithm is constructed by using the radical inverse function. Any integer n can be representedas a sequence of digits n0, n1, n2, ..., nm by the following equation:

(1)n n n n n nm= 0 1 2 3⋯

For example, consider the integer 687459, which can be represented this way as n0 = 6, n1 = 8, and so on.Because this integer is represented with radix 10, we can write it as 687459 = 9 + 5 * 10 + 4 * 100 and soon. In general, for a radix R representation, the equation is:

(2)n n n R nm m= + + +−1 0* ⋯

The inverse radical function is defined as the function which generates a fraction in (0, 1) by reversing theorder of the digits in Equation 2 (p. 71) about the decimal point, as shown below.

(3)ΦR m m m

m mm

n n n n n

n R n R n R

( )

* * * ( )

=

= + + +

− −−

−− − −

0 1 2 0

11

20

1

⋯

⋯

Thus, for a k-dimensional search space, the Hammersley points are given by the following expression:

(4)H i i N i i ik R R Rk( ) [ / , ( ), ( ), , ( )]= −Φ Φ Φ1 2 1⋯

where i = 0, ..., N indicates the sample points. Now, from the plot of these points, it is seen that the first row(corresponding to the first sample point) of the Hammersley matrix is zero and the last row is not 1. Thisimplies that, for the k-dimensional hypercube, the Hammersley sampler generates a block of points that are



Shifted Hammersley Sampling Method

skewed more toward the origin of the cube and away from the far edges and faces. To compensate for thisbias, a point-shifting process is proposed that shifts all Hammersley points by the amount below:

(5)∆ =1

2N

This moves the point set more toward the center of the search space and avoids unnecessary bias. Thus,the initial population always provides unbiased, low-discrepancy coverage of the search space.

Pareto Dominance in Multi-Objective Optimization

The concept of Pareto dominance is of extreme importance in multi-objective optimization, especially wheresome or all of the objectives are mutually conflicting. In such a case, there is no single point that yields the"best" value for all objectives (i.e., the Utopia Point). Instead, the best solutions, often called a Pareto or non-dominated set, are a group of solutions such that selecting any one of them in place of another will alwayssacrifice quality for at least one objective, while improving at least one other. Formally, the description ofsuch Pareto optimality for generic optimization problems can be formulated as in the following equations.

Taking a closer, more formal look at the multi-objective optimization problem, let the following denote theset of all feasible (i.e., do not violate constraints) solutions:

(6)X x g x h x x x xi u: { : ( ) , ( ) , }= ∈ ′′ℜ ≥ = ≤ ≤0 0

The problem can then be simplified to:

(7)min ( )rf x

x X∈

If there exists x' X such that for all objective functions x* is optimal. This, for i = 1, ..., k, is expressed:

(8)f x f x x Xi i( *) ( ) ,≤ ∀ ∈ ∀

This indicates that x* is certainly a desirable solution. Unfortunately, this is a utopian situation that rarelyexists, as it is unlikely that all f1(x) will have minimum values for X at a common point (x*). The question isleft: What solution should be used? That is, how should an "optimal" solution be defined? First, consider theso-called ideal (utopian) solution. In order to define this solution, separately attainable minima must befound for all objective functions. Assuming there is one, let x'1 be the solution of the scalar optimizationproblem:

(9)min ( ) *f x fx X

i i=∈

Here f1' is called the individual minimum for the scalar problem i; the vector f f fi ki* * *( , , )= … is called ideal

for a multi-objective optimization problem; and the points in X which determined this vector is the idealsolution.



It is usually not true that Equation 10 (p. 73) holds, although it would be useful, as the multi-objectiveproblem would have been solved by considering a sequence for scalar problems. It is necessary to define anew form of optimality, which leads to the concept of Pareto Optimality. Introduced by V. Pareto in 1896,it is still the most important part of multi-objective optimization.

(10)f f fi ki* * *( , , )= …

A point x' X is said to be Pareto Optimal for the problem if there is no other vector x X such that for alli = 1, ..., k,

(11)f x f xi i( ) ( )*≤

and, for at least one i {1, ..., k}:

(12)f x f Xi i( ) ( )*<

This definition is based on the intuitive conviction that the point x* X is chosen as the optimal if no criterioncan be improved without worsening at least one other criterion. Unfortunately, the Pareto optimum almostalways gives not a single solution, but a set of solutions. Usually Pareto optimality is spoken of as beingglobal or local depending on the neighborhood of the solutions X, and in this case, almost all traditionalalgorithms can at best guarantee a local Pareto optimality. However, this MOGA-based system, which incor-porates global Pareto filters, yields the global Pareto front.

MOGA (Multi-Objective Genetic Algorithm)

The MOGA used in GDO is a hybrid variant of the popular NSGA-II (Non-dominated Sorted Genetic Algorithm-II) based on controlled elitism concepts. Currently, only continuous problems can be solved. The Paretoranking scheme is done by a fast, non-dominated sorting method that is an order of magnitude faster thantraditional Pareto ranking methods. The constraint handling uses the same non-dominance principle as theobjectives, thus penalty functions and Lagrange multipliers are not needed. This also ensures that the feasiblesolutions are always ranked higher than the infeasible solutions.

The first Pareto front solutions are archived in a separate sample set internally and are distinct from theevolving sample set. This ensures minimal disruption of Pareto front patterns already available from earlieriterations. You can control the selection pressure (and, consequently, the elitism of the process) to avoidpremature convergence by altering the parameter Percent Pareto.

Decision Support Process

The decision support process is a goal-based, weighted, aggregation-based design ranking technique.

Given n input parameters, m output parameters, and their individual targets, the collection of objectives iscombined into a single, weighted objective function, Φ, which is sampled by means of a direct Monte Carlomethod using uniform distribution. The candidate designs are subsequently ranked by ascending magnitudesof the values of Φ. The proposed function for Φ (where all continuous input parameters have usable valuesof type "continuous") is given by the following:




(13)Φ ≡ ∑ + ∑= =

w N w Mi ii

n

j jj

m

1 1

where:

wi and wj = weights defined in Equation 16 (p. 74)Ni and Mi = normalized objectives for input and output parameters, respectively

The normalized objectives (metrics) are:

(14)Nx x

x xi

t

u l i

=−

−

(15)My y

y yj

t

j

=−

−

max min

where:

x = current value for input parameter ixt, yt = corresponding "target value"y = current value for output parameter jxl and xu = lower and upper values, respectively, for input parameter iymin and ymax = corresponding lower and upper bounds, respectively, for output parameter j

The fuzziness of the combined objective function derives from the weights w, which are simply defined asfollows:

(16)w wi j= =1 000

0 666

.

.

,

,

if the Importance is "Higher"

if the Impoortance is "Default"

if the Importance is "Lower",0 333.

The labels used are defined in Defining Design Objectives (p. 59).

The targets represent the desired values of the parameters, and are defined for the continuous input para-meters as follows:

(17)x

x

xt

l=

,

,

if Objective is "No Objective"

if Objective is "Minnimize"

f Objective is "Seek Midpoint"

if Ob

12 ,

,

x x

x

l u

u

i+( )

jjective is "Maximize"

and, for the output parameters we have the following desired values:



(18)y

y

y

t =

,

,


if Objective is "min MMinimize" and a "Target" value is not defined

if Objec,yt*

ttive is "Values < = Target" and "Target" is defined and y ≥≥ yt

y

*

, if Objective is "Values < = Target" and "Target" is defined and

if Objective is "Seek Target" or "Vy ,t*

y yt≤ *

aalues = Target"

f Objective is "Values > = Target" and ,y i ""Target" is defined and

if Objective is "Values ,

y yt

yt

≥ *

*>> = Target" and "Target" is defined and

if Obje,

y yt

y

≤ *

max cctive is "Maximize" and a "Target" value is not defined

where:

y t*

= user-specified target value

Thus, Equation 16 (p. 74) and Equation 17 (p. 74) constitute the input parameter goals for the continuousinput parameters and Equation 16 (p. 74) and Equation 18 (p. 75) constitute the output parameter goals.

The following section considers the case where the continuous input parameters have discrete Usabilityvalues ("Levels" in the parameters' Table view) and there may be discrete input parameters of type "Discrete."Let us consider the case where the Usable Values of the continuous input parameter are defined as the fol-lowing:

(19){ } { , ,..., }X x x xl= −0 1 1

with l usable values, the following metric is defined:

(20)Px x

x xk

t

MAX MIN

=−

−

where, as before,

xMAX = upper bound of the usable valuesxMIN = lower bound of the usable values

The target value xt is given by the following:




(21)x

x

t

t

=

x,

,


if Objective is "Va*

llues <= Target" and "Target" is defined and x

if Obj

*

x,

≥ xt

eective is "Values <= Target" and "Target" is defined and xx

if Objective is "Seek Target"

if Objective is "

*

,

x,

≤ xt

xt*

VValues >= Target" and "Target" is defined and x

if

*

*,

≥ xt

xt Objective is "Values >= Target" and "Target" is defined aand x ≤

xt

*

Thus, the GDO objective equation becomes the following (for parameters with discrete Usable Values).

(22)Φ ≡ + += = =∑ ∑ ∑w N w M w Pi ii

n

j jj

m

l ll

p

1 1 1

Therefore, Equation 15 (p. 74), Equation 16 (p. 74), and Equation 20 (p. 75) constitute the input parametergoals for parameters which may be continuous or possess discrete usable values.

The norms and goals as in equations Equation 20 (p. 75) and Equation 21 (p. 76) are also adopted to definethe input goals for the discrete input parameters of the type "Discrete"; i.e., those discrete parameters whoseUsable Alternatives indicate a whole number of some particular design feature (number of holes in a plate,number of stiffeners, etc.).

Thus, equations Equation 15 (p. 74), Equation 16 (p. 74), and Equation 20 (p. 75) constitute the input para-meter goals for "Discrete" parameters.

Therefore, the GDO objective function equation for the most general case (where there are continuous anddiscrete parameters) can be written as the following:

(23)

Φ ≡ += =∑ ∑w N w Mi ii

n

j jj

m

1 1( ) ( )Continuous Input Continuous Output

++=∑ w Pf ff

l

1 ( )Usability + Discrete

where:

n = number of Continuous Input parametersm = number of Continuous Output parametersl = number of Continuous Input parameters with Usability Levels and "Discrete" parameters

From the normed values it is obvious that the lower the value of Φ, the better the design with respect tothe desired values and importances. Thus, a quasi-random uniform sampling of design points is done by aHammersley algorithm and the samples are sorted in ascending order of Φ. The desired number of designsare then drawn from the top of the sorted list. A crowding technique is employed to ensure that any twosampled design points are not very close to each other in the space of the input parameters.



Non-linear Programming by Quadratic Lagrangian (NLPQL)

NLPQL (Non-linear Programming by Quadratic Lagrangian) is a mathematical optimization algorithm as de-veloped by Klaus Schittkowski. This method solves constrained nonlinear programming problems of theform

Minimize:

f f x= ({ })

Subject to:

g x k Kk ({ }) , , ,...,≤ ∀ =0 1 2

h x l Ll({ }) , , ,...,≤ ∀ =0 1 2

where

{ } { } { }x x xL U≤ ≤

It is assumed that objective function and constraints are continuously differentiable. The idea is to generatea sequence of quadratic programming subproblems obtained by a quadratic approximation of the Lagrangianfunction and a linearization of the constraints. Second order information is updated by a quasi-Newton formulaand the method is stabilized by an additional (Armijo) line search.

The method presupposes that the problem size is not very large (less than 2000 input variables) and that itis well scaled. Also, the accuracy of the methods depends on the accuracy of the gradients. Since, for mostpractical problems, analytical gradients are unavailable, it is imperative that the numerical (finite differencebased) gradients are as accurate as possible.

Newton's iterative method

Before the actual derivation of the NLPQL equations, Newton’s iterative method for the solution of nonlinearequation sets is reviewed. Let f(x) be a multivariable function such that it can be expanded about the pointx in a Taylor’s series.

(24)f x x f x x f x x f x xT T( ) ( ) { } { ( )} { } [ ( )]{ }+ ∆ ≈ + ∆ ′ +

∆ ′′ ∆

1

2

where, it is assumed that the Taylor series actually models a local area of the function by a quadratic approx-imation. The objective is to devise an iterative scheme by linearizing the vector Equation 24 (p. 77). To thisend, it is assumed that at the end of the iterative cycle, the Equation 24 (p. 77) would be exactly valid. Thisimplies that the first variation of the following expression with respect to ∆x must be zero.

(25)φ( ) ( ) ( ) { } { ( )} { } [ ( )]{ }∆ = + ∆ − + ∆ ′ +

∆ ′′ ∆

x f x x f x x f x x f x xT T1

2

which implies that



Newton's iterative method

(26)f x x f x f x xx( ), ({ ( )} [ ( )]{ })+ ∆ − ′ + ′′ ∆ =∆ 0

The first expression indicates the first variation of the converged solution with respect to the increment inthe independent variable vector. This gradient is necessarily zero since the converged solution clearly doesnot depend on the step-length. Thus, Equation 26 (p. 78) can be written as the following:

(27){ } { } [ ( )] { ( )}x x f x f xj j j j+−= − ′′ ′11

where, the index "j" indicates the iteration. Equation 27 (p. 78) is thus used in the main quadratic programmingscheme.

NLPQL derivation:

Consider the following single-objective nonlinear optimization problem. It is assumed that the problem issmooth and analytic throughout and is a problem of N decision variables.

Minimize:

f f x= ({ })

Subject to:

g x k Kk ({ }) , , ,...,≤ ∀ =0 1 2

h x l Ll({ }) , , ,...,≤ ∀ =0 1 2

where

(28){ } { } { }x x xL U≤ ≤

Where, K and L are the numbers of inequality and equality constraints. In many cases the inequality constraintsare bound above and below, in such cases, it is customary to split the constraint into two inequality con-straints. In order to approximate the quadratic sub-problem assuming the presence of only equality constraints,the Lagrangian for Equation 28 (p. 78) as:

(29)Λ({ },{ }) ({ }) { } { ({ })}x f x h xTλ λ= +

Where, {λ} is a L dimensional vector (non-zero) which are used as Lagrange multipliers. Thus, Equa-

tion 29 (p. 78) becomes a functional, which depends on two sets of independent vectors. In order to minimizethis expression, we seek the stationarity of this functional with respect to the two sets of vectors. These ex-pressions give rise to two sets of vector expressions as the following:



(30)Λ Λ

Λ

({ },{ }), { } { } [ ]{ } { }

({ },{ }), { ({ })}

{ }

{ }

x f H

x h x

xλ λ

λ λ

= ∇ = ∇ + =

=

0

== { }0

Equation 30 (p. 79) defines the Karush-Kuhn-Tucker (KKT) conditions which are the necessary conditions forthe existence of the optimal point. The first equation comprises "N" nonlinear algebraic equations and thesecond one comprises "L" nonlinear algebraic equations. The matrix [H] is a (N*L) matrix defined as the fol-lowing:

(31)[ ] [{ ({ })}, { ({ })}, { ({ })}, ...]( * )H h x h x h xN L = ∇ ∇ ∇1 2 3

Thus, in Equation 31 (p. 79), each column is a gradient of the corresponding equality constraint. For ourconvenience, the nonlinear equations of Equation 30 (p. 79) can be written as the following:

(32){ ({ })} { }F Y = 0

Where, Equation 32 (p. 79) is a (N+L) system of nonlinear equations. The independent variable set {Y} canbe written as:

(33){ }{ }

{ }Y

x=

λ

while the functional set {F} is written as the following:

(34){ }{ }

{ }F

h=

∇∧

Referring to the section on Newton based methods, the vector {Y} in Equation 33 (p. 79) is updated as thefollowing:

(35){ } { } { )Y Y Yj j j+ = + ∆1

The increment of the vector is given by the iterative scheme in Equation 27 (p. 78). Referring to Equa-

tion 32 (p. 79), Equation 33 (p. 79), and Equation 34 (p. 79), the iterative equation is expressed as the following:

(36)∇ ∆ = −F Y F Yj j j{ } { ({ })}

Note that this is only a first order approximation of the Taylor expansion of Equation 32 (p. 79). This is incontrast to Equation 27 (p. 78) where a quadratic approximation is done. This is because in Equation 33 (p. 79),a first order approximation has already been done. The matrices and the vectors in Equation 36 (p. 79) canbe expanded as the following:



NLPQL derivation:

(37){ }{ }

{ }∆ =

∆∆

Yx

λ

and

(38)[ ][ ] [ ]

[ ] [ ]

( * ) ( * )

( * ) ( * ) (( *(

∇ =∇

+ +

FH

H

N N N L

TL N L L N L N L

2

0

Λ

)))

This is obtained by taking the gradients of Equation 34 (p. 79) with respect to the two variable sets. The sub-

matrix [ ]( * )∇2Λ N N is the (N*N) Hessian of the Lagrange function in implicit form.

To demonstrate how Equation 38 (p. 80) is formed, let us consider the following simple case. Given a vectorof two variables x and y, we write:

{ }{ ( , )}

{ ( , )}

( , )

( , )

( , )

( , )

Va x y

b x y

a x y

a x y

b x y

b x y

=

=

1

2

1

2

It is required to find the gradient (i.e. Jacobian of the vector function V ). To this effect, the derivation of theJacobian is evident because in the present context, the vector V indicates a set of nonlinear (algebraic)equations and the Jacobian is the coefficient matrix which "links" the increment of the independent variablevector to the dependent variable vector. Thus, we can write:

{ }{ ( , )}

{ ( , )}∆ =

∆∆

=

∂∂

∂∂∂∂

∂∂

∂∂

∂

Va x y

b x y

a

x

a

x

b

x

b

x

a

y

a

1

2

1

2

1

2

∂∂∂∂

∂∂

∆∆

y

b

y

b

y

x

y1

2

Thus, the Jacobian matrix is formed. In some applications, the equation is written in the following form:

{ }{ ( , )}

{ ( , )}∆ =

∆∆

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂V

a x y

b x y

a

x

a

x

b

x

b

x

a

y

a

1 2 1 2

1 2

∂∂∂∂

∂∂

∆∆

y

b

y

b

y

x

y

T

1 2

Where, the column "i" indicates the gradient of the "i-th" component of the dependent variable vector withrespect to the independent vector. This is the formalism we use in determining Equation 38 (p. 80).



Equation 36 (p. 79) may be rewritten as the following:

(39)[ ] [ ]

[ ] [ ]

{ }

{ }

( * ) ( * )

( * ) ( * )

( )∇

∆∆

2

0

Λ N N N L

TL N L L

jH

H

x

λ

= −∇

( ) ( ){ }

{ }

j j

h

Λ

Solution of Equation 39 (p. 81) iteratively will solve Equation 40 (p. 81) in a series of linear steps till the pointwhen the increment is negligible. The update schemes for the independent variable and Lagrange multipliervectors x and λ are written as the following:

(40){ } { } { }

{ } { } { }

x x xj j j

j j j

+

+

= + ∆

= + ∆1

1λ λ λ

The individual equations of the Equation 39 (p. 81) are written separately now. The first equation (corres-ponding to minimization with respect to x) may be written as:

(41)

[ ] { } [ ] { } { }

[ ] { } [ ] { } { } [

∇ ∆ + ∆ = − ∇

∇ ∆ + ∆ = − ∇ −

2

2

Λ Λ

Λ

j j j j j

j j j j j

x H

x H f

λ

λ HH

x H f

j j

j j j j j

] { }

[ ] { } [ ] { } { }

λ

λ∇ ∆ + = − ∇+2

1Λ

The last step in Equation 41 (p. 81) is done by using Equation 40 (p. 81). Thus, using Equation 41 (p. 81) andEquation 39 (p. 81), the iterative scheme can be rewritten in a simplified form as:

(42)[ ] [ ]

[ ] [ ]

{ }

{ }

( * ) ( * )

( * ) ( * )

( )( )

(

∇

∆2

0

Λ N N N L

TL N L L

jj

j

H

H

x

λ ++

= −

∇

1)

( ){ }

{ }

f

h

j

Thus, Equation 42 (p. 81) can be used directly to compute the (j+1)th value of the Lagrange multiplier vector.Note that by using Equation 42 (p. 81), it is possible to compute the update of x and the new value of theLagrange multiplier vector λ in the same iterative step. Equation 42 (p. 81) shows the general scheme bywhich the KKT optimality condition can be solved iteratively for a generalized optimization problem.

Now, consider the following quadratic approximation problem as given by:

(43)Q f x x xT T= ∇ ∆ +

∆ ∇ ∆{ } { } { } [ ]{ }

1

2

2Λ

Subject to the equality constraints given by the following:



NLPQL derivation:

(44){ } [ ] { } { }( * ) ( * )h H xLT

L N1 0+ ∆ =

Where the definition of the matrices in Equation 43 (p. 81) and Equation 44 (p. 82) are given earlier. In orderto solve the quadratic minimization problem let us form the Lagrangian as:

(45)Γ Λ= ∇ ∆ +

∆ ∇ ∆ + + ∆{ } { } { } [ ]{ } { } { } { } [ ] { }f x x x h H xT T T T T1

2

2 λ λ

Now, the KKT conditions may derived (as done earlier) by taking gradients of the Lagrangian in Equa-

tion 45 (p. 82) as the following:

(46)Γ Λ

Γ

, { } [ ]{ } [ ]{ } { }

, { } [ ] { } { }

{ }

{ }

∆ = ∇ + ∇ ∆ + =

= + ∆ =

x

T

f x H

h H x

2 0

0

λ

λ

In a condensed matrix form Equation 46 (p. 82) may be written as the following:

(47)[ ] [ ]

[ ] [ ]

{ }

{ }

{ }

{ }

∇

∆

= −∇

2

0

Λ H

H

x f

hT λ

Equation 47 (p. 82) is the same as Equation 42 (p. 81) – this implies that the iterative scheme in Equa-

tion 47 (p. 82) which actually solves a quadratic subproblem (Equation 43 (p. 81) and Equation 44 (p. 82)) inthe domain ∆x. In case the real problem is quadratic then this iterative scheme solves the exact problem.

On addition of inequality constraints, the Lagrangian of the actual problem can be written as the following:

(48)Λ({ },{ },{ },{ }) ({ }) { } { ({ })} { } { ({ })} { } {x y f x h x g x yT T Tλ µ λ µ µ= + + + 22}

Note that the inequality constraints have been converted to equality constraints by using a set of slackvariables y as the following:

(49)g x y k Mk k({ }) ( ) , , ,...,+ = ∀ =2 0 1 2

The squared term is used to ensure that the slack variable remains positive which is required to satisfyEquation 49 (p. 82). The Lagrangian in Equation 48 (p. 82) acts as an enhanced objective function. It is seenthat the only case where the additional terms may be active is when the constraints are not satisfied.

The KKT conditions as derived from Equation 48 (p. 82) (by taking first variations with respect to the inde-pendent variable vectors) are:



(50)

Λ Λ

Λ

, { } { } [ ]{ } [ ]{ } { }

, { ({ })} { }

{ } ( * )

{ } (

x N

L

f H G

h x

= ∇ = ∇ + + =

= =

λ µ

λ

0

0

1

** )

{ } ( * )

{ } ( * )

, { ({ })} { } { }

, { } { }

1

21

1

0

2 0

Λ

Λµ

µ

= = =

= =

g x y

y

M

y M

From the KKT conditions in Equation 50 (p. 83), it is evident that there are (N + L + 2*M) equations for asimilar number of unknowns, thus this equation set possesses an unique solution. Let this (optimal) solutionbe marked as x. At this point, a certain number of constraints will be active and some others will be inactive.Let the number of active inequality constraints be p and the total number of active equality constraints beq. By an active constraint it is assumed that the constraint is at its threshold value of zero. Thus, let J1 andJ2 be the sets of active and inactive equality constraints (respectively) and K1 and K2 be the sets of activeand inactive inequality constraints respectively. Thus, we can write the following relations:

(51)J J meas J J L meas J q meas J L q

K K meas

1 2 1 2 1 2

1 2

∩ ∪

∩

= ∅ = = = −

= ∅

; ( ) ; ( ) ; ( ( )

; (( ) ; ( ) ; ( ( )K K M meas K p meas K M p1 2 1 2∪ = = = −

Where, meas() indicates the count of the elements of the set under consideration. These sets thus partitionthe constraints into active and inactive sets. Thus, we can write:

(52)

h x k J

g x y K

g x y

k

j j j j

j j j

({ }) ,

({ }) , ; ;

({ }) , ;

= ∀ ∈

= = ≠ ∀ ∈

< ≠ =

0

0 0 0

0 0

2

1µ

µ 00 1;∀ ∈j K

Thus, it can be stated that the last 3 equations in Equation 50 (p. 83) may be represented by the Equa-

tion 52 (p. 83). These are the optimality conditions for constraint satisfaction. From these equations, we cannow eliminate y such that the Lagrangian in Equation 48 (p. 82) will depend on only 3 independent variablevectors. From the last two conditions in Equation 52 (p. 83), we can write the following condition, which isalways valid for a optimal point:

(53)µ j jg x({ }) = 0

Using Equation 53 (p. 83) in the set Equation 50 (p. 83), the KKT optimality conditions may be written as thefollowing:

(54)

{ } { } [ ]{ } [ ]{ } { }

{ ({ })} { }

{ ({ })}

( * )

( * )

∇∧ = ∆ + + =

=

f H G

h x

g x

N

L

λ µ

µ

0

0

1

1

== { }( * )0 1M

Thus, the new set contains only (N + L + M) unknowns. Now, following the same logic as in Equation 33 (p. 79)as done earlier—let us express Equation 54 (p. 83) in the same form as in Equation 32 (p. 79). This representsa (N + L + M) system of nonlinear equations. The independent variable set may be written in vector for asthe following:



NLPQL derivation:

(55){ }

{ }

{ }

{ }

Y

x

=

λµ

Newton’s iterative scheme is also used here, thus, the same equations as in Equation 35 (p. 79) and Equa-

tion 36 (p. 79) also apply here. Thus, following Equation 36 (p. 79), we can write:

(56){ }

{ }

{ }

{ }

∆ =∆∆∆

Y

x

λµ

Taking the first variation of the KKT equations in Equation 54 (p. 83) and equating to zero, the sub-quadraticequation is formulated as the following:

(57)

[ ] [ ] [ ]

[ ]

[ ]

∇

2

0 0

0

Λ H G

H

G g

xT T

µ

δδλδµ

= −

∇

{ }

{ }

{ }

Λh

gµ

Where µ indicates a diagonal matrix.

At the jth step, the first equation can be written as (by linearization)

(58)[ ] { } [ ] { } [ ] { } { } [ ] { } [ ] { }∇ ∆ + ∆ + ∆ = − ∇ − −2Λ j j j j j j j j j jx H G f H Gj

λ µ λ µ

Which simplifies to

(59)[ ] { } [ ] { } [ ] { } { }∇ ∆ + + = − ∇+ +2

1 1Λ j j j j j jx H G fj

λ µ

Thus, the linearized set of equations for Newton’s method to be applied can be written in an explicit formas:

(60)

[ ] [ ] [ ]

[ ]

[ ]

{ }

{ }

∇

∆2

0 0

0

Λ H G

H

G g

xjT

jT

j

j j j

j

µ

λ jj

j

j

j

j

f

h

g

+

+

= −

∇

1

1 2{ }

{ }

{ }

{ }µ µ

So, in presence of both equality and inequality constraints, Equation 60 (p. 84) can be used in a quasi Newtonframework to determine the increments {∆x}j and Lagrange multipiers {λ}j+1 and {µ}j+1 when stepping fromiteration j to j+1.



The Hessian matrix [ ]∇2Λ is not computed directly but is estimated and updated in a BFGS type line search.



NLPQL derivation:

Understanding Six Sigma Analysis

A Six Sigma Analysis allows you to determine the extent to which uncertainties in the model affect the resultsof an analysis. An uncertainty (or random quantity) is a parameter whose value is impossible to determineat a given point in time (if it is time-dependent) or at a given location (if it is location-dependent). An exampleis ambient temperature; you cannot know precisely what the temperature will be one week from now in agiven city.

A Six Sigma Analysis uses statistical distribution functions (such as the Gaussian, or normal, distribution, theuniform distribution, etc.) to describe uncertain parameters.

Six Sigma Analysis allows you to determine whether your product satisfies Six Sigma quality criteria. Aproduct has Six Sigma quality if only 3.4 parts out of every 1 million manufactured fail. This quality definitionis based on the assumption that an output parameter relevant to the quality and performance assessmentfollows a Gaussian distribution, as shown below.

An output parameter that characterizes product performance is typically used to determine whether aproduct's performance is satisfactory. The parameter must fall within the interval bounded by the lowerspecification limit (LSL) and the upper specification limit (USL). Sometimes only one of these limits exists.

An example of this is a case when the maximum von Mises stress in a component must not exceed the yieldstrength. The relevant output parameter is, of course, the maximum von Mises stress and the USL is theyield strength. The lower specification limit is not relevant. The area below the probability density functionfalling outside the specification interval is a direct measure of the probability that the product does notconform to the quality criteria, as shown above. If the output parameter does follow a Gaussian distribution,then the product satisfies a Six Sigma quality criterion if both specification limits are at least six standarddeviations away from the mean value.

In reality, an output parameter rarely exactly follows a Gaussian distribution. However, the definition of SixSigma quality is inherently probabilistic -- it represents an admissible probability that parts do not conformto the quality criteria defined by the specified limits. The nonconformance probability can be calculated nomatter which distribution the output parameter actually follows. For distributions other than Gaussian, the



Six Sigma level is not really six standard deviations away from the mean value, but it does represent aprobability of 3.4 parts per million, which is consistent with the definition of Six Sigma quality.

This section describes the Six Sigma Analysis tool and how to use it to perform a Six Sigma Analysis.

Principles (SSA)

Computer models are described with specific numerical and deterministic values: material properties areentered using certain values, the geometry of the component is assigned a certain length or width, etc. Ananalysis based on a given set of specific numbers and values is called a deterministic analysis. The accuracyof a deterministic analysis depends upon the assumptions and input values used for the analysis.

While scatter and uncertainty naturally occur in every aspect of an analysis, deterministic analyses do nottake them into account. To deal with uncertainties and scatter, use Six Sigma Analysis, which allows you toanswer the following questions:

• If the input variables of a finite element model are subject to scatter, how large is the scatter of theoutput parameters? How robust are the output parameters? Here, output parameters can be any para-meter that ANSYS Workbench can calculate. Examples are the temperature, stress, strain, or deflectionat a node, the maximum temperature, stress, strain, or deflection of the model, etc.

• If the output is subject to scatter due to the variation of the input variables, then what is the probabilitythat a design criterion given for the output parameters is no longer met? How large is the probabilitythat an unexpected and unwanted event takes place (i.e., what is the failure probability)?

• Which input variables contribute the most to the scatter of an output parameter and to the failureprobability? What are the sensitivities of the output parameter with respect to the input variables?

Six Sigma Analysis can be used to determine the effect of one or more variables on the outcome of theanalysis. In addition to the Six Sigma Analysis techniques available, ANSYS Workbench offers a set of strategictools to enhance the efficiency of the Six Sigma Analysis process. For example, you can graph the effects ofone input parameter versus an output parameter, and you can easily add more samples and additionalanalysis loops to refine your analysis.

In traditional deterministic analyses, uncertainties are either ignored or accounted for by applying conservativeassumptions. You would typically ignore uncertainties if you know for certain that the input parameter hasno effect on the behavior of the component under investigation. In this case, only the mean values or somenominal values are used in the analysis. However, in some situations, the influences of uncertainties existsbut is still neglected, as for the thermal expansion coefficient, for which the scatter is usually ignored.

Example 1 Accounting for Uncertainties

If you are performing a thermal analysis and want to evaluate the thermal stresses, the equation is:

σtherm = E α ∆T

because the thermal stresses are directly proportional to the Young's modulus as well as to the thermal ex-pansion coefficient of the material.

The table below shows the probability that the thermal stresses will be higher than expected, taking uncer-tainty variables into account.



Probability that the

thermal stresses are more

than 10% higher than ex-

pected

Probability that the

thermal stresses are more

than 5% higher than expec-

ted

Uncertainty variables taken into

account

~2.3%~16%Young's modulus (Gaussian distribu-tion with 5% standard deviation)

~8%~22%Young's modulus and thermal expan-sion coefficient (each with Gaussiandistribution with 5% standard devi-ation)

Reliability, Quality, and Safety Issues

Use Six Sigma Analysis when issues of reliability, quality, and safety are paramount.

Reliability is typically a concern when product or component failures have significant financial consequences(costs of repair, replacement, warranty, or penalties) or worse, can result in injury or loss of life.

If you use a conservative assumption, the difference in thermal stresses shown above tells you that uncertaintyor randomness is involved. Conservative assumptions are usually expressed in terms of safety factors.Sometimes regulatory bodies demand safety factors in certain procedural codes. If you are not faced withsuch restrictions or demands, then using conservative assumptions and safety factors can lead to inefficientand costly over-design. By using Six Sigma Analysis methods, you can avoid over-design while still ensuringthe safety of the component.

Six Sigma Analysis methods even enable you to quantify the safety of the component by providing a prob-ability that the component will survive operating conditions. Quantifying a goal is the necessary first steptoward achieving it.

Guidelines for Selecting SSA Variables

This section presents useful guidelines for defining your Six Sigma Analysis variables.

Choosing and Defining Uncertainty Variables

First, you should:

• Specify a reasonable range of values for each uncertainty variable.

• Set reasonable limits on the variability for each uncertainty variable.

More information about choosing and defining uncertainty variables can be found in the following sections.Uncertainty Variables for Response Surface AnalysesChoosing a Distribution for a Random VariableDistribution Functions

Uncertainty Variables for Response Surface Analyses

The number of simulation loops that are required for a Response Surface analysis depends on the numberof uncertainty variables. Therefore, you want to select the most important input variable(s), the ones youknow have a significant impact on the result parameters. If you are unsure which uncertainty variables areimportant, include all of the random variables you can think of and then perform a Monte Carlo analysis.



Choosing and Defining Uncertainty Variables

After you learn which uncertainty variables are important and should be included in your Response SurfaceAnalysis, you can eliminate those that are unnecessary.

Choosing a Distribution for a Random Variable

The type and source of the data you have determines which distribution functions can be used or are bestsuited to your needs.

Measured DataMean Values, Standard Deviation, Exceedance ValuesNo Data

Measured Data

If you have measured data, then you must first know how reliable that data is. Data scatter is not just aninherent physical effect, but also includes inaccuracy in the measurement itself. You must consider that theperson taking the measurement might have applied a "tuning" to the data. For example, if the data measuredrepresents a load, the person measuring the load may have rounded the measurement values; this meansthat the data you receive are not truly the measured values. The amount of this tuning could provide a de-terministic bias in the data that you need to address separately. If possible, you should discuss any bias thatmight have been built into the data with the person who provided that data to you.

If you are confident about the quality of the data, then how you proceed depends on how much data youhave. In a single production field, the amount of data is typically sparse. If you have only a small amount ofdata, use it only to evaluate a rough figure for the mean value and the standard deviation. In these cases,you could model the uncertainty variable as a Gaussian distribution if the physical effect you model has nolower and upper limit, or use the data and estimate the minimum and maximum limit for a uniform distri-bution.

In a mass production field, you probably have a lot of data. In these cases you could use a commercialstatistical package that will allow you to actually fit a statistical distribution function that best describes thescatter of the data.

Mean Values, Standard Deviation, Exceedance Values

The mean value and the standard deviation are most commonly used to describe the scatter of data. Fre-quently, information about a physical quantity is given as a value such as "100±5.5." Often, this form meansthat the value "100" is the mean value and "5.5" is the standard deviation. Data in this form implies a Gaus-sian distribution, but you must verify this (a mean value and standard deviation can be provided for anycollection of data regardless of the true distribution type). If you have more information, for example, youknow that the data is lognormal distributed, then Six Sigma Analysis allows you to use the mean value andstandard deviation for a lognormal distribution.

Sometimes the scatter of data is also specified by a mean value and an exceedance confidence limit. Theyield strength of a material is sometimes given in this way; for example, a 99% exceedance limit based ona 95% confidence level is provided. This means that, from the measured data, we can be sure by 95% thatin 99% of all cases the property values will exceed the specified limit and only in 1% of all cases will theydrop below the specified limit. The supplier of this information is using the mean value, the standard deviation,and the number of samples of the measured data to derive this kind of information. If the scatter of thedata is provided in this way, the best way to pursue this further is to ask for more details from the datasupplier. Because the given exceedance limit is based on the measured data and its statistical assessment,the supplier might be able to provide you with the details that were used.



If the data supplier does not give you any further information, then you could consider assuming that thenumber of measured samples was large. If the given exceedance limit is denoted with x1 - α/2 and the givenmean value is denoted with xµ, then the standard deviation can be derived from the equation:

σµα

=

x x

C

1 2− −/

where the values for the coefficient C are:

CExceedance Probability

2.575899.5%

2.326399.0%

1.960097.5%

1.644995.0%

1.281690.0%

No Data

In situations where no information is available, there is never just one right answer. Below are hints aboutwhich physical quantities are usually described in terms of which distribution functions. This informationmight help you with the particular physical quantity you have in mind. Also below is a list of which distributionfunctions are usually used for which kind of phenomena. Keep in mind that you might need to choose frommultiple options.

Geometric Tolerances

• If you are designing a prototype, you could assume that the actual dimensions of the manufacturedparts would be somewhere within the manufacturing tolerances. In this case it is reasonable to use auniform distribution, where the tolerance bounds provide the lower and upper limits of the distributionfunction.

• If the manufacturing process generates a part that is outside the tolerance band, one of two things mayhappen: the part must either be fixed (reworked) or scrapped. These two cases are usually on oppositeends of the tolerance band. An example of this is drilling a hole. If the hole is outside the toleranceband, but it is too small, the hole can just be drilled larger (reworked). If, however, the hole is largerthan the tolerance band, then the problem is either expensive or impossible to fix. In such a situation,the parameters of the manufacturing process are typically tuned to hit the tolerance band closer to therework side, steering clear of the side where parts need to be scrapped. In this case, a Beta distributionis more appropriate.

• Often a Gaussian distribution is used. The fact that the normal distribution has no bounds (it spansminus infinity to infinity), is theoretically a severe violation of the fact that geometrical extensions aredescribed by finite positive numbers only. However, in practice, this lack of bounds is irrelevant if thestandard deviation is very small compared to the value of the geometric extension, as is typically truefor geometric tolerances.

Material Data

• Very often the scatter of material data is described by a Gaussian distribution.



Material Data

• In some cases the material strength of a part is governed by the "weakest-link theory." The "weakest-link theory" assumes that the entire part will fail whenever its weakest spot fails. For material propertieswhere the "weakest-link" assumptions are valid, the Weibull distribution might be applicable.

• For some cases, it is acceptable to use the scatter information from a similar material type. For example,if you know that a material type very similar to the one you are using has a certain material propertywith a Gaussian distribution and a standard deviation of ±5% around the measured mean value, thenyou can assume that for the material type you are using, you only know its mean value. In this case,you could consider using a Gaussian distribution with a standard deviation of ±5% around the givenmean value.

Load Data

For loads, you usually only have a nominal or average value. You could ask the person who provided thenominal value the following questions: Out of 1000 components operated under real life conditions, whatis the lowest load value any one of the components sees? What is the most likely load value? That is, whatis the value that most of these 1000 components are subject to? What is the highest load value any onecomponent would be subject to? To be safe you should ask these questions not only of the person whoprovided the nominal value, but also to one or more experts who are familiar with how your products areoperated under real-life conditions. From all the answers you get, you can then consolidate what the min-imum, the most likely, and the maximum value probably is. As verification, compare this picture with thenominal value that you would use for a deterministic analysis. The nominal value should be close to themost likely value unless using a conservative assumption. If the nominal value includes a conservative as-sumption (is biased), then its value is probably close to the maximum value. Finally, you can use a triangulardistribution using the minimum, most likely, and maximum values obtained.

Distribution Functions

Beta Distribution

fX(x)

x

xmin

xmax

r,t

You provide the shape parameters r and t and the lower and the upper limit xmin and xmax of the randomvariable x.

The Beta distribution is very useful for random variables that are bounded at both sides. If linear operationsare applied to random variables that are all subjected to a uniform distribution, then the results can usuallybe described by a Beta distribution. For example, if you are dealing with tolerances and assemblies wherethe components are assembled and the individual tolerances of the components follow a uniform distribution(a special case of the Beta distribution), the overall tolerances of the assembly are a function of adding orsubtracting the geometrical extension of the individual components (a linear operation). Hence, the overalltolerances of the assembly can be described by a Beta distribution. Also, as previously mentioned, the Betadistribution can be useful for describing the scatter of individual geometrical extensions of components aswell.



Exponential Distribution

fX(x)

x

xmin

λ

You provide the decay parameter λ and the shift (or lower limit) xmin of the random variable x.

The exponential distribution is useful in cases where there is a physical reason that the probability densityfunction is strictly decreasing as the uncertainty variable value increases. The distribution is mostly used todescribe time-related effects; for example, it describes the time between independent events occurring ata constant rate. It is therefore very popular in the area of systems reliability and lifetime-related systems re-liability, and it can be used for the life distribution of non-redundant systems. Typically, it is used if the lifetimeis not subjected to wear-out and the failure rate is constant with time. Wear-out is usually a dominant life-limiting factor for mechanical components that would preclude the use of the exponential distribution formechanical parts. However, where preventive maintenance exchanges parts before wear-out can occur, thenthe exponential distribution is still useful to describe the distribution of the time until exchanging the partis necessary.

Gaussian (Normal) Distribution

fX(x)

2

µx

σ

You provide values for the mean value µ and the standard deviation σ of the random variable x.

The Gaussian, or normal, distribution is a fundamental and commonly-used distribution for statistical matters.It is typically used to describe the scatter of the measurement data of many physical phenomena. Strictlyspeaking, every random variable follows a normal distribution if it is generated by a linear combination ofa very large number of other random effects, regardless which distribution these random effects originallyfollow. The Gaussian distribution is also valid if the random variable is a linear combination of two or moreother effects if those effects also follow a Gaussian distribution.



Gaussian (Normal) Distribution

Lognormal Distribution

fX(x)

ξδ

x

You provide values for the logarithmic mean value ξ and the logarithmic deviation δ. The parameters ξ andδ are the mean value and standard deviation of ln(x):

f x, , =1

2-

1

2

ln -( ) expξ δ

π σξ

δ⋅ ⋅⋅

x

x2

The lognormal distribution is another basic and commonly-used distribution, typically used to describe thescatter of the measurement data of physical phenomena, where the logarithm of the data would follow anormal distribution. The lognormal distribution is suitable for phenomena that arise from the multiplicationof a large number of error effects. It is also used for random variables that are the result of multiplying twoor more random effects (if the effects that get multiplied are also lognormally distributed). It is often usedfor lifetime distributions such as the scatter of the strain amplitude of a cyclic loading that a material canendure until low-cycle-fatigue occurs.

Uniform Distribution

fX(x)

x

xmin xmax

You provide the lower and the upper limit xmin and xmax of the random variable x.

The uniform distribution is a fundamental distribution for cases where the only information available is alower and an upper limit. It is also useful to describe geometric tolerances. It can also be used in cases whereany value of the random variable is as likely as any other within a certain interval. In this sense, it can beused for cases where "lack of engineering knowledge" plays a role.



Triangular Distribution

fX(x)

x

xmin xmaxxmlv

You provide the minimum value xmin, the most likely value limit xmlv and the maximum value xmax.

The triangular distribution is most helpful to model a random variable when actual data is not available. Itis very often used to capture expert opinions, as in cases where the only data you have are the well-foundedopinions of experts. However, regardless of the physical nature of the random variable you want to model,you can always ask experts questions like "Out of 1000 components, what are the lowest and highest loadvalues for this random variable?" and other similar questions. You should also include an estimate for therandom variable value derived from a computer program, as described above. For more details, see Choosing

a Distribution for a Random Variable (p. 90).

Truncated Gaussian (Normal) Distribution

fX(x)

2σG

µGx

xmin xmax

You provide the mean value µ and the standard deviation σ of the non-truncated Gaussian distribution andthe truncation limits xmin and xmax.

The truncated Gaussian distribution typically appears where the physical phenomenon follows a Gaussiandistribution, but the extreme ends are cut off or are eliminated from the sample population by quality controlmeasures. As such, it is useful to describe the material properties or geometric tolerances.



Truncated Gaussian (Normal) Distribution

Weibull Distribution

fX(x)

x

xmin

m,xchr

You provide the Weibull characteristic value xchr , the Weibull exponent m, and the minimum value xmin.There are several special cases. For xmin = 0 the distribution coincides with a two-parameter Weibull distri-bution. The Rayleigh distribution is a special case of the Weibull distribution with α = xchr - xmin and m = 2.

In engineering, the Weibull distribution is most often used for strength or strength-related lifetime parameters,and is the standard distribution for material strength and lifetime parameters for very brittle materials (forthese very brittle materials, the "weakest-link theory" is applicable). For more details, see Choosing a Distri-

bution for a Random Variable (p. 90).

Sample Generation

For Six Sigma Analysis, the sample generation by default is based on the Latin Hypercube Sampling (LHS)technique. (You can also select Weighted Latin Hypercube Sampling (WLHS) as the Sampling Type in theSix Sigma Properties view.) The LHS technique is a more advanced and efficient form of Monte Carlo ana-lysis methods. The only difference between LHS and the Direct Monte Carlo Sampling technique is that LHShas a sample "memory," meaning it avoids repeating samples that have been evaluated before (it avoidsclustering samples). It also forces the tails of a distribution to participate in the sampling process. Generally,the LHS technique requires 20% to 40% fewer simulations loops than the Direct Monte Carlo analysis techniqueto deliver the same results with the same accuracy. However, that number is largely problem-dependent.

Weighted Latin Hypercube Sampling

In some cases, a very small probability of failure (Pf ), e.g., in the order of 10-6, is of interest in engineeringdesign. It would require 100/Pf simulations/runs of regular Latin Hypercube samples. To reduce number ofruns, the Weighted Latin Hypercube Sampling (WLHS) may be used to generate biased/more samples in theregion.

In WLHS, the input variables are discretized unevenly/unequally in their design space. The cell (or hypercube,in multiple dimensions) size of probability of occurrence is evaluated according to topology of output/re-sponse. The cell size of probability of occurrence is discretized such that it is smaller (relatively) aroundminimum and maximum of output/response. As evaluation of cell size is output/response oriented, WLHSis somewhat unsymmetrical/biased.

In general, the WLHS is intended to stretch distribution farther out in the tails (lower and upper) with lessnumber of runs than Latin Hypercube Sampling (LHS). In other words, given same number of runs, WLHS isexpected to reach a smaller Pf when compared to LHS. Due to biasness, however, the evaluated Pf maysubject to some difference compared to LHS.



Postprocessing SSA Results

• Histogram (p. 97)

• Cumulative Distribution Function (p. 97)

• Probability Table (p. 98)

• Statistical Sensitivities in a Six Sigma Analysis (p. 99)

• Six Sigma Analysis Theory (p. 101)

Histogram

A histogram plot is most commonly used to visualize the scatter of a Six Sigma Analysis variable. A histogramis derived by dividing the range between the minimum value and the maximum value into intervals of equalsize. Then Six Sigma Analysis determines how many samples fall within each interval, that is, how many"hits" landed in each interval.

Six Sigma Analysis also allows you to plot histograms of your uncertainty variables so you can double-checkthat the sampling process generated the samples according to the distribution function you specified. Foruncertainty variables, Six Sigma Analysis not only plots the histogram bars, but also a curve for values derivedfrom the distribution function you specified. Visualizing histograms of the uncertainty variables is anotherway to verify that enough simulation loops have been performed. If the number of simulation loops is suffi-cient, the histogram bars will:

• Be close to the curve that is derived from the distribution function

• Be "smooth" (without large "steps")

• Not have major gaps (no hits in an interval where neighboring intervals have many hits)

However, if the probability density function is flattening out at the far ends of a distribution (for example,the exponential distribution flattens out for large values of the uncertainty variable) then there might logicallybe gaps. Hits are counted only as positive integer numbers and as these numbers gradually get smaller, azero hit can happen in an interval.

Cumulative Distribution Function

The cumulative distribution function is a primary review tool if you want to assess the reliability or the failureprobability of your component or product. Reliability is defined as the probability that no failure occurs.Hence, in a mathematical sense, reliability and failure probability are different ways of examining the sameproblem and numerically they complement each other (they sum to 1.0). The cumulative distribution functionvalue at any given point expresses the probability that the respective parameter value will remain belowthat point.

The value of the cumulative distribution function at the location x0 is the probability that the values of X

stay below x0. Whether this probability represents the failure probability or the reliability of your componentdepends on how you define failure; for example, if you design a component such that a certain deflectionshould not exceed a certain admissible limit, then a failure event occurs if the critical deflection exceeds thislimit. Thus, for this example, the cumulative distribution function is interpreted as the reliability curve of thecomponent. On the other hand, if you design a component such that the eigenfrequencies are beyond acertain admissible limit, then a failure event occurs if an eigenfrequency drops below this limit. So for thisexample, the cumulative distribution function is interpreted as the failure probability curve of the component.

The cumulative distribution function also lets you visualize what the reliability or failure probability wouldbe if you chose to change the admissible limits of your design.



Cumulative Distribution Function

A cumulative distribution function plot is an important tool to quantify the probability that the design ofyour product does or does not satisfy quality and reliability requirements. The value of a cumulative distri-bution function of a particular output parameter represents the probability that the output parameter willremain below a certain level as indicated by the values on the x-axis of the plot.

Example 2 Illustration of Cumulative Distribution Function

The probability that the Shear Stress Maximum will remain less than a limit value of 1.71E+5 is about 93%,which also means that there is a 7% probability that the Shear Stress Maximum will exceed the limit valueof 1.71E+5.

Figure: Cumulative Distribution Function

For more information, see Six Sigma Analysis Theory (p. 101).

Probability Table

Instead of reading data from the cumulative distribution chart, you can also obtain important informationabout the cumulative distribution function in tabular form. A Probability Table is available that is designedto provide probability values for an even spread of levels of an input or output parameter. You can viewthe table in either Quantile-Percentile (Probability) mode or Percentile-Quantile (Inverse Probability) mode.The Probability Table lets you find out the parameter levels corresponding to probability levels that aretypically used for the design of reliable products. If you want to see the probability of a value that is notlisted, you can add it to the table. Likewise, you can add a probability or sigma-level and see the correspondingvalues. You can also delete values from the table. For more information, see Using Statistical Postpro-

cessing (p. 63).



Figure: Probability Tables

Note

Both tables will have more rows, i.e., include more data, if the number of samples is increased. Ifyou are designing for high product reliability, i.e., a low probability that it does not conform toquality or performance requirements, then the sample size must be adequately large to addressthose low probabilities. Typically, if your product does not conform to the requirements denotedwith "Preq," then the minimum number of samples should be determined by: Nsamp = 10.0 /Preq.

For example, if your product has a probability of Preq=1.0 e-4 that it does not conform to therequirements, then the minimum number of samples should be Nsamp = 10.0 / 1.0e-4 = 10e+5.

Statistical Sensitivities in a Six Sigma Analysis

The available sensitivities plots allow you to efficiently improve your design toward a more reliable andbetter quality design, or to save money in the manufacturing process while maintaining the reliability and



Statistical Sensitivities in a Six Sigma Analysis

quality of your product. You can view a sensitivities plot for any output parameter in your model. See Sens-

itivities Chart (SSA) (p. 29) for more information.

The sensitivities available under the Six Sigma Analysis and the Goal Driven Optimization views are stat-istical sensitivities. Statistical sensitivities are global sensitivities, whereas the single parameter sensitivitiesavailable under the Responses view are local sensitivities. The global, statistical sensitivities are based on acorrelation analysis using the generated sample points, which are located throughout the entire space ofinput parameters. The local single parameter sensitivities are based on the difference between the minimumand maximum value obtained by varying one input parameter while holding all other input parametersconstant. As such, the values obtained for single parameter sensitivities depends on the values of the inputparameters that are held constant. Global, statistical sensitivities do not depend on the settings for the inputparameters, because all possible values for the input parameters are already taken into account when de-termining the sensitivities.

Design Exploration displays sensitivities as both a bar chart and pie chart. The charts describe the sensitivitiesin an absolute fashion (taking the signs into account); a positive sensitivity indicates that increasing thevalue of the uncertainty variable increases the value of the result parameter for which the sensitivities areplotted. Conversely, a negative sensitivity indicates that increasing the uncertainty variable value reducesthe result parameter value.

Using a sensitivity plot, you can answer the following important questions.

How can I make the component more reliable or improve its quality?

If the results for the reliability or failure probability of the component do not reach the expected levels, orif the scatter of an output parameter is too wide and therefore not robust enough for a quality product,then you should make changes to the important input variables first. Modifying an input variable that isinsignificant would be a waste of time.

Of course, you are not in control of all uncertainty parameters. A typical example where you have very limitedmeans of control involves material properties. For example, if it turns out that the environmental temperature(outdoor) is the most important input parameter, then there is probably nothing you can do. However, evenif you find out that the reliability or quality of your product is driven by parameters that you cannot control,this data has importance -- it is likely that you have a fundamental flaw in your product design! You shouldwatch for influential parameters like these.

If the input variable you want to tackle is a geometry-related parameter or a geometric tolerance, then im-proving the reliability and quality of your product means that it might be necessary to change to a moreaccurate manufacturing process or use a more accurate manufacturing machine. If it is a material property,then there might be nothing you can do about it. However, if you only had a few measurements for a ma-terial property and consequently used only a rough guess about its scatter, and the material property turnsout to be an important driver of product reliability and quality, then it makes sense to collect more raw data.

How can I save money without sacrificing the reliability or the quality of the product?

If the results for the reliability or failure probability of the component are acceptable or if the scatter of anoutput parameter is small and therefore robust enough for a quality product, then there is usually thequestion of how to save money without reducing the reliability or quality. In this case, you should first makechanges to the input variables that turned out to be insignificant, because they do not effect the reliabilityor quality of your product. If it is the geometrical properties or tolerances that are insignificant, you canconsider applying a less expensive manufacturing process. If a material property turns out to be insignificant,then this is not typically a good way to save money, because you are usually not in control of individual



material properties. However, the loads or boundary conditions can be a potential for saving money, but inwhich sense this can be exploited is highly problem-dependent.

Six Sigma Analysis Theory

The purpose of a Six Sigma Analysis is to gain an understanding of the impact of uncertainties associatedwith the input parameter of your design. This goal is achieved using a variety of statistical measures andpostprocessing tools.

Statistical Postprocessing

Convention: Set of data xi.

1. Mean Value

Mean is a measure of average for a set of observations. The mean of a set of n observations is definedas follows:

(1)µ^ ==∑

1

1nyi

i

n

2. Standard Deviation

Standard deviation is a measure of dispersion from the mean for a set of observations. The standarddeviation of a set of n observations is defined as follows:

(2)σ µ^ ^

( )( )=

−−

=∑

1

1

2

1nyi

i

n

3. Sigma Level

Sigma level is calculated as the inverse cumulative distribution function of a standard Gaussian distri-bution at a given percentile. Sigma level is used in conjunction with standard deviation to measuredata dispersion from the mean. For example, for a pair of quantile X and sigma level nα, it means thatvalue X is about nα standard deviations away from the sample mean.

4. Skewness

Skewness is a measure of degree of asymmetry around the mean for a set of observations. The obser-vations are symmetric if distribution of the observations looks the same to the left and right of themean. Negative skewness indicates the distribution of the observations being left-skewed. Positiveskewness indicates the distribution of the observations being right-skewed. The skewness of a set ofn observations is defined as follows:




(3)γµ

σ^

^

( )( )=

− −−

=

∑n

n n

yi

i

n

1 2 1

3

5. Kurtosis

Kurtosis is a measure of relative peakedness/flatness of distribution for a set of observations. It isgenerally a relative comparison with the normal distribution. Negative kurtosis indicates a relativelyflat distribution of the observations compared to the normal distribution, while positive kurtosis indicatesa relatively peaked distribution of the observations. As such, the kurtosis of a set of n observations isdefined with calibration to the normal distribution as follows:

(4)κµ

σ^

^( )

( )( )( )=

+− − −

−

=∑

n n

n n n

yi

i

n1

1 2 3 1

4

−−

− −3 1

2 3

2( )

( )( )

n

n n

where µ^ and σ^ represent mean and standard deviation, respectively.

Table 1 Different Types of Kurtosis

κ > 0κ = 0κ < 0

κ < 0 for flat distributionκ = 0 for normal distributionκ > 0 for peaked distribution

6. Shannon Entropy

Entropy is a concept first introduced in classical thermodynamics. It is a measure of disorder in a system.The concept of entropy was later extended and recognized by Claude Shannon in the domain of in-formation/communication as a measure of uncertainty over the true content of a message. In statistics,the entropy is further reformulated as a function of mass density, as follows:

(5)H P Pi i= −∑ ln

where Pi represents mass density of a parameter. In the context of statistics/probability, entropy becomesa measure of complexity and predictability of a parameter. A more complex, and less predictable,parameter carries higher entropy, and vice versa. For example, a parameter characterized with uniformdistribution has higher entropy than that with truncated triangular distribution (within the same



bounds). Hence, the parameter characterized with uniform distribution is less predictable comparedto the triangular distribution.

(6)PN

N wi

i=⋅ ∆

The probability mass is used in a normalized fashion, such that not only the shape, but the range ofvariability, or the distribution is accounted for. This is shown in Equation 6 (p. 103), where Ni/N is therelative frequency of the parameter falling into a certain interval, and ∆w is the width of the interval.As a result, Shannon entropy can have a negative value. Below are some comparisons of the Shannonentropy, where S2 is smaller than S1.

7. Taguchi Signal-to-Noise Ratios

Three signal-to-noise ratios are provided in the statistics of each output in your SSA. These ratios areas follows:

• Nominal is Best

• Smaller is Better

• Larger is Better

Signal-to-noise (S/N) ratios are measures used to optimize control parameters in order to achieve arobust design. These measures were first proposed by Dr. Taguchi of Nippon Telephone and TelegraphCompany, Japan, to reduce design noises in manufacturing processes. These design noises are normallyexpressed in statistical terms such as mean and standard deviation (or variance). In computer aidedengineering (CAE), these ratios have been widely used to achieve a robust design in computer simula-tions. For a design to be robust, the simulations are carried out with an objective to minimize thevariances. The minimum variance of designs/simulations can be done with or without targeting acertain mean. In Design Exploration, minimum variance targeted at a certain mean (called Nominal isBest) is provided, and is given as follows:

(7)ηµσ

= 20log^

^

where µ^ and σ^ represent mean and standard deviation, respectively.

Nominal is Best is a measure used for characterizing design parameters such as model dimension ina tolerance design, in which a specific dimension is required, with an acceptable standard deviation.

In some designs, however, the objectives are to seek a minimum or a maximum possible at the priceof any variance.




• For the cases of minimum possible (Smaller is Better), which is a measure used for characterizingoutput parameters such as model deformation, the S/N is expressed as follows:

(8)η = − ∑

=10

1 2

1log

nyi

i

n

• For the cases of maximum possible (Larger is Better), which is a measure used for characterizingoutput parameters such as material yield, the S/N is formulated as follows:

(9)η = − ∑

=

101 1

21log

n yii

n

These three S/N ratios are mutually exclusive. Only one of the ratios can be optimized for any givenparameter. For a better design, these ratios should be maximized.

8. Minimum and Maximum Values

The minimum and maximum values of a set of n observations are:

(10)x x x xnmin min( , ,..., )= 1 2

(11)x x x xnmax max( , ,..., )= 1 2

Note

The minimum and maximum values strongly depend on the number of samples. If yougenerate a new sample set with more samples, then chances are that the minimum valuewill be lower in the larger sample set. Likewise, the maximum value of the larger sampleset will most likely be higher than for the original sample set. Hence, the minimum andmaximum values should not be interpreted as absolute physical bounds.

9. Statistical Sensitivity Measures

The sensitivity charts displayed under the Six Sigma Analysis view are global sensitivities based onstatistical measures (see Single Parameter Sensitivities). Generally, the impact of an input parameteron an output parameter is driven by:

• The amount by which the output parameter varies across the variation range of an input parameter.

• The variation range of an input parameter. Typically, the wider the variation range is, the largerthe impact of the input parameter will be.

The statistical sensitivities used under Six Sigma Analysis are based on the Spearman-Rank Order Cor-relation coefficients that take both those aspects into account at the same time.



Basing sensitivities on correlation coefficients follows the concept that the more strongly an outputparameter is correlated with a particular input parameter, the more sensitive it is with respect tochanges of that input parameter.

10. Spearman Rank-Order Correlation Coefficient

The Spearman rank-order correlation coefficient is:

(12)r

R R S S

R R S S

s

i ii

n

ii

n

ii

n=

− −

− −

∑

∑ ∑

( )( )

( ) ( )2 2

where:

Ri = rank of xi within the set of observations x x xn

T1 2 ...

Si = rank of yi within the set of observations y y yn

T1 2 ...

R S, = average ranks of a Ri and Si respectively

11. Significance of Correlation Coefficients

Since the sample size n is finite, the correlation coefficient rp is a random variable itself. Hence, thecorrelation coefficient between two random variables X and Y usually yields a small, but nonzero value,even if X and Y are not correlated at all in reality. In this case, the correlation coefficient would be in-significant. Therefore, we need to find out if a correlation coefficient is significant or not. To determinethe significance of the correlation coefficient, we assume the hypothesis that the correlation betweenX and Y is not significant at all, i.e., they are not correlated and rp = 0 (null hypothesis). In this casethe variable:

(13)t rn

rP

P

=−

−

2

1 2

is approximately distributed like the student's t-distribution with ν = n - 2 degrees of freedom. Thecumulative distribution function student's t-distribution is:

(14)A t

B

xdx

t

t

( | )

,

νν

ν ν

ν

=

+

−+

−∫

1

1

2 2

12

1

2

where:

B(...) = complete Beta function




There is no closed-form solution available for Equation 14 (p. 105). See Abramowitz and Stegun (Pocket-

book of Mathematical Functions, abridged version of the Handbook of Mathematical Functions, HarryDeutsch, 1984) for more details.

The larger the correlation coefficient rp, the less likely it is that the null hypothesis is true. Also, thelarger the correlation coefficient rp, the larger is the value of t from Equation 13 (p. 105) and consequentlyalso the probability A(t|ν) is increased. Therefore, the probability that the null hypothesis is true isgiven by 1-A(t|ν). If 1-A(t|ν) exceeds a certain significance level, for example 2.5%, then we can assumethat the null hypothesis is true. However, if 1-A(t|ν) is below the significance level, then it can be as-sumed that the null hypotheses is not true and that consequently the correlation coefficient rp is sig-nificant. This limit can be changed in Design Exploration Options (p. 11).

12. Cumulative Distribution Function

The cumulative distribution function of sampled data is also called the empirical distribution function.To determine the cumulative distribution function of sampled data, you need to order the samplevalues in ascending order. Let xi be the sampled value of the random variable X having a rank of i,i.e., being the ith smallest out of all n sampled values. The cumulative distribution function Fi thatcorresponds to xi is the probability that the random variable X has values below or equal to xi. Sincewe have only a limited amount of samples, the estimate for this probability is itself a random variable.According to Kececioglu (Reliability Engineering Handbook, Vol. 1, 1991, Prentice-Hall, Inc.), the cumu-lative distribution function Fi associated with xi is:

(15)n

n k kF Fi

ki

n k

k i

n !

( )! !( ) %

−− =−

=∑ 1 50

Equation 15 (p. 106) must be solved numerically.

13. Probability and Inverse Probability

The cumulative distribution function of sampled data can only be given at the individual sampledvalues x1, x2, ..., xi, xi+1, ..., xn using Equation 15 (p. 106). Hence, the evaluation of the probability thatthe random variable is less than or equal to an arbitrary value x requires an interpolation between theavailable data points.

If x is, for example, between xi and xi+1, then the probability that the random variable X is less or equalto x is:

(16)P X x F F Fx x

x xi i i

i

i i

( ) ( )≤ = + −−

−++

11

The cumulative distribution function of sampled data can only be given at the individual sampledvalues x1, x2, ..., xi, xi+1, ..., xn using Equation 15 (p. 106). Hence, the evaluation of the inverse cumulativedistribution function for any arbitrary probability value requires an interpolation between the availabledata points.

The evaluation of the inverse of the empirical distribution function is most important in the tails ofthe distribution, where the slope of the empirical distribution function is flat. In this case, a direct in-terpolation between the points of the empirical distribution function similar to Equation 16 (p. 106) canlead to inaccurate results. Therefore, the inverse standard normal distribution function Φ-1 is appliedfor all probabilities involved in the interpolation. If p is the requested probability for which we are



looking for the inverse cumulative distribution function value, and p is between Fi and Fi+1, then theinverse cumulative distribution function value can be calculated using:

(17)x x x xp F

F Fi i i

i

i i

= + −−

+ −+

− −

− −( )

( ) ( )

( ) ( )1

1 1

1 11

Φ Φ

Φ Φ




Troubleshooting

Table view and Charts view become unreadable because of the number of parameters and the length oftheir names

In order to make your charts and tables more readable, consider using the short parameter name option:Open the Tools>Options dialog, select Design Exploration, and uncheck the Display Parameter Full

Name preference.

A Design Exploration Update operation returns an error message saying that one or several Design Pointshave failed to update

Press the Show Details button in the error dialog to see the list of Design Points that failed to updateand the associated error messages. Each failed Design Point is automatically preserved for you at theproject level, so you can Return to the Project and Edit the Parameter Set cell in order to see the failedDesign Points and investigate the reason for the failures.

This error means that the project failed to update correctly for the parameter values defined in the listedDesign Points. There may be a variety of failure reasons, from a lack of a license to a CAD model gener-ation failure, for instance. If you try to update the system again, only the failed Design Points will beattempted for a new Update, and if the update is successful, the operation will complete. If there arepersistent failures with a Design Point, copy its values to the Current Design Point, attempt a ProjectUpdate, and edit the cells in the project that are not correctly updated in order to investigate.

Table view disappears when all Response Points are deleted from the tableWhile editing a Response Surface, if there is only one Response Point in the Table view and you rightclick on it and select Delete from the menu, instead of an empty table remaining in table view, the tabledisappears completely and the reference to all response points (including any charts) is removed fromthe Outline. However, your Response Surface still exists and the table will reappear as soon as you createa new Response Point. This can be done from the Outline view. To get the Response Table to reappear:

1. In the Outline view, click on the New Response Point cell, type in a name for the Response Point,and press the Enter key to create a new Response Point with default values. The table will reappear,showing the Response Point that you just created.

2. Click on the new Response Point cell in the Outline and create charts for this Response Point asdescribed in Response Surface Charts (p. 49)



Appendices

• Glossary of General Terms (p. 111)

Glossary of General Terms

Design of Experiments(DOE)

Optimization method (used in Design Exploration) where each change of thevalue of any input variable requires a new analysis. To perform a "what-ifstudy" where several input variables are varied in a certain range, a consider-able number of analyses may be required to satisfactorily evaluate the resultsover the space of the input variables. The DOE method generates a ResponseSurface using curve and surface fitting algorithms to "fit" output data as afunction of input data. This requires a group of design points where eachpoint is generated via a solve.

Design Point A design point in which Parameter values were calculated directly in anAnalysis System, a CAD system, or DesignModeler.

Input Parameters Those model parameters exposed by the Analysis System (geometry, boundaryconditions, materials, etc.) for Design Exploration to use in the parametricanalysis.

Mean Value A measure of location often used to describe the general location of the bulkof the scattering data of a random output parameter or of a statistical distri-bution function.

Mathematically, the mean value is the arithmetic average of the data. Themean value also represents the center of gravity of the data points. Anothername for the mean value is the expected value.

Median Value The statistical point where 50% of the data is below the value and 50% isabove.

For symmetrical distribution functions (Gaussian, uniform, etc.) the medianvalue and the mean value are identical, but for nonsymmetrical distributions,they are different.

Output Parameters Parameters that define the response outputs (e.g., volume) from the analysis.

Pareto Set A concept used in multi-objective optimization, especially where some or allof the objectives are mutually conflicting. In such a case, there is no singlepoint which simultaneously yields the "best" value of all the objectives. Instead,the best solutions, called a Pareto or non-dominated set, are a group ofsolutions such that selecting any one of them in place of another will alwayssacrifice the quality of at least one objective while improving at least oneother.



Response Point A design point which has parameter values calculated in Design Exploration.As such, the parameter values are approximate and calculated from ResponseSurfaces.

Response Surface The empirical relationship between a variable of interest, y, and a set of inde-pendent variables, x1, x2, x3, ... Usually the function y = F(x1, x2, x3, . . . ) is apolynomial or some other well-defined relationship, which forms the responsesurface model of y. When applied to design and analysis in the realm of CADand CAE, a response surface is the representation of the physical behavior ofa structure in terms of its independent variables. For example, a responsesurface can be devised for the fundamental frequency (F) of a structure as afunction of CAD geometry parameters (R, L) and the modulus of elasticity (E);that is, F = G(R,L,E).

Sample A unique set of parameter values that represents a particular model configur-ation.

A sample is characterized by uncertainty variable values. Think of a sampleas one virtual prototype. Every component manufactured represents onesample, because you can measure its particular properties (material, geometry,etc.) and obtain specific values for each.

In statistics, however, sample also has a wider and more general use. For ex-ample, any single measured value of any physical property is considered tobe one sample. Because a Six Sigma Analysis is based on a statistical evaluationof the result parameters, the values of the result parameters are also calledsamples.

Standard Deviation A measure of variability (i.e., dispersion or spread) about the arithmetic meanvalue, often used to describe the width of the scatter of a random outputparameter or of a statistical distribution function.

The larger the standard deviation, the wider the scatter, and the more likelyit is that there are data values further apart from the mean value.

Uncertainty Variables Quantities that influence the result of an analysis.

In a Six Sigma Analysis, uncertainty variables are often called "drivers" becausethey drive the result of an analysis. You must specify the type of statisticaldistribution the uncertainty variables follow and the parameter values of theirdistribution functions.


Appendices

Index

Symbols2-D contour graph , 503-D contour graph , 51

Ccontour graphs

2-D, 503-D, 51

correlationparameters, 20

correlation matrixobject reference, 21

correlation scatter chartobject reference, 21

correlation sensitivities chartobject reference, 21

cumulative distribution function, 97

Ddecision support, 73Design Exploration, 17

introduction, 1options, 11overview, 17user interface, 5workflow, 9

design goalsdefining, 59

design of experiments, 18definition, 111six sigma, 27

design points, 8adding, 54, 61definition, 111uses, 54using, 33

design points vs parameter plotobject reference, 20

design values, 31distribution

choosing for random variables, 90data scatter values, 90measured data, 90no data, 91types, 92

DOEdefinition, 111

DOE solution, 34

Eerror messages, 109

Ggenetic algorithm, 73glossary, 111goal driven optimization, 65

creating design points, 61decision support, 73defining goals, 59genetic algorithm, 73guidelines, 66MOGA option, 57NLPQL option, 58options, 56Pareto dominance, 72postprocessing, 68principles, 65sampling method, 71screening option, 56sensitivities, 61theory, 71tradeoff studies, 62using, 56

goodness of fit, 49

Hhistogram , 97

Iinput parameters

definition, 111using, 30

Llocal sensitivities

description, 52local sensitivity chart

description, 52object reference, 24

Mmean value

definition, 111median value

definition, 111

Ooptimization, 56

defining goals, 59



MOGA option, 57NLPQL option, 58object reference, 24screening option, 56

options, 11output parameters, 32

definition, 111

Pparallel coordinate plot

description, 70parameters, 7, 29

design variables, 31input, 30output, 32uncertainty variables, 31

parameters correlation, 20, 54parameters parallel plot

object reference, 19Pareto dominance, 72Pareto set

definition, 111platform, 4postprocessing overview, 69postprocessing six sigma analysis results , 97

cumulative distribution function, 97histogram, 97probability table, 98sensitivities, 99

probability table, 98

Rresponse chart

object reference, 23response charts, 49

2-D contour graph, 503-D contour graph, 51goodness of fit, 49local sensitivities, 52spider charts, 51x-y graph, 50

response points, 8adding, 54, 61definition, 111

response surfacedefinition, 111object reference, 22

responsesoverview, 39response charts, 49

result parameterdefinition, 111

Ssample

definition, 111sample generation

MOGA, 57NLPQL, 58screening, 56, 71six sigma analysis, 96

samples chartobject reference, 26

sensitivitiesglobal, 53, 61, 63goal driven optimization, 61local, 52–53six sigma analysis, 63, 99

sensitivities chartglobal, 26, 29goal driven optimization, 26, 61object reference, 26, 29six sigma, 29six sigma analysis, 63

sensitivity chartdescription, 69local, 24, 52response surface, 24, 52

six sigmaobject reference, 26

six sigma analysis, 28, 87choosing variables, 89guidelines, 89performing, 63postprocessing, 63, 97sample generation, 96statistical measures, 64theory, 101understanding, 88using, 62

six sigma analysis postprocessing, 63charts, 63sensitivities, 63tables, 63

solutionDOE, 34

spider chartdescription, 51object reference, 24

standard deviationdefinition, 111

Ttradeoff chart

description, 69


Index

object reference, 25tradeoff studies, 62troubleshooting, 109

Uuncertainty variable

definition, 111uncertainty variables , 31

choosing, 89choosing distribution, 90response surface analyses, 89

user interface, 5

Wworkflow, 9

Xx-y graph, 50



Index

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