Optimization and Analysis of Doubly-Curved Kirigami Space ...

Optimization and Analysis of Doubly-Curved

Kirigami Space Frames by

Michael Ramirez

B.S. in Civil and Environmental Engineering

The University of Texas at Austin, 2019

Submitted to the Department of Civil and Environmental Engineering

in Partial Fulfillment of the Requirements for the degree of

MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

May 2020

©2020 Michael R. Ramirez. All rights reserved.

The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic

copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature of Author:

Michael Ramirez

Department of Civil and Environmental Engineering

May 8, 2020

Certified by:

Gordana Herning

Lecturer in Civil and Environmental Engineering

Thesis Supervisor

Accepted by:

Colette L. Heald

Professor of Civil and Environmental Engineering

Chair, Graduate Program Committee

3

Optimization and Analysis of Doubly-Curved

Kirigami Space Frames

by

Michael Ramirez

Submitted to the Department of Civil and Environmental Engineering

on May 8, 2020 in Partial Fulfillment of the Requirements for the

Degree of Master of Engineering in Civil and Environment Engineering

Abstract

Inspired by the Japanese art of origami and kirigami, the concept of Spin-Valence is used to

transform a two-dimensional sheet of metal into a three-dimensional spatial frame. Previous

iterations of this design were developed by Emily Baker of the University of Arkansas, mainly by

making scaled models. Recently a frame assembly of Spin-Valence units was analyzed

experimentally and computationally to characterize the structural behavior of the system.

Transitioning from flat to curved systems and building a full-scale pavilion motivates this study.

This thesis focuses on using optimization to computationally construct doubly-curved

configurations of the Spin-Valence pattern logic from input surfaces. To accomplish this,

optimization in Rhinoceros v6 and Python v3.7 is used to create a coherent primary and secondary

surface. The final structure is then subjected to finite element analysis using Abaqus 2017.

Throughout history, spanning structures have evolved from linear elements such as beams to

arches and finally to spatial systems. Each iteration manipulates form to counterbalance internal

element forces with better material efficiency and architectural flexibility. Doubly-curved Spin-

Valence surfaces are developed to allow greater versatility of form and frame characteristics.

Thesis Supervisor: Gordana Herning

Title: Lecturer in Civil and Environmental Engineering

5

Acknowledgements

I place my deepest gratitude for my thesis advisor, Dr. Gordana Herning, who has convincingly

guided and encouraged me even when the road became tough. Without her persistent help, the goal

of this project would not have been realized.

For Emily Baker and Edmund Harriss from the University of Arkansas and Nick Sherrow-Groves

from Arup, I would like to recognize the invaluable assistance and mentorship that you have given

me. Becoming part of the Spin-Valence team has been one of the most fruitful experiences of my

academic career.

I am indebted to the Massachusetts Institute of Technology’s Department of Civil and

Environmental Engineering for giving me the opportunity to explore structural engineering in a

new light and supporting me through the journey.

I would like to thank all my schoolmates and friends from MIT for making this year an

unforgettable experience, especially, Camp Seats, Yijiang Huang, and Mohamed Ismail, who

helped with the early configurations of the Spin-Valence deployment algorithm.

Finally, I would like to pay special regards to my lifelong friends and my family, my parents,

Rolando and Paz; my sister, Elaine; and my brother, Matthew. Your loving and continued support

has kept me going this past year.

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Contents

Abstract ......................................................................................................................................3

Acknowledgements .....................................................................................................................5

Contents .....................................................................................................................................7

List of Figures .......................................................................................................................... 11

List of Tables ............................................................................................................................ 13

1. Introduction.......................................................................................................................... 15

1.1. Characterizing the Spin-Valence Pattern-Logic ................................................................ 15

1.2. Qualitative Modifications to the RES ................................................................................. 17

1.3. Past Work ........................................................................................................................... 18 1.3.1. Structural Analysis by MIT and Arup ............................................................................................. 18 1.3.2. Doubly-Curved Variants of Spin-Valence ...................................................................................... 19

1.4. Motivation ........................................................................................................................... 19

1.5. Thesis Objectives: Computational Formulation and Analysis .......................................... 20

2. Spatial and Deployable Structures ....................................................................................... 21

2.1. Spatial Structures ............................................................................................................... 21

2.2. Deployable Structures ........................................................................................................ 23 2.2.1. Application of Deployable Structures ............................................................................................. 23 2.2.2. Classification of Deployable Structures .......................................................................................... 24

3. Previous Analyses of Kirigami Space Frames ...................................................................... 27

3.1. Motivation ........................................................................................................................... 27

3.2. Preliminary Structural Modifications by Baker ................................................................ 28

3.3. Initial Structural Characterization of the Spin-Valence Frame ........................................ 29 3.3.1. Strain Hardening of Nodal Components ......................................................................................... 29 3.3.2. Self-contact Between System Elements .......................................................................................... 30 3.3.3. Local Buckling Failure at Primary and Secondary Surfaces ............................................................ 30

3.4. Structural Recommendations ............................................................................................. 31

CONTENTS

8

4. Preliminary Computational Modeling for Deployment ........................................................ 33

4.1. Modeling Software .............................................................................................................. 33

4.2. Motivation ........................................................................................................................... 33

4.3. Base Unit Deployment Logic .............................................................................................. 34 4.3.1. Constraints to the Problem ............................................................................................................. 35 4.3.2. Three-Dimensional Configuration-Space ........................................................................................ 35

4.4. Algorithm Limitations and Transition ............................................................................... 38

5. Mathematic Definition for Spin-Valence Geometry Deployment ......................................... 39

5.1. Motivation ........................................................................................................................... 39

5.2. New Base Unit Deployment Logic ...................................................................................... 40 5.2.1. Notation ........................................................................................................................................ 40 5.2.2. Mathematic Constraints ................................................................................................................. 41 5.2.3. Solving the Deployment Equation for Parameters ........................................................................... 41

5.2.3.1: Case 1 (θ = 0): .......................................................................................................................... 46 5.2.3.2: Cases 2 and 3 (𝑡. 𝑟1 = 0) or (𝑡. 𝑟2 = 0) ...................................................................................... 46

6. Algorithms for Computational Primary and Secondary Surface Generation ....................... 49

6.1. Modeling Software .............................................................................................................. 49

6.2. Motivation ........................................................................................................................... 50

6.3. Proposed Workflow for Constructing a Doubly-Curved Pavilion ................................... 50 6.3.1. User-generated surfaces ................................................................................................................. 51 6.3.2. Grasshopper-Produced Secondary Surface ..................................................................................... 52 6.3.3. Optimized Deployed Primary Surface ............................................................................................ 54 6.3.4. Final Meshed Model in Rhino ........................................................................................................ 54 6.3.5. Finite Element Analysis ................................................................................................................. 55

6.4. Computationally Produced Secondary Surface ................................................................. 56 6.4.1. Constraints .................................................................................................................................... 57 6.4.2. Transition to Python-produced Primary Surface ............................................................................. 58

6.5. Optimized Deployment of Primary Surface ....................................................................... 59 6.5.1. Imported Python libraries ............................................................................................................... 59 6.5.2. Notation ........................................................................................................................................ 59 6.5.3. Parameters ..................................................................................................................................... 61 6.5.4. Optimization.................................................................................................................................. 61 6.5.5. File_read ....................................................................................................................................... 62 6.5.6. Objective ....................................................................................................................................... 63 6.5.7. Kirigami ........................................................................................................................................ 63 6.5.8. Distance ........................................................................................................................................ 68 6.5.9. Plot ............................................................................................................................................... 75 6.5.10. Results ............................................................................................................................................. 75

6.5.10.1. Nelder-Mead ............................................................................................................................ 76 6.5.10.2. Powell ...................................................................................................................................... 77 6.5.10.3. TNC (Truncated Newton Conjugate-Gradient) .......................................................................... 79 6.5.10.4. SLSQP (Sequential Least Squares Programming) ...................................................................... 80

CONTENTS

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6.5.10.5. Summary of 3x14 Assembly Optimization Case Study .............................................................. 82 6.5.10.6. Final Pavilion ........................................................................................................................... 82

6.5.11. Improvements for the Future ............................................................................................................ 84

7. Finite Element Analysis of Proposed Pavilion Design ......................................................... 85

8. Conclusion ........................................................................................................................... 87

8.1. Summary of Contributions ................................................................................................. 87

8.2. Future Work ....................................................................................................................... 88

8.3. Concluding Remarks and Potential Impacts ..................................................................... 89

9. References ............................................................................................................................ 91

Appendix A: Individual Deployment Logic in Mathematica .................................................... 93

Appendix B: Sample Calculation of Deployment in Mathematica ........................................... 99

Appendix C: Computational Primary Surface Python Script ................................................. 105

Objective ........................................................................................................................................ 106

File_Read ....................................................................................................................................... 109

Kirigami......................................................................................................................................... 110

Distance ......................................................................................................................................... 114

Plot ................................................................................................................................................. 118

Appendix D: Additional Images from Chapter 6 Results ....................................................... 121

3x14 Assembly ............................................................................................................................... 122

Nelder-Mead Optimization ........................................................................................................... 123

Powell Optimization ...................................................................................................................... 123

TNC Optimization ......................................................................................................................... 124

SLSQP Optimization ..................................................................................................................... 125

Non-Optimized Pavilion ................................................................................................................ 126

TNC Optimized Pavilion ............................................................................................................... 127

SLSQP Optimized Pavilion ........................................................................................................... 128

CONTENTS

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11

List of Figures

Figure 1.1: Deployment of Spin-Valence assembly: primary surface shown in blue;

secondary surface shown in red (Baker, 2018) ........................................................ 16

Figure 1.2: Miyamoto Tower, designed via the RES, by Yoshinobu Miyamoto

(Miyamoto, 2015) .................................................................................................... 16

Figure 1.3: Pythagorean tiling techniques for unit arrangement in CNC sheet

(Baker and Harriss, n.d.) .......................................................................................... 17

Figure 1.4: Individual structural unit modifications (Baker, 2018) ............................................. 18

Figure 1.5: NPPUs to create spherical double-curved surfaces (Baker and Harriss, n.d.) ............ 19

Figure 1.6: Aesthetic displays of Spin-Valence on University of Arkansas campus

(Baker, 2018) ........................................................................................................... 19

Figure 2.1: Crystal Palace roof spatial frame geometry (Addis, 2006) ....................................... 22

Figure 2.2: Spatial structures examples – San Francisco International Airport (left),

Singapore Jewel Changi airport (middle), Munich Olympiastadion (right)

(SOM, 2000; Saiidi, 2019; Rawn, 2019) .................................................................. 22

Figure 2.3: Origami mobile bridge (left), Rapid deployable military tent (right)

(Hiroshima University, 2015) (Eureka, 2014) .......................................................... 24

Figure 2.4: Deployable structures classification examples (Hanaor and Levy, 2001) ................. 25

Figure 3.1: Baker’s modifications to the initial RES pattern (Baker, 2018) ................................ 28

Figure 3.2: Strain hardening at kirigami unit nodes (Sherrow-Groves and Baker, 2019) ............ 29

Figure 3.3: Contact between angle-legs and primary surface

(Sherrow-Groves and Baker, 2019) .......................................................................... 30

Figure 3.4: Primary surface buckling failure (Sahuc et al., 2019) ............................................... 31

Figure 4.1: Parallel plane units (PPUs) vs. non-parallel plane units (NPPUs) ............................. 34

Figure 4.2: Center line equivalent length pairings for Spin-Valence geometry ........................... 35

Figure 4.3: Assigning S1 through parameters ϴ and ϕ ................................................................ 36

Figure 4.4: Assigning S2 from parameter, τ ............................................................................... 36

Figure 4.5: Locating S4 from intersection of three spheres (left); locating S3 from planar

relationships (right) ................................................................................................. 37

Figure 5.1: Notation for deployment algorithm .......................................................................... 40

Figure 5.2: Rodrigues Rotation Formula notation (Dale, 2015) .................................................. 45

Figure 5.3: Notation for r1 and r2 axes (left); Non-deployed tilting along r1 axis (middle);

tilting along r2 axis (right) ....................................................................................... 46

Figure 6.1: Input curves for surface generation .......................................................................... 51

Figure 6.2: Formulated surface from input curves...................................................................... 52

LIST OF FIGURES

12

Figure 6.3: Boundary conditions for input surface (highlighted in blue)..................................... 52

Figure 6.4: Grid-pattern of secondary surface ............................................................................ 53

Figure 6.5: Optimized Pythagorean tiling of secondary surface ................................................. 53

Figure 6.6: Optimized primary surface ...................................................................................... 54

Figure 6.7: Point connections in the primary surface created by E. Baker .................................. 55

Figure 6.8: Preliminary meshing of primary and secondary surface created by E. Baker (left) ... 55

Figure 6.9: Rendering of the proposed pavilion created by E. Baker (right) .............................. 55

Figure 6.10: Abaqus finite element model of 3x3 assembly ....................................................... 56

Figure 6.11: Secondary surface generated from input surface .................................................... 57

Figure 6.12: Global tiling specifications .................................................................................... 58

Figure 6.13: Labeled primary and secondary vertices ................................................................ 60

Figure 6.14: Offset center-line geometry ................................................................................... 61

Figure 6.15: Orient matrix numbering notation .......................................................................... 68

Figure 6.16: Notation for assigning sides to columns in xn matrix ............................................. 69

Figure 6.17: Sample orient (left) and xn (right) matrices for a 4x4 assembly ............................. 70

Figure 6.18: Flowchart for finding adjacent units ...................................................................... 71

Figure 6.19: pA and pB combinations including segment boundary conditions .......................... 74

Figure 6.20: Plot example using Axis3d .................................................................................... 75

Figure 6.21: Sample 3x14 assembly for optimization analysis ................................................... 76

Figure 6.22: Nelder-Mead optimization results .......................................................................... 77

Figure 6.23: Powell optimization results .................................................................................... 78

Figure 6.24: Steepest descent (blue) vs. Newton method (black) for locating local minima........ 79

Figure 6.25: TNC 1: Theta [.301,.800], Phi [.001,.5] (left), TNC 2: Theta [.301,.900],

Phi [.001,.6] (middle), TNC 3: Theta [.401,.900], Phi [.001,.5] (right) .................. 80

Figure 6.26: SLSQP 1: Theta [.301,.800], Phi [.001,.5] (left), SLSQP 2: Theta [.301,.900],

Phi [.001,.6] (middle), SLSQP 3: Theta [.401,.900], Phi [.001,.5] (right) .............. 81

Figure 6.27: Pavilion design prior to optimization ..................................................................... 82

Figure 6.28: TNC optimization for proposed pavilion design..................................................... 83

Figure 6.29: SLSQP optimization for proposed pavilion design ................................................. 83

Figure 7.1: 3x3 unit assembly for detailed finite element analysis ............................................. 86

13

List of Tables

Table 1: Sphere specifications to locate S4................................................................................. 37

Table 2: Simplified legend for Figure 6.16 ................................................................................ 69

Table 3: Xn matrix rules ............................................................................................................ 70

Table 4: Nelder-Mead Optimization Results .............................................................................. 77

Table 5: Powell Optimization Results........................................................................................ 78

Table 6: TNC Optimization Results........................................................................................... 80

Table 7: SLSQP Optimization Results ....................................................................................... 81

Table 8: Summary of 3x14 Assembly Optimization Results ...................................................... 82

Table 9: Proposed Pavilion Optimization Results ...................................................................... 83

LIST OF TABLES

14


15

Chapter 1

Introduction

This thesis presents new work in the computational design of doubly-curved kirigami space

frames, expanding upon previous research on deployable structures by Emily Baker (Baker,

2014, 2018). This chapter introduces the topic of Spin-Valence, past work on the subject, and

key applications for a computationally generated Spin-Valence frame from input surfaces.

1.1. Characterizing the Spin-Valence Pattern-Logic

Spin-Valence is a kirigami (cut and fold) pattern-logic that creates a deployable structural system

using a single sheet of CNC plasma cut steel, folded and reconnected to itself, to develop a double-

layer space frame (Baker, 2014, 2018). The system is devised to expand from the thickness of a

single sheet to dual surfaces (primary and secondary), that are connected with continuous

triangulated legs. Figure 1.1 depicts its deployment from a two-dimensional (2D) sheet to a three-

dimensional (3D) spatial frame.

16

Figure 1.1: Deployment of Spin-Valence assembly: primary surface shown in blue; secondary surface shown in red

(Baker, 2018)

Spin-Valence seeks to incorporate the advancements in deployable flat-pack design and

CAD/CAM software with building construction (Beorkem, 2013). The Spin-Valence pattern itself

takes inspiration from the spin-fold technique of Yoshinobu Miyamoto’s Rotational Erection

System (RES), shown in Figure 1.2. The RES on its own is unable to perform as a structural system

due to its tendency to fold back in on itself. However, arranging the units in an “assembly”, or an

array, as seen in Figure 1.1, provides that necessary stability by reconnecting with adjacent units

after deployment. By reconnecting, the frame creates triangulating legs between the two surfaces

that behave similar to a spatial truss.

Figure 1.2: Miyamoto Tower, designed through the RES, by Yoshinobu Miyamoto (Miyamoto, 2015)

CHAPTER 1. INTRODUCTION

17

The first Spin-Valence designs use the method of Pythagorean tiling to arrange units on a flat sheet

of material. Pythagorean tiling assigns a grid pattern to a surface and rotates the created quads

around their centroid. During rotation, each unit is scaled so its sides connect with adjacent units

at the midpoints of their original quad’s sides. These unique patterns are integral to conceptualizing

flat and singly-curved kirigami frames. As seen in Figure 1.3, rotating the primary and secondary

surface units, 15° and 45° respectively, allows for the appropriate reconnection in the deployed

surface (Baker and Harriss, n.d.).

Figure 1.3: Pythagorean tiling techniques for unit arrangement in CNC sheet (Baker and Harriss, n.d.)

1.2. Qualitative Modifications to the RES

The original RES geometry has since been modified by Emily Baker to add structural capacity to

an individual unit. Baker employs an attitude of “serious play” during this investigation with

iterative physical prototyping, adjusting the geometry based on qualitative observations (Gore,

2004). Figure 1.4 illustrates the modifications from the initial RES system. Miyamoto’s original

straight connecting legs now incorporate a slight convex curvature, being the widest at their

midsections. Moreover, angled legs are introduced to the frame for additional cross-sectional area

and buckling capacity (Baker, 2018).

1.2. QUALITATIVE MODIFICATIONS TO THE RES

18

Figure 1.4: Individual structural unit modifications (Baker, 2018)

1.3. Past Work

1.3.1. Structural Analysis by MIT and Arup

During the International Association for Shell and Spatial Structures (IASS) Annual Symposium

in 2019, Baker established a team of engineers from MIT and Arup to start initial structural

computational analysis for the Spin-Valence system. This collaboration led to physical testing of

a single kirigami unit and of a 3x3 assembly, coupled with a finite element model created in LS-

DYNA (LSTC, 1978) that was validated using the experimental testing results. Both approaches

explored the structural limitations of the spatial frame in terms of strength, stiffness, ductility, and

buckling, and proposed improvements to the design (Sherrow-Groves and Baker, 2019; Sahuc et

al. 2019)


19

1.3.2. Doubly-Curved Variants of Spin-Valence

To explore possible configurations of Spin-Valence, double-curved variants have been produced,

making use of non-parallel plane units (NPPUs) as seen in Figure 1.5. However, these

arrangements are limited to geometric aggregation instead of user-input surfaces (Baker and

Harriss, n.d.). The next steps for the Spin-Valence system explore creating doubly-curved kirigami

frames that would allow additional geometric configurations.

Figure 1.5: NPPUs to create spherical doubly-curved surfaces (Baker and Harriss, n.d.)

1.4. Motivation

With increased architectural variability and structural capacity, the Spin-Valence system has the

potential to be extended to wider applications including structural installations.

Figure 1.6: Aesthetic displays of Spin-Valence on University of Arkansas campus (Baker, 2018)

1.4. MOTIVATION

20

Extension to various input surfaces would allow Spin-Valence to handle multiple boundary

conditions and loading scenarios. This would yield opportunities for use as long-spanning spatial

structures for temporary bridges, curved vaults, or gabled frames. In addition, its deployable flat-

pack design gives the system enhanced utility for transportation and eliminates the need for

complex assembly of discrete members.

These characteristics differentiate the Spin-Valence frames amongst the established deployable

methods, such as scissor mechanisms. Introducing computational programming to the kirigami

frames can further strengthen the value of the system by highlighting its adaptability.

1.5. Thesis Objectives: Computational Formulation

and Analysis

To produce a workflow for generating variable kirigami frames, a team including Emily Baker, an

architect from the University of Arkansas, Edmund Harriss, a mathematician from the University

of Arkansas, Nick Sherrow-Groves, a structural engineer from Arup, Gordana Herning, a lecturer

at MIT, and the author was formed. Each member contributes their strengths, with Baker and

Harriss focusing on computational geometry and Sherrow-Groves, Herning, and Ramirez

exploring the structural analysis, while still engaging in interdisciplinary conversation. To begin,

a preliminary algorithm for mapping input surfaces has been created, using Rhinoceros v6

(McNeel, 1980) and the Pythagorean tiling method referenced in Section 1.1. Baker and Harriss

progressed the design by producing a computationally formed secondary surface with the Rhino

plug-in, Grasshopper (McNeel, 2007).

This thesis focuses on improving Baker and Harriss’s initial algorithm and creating a script that

mathematically formulates a primary surface from the data obtained for the secondary surface.

Moreover, this paper will investigate the formation of an efficient workflow from input surface

conception to finite element analysis of a completed doubly-curved kirigami frame.


21

Chapter 2

Spatial and Deployable Structures

This chapter discusses a brief literature review of spatial and deployable structures. First, their

features and applications are characterized, and then their respective categorizations are listed.

Deployable structures are a subcategory of spatial structures. The kirigami spatial frame is a new

methodology for deployable structures that takes inspiration from spatial structure precedents.

2.1. Spatial Structures

Prior to the 19th century, horizontal spanning structures, composed of interconnected flat planes,

were the preferred methods for roof and bridge construction. However, deficiencies with out-of-

plane stiffness and stability limit their applications, both structurally and aesthetically. Spatial

structures improve upon these limitations by incorporating three-dimensionality to their restraints

and internal networks of members (Lin and Huang, 2016). The roof structure of London's Crystal

Palace shown in Figure 2.1 is an early example of spatial frame development. Joseph Paxton, the

architect, through prefabrication, efficient geometric force transfers, and pre-tensioning,

significantly reduced construction time and material costs with his design (Addis, 2006). Projects

similar to Paxton’s increased the renown of spatial structures for both their elegant appearance and

economic value.

22

Figure 2.1: Crystal Palace roof spatial frame geometry (Addis, 2006)

Spatial structures can be divided into three categories: spatial truss/grid structures, spatial latticed

shell, and membrane. Figure 2.2 provides an example of each system from left to right. Each

category exploits introducing three-dimensionality to the system in different ways for additional

structural capacity.

Figure 2.2: Spatial structures examples – San Francisco International Airport (left),

Singapore Jewel Changi Airport (middle), Munich Olympiastadion (right)

(SOM, 2000; Saiidi, 2019; Rawn, 2019)

The traditional kirigami frame, once fully deployed, is modeled after the spatial truss system. This

structural method incorporates out-of-plane connections to the traditional planar truss design. 3D

connections establish efficient force transfers, where members are arranged to mutually support

one another against in-plane and out-of-plane loading (Lin and Huang, 2016). The triangulating

legs of the kirigami frame enable these productive force transfers.

CHAPTER 2. SPATIAL AND DEPLOYABLE STRUCTURES

23

Introducing double-curvature to the Spin-Valence design rewards the benefits from latticed shell

structures. Latticed shell structures are composed of a geometric array of singly or doubly-curved

members that interconnect to generate a rigid shell framework (Lin and Huang, 2016).

Throughout history spanning structures have evolved from linear elements such as beams and

columns to arches and finally to spatial systems. Each iteration showcased new ways of

counterbalancing internal element forces and bending moments with higher material efficiency

and greater architecturally variability. Doubly-curved Spin-Valence frames attempts to innovate

the previously flat configuration. Introducing curvature provides greater material performance,

capability for longer spans, and visually appealing designs.

2.2. Deployable Structures

Deployable structures are a subcategory of spatial structures where systems transform from a

compacted, stowed state to an expanded and functional form.

2.2.1. Application of Deployable Structures

Typically, deployable structures are used in two types of applications: “temporary structures…

[or] inaccessible or remote places, such as outer space” (Hanaor and Levy, 2001). Moreover,

deployable structures can be implemented in emergency-relief situations, such as natural disaster

infrastructure or transportable military shelters. Examples are displayed in Figure 2.3.

2.2. DEPLOYABLE STRUCTURES

24

Figure 2.3: Origami mobile bridge (left), Rapid deployable military tent (right)

(Hiroshima University, 2015) (Eureka, 2014)

The kirigami space frame has the potential to satisfy both applications. Baker’s flat pack design

enables the system to have all structural components inscribed computationally into a sheet of

material. This facilitates easier transportation and implementation to areas with little resources.

Currently, reconnection in the deployed surface requires tack-welding for stability, making it

irreversible. However, Baker is exploring other methods such as bolting to make the system more

adaptable to potential construction constraints.

2.2.2. Classification of Deployable Structures

Deployable structures can be classified under two tiers: morphology and kinematics. Morphology

of these systems can either be skeletal (lattice, scissor mechanism), continuous (surface or stressed-

skin), or a hybrid of the two. Kinematics refers to the method of deployment, whether it is driven

by rigid linkages or deformables (cables or fabrics) (Hanaor and Levy, 2001). Figure 2.4 visually

explains both tiers with examples.


25

Figure 2.4: Deployable structures classification examples (Hanaor and Levy, 2001)

2.2. DEPLOYABLE STRUCTURES

26



27

Chapter 3

Previous Analyses of Kirigami Space Frames

This chapter summarizes prior analysis of the Spin-Valence kirigami space frames. These include

qualitative analyses through creating and refining the physical models of various configurations

for the frame (Baker, 2014, 2018), computational form finding (Baker and Harriss, n.d.), and

characterization of structural behavior of frame elements and assemblies (Sherrow-Groves and

Baker, 2019; Sahuc et al., 2019).

3.1. Motivation

As mentioned in Section 1.2, the previous iterations of Spin-Valence units were created through

physical prototyping and qualitative observations of their structural characteristics by Baker.

Creating a computational model for the kirigami frame is an important next step to advance the

Spin-Valence geometry.

With an experimentally validated computational model, structural testing and geometric

implications for strength and stability could be investigated without producing physical models,

which would save time and materials. Physical model making remains valuable and would be

focused on anticipating and resolving construction-related features for specific configurations.

Preemptively seeing stress distributions or connections prior to construction allows designers to

28

fine-tune the system according to exact scenarios. This additional flexibility enables exploration

into new materials, such as carbon fiber, or complex shapes like doubly-curved surfaces.

3.2. Preliminary Structural Modifications by Baker

In order to achieve increased structural capacity, Baker modified the initial RES pattern, as seen

in Figure 1.4, with three notable changes:

1. Adding curvature to all legs

2. Designing legs that have inscribed folds, named “angle-legs”

3. Designing “plate-legs” that have no longitudinal folds

These elements of design are highlighted in Figure 3.1. Methods listed above provide an added

cross-sectional area for the diagonal elements where it is needed, giving them greater buckling

capacity. They also prevent the deployed surface from folding back in on itself (Baker, 2018).

Figure 3.1: Baker’s modifications to the initial RES pattern (Baker, 2018)

Every structural unit in an assembly includes angle-legs while plate-legs are reserved for specific

locations at the perimeter of an assembly. Triangulation by connecting adjacent units in the

deployed frame generates added strength and stiffness.

CHAPTER 3. PREVIOUS ANALYSES OF KIRIGAMI SPACE FRAMES

29

3.3. Initial Structural Characterization of the

Spin-Valence Frame

The experimental testing and computational modeling yielded three significant behaviors for the

modified geometry:

1. Strain hardening of nodal components

2. Self-contact between Spin-Valence elements

3. Local buckling failure at primary and secondary surfaces

3.3.1. Strain Hardening of Nodal Components

Strain hardening, occurring at the ends of the angle-legs, contributed considerably to structural

capacity during loading. Angle-legs have their lowest cross-sectional areas in those sections,

causing them to be the first elements to yield, or reach stress values beyond the elastic region,

when the frame is subjected to progressively increasing loads. A unique property of steel is when

loaded beyond its elastic region, steel provides additional resistance until failure, a phenomenon

known as plastic strain hardening (Sherrow-Groves and Baker, 2019). Figure 3.2 illustrates an

approximated zone where plasticity occurs.

Figure 3.2: Strain hardening at kirigami unit nodes (Sherrow-Groves and Baker, 2019)

3.3. INITIAL STRUCTURAL CHARACTERIZATION OF THE SPIN-

VALENCE FRAME

30

Experimental testing observed that the redundancy of the triangulating legs will redistribute loads

when these nodes reach maximum capacity (Sahuc et al. 2019).

3.3.2. Self-contact Between System Elements

At the same interface mentioned above, another observation occurred post-buckling, instead of

post-yielding. Because of their lower cross-sectional area, those portions have a tendency to buckle

under axial loading. Sometimes, post-buckled angle-legs make contact with the primary surface

during loading as seen in Figure 3.3, adding significant strength to a yielded kirigami frame.

Contact between the angle-legs and the secondary surface, however, is more uncertain and cannot

be relied on (Sherrow-Groves and Baker, 2019).

Figure 3.3: Contact between angle-legs and primary surface (Sherrow-Groves and Baker, 2019)

3.3.3. Local Buckling Failure at Primary and Secondary Surfaces

Element buckling occurred mostly in the primary and secondary surfaces. Tensile and compressive

axial forces from the trusses activated this failure. This indicates that the angle-legs were over-

designed, as the distribution of stresses to the folded region was not as significant as anticipated.

Improved material allocations between the legs and surface elements were recommended for future

iterations (Sherrow-Groves and Baker, 2019).


31

Figure 3.4: Primary surface buckling failure (Sahuc et al., 2019)

3.4. Structural Recommendations

Prior analyses indicated that more efficient material allocation in the Spin-Valence design may be

possible. As mentioned in the previous section, angle-legs were slightly over-designed while the

primary and secondary surfaces experienced buckling. Therefore, future iterations of Spin-Valence

should CNC cut more steel into the surfaces instead of the legs. Moreover, modifying the geometry

to reliably force post-buckling self-contact from Section 3.2 will increase strength. Concluding

remarks from the laboratory and computational data also suggest that the tested system could

successfully be used in artistic structural installations but not yet suitable for spanning structures

(Sahuc et al., 2019; Sherrow-Groves and Baker, 2019).

To explore its architectural and structural benefits, this study looks into introducing double-

curvature to the Spin-Valence geometry. Similar to the spatial frame evolution from Section 2.1,

three-dimensionality or curvature traditionally enables more architectural versatility and structural

capacity to designs. Coupled with allowing input surfaces, the Spin-Valence system has the

potential to easily transport and construct structurally viable complex shells.

3.4. STRUCTURAL RECOMMENDATIONS

32



33

Chapter 4

Preliminary Computational Modeling for

Deployment

This chapter examines Ramirez’s initial attempt at computationally modeling the deployment of

an individual Spin-Valence unit. The motivations behind computational modeling, the written

Grasshopper-based algorithm, and the results are discussed in this section.

4.1. Modeling Software

Geometry is defined in the CAD software, Rhinoceros v6 (Rhino), using the computational plug-

in Grasshopper for parametric design.

4.2. Motivation

As mentioned in Section 3.1, computational models enable users to discover more variable designs

without having to physically construct them. Prior to the following algorithms in Chapters 4-6, all

CAD configurations of Spin-Valence have been produced iteratively in Rhino. The methods

discussed below focus on creating a dynamic model from user-input parameters. Chapter 4

34

specifically explores generating a Spin-Valence unit using Grasshopper, allowing for adjustable

parameters such as unit sizing and deployment position.

Advancing from the traditionally flat kirigami frame to a doubly-curved surface requires

understanding NPPUs, or non-parallel plane units from Section 1.3.2. Flat or singly-curved

surfaces allow for both planes to remain relatively parallel, which can be easily modeled in a CAD

software. However, doubly-curved surfaces force the planes to be non-parallel, creating the tilting

effect seen in Figure 4.1. Figure 4.1 displays NPPUs compared to their counterparts PPUs, or

parallel plane units, using a 3D printed Spin-Valence model created by Emily Baker. Figure 5.3

showcases this model as well.

Figure 4.1: Parallel plane units (PPUs) vs. non-parallel plane units (NPPUs)

Therefore, properly understanding the deployment of a single unit is paramount to creating a fully

computational model. For user-variability and future optimization, unit deployment needs to be

controllable by modifiable parameters. The algorithm below describes a preliminary method for

solving the deployment of a Spin-Valence unit, using Grasshopper’s object-oriented software.

4.3. Base Unit Deployment Logic

This deployment logic is a 3D configuration-space for an individual unit. Input variables for this

CHAPTER 4. PRELIMINARY COMPUTATIONAL MODELING FOR

DEPLOYMENT

35

algorithm are polygon size, which can be adjusted freely using Grasshopper’s built-in sliders, and

the deployment parameters: ϴ, ϕ, and τ. The derivation for these parameters will be described in

Section 4.3.2. Moreover, polygons for this algorithm are limited to quads for simplification.

4.3.1. Constraints to the Problem

Spin-Valence’s design is deeply rooted in kirigami, so the computational model must adhere to

the real-world conditions of this system. To simulate this constraint, center-line lengths for all

pairings of primary surface, secondary surface, and leg elements must be equivalent before, during,

and after deployment. These pairings are illustrated in Figure 4.2.

Figure 4.2: Center line equivalent length pairings for Spin-Valence geometry

4.3.2. Three-Dimensional Configuration-Space

Unit deployment varies according to three user-defined parameters: ϴ, ϕ, and τ. These variables

are derived using Grasshopper’s visual coding and object-based programming. Grasshopper is a

convenient medium since it can calculate relationships between CAD objects, such as intersections

and distances, without needing to specify a mathematic equation. This method is highly systematic,

starting with prescribing a ϴ and ϕ and ending with defining a τ based on the previous variables.

The following pictures and commentary will provide a visual instruction of the deployment logic.

4.3. BASE UNIT DEPLOYMENT LOGIC

36

The process begins by choosing a point on the primary surface as a reference point, denoted 𝑃1.

𝑆1, its corresponding secondary surface point is found by creating a sphere with radius of length,

𝐿1 as referenced in Figure 4.2. Using polar coordinates with the origin at 𝑃1, defining a ϴ (x-y

plane rotation) and a ϕ (z-axis rotation) yields the point 𝑆1, as shown in Figure 4.3.

Figure 4.3: Assigning S1 through parameters ϴ and ϕ

Referring to Figure 4.2, 𝑆2 must be 𝐿2 away from two points: 𝑃2 and the previously defined 𝑆1.

Following a similar sequence as Figure 4.3, 𝑆2 is found by calculating the intersection of two

spheres of radius, 𝐿2, centered at 𝑆1 and 𝑃2.

The intersection of two spheres is a circle, so the possible locations for 𝑆2 are found in the

calculated circle’s circumference. Parameter, τ, ranged [0,1], defines a point on the circle based on

the completion around the circumference. Figure 4.4 illustrates this logic.

Figure 4.4: Assigning S2 from parameter, τ


DEPLOYMENT

37

𝑆4 is found using the length relationships from Figure 4.2 and the same methodology of Figure

4.4. However, in this case, it includes the intersection of three spheres instead of two. Table 1

summarizes the specifications for all three spheres.

Table 1: Sphere specifications to locate S4

Sphere Center point Radius

1 𝑃4 𝐿4

2 𝑆1 𝐿1

3 𝑆2 𝐿2→4

First, the intersection of Spheres 2 and 3 are found, and once more, this yields a circle.

Radius, 𝐿2→4, is the distance between 𝑃2 and 𝑃4 on the primary surface. Finding where this newly

defined circle and Sphere 1 converge provides two point solutions. 𝑆4 is assigned to the point with

the greatest z-value because the other solution creates a non-constructible Spin-Valence

configuration.

Having 𝑆1, 𝑆2, and 𝑆4 generates the plane for the secondary surface. Using Grasshopper’s built-in

programs, creating circles around 𝑆2 and 𝑆4 with radii, 𝐿3 and 𝐿4 respectively, on the newly

constructed plane of the secondary surface, gives the coordinates for the final point, 𝑆3. Locating

the final points, 𝑆3 and 𝑆4 are illustrated in Figure 4.5.

Figure 4.5: Locating S4 from intersection of three spheres (left); locating S3 from planar relationships (right)

4.3. BASE UNIT DEPLOYMENT LOGIC

38

4.4. Algorithm Limitations and Transition

The algorithm, although functional, is restricted to a non-linear and discrete configuration-space.

Qualitative observations suggest that parameter, τ, is highly dependent on the combination of ϴ

and ϕ. Keeping τ consistent while changing ϴ and ϕ has the potential to create a non-constructible

or imaginary surface. This makes performing future optimization difficult due to the system’s non-

predictability.

Moreover, the algorithm is reliant on Grasshopper’s object-oriented programming. Translating

these intersections of 3D objects to mathematic equations becomes more complicated with every

equation introduced. For these reasons, research into finding a more mathematic-based deployment

algorithm is necessary to use optimization for generating a primary or secondary surface.


DEPLOYMENT

39

Chapter 5

Mathematic Definition for Spin-Valence

Geometry Deployment

This chapter explores finding a mathematic definition for the deployment of an individual Spin-

Valence unit. Mathematic equations enable a better understanding of the system and have the

ability to be translated across software. The section below discusses the motivations for moving

past the initial Grasshopper algorithm and the derivation of the 2D configuration-space.

5.1. Motivation

As discussed in Section 4.2, understanding the deployment of a single Spin-Valence unit is integral

to generating a fully computational doubly-curved frame. Doubly-curved surfaces force their

primary and secondary planes to be non-parallel, so computational models must support these

configurations. Explained in Section 4.4, the previous method is unideal because it creates a

disjointed configuration-space and is too difficult to translate into other programs besides

Grasshopper.

Therefore, an entirely mathematic approach to solving deployment is optimal. These equations

would establish a more predictable configuration-space while being easily adapted to any matrix-

based coding language.

40

The following linear algebra proof is completed through collaborations between the University of

Arkansas and MIT. This derivation solves deployment as a 2D configuration-space instead of 3D

as first seen in Chapter 4 and works efficiently with rhombus shapes. Systems of equations and

simplifications are solved using Wolfram Mathematica v12.0 (Wolfram, 1988).

5.2. New Base Unit Deployment Logic

5.2.1. Notation

The initial inputs for the algorithm are the coordinates of the primary surface, denoted as:

𝑟1, 𝑟2, 𝑟3, 𝑟4

Similarly, the coordinates of the deployed surface are denoted as:

𝑠1, 𝑠2, 𝑠3, 𝑠4

These points can be thought of as 3D vectors starting at the origin. The vector, P, in Figure 5.1,

gives the centroid of the secondary surface. In this derivation, finding the coordinates of P solves

all points on the secondary surface.

Figure 5.1: Notation for deployment algorithm

CHAPTER 5. MATHEMATIC DEFINITION FOR SPIN-VALENCE

GEOMETRY DEPLOYMENT

41

The derivation in Section 5.2.3 refers to dot and cross products with the following notation:

Dot product: 𝑎. 𝑏

Cross: 𝑎 𝑥 𝑏

5.2.2. Mathematic Constraints

For consistency with real-world kirigami logic, the algorithm adheres to the same kirigami

constraints as Section 4.3.1. Center-line lengths in the primary surface, secondary surface, and leg

elements must be equivalent. Therefore, the following relationships, (1) and (2), hold true for all

length pairings in Figure 4.2.

Equation (1) ensures that the original side lengths are consistent in the secondary surface.

(𝑟1 − 𝑟2). (𝑟1 − 𝑟2) = (𝑠1 − 𝑠2). (𝑠1 − 𝑠2) = 𝑙1 (1)

Equation (2) forces diagonal leg lengths and side lengths to be equivalent.

(𝑠1 − 𝑟2). (𝑠1 − 𝑟2) = 𝑙1 (2)

Moreover, the algorithm currently works smoothly with only squares and rhombuses. Therefore,

𝑟1 = −𝑟3; 𝑟2 = −𝑟4 (3)

5.2.3. Solving the Deployment Equation for Parameters

The following proof solves the Spin-Valence deployment for a defined rhomb in a “local

coordinate system”. In the local system, the base geometry has its centroid at the origin, with its

rhomboidal primary and secondary axes along the x and y axes, respectively.

5.2. NEW BASE UNIT DEPLOYMENT LOGIC

42

The deployment relationship between the primary and secondary surface can be described with:

Side 1: 𝑠1 = 𝑅[𝑟1]

Side 2: 𝑠2 = 𝑅[𝑟2]

Side 3: 𝑠3 = 𝑅[𝑟3]

Side 4: 𝑠4 = 𝑅[𝑟4] (4)

Combining and simplifying (2) and (4) with Side 1:

(𝑠1 − 𝑟2). (𝑠1 − 𝑟2) =

(𝑅[𝑟1] − 𝑟2). (𝑅[𝑟1] − 𝑟2) =

𝑅[𝑟1]. 𝑅[𝑟1] − 2𝑅[𝑟1]. 𝑟2 + 𝑟2. 𝑟2 = 𝑙1 (5)

Transformation, 𝑅, is assumed to be an isometry, or a transformation that remains invariant with

respect to internal distances. In other words, applying 𝑅 does not change 𝑟1 and 𝑟2’s relationship

with one another. Now, (5) can be simplified with (1).

𝑅[𝑟1]. 𝑅[𝑟1] − 2𝑅[𝑟1]. 𝑟2 + 𝑟2. 𝑟2 = (𝑟1 − 𝑟2). (𝑟1 − 𝑟2)

𝑅[𝑟1]. 𝑅[𝑟1] − 2𝑅[𝑟1]. 𝑟2 + 𝑟2. 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟2. 𝑟1) + 𝑟2. 𝑟2

𝑅[𝑟1]. 𝑅[𝑟1] − 2𝑅[𝑟1]. 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟2. 𝑟1) (5a)

Applying (5a) to all sides in (4):

Side 1: 𝑅[𝑟1]. 𝑅[𝑟1] − 2𝑅[𝑟1]. 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟2. 𝑟1)



Side 4: 𝑅[𝑟4]. 𝑅[𝑟4] − 2𝑅[𝑟4]. 𝑟1 = 𝑟4. 𝑟4 − 2(𝑟1. 𝑟4) (6)


GEOMETRY DEPLOYMENT

43

Combining (3) and (6),


Side 2: 𝑅[𝑟2]. 𝑅[𝑟2] + 2𝑅[𝑟2]. 𝑟1 = 𝑟2. 𝑟2 + 2(𝑟1. 𝑟2)

Side 3: 𝑅[−𝑟1]. 𝑅[−𝑟1] + 2𝑅[−𝑟1]. 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟1. 𝑟2)

Side 4: 𝑅[−𝑟2]. 𝑅[−𝑟2] − 2𝑅[−𝑟2]. 𝑟1 = 𝑟2. 𝑟2 + 2(𝑟1. 𝑟2) (7)

Now, as 𝑅[𝑥] is an isometry, it can be replaced with the function, 𝑅 ∗ 𝑥 + 𝑃, where 𝑃 is the vector

to the centroid of the secondary surface. Moreover, since R is defined as a linear rotation matrix,

𝑅[−𝑥] = −𝑅 ∗ 𝑥 + 𝑃. Substituting into (7),

Side 1: (𝑅𝑟1 + 𝑃). (𝑅𝑟1 + 𝑃) − 2(𝑅𝑟1 + 𝑃). 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟1. 𝑟2)

Side 2: (𝑅𝑟2 + 𝑃). (𝑅𝑟2 + 𝑃) + 2(𝑅𝑟2 + 𝑃). 𝑟1 = 𝑟2. 𝑟2 + 2(𝑟1. 𝑟2)

Side 3: (−𝑅𝑟1 + 𝑃). (−𝑅𝑟1 + 𝑃) + 2(−𝑅𝑟1 + 𝑃). 𝑟2 = 𝑟1. 𝑟1 − 2(𝑟1. 𝑟2)

Side 4: (−𝑅𝑟2 + 𝑃). (−𝑅𝑟2 + 𝑃) − 2(−𝑅𝑟2 + 𝑃). 𝑟1 = 𝑟2. 𝑟2 + 2(𝑟1. 𝑟2) (7a)

Where,

𝑅𝑟1 = 𝑅 ∗ 𝑟1 and 𝑅𝑟2 = 𝑅 ∗ 𝑟2

Expanding Side 1 in (7a),

𝑅𝑟1. 𝑅𝑟1 + 2(𝑃. 𝑅𝑟1) + 𝑃. 𝑃 − 2(𝑅𝑟1. 𝑟2) − 2(𝑃. 𝑟2) = 𝑟1. 𝑟1 − 2(𝑟1. 𝑟2) (7b)

To comply with kirigami constraints and during deployment, 𝑅𝑟1. 𝑅𝑟1 = 𝑟1. 𝑟1. This makes sure

that all side lengths pairings are equivalent in both surfaces.

2(𝑃. 𝑅𝑟1) + 𝑃. 𝑃 − 2(𝑅𝑟1. 𝑟2) − 2(𝑃. 𝑟2) + 2(𝑟1. 𝑟2) = 0 (7c)


44

Applying this constraint to all sides in (7):

Side 1: 2(𝑃. 𝑅𝑟1) + 𝑃. 𝑃 − 2(𝑅𝑟1. 𝑟2) − 2(𝑃. 𝑟2) + 2(𝑟1. 𝑟2) = 0

Side 2: 2(𝑃. 𝑅𝑟2) + 𝑃. 𝑃 + 2(𝑅𝑟2. 𝑟1) + 2(𝑃. 𝑟1) − 2(𝑟1. 𝑟2) = 0

Side 3: −2(𝑃. 𝑅𝑟1) + 𝑃. 𝑃 − 2(𝑅𝑟1. 𝑟2) + 2(𝑃. 𝑟2) + 2(𝑟1. 𝑟2) = 0

Side 4: −2(𝑃. 𝑅𝑟2) + 𝑃. 𝑃 + 2(𝑅𝑟2. 𝑟1) − 2(𝑃. 𝑟1) − 2(𝑟1. 𝑟2) = 0 (8)

Combining and simplifying the Sides in (8):

Eq. 1: (Side 1 - Side 3) 𝑃. 𝑅𝑟1 − 𝑃. 𝑟2 = 0

Eq. 2: (Side 2 - Side 4) 𝑃. 𝑅𝑟2 + 𝑃. 𝑟1 = 0

Eq. 3: (Side 1+ Side 3) 𝑃. 𝑃 + 2(𝑟1. 𝑟2) − 2(𝑅𝑟1. 𝑟2) = 0

Eq. 4: (Side 2+ Side 4) 𝑃. 𝑃 − 2(𝑟1. 𝑟2) + 2(𝑅𝑟2. 𝑟1) = 0 (9)

By restricting the solution to rhombuses, 𝑟1. 𝑟2 = 0. This simplifies (9) to:

Eq. 1: 𝑃. 𝑅𝑟1 − 𝑃. 𝑟2 = 0

Eq. 2: 𝑃. 𝑅𝑟2 + 𝑃. 𝑟1 = 0

Eq. 3a: 𝑃. 𝑃 − 2(𝑅𝑟1. 𝑟2) = 0

Eq. 4a: 𝑃. 𝑃 + 2(𝑅𝑟2. 𝑟1) = 0 (9a)

The linear transformation matrix, 𝑅, can be described using the Rodrigues’ Rotation Formula

(RRF) (Belongie, n.d.). This is an efficient algorithm that rotates a vector in space, given an axis

and angle of rotation. The equation is the following:

𝑅[𝑎] = 𝑎 ∗ 𝑐𝑜𝑠[𝜃] + (𝑡 𝑥 𝑎) ∗ 𝑠𝑖𝑛[𝜃] + 𝑡(𝑡. 𝑎)(1 − 𝑐𝑜𝑠[𝜃])

Where,

𝑎 = vector to be rotated

𝑡 = unit vector for axis of rotation

𝜃 = angle of rotation


GEOMETRY DEPLOYMENT

45

Figure 5.2 labels these components and helps visualize the equation.

Figure 5.2: Rodrigues Rotation Formula notation (Dale, 2015)

Applying this formula to (9a),

Eq. 1: 𝑃. (−𝑟2 + 𝑟1 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1) + (𝑡 𝑥 𝑟1) ∗ 𝑠𝑖𝑛[𝜃]) = 0

Eq. 2: 𝑃. (𝑟1 + 𝑟2 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟2) + (𝑡 𝑥 𝑟2) ∗ 𝑠𝑖𝑛[𝜃]) = 0

Eq. 3a: 𝑃. 𝑃 − 2 ∗ 𝑟2. (𝑟1 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1) + (𝑡 𝑥 𝑟1) ∗ 𝑠𝑖𝑛[𝜃]) = 0

Eq. 4a: 𝑃. 𝑃 + 2 ∗ 𝑟1. (𝑟2 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟2) + (𝑡 𝑥 𝑟2) ∗ 𝑠𝑖𝑛[𝜃]) = 0 (10)

Combining Eq. 3a and Eq. 4a in (10) yields the following characteristic relationships:

Eq. 3a - Eq. 4a: (−1 + 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1)(𝑡. 𝑟2) = 0 (11)

Eq. 3a + Eq. 4a: 𝑃. 𝑃 − 2 ∗ 𝑟2. (𝑡 𝑥 𝑟1 ∗ 𝑠𝑖𝑛[𝜃]) = 0 (12)

(11) implies 3 cases: 𝜃 = 0, 𝑡. 𝑟1 = 0, or 𝑡. 𝑟2 = 0.


46

5.2.3.1: Case 1 (𝜃 = 0):

If 𝜃 = 0, then (12) would simplify to the following:

𝑃. 𝑃 = 2 ∗ 𝑟2. (𝑡 𝑥 𝑟1 ∗ 𝑠𝑖𝑛[0])

𝑃. 𝑃 = 0

Recall that 𝑃. 𝑃 is the squared length of the vector to the centroid of the secondary surface. If

𝑃. 𝑃 = 0, then 𝜃 = 0 implies the non-deployed case, where the secondary surface is at the origin

while allowing the surface to tilt along its primary and secondary axes. Figure 5.3 depicts the 𝑟1

and 𝑟2 axes notation on a Spin-Valence unit. It also shows the tilting configurations in the non-

deployed case.

Figure 5.3: Notation for r1 and r2 axes (left); Non-deployed tilting along r1 axis (middle); tilting along r2 axis (right)

Therefore, fixing 𝜃 ≠ 0 and allowing 𝜃 to be an adjustable parameter enables the system to deploy

as the Spin-Valence geometry intended.

5.2.3.2: Cases 2 and 3 (𝑡. 𝑟1 = 0) or (𝑡. 𝑟2 = 0)

Cases 2 and 3 explore the relationships of 𝑡. 𝑟1 = 0 or 𝑡. 𝑟2 = 0. Referring to the notation in Figure

5.3, Case 2 allows for tilting along the 𝑟1 axis, while Case 3 accomplishes the same for the 𝑟2 axis.

The proofs for Case 2 and Case 3 are almost equivalent with the only notable differences, being


GEOMETRY DEPLOYMENT

47

(13) and (14). Therefore, the following derivation only looks into Case 2, providing supplemental

commentary for Case 3. Case 3 is referred as the “b” equation.

𝑡. 𝑟1 = 0 (13)

𝑡. 𝑟2 = 0 (13b)

For (13) to be valid, 𝑡 must be a 1x3 matrix or vector that is perpendicular to 𝑟1. Since 𝑟1 is defined

is defined in the local coordinate system, 𝑡 can be denoted as follows:

𝑡 = [0 𝑠𝑖𝑛 [𝜑] 𝑐𝑜𝑠 [𝜑]] (14)

𝑡𝑏 = [𝑠𝑖𝑛 [𝜑] 0 𝑐𝑜𝑠 [𝜑]] (14b)

Defining 𝑡 with 𝑠𝑖𝑛 [𝜑] and 𝑐𝑜𝑠 [𝜑] allows for 𝑡 to be consistent as a unit vector, while introducing

variability for the y and z directions with only one parameter, 𝜑.

Plugging (14) into (12) and assigning a 𝜃 and φ gives the squared length of 𝑃, the distance to the

deployed surface’s centroid. Vector, 𝑃, can be solved through a system of equations, using 𝑃. 𝑃

and Equations 1 and 2 of (10). Section 6.5.7 and the Mathematica file in Appendix A detail the

exact methodology for solving the system.

Modifying (4) with the RRF and the isometry relationship, the exact secondary surface vertices

can be derived.

Side 1: 𝑠1 = 𝑅[𝑟1] = 𝑟1 ∗ 𝑐𝑜𝑠[𝜃] + (𝑡 𝑥 𝑟1) ∗ 𝑠𝑖𝑛[𝜃] + 𝑡(𝑡. 𝑟1)(1 − 𝑐𝑜𝑠[𝜃]) + 𝑃



Side 4: 𝑠4 = 𝑅[𝑟4] = 𝑟4 ∗ 𝑐𝑜𝑠[𝜃] + (𝑡 𝑥 𝑟4) ∗ 𝑠𝑖𝑛[𝜃] + 𝑡(𝑡. 𝑟4)(1 − 𝑐𝑜𝑠[𝜃]) + 𝑃 (4a)


48

To summarize, these vertices can be found by giving the parameters, θ and φ, and specifying

whether performing Case 2 or 3.

A sample calculation using the parameters:

θ = 0.5

φ = 0.25

is available in Appendix B, using Mathematica.


GEOMETRY DEPLOYMENT

49

Chapter 6

Algorithms for Computational Primary and

Secondary Surface Generation

With a mathematic algorithm for deployment, doubly-curved Spin-Valence frames can now be

modeled computationally. To review, introducing double-curvature to Spin-Valence forces its

primary and secondary surfaces to become non-parallel. Therefore, an algorithm that can create

NPPUs through adjustable parameters is necessary.

Having equations to describe deployment produces a more predictable configuration-space for

optimization. Creating the primary surface through optimization, covered in Section 6.5, is

contingent on this more navigable space.

This chapter overviews a workflow for generating a complete Spin-Valence frame from input

surfaces, based on research by a team of collaborators at MIT, the University of Arkansas (Emily

Baker and Edmund Harriss), and Arup (Nick Sherrow-Groves).

6.1. Modeling Software

Input surfaces are defined in Rhinoceros v6 (Rhino), using the computational plug-in Grasshopper

for parametric design. Grasshopper’s Kangaroo physics simulator is used for the secondary surface

50

generation. Python v3.7 is used to calculate an optimized primary surface that is deployed from

data obtained for the secondary surface. The library SciPy is used for optimization in this step.

6.2. Motivation

As introduced in Section 1.4, accurate computational models extend the Spin-Valence geometry

to allow input surfaces. This design freedom unlocks the system’s capabilities to handle various

boundary conditions and loading scenarios. Double-curvature, as seen in Chapter 2, strengthens

spanning systems, giving Spin-Valence more opportunity for use as a long-spanning spatial

structure for temporary bridges, vaults, or roofing.

The workflow is this chapter presents a systematic procedure to create complex geometries from

Spin-Valence. Surface generation is semi-automatic, letting users pursue creativity while requiring

little to no effort.

6.3. Proposed Workflow for Constructing a

Doubly-Curved Pavilion

The project team has created a workflow to generate a fully constructed doubly-curved pavilion,

which was proposed to be displayed at the International Shell and Spatial Structures (IASS) 2020

Symposium. The abstract for the related publication, co-authored by Baker, Harriss, Herning,

Ramirez, and Sherrow-Groves and entitled "DCSV Pavilion: Double-Curved Spin-Valence Space

Frame," has been accepted by the IASS 2020 Symposium committee. However, the IASS 2020

Symposium was postponed due to a global health crisis (IASS, 2020). It is expected that the paper

will be presented at the IASS 2021 Symposium.

CHAPTER 6. ALGORITHMS FOR COMPUTATIONAL PRIMARY AND

SECONDARY SURFACE GENERATION

51

Design and construction of the pavilion aims to showcase the advancements in Spin-Valence,

featuring computational modeling, finite element analysis, and doubly-curved surface

construction.

The proposed workflow for creating the pavilion is as follows:

1. User-generated surface in Rhino

2. Secondary surface generation using Grasshopper

3. Primary surface optimization in Python

4. Final meshing in Rhino

5. Finite Element Analysis of Pavilion

6.3.1. User-generated surfaces

Input surfaces are created by manipulating the geometry of 6 curves in Rhino. Each of these curves

are modifiable by control points, able to be moved anywhere in the 3D space. The group of curves

are then lofted in Grasshopper, or approximated into a surface. Figure 6.1 displays the proposed

input curves for the IASS design, and Figure 6.2 depicts the final approximated surface from the

input curves.

Figure 6.1: Input curves for surface generation

6.3. PROPOSED WORKFLOW FOR CONSTRUCTING A DOUBLY-CURVED

PAVILION

52

Figure 6.2: Formulated surface from input curves

In Section 6.3.2, a hard constraint generating the secondary surface are the boundaries highlighted

in blue in Figure 6.3. Physically, these two edge curves represent the structural supports in the

system. The left most curve is pin-supported throughout its entire length while the right-most curve

is pin-supported only along the units in front. This poses an interesting stress distribution

throughout the space frame which will be investigated in the finite element analysis in Chapter 7.

Figure 6.3: Boundary conditions for input surface (highlighted in blue)

6.3.2. Grasshopper-Produced Secondary Surface

Referenced in Chapter 1, the kirigami frame receives its rigidity from its triangulating legs,

resembling that of a spatial truss. Building this formation requires adjacent units to be connected

on both surfaces. The secondary surface due to the Spin-Valence geometry has a fixed amount of

material. Meanwhile, the primary surface has more flexibility because material can always be

added to its exterior.



53

Therefore, the more constrained secondary surface will be created first, directly from the input

geometry. The primary surface will then be deployed from the secondary, using the algorithm from

Chapter 5. Optimization is performed to make these deployed units infinitesimally close to one

another. Final meshing in Rhino appends the needed material to create a cohesive layer, forming

the internal truss network.

To produce the secondary surface, a Grasshopper script is created to decompose the input geometry

in Figure 6.2 into a grid as seen in Figure 6.4.

Figure 6.4: Grid-pattern of secondary surface

Next, the gridded surface is converted to a highly controlled, secondary surface configuration

through optimization in Kangaroo. The optimization, explained in Section 6.4, aims to create an

accurate 45° Pythagorean tiling while staying consistent with Figure 6.2. Individual unit data and

the dimensions of the pavilion in Figure 6.5 are then sent to Python to calculate a primary surface,

using optimization once more.

Figure 6.5: Optimized Pythagorean tiling of secondary surface


PAVILION

54

6.3.3. Optimized Deployed Primary Surface

The Grasshopper data is then translated into Python, where the following process is explained in

more detail in Section 6.5. Individual primary surface units are deployed from their secondary

counterparts using the algorithm in Chapter 5. To summarize, parameters, θ and φ, dictate each

unit’s final position according to mathematic relationships.

Optimization, using the SciPy library in Python, is used to construct a buildable primary surface.

The objective value for this optimization are the distances between deployed units. The optimizer

minimizes this value by altering the parameter matrix of θ’s and φ’s to find a local minimum in

the 2N configuration-space, with N being the total number of units. Figure 6.6 displays an example

primary surface that will be exported to Rhino for final meshing.

Figure 6.6: Optimized primary surface

6.3.4. Final Meshed Model in Rhino

A new Grasshopper script receives the Python data for both surfaces and creates the pavilion using

those returned points. This last step in Rhino unites any separated units through steel point

connections as seen in Figure 6.7.



55

Figure 6.7: Point connections in the primary surface created by E. Baker

Moreover, triangulating legs are meshed in this final step to connect both surfaces. Figure 6.8

displays a preliminary model of the final meshing, and Figure 6.9 illustrates a completed rendering

of the final proposed IASS 2020 pavilion design.

Figure 6.8: Preliminary meshing of primary and secondary surface created by E. Baker (left)

Figure 6.9: Rendering of the proposed pavilion created by E. Baker (right)

6.3.5. Finite Element Analysis

The final meshing in Rhino is exported to a finite element software for structural analysis on the

global and local systems. Sherrow-Groves is examining the global structure, analyzing the flow of

stresses throughout all units. Ramirez and Herning from MIT are investigating a detailed 3x3


PAVILION

56

assembly of the most stressed boundary condition. Figure 6.10 showcases a preliminary

computational test in Abaqus 2017 (ABAQUS, 1978).

Figure 6.10: Abaqus finite element model of 3x3 assembly

6.4. Computationally Produced Secondary Surface

The preliminary version for modeling kirigami frames from input surfaces was created by Baker

and Harriss (Baker and Harriss, n.d.). Their work references initial influences from Song’s study

of modeling reciprocal frames from user generated surfaces (Song et al., 2014). However, the

algorithm’s main inspiration comes from the original Pythagorean tiling method mentioned in

Chapter 1.

The original Grasshopper algorithm has since been modified through collaborations with Ramirez.

After creating an input surface, the program divides the mesh into a grid. These individual quads

are the foundational pieces to perform optimization.

Forming a cohesive secondary surface requires hard constraints detailed in Section 6.4.1. The

constraints for this system can be thought of as springs – “the further they are from the ideal

position, the harder they pull” (Baker and Harriss, n.d.). That allows this problem to be



57

reconfigured for Kangaroo, a physics simulator in Grasshopper. Kangaroo uses optimization under

these constraints to convert the gridded surface into the prescribed 45° Pythagorean tiling, as

illustrated in Figure 6.11. More details on the optimization are described in the ongoing paper,

titled “Double-Curved Spin-Valence: Geometric and Computational Basis,” for the upcoming

Advances in Architectural Geometry (AAG) 2020 conference.

Figure 6.11: Secondary surface generated from input surface (Baker and Harriss, n.d.)

6.4.1. Constraints

The constraints in the Kangaroo script originate from qualitative observations of the doubly-curved

frames. As referenced in Section 4.2, doubly-curved Spin-Valence units have a non-parallel

internal geometry. However, having near parallel units is preferable for strength and structural

redundancy. To achieve this while respecting the input geometry, connected units must spread

local deformations smoothly along the surface while remaining somewhat parallel.

This can be accomplished through optimization in Kangaroo by having strict constraints for the

positions of each unit’s vertices. These mathematic constraints can be summarized as:

A) Neighboring units meet at their corners

B) Vertices must stay close to the target surfaces

C) Edge curves (Figure 6.3) are hard boundaries that cannot be passed

D) Individual units must be rhomboidal

6.4. COMPUTATIONALLY PRODUCED SECONDARY SURFACE

58

More details regarding the derivation of constraints A, B, and C are found in the ongoing AAG

2020 paper. Constraint D is a revision to Harriss’s initial Grasshopper script due to the deployment

logic in Chapter 5 requiring rhomboidal units.

Figures 6.4 and 6.5 depict the input surface before and after optimization with Kangaroo.

6.4.2. Transition to Python-produced Primary Surface

After optimization, individual unit data is transferred to Python to compute the primary surface.

The exported data includes two pieces of information:

1. Global tiling data

2. Unit rhombus specifications

Figure 6.12: Global tiling specifications

Global tiling data details the number of units in all rows, P, and columns, Q. This gives the total

number of units, N, by multiplying P with Q. Unit rhombus specifications are the center points,

vectors to units’ first corners, and vectors to units’ second corners for every rhombus in the

pavilion.



59

6.5. Optimized Deployment of Primary Surface

The primary surface deployment script in Python is produced at MIT with coordination from PI,

Gordana Herning. It can be subdivided into five main functions: File_read, Objective, Kirigami,

Distance, and Plot.

To summarize, File_read imports Grasshopper data and divides it into usable matrices. Objective

loops through all units and runs the Kirigami and Distance functions. Kirigami simulates local unit

deployment according to parameters, θ and φ. Next, Objective stores all units’ final primary and

secondary surface positions and sends this information to Distance. Distance returns the objective

value for optimization by summing the distances between all adjacent units.

SciPy’s minimize function performs optimization on the structure. It modifies the Distance outputs

by incrementally changing the θ and φ parameters for each unit. Plot illustrates the final

configuration.

6.5.1. Imported Python libraries

The Python libraries used in this section are: NumPy, SciPy, and Matplotlib. NumPy and SciPy

are used for basic math functions in Python. SciPy has a built in optimizer that will be used in

Section 6.5.4. Matplotlib plots the final outcome of the optimization.

6.5.2. Notation

All variables in the Python script are located in Appendix C and defined by comments. There are

multiple 4x3 matrices that represent different versions of the primary and secondary surface

vertices. Figure 6.13 illustrates the notation of an individual unit. Each corner is represented by a

row in their respective 4x3 matrix.

6.5. OPTIMIZED DEPLOYMENT OF PRIMARY SURFACE

60

Figure 6.13: Labeled primary and secondary vertices

Notice the differences between Figures 5.1 and 6.13. Figure 6.13 shows the primary surface

deploying from its secondary counterpart, as mentioned in Section 6.3.2.

For this script, the points switch between a global and an arbitrary local coordinate system. The

deployment algorithm from Chapter 5 is successful if the non-deployed rhombs meet two

conditions:

1. Non-deployed rhombs are centered at the origin

2. Primary and secondary rhomboidal axes are along the x and y axes

To distinguish between the matrices in the different coordinate axes, matrices are labeled the

following:

pl = primary surface in local

pgb = primary surface in global before expansion factor

pg = primary surface in global after expansion factor

sl = secondary surface in local

sgd = secondary surface input data for Kirigami

sg = secondary surface in global



61

The vector, P, remains the same as Chapter 5, as seen in Figure 5.1. It has been renamed to 𝑝𝑙𝑐𝑒𝑛

in the Python script for clarity. Similar to Chapter 5, solving for 𝑝𝑙𝑐𝑒𝑛 gives the deployed primary

surface vertices in the local system.

6.5.3. Parameters

Parameters for the algorithm are the Grasshopper data file, the θ and φ matrices, and the primary

surface expansion factor, exp.

Section 6.5.4 discusses deconstructing the Grasshopper inputs into the pavilion dimensions and

individual unit data. θ and φ are arbitrary Nx1 matrices and are the same parameters derived from

Chapter 5. These will be used as the base parameters for optimization. Exp enlarges the deployed

rhomb by an assigned ratio after reaching its final position. This attempts to accurately represent

the shifted center-line geometries due to real-life leg thicknesses. Figure 6.14 depicts this offset.

Figure 6.14: Offset center-line geometry

6.5.4. Optimization

For specific surfaces, like spheres, connections between adjacent units can be investigated

analytically and adjusted to form perfect solutions. However, the more complex, input surfaces

cannot yield a direct solution due to the amount of variation involved (Gelfand and Silverman,


62

2000). Therefore, optimization is used to incrementally find a near-ideal solution. Simple

unconstrained optimization techniques require two inputs: an objective function, f, and adjustable

parameters. More complex forms involve the Jacobian (first derivative of f), Hessian (second

derivative of f), and parameter constraints.

The main method for this script’s optimization is gradient descent, which modifies its parameters

based on the local slope of the configuration-space. It makes small improvements until it reaches

a threshold value, ε, or exceeds the set number of iterations. If the objective value differs by less

than ε per iteration, the optimizer reaches convergence and returns the final objective value.

In this script, f is defined as the Objective function, where the objective value is the summation of

distances between adjacent units, which is calculated in Distance. The optimizer minimizes these

distances by modifying the parameters θ and φ for each unit.

6.5.5. File_read

Before use in Python, the exported Grasshopper data from Section 6.4 is slightly modified. All

data from Grasshopper is exported as one text file for simplicity. File_Read decomposes and

reformats this data into three matrices, as referenced in Section 6.4.2.

The first two rows of exported data are P and Q, which detail the number of units in each row and

column in the pavilion. N is the total number of units, found by multiplying P and Q. Being a 2D

configuration-space, this script contains 2N number of parameters.

The remaining rows are the rhombus data for every unit in the pavilion. Every 3 rows in the matrix

represents a new unit, specifying global vectors for the base polygon’s origin, primary axis, and

secondary axis. The data for an individual unit is termed, sgd, which stands for “secondary global

unit data”.



63

6.5.6. Objective

As referenced in Section 6.5.4, the objective function, f, minimizes the total distance between

adjacent units by modifying the parameters, θ and φ. Objective organizes the received data from

File_Read into N individual units with matrix size 3x3. Objective then loops N number of times

and transfers the individual unit data, sgd, alongside that unit’s assigned θ and φ values to the

function Kirigami. Afterwards, Objective compiles a matrix of all global primary and secondary

coordinates returned from Kirigami.

These final global positions are stored in two 4Nx3 matrices, denoted as primary and secondary.

The primary matrix is transferred to Distance to calculate the optimizer’s objective value. This

final value is returned to the SciPy optimizer where the total distance will be minimized. The

final product will create a more cohesive, constructible primary surface.

6.5.7. Kirigami

The function, Kirigami, models single unit deployment according to the following variables: 𝑠𝑔𝑑,

θ, and φ. Referenced in Section 6.5.5, 𝑠𝑔𝑑 is a matrix composed of a rhomb’s global centroid

position and vectors for its primary and secondary axes, titled 𝑠𝑔𝑝 and 𝑠𝑔𝑠.

𝑠𝑔𝑑 = [

𝑠𝑔𝑐𝑒𝑛

𝑠𝑔𝑝

𝑠𝑔𝑠

] (15)

Where,

𝑠𝑔𝑐𝑒𝑛 = coordinates of polygon centroid

𝑠𝑔𝑝 = vector along rhombus primary axis

𝑠𝑔𝑠 = vector along rhombus secondary axis

The global coordinates of the secondary surface are calculated and stored in the matrix, 𝑠𝑔.


64

𝑠𝑔 = [

𝑠𝑔𝑐𝑒𝑛 + 𝑠𝑔𝑝

𝑠𝑔𝑐𝑒𝑛 + 𝑠𝑔𝑠

𝑠𝑔𝑐𝑒𝑛 − 𝑠𝑔𝑝

𝑠𝑔𝑐𝑒𝑛 − 𝑠𝑔𝑠

]

(16)

As stated in Section 6.5.2, the Python script switches between a global and an arbitrary local

coordinate system. The algorithm from Chapter 5 is successful only in this newly created local

system. Therefore, 𝑠𝑔 must be converted into a 4x3 matrix, named 𝑠𝑙, containing the local

coordinates of the secondary surface’s corners. To do this, the Euclidean distances of vectors 𝑠𝑔𝑝

and 𝑠𝑔𝑠 are calculated and stored as 𝑝𝑙𝑒𝑛 and 𝑠𝑙𝑒𝑛 respectively. The 𝑠𝑙 matrix is calculated in (17).

𝑠𝑙 = [

𝑝𝑙𝑒𝑛 0 00 𝑠𝑙𝑒𝑛 0

−𝑝𝑙𝑒𝑛 0 00 −𝑠𝑙𝑒𝑛 0

]

(17)

From Chapter 5, calculating the deployment of the primary surface requires 3 variables: a base

polygon, and parameters θ and φ. To recap, these variables solve a system of equations that gives,

the vector, P, renamed 𝑝𝑙𝑐𝑒𝑛, as seen in Figure 5.1. The Rodrigues Rotation Formula (RRF)

alongside 𝑝𝑙𝑐𝑒𝑛 equates the final local position of the deployed primary surface. Equations (10),

(11), (12), and (14) from Chapter 5 are used in the following algorithm.

First, using the Case 2 assumption of 𝑡. 𝑟1 = 0 in (11), (14) is used to establish the unit vector for

the axis of rotation in the RRF.

(−1 + 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1)(𝑡. 𝑟2) = 0 (11)

𝑡 = [0 𝑠𝑖𝑛 [𝜑] 𝑐𝑜𝑠 [𝜑]] (14)

Where,

r1 = [𝑠𝑙11 𝑠𝑙12 𝑠𝑙13]

r2 = [𝑠𝑙21 𝑠𝑙22 𝑠𝑙23]



65

Rearranging (12) equates the magnitude of vector 𝑝𝑙𝑐𝑒𝑛, ‖𝑝𝑙𝑐𝑒𝑛‖:

‖𝑝𝑙𝑐𝑒𝑛‖ = √𝑝𝑙𝑐𝑒𝑛 . 𝑝𝑙𝑐𝑒𝑛 = √2 ∗ 𝑟2. (𝑡 𝑥 𝑟1 ∗ 𝑠𝑖𝑛[𝜃]) (18)

To solve for the vector, 𝑝𝑙𝑐𝑒𝑛, finding its i, j, and k components is necessary. This requires a

system of three equations to solve completely.

One equation can be derived through the distance formula.

𝑖2 + 𝑗2 + 𝑘2 = ‖𝑝𝑙𝑐𝑒𝑛‖2 (19)

The other two can be obtained from (10).

𝑝𝑙𝑐𝑒𝑛 . (−𝑟2 + 𝑟1 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1) + (𝑡 𝑥 𝑟1) ∗ 𝑠𝑖𝑛[𝜃]) = 0 (10a)

𝑝𝑙𝑐𝑒𝑛 . (𝑟1 + 𝑟2 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟2) + (𝑡 𝑥 𝑟2) ∗ 𝑠𝑖𝑛[𝜃]) = 0 (10b)

Since 𝑟2, 𝑟1, θ, and 𝑡 are known, the parenthetical regions can be rewritten as constants:

𝐴 = −𝑟2 + 𝑟1 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟1) + (𝑡 𝑥 𝑟1) ∗ 𝑠𝑖𝑛[𝜃] (20)

𝐵 = 𝑟1 + 𝑟2 ∗ 𝑐𝑜𝑠[𝜃] + 𝑡(1 − 𝑐𝑜𝑠[𝜃])(𝑡. 𝑟2) + (𝑡 𝑥 𝑟2) ∗ 𝑠𝑖𝑛[𝜃] (21)

Substituting,

𝑝𝑙𝑐𝑒𝑛 . 𝐴 = 0 (10ai)

𝑝𝑙𝑐𝑒𝑛 . 𝐵 = 0 (10bi)

Where vectors, A and B, are notated,

A = [𝐴𝑥 𝐴𝑦 𝐴𝑧]

B = [𝐵𝑥 𝐵𝑦 𝐵𝑧]


66

Expanding (10ai) and (10bi) gives the last two equations to solve for i, j, and k,

𝐴𝑥 ∗ 𝑖 + 𝐴𝑦 ∗ 𝑗 + 𝐴𝑧 ∗ 𝑘 = 0 (10aii)

𝐵𝑥 ∗ 𝑖 + 𝐵𝑦 ∗ 𝑗 + 𝐵𝑧 ∗ 𝑘 = 0 (10bii)

This system of equations is solved in Wolfram Mathematica and the exact solution for i, j, and k

is located in Appendix A.

The locations of each vertex in the local coordinate system can be found by modifying (4) with

the isometry relationship and the RRF.

Side 1: 𝑝1 = 𝑅[𝑠1] = 𝑠1 ∗ 𝑐𝑜𝑠[𝜃] + (𝑡 𝑥 𝑠1) ∗ 𝑠𝑖𝑛[𝜃] + 𝑡(𝑡. 𝑠1)(1 − 𝑐𝑜𝑠[𝜃]) + 𝑝𝑙𝑐𝑒𝑛



Side 4: 𝑝4 = 𝑅[𝑠4] = 𝑠4 ∗ 𝑐𝑜𝑠[𝜃] + (𝑡 𝑥 𝑠4) ∗ 𝑠𝑖𝑛[𝜃] + 𝑡(𝑡. 𝑠4)(1 − 𝑐𝑜𝑠[𝜃]) + 𝑝𝑙𝑐𝑒𝑛 (4a)

This is stored in the matrix, pl:

𝑝𝑙 = [

𝑝1

𝑝2

𝑝3

𝑝4

]

(22)

To convert these newly found points from the local to global coordinate systems, another

transformation matrix, G, must be calculated. This is found by compiling the unit vectors that make

up the desired global axis and organizing them in the following manner:

𝐺 = [

𝑠𝑔𝑝

𝑠𝑔𝑠

𝑠𝑔𝑝 𝑥 𝑠𝑔𝑠

] (23)



67

Performing the transformation:

Side 1: 𝑝𝑔𝑏1 = 𝑝1 ∗ 𝐺 + 𝑠𝑔𝑐𝑒𝑛



Side 4: 𝑝𝑔𝑏4 = 𝑝4 ∗ 𝐺 + 𝑠𝑔𝑐𝑒𝑛 (24)

Similarly,

𝑝𝑔𝑐𝑒𝑛 = 𝑝𝑙𝑐𝑒𝑛 ∗ 𝐺 + 𝑠𝑔𝑐𝑒𝑛 (25)

For the primary surface to accurately represent real-world construction, a primary surface

expansion factor, exp, is introduced. This ratio takes into account the offset between the secondary

surface and primary surface that is created during the CNC cut. Figure 6.14 shows a graphical

depiction of this. To solve for the new vertex positions, the linear relationship below is used:

Side 1: 𝑝𝑔1 = 𝑝𝑔𝑏1 + 𝑒𝑥𝑝 ∗𝑝𝑔𝑏1 − 𝑝𝑔𝑐𝑒𝑛

‖𝑝𝑔𝑏1 − 𝑝𝑔𝑐𝑒𝑛‖







(26)

Aggregating gives the final global coordinates for the primary surface,

𝑝𝑔 = [

𝑝𝑔1

𝑝𝑔2

𝑝𝑔3

𝑝𝑔4

]

(27)


68

6.5.8. Distance

Distance calculates the objective value for optimization by summing the distances between

adjacent units. The function requires three parameters: P, Q, and primary. It will receive these

parameters from Objective after all units have undergone the Kirigami function.

There are two principal matrices in this operation: orient and xn. Orient stands for orientation and

is a QxP matrix that numbers off unit positions in the global structure. Figure 6.15 illustrates the

numbering of the matrix, orient.

Figure 6.15: Orient matrix numbering notation

𝑜𝑟𝑖𝑒𝑛𝑡 = [

1 ⋯ 𝑃⋮ ⋱ ⋮

𝑄 ∗ (𝑃 − 1) + 1 ⋯ 𝑄 ∗ 𝑃]

(28)

Xn is a Nx4 matrix that determines the adjacent units for an individual rhomb based on its position

in orient. To begin, the algorithm operates on an arbitrary notation that assigns a matrix column to

a specific side on the rhomb. This is depicted in Figure 6.16.



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Figure 6.16: Notation for assigning sides to columns in xn matrix

To simplify, in the xn matrix, columns correspond to the following sides:

Table 2: Simplified legend for Figure 6.16

Column Side Segment

1 Bottom P1 P2

2 Left P2 P3

3 Top P3 P4

4 Right P4 P1

The script loops through all of the units in orient and examines its current position. From here, the

program finds the numbers adjacent to its current position and assigns those adjacent numbers to

a column based on the notation in Figure 6.16.

Noticing patterns in the connections of the orient matrix, xn follows 5 basic rules. These rules are

shown in Table 3.


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Table 3: Xn matrix rules

Position in orient Column

in xn

Row of

Adjacent Unit

Column of

Adjacent Unit

Rows after the first 3 Row above Same

Rows before the last 1 Row below Same

Columns right of the first 2 Same Column left

Columns left of the last 4 Same Column right

Moreover, sides without an adjacent unit are assigned 0 in their respective columns. Figure 6.17

displays an example output script in Windows Powershell with a sample orient and xn for more

clarification.

Figure 6.17: Sample orient (left) and xn (right) matrices for a 4x4 assembly

After setting up xn, the script establishes a relationship between the xn and primary matrices to

calculate adjacent unit distances. It accomplishes this by following the flowchart in Figure 6.18.



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Figure 6.18: Flowchart for finding adjacent units

To create line segments in the pavilion, the sides in xn are expanded to two end points found in

primary. Primary, to reiterate, is a 4Nx3 matrix with the global primary coordinates, and every

four rows in primary details an individual unit. Side 1 in xn corresponds to points 1 and 2 in a unit

grouping, side 2 corresponds to points 2 and 3, and so on. Figure 6.16 helps visualize the points

and the directionality of the created vectors.

An internal function, closestDistance, calculates the distance between line segments. For a typical

case in a doubly-curved surface, line segments are most likely non-parallel, or askew. Although


72

the parallel case is coded in the script for security purposes, it will not be explored in this paper.

The typical non-parallel case in closestDistance is explained below.

For this algorithm, the current side, titled A, will have end points, 𝑎0 and 𝑎1, and the adjacent side,

titled B, will be have end points, 𝑏0 and 𝑏1. This allows us to represent A and B as:

𝐴 = 𝑎1 − 𝑎0 (29a)

𝐵 = 𝑏1 − 𝑏0 (29b)

The strategy behind finding the distance between 2 askew lines is to find two parallel planes

passing through each line. Distance between the planes is calculated using vector projection as

seen in equation (30).

𝑑𝑖𝑠𝑡 =

(𝐴 𝑥 𝐵). (𝑏0 − 𝑎0)

‖𝐴 𝑥 𝐵‖

(30)

However, equation (30) assumes infinite length vectors instead of finite segments like A and B.

Therefore, for the closest points, 𝑐1 (line A) and 𝑐2 (line B), it has to be verified whether they are

within the bounds of the segments. If not, then the endpoints of the line segments are assumed as

the new closest points.

The closest points, 𝑐1 and 𝑐2, are calculated in equations (31a) and (31b).

𝑐1 = 𝑎0 +

(𝑏0 − 𝑎0). (𝐵𝑢 𝑥 𝑛)

𝐴𝑢 . (𝐵𝑢 𝑥 𝑛)∗ 𝐴𝑢

(31a)

𝑐2 = 𝑏0 +

(𝑎0 − 𝑏0). (𝐴𝑢 𝑥 𝑛)

𝐵𝑢 . (𝐴𝑢 𝑥 𝑛)∗ 𝐵𝑢

(31b)

Where,

𝐴𝑢 = unit vector of segment A



73

𝐵𝑢 = unit vector of segment B

𝑛 = cross product of 𝐴𝑢 and 𝐵𝑢

To determine whether 𝑐1 and 𝑐2 are within Sides A and B, the script uses a mathematic relationship,

knowing that they are points along their corresponding infinite vectors.

𝑐1 = 𝑎0 + 𝑡1 ∗ 𝐴𝑢 (32a)

𝑐2 = 𝑏0 + 𝑡2 ∗ 𝐵𝑢 (32b)

Where,

𝑡1 = constant to describe position along Side A

𝑡2 = constant to describe position along Side B

Since 𝑐1 − 𝑎0 is proportional to 𝐴𝑢, any coordinate in both matrices are proportional to each other.

This yields two ratios to solve for 𝑐1 and 𝑐2, where subscript, x, represents the x-coordinate.

𝑡1 =

(𝑐1 − 𝑎0)𝑥

𝐴𝑢𝑥

(32ai)

𝑡2 =

(𝑐2 − 𝑏0)𝑥

𝐵𝑢𝑥

(32bi)

Figure 6.19 distinguishes the different possibilities for the final closest points pA and pB, with the

boundary conditions included.


74

Figure 6.19: pA and pB combinations including segment boundary conditions

If 𝑡1 and 𝑡2 are within 0 to ‖𝐴‖ or ‖𝐵‖ respectively, then:

pA = c1 = a0 + t1 ∗ Au (33a)

pB = c2 = b0 + t2 ∗ Bu (33b)

The final distance is calculated by taking the norm of (pB − pA). This number is returned back to

Objective for later optimization.

t1

< 0 pA = a0 B.(a0-b0)

< 0 pB = 0

> 0 , < ||B|| pB = b0 + B*B.(a0-b0)

> ||B|| pB = b0 + B*||B||

> ||A|| pA = a1 B.(a1-b0)

< 0 pB = 0

> 0 , < ||B|| pB = b0 + B*B.(a1-b0)

> ||B|| pB = b0 + B*||B||

t2

< 0 pB = b0 A.(b0-a0)

< 0 pA = 0

> 0 , < ||A|| pA = a0 + A*A.(b0-a0)

> ||A|| pA = a0 + A*||A||

> ||B|| pB = b1 A.(b1-a0)

< 0 pA = 0

> 0 , < ||A|| pA = a0 + A*A.(b1-a0)

> ||A|| pA = a0 + A*||A||



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6.5.9. Plot

Plot is a 3D plotter using the Axis3d plug-in within the Matplotlib library. This displays the

global secondary surface, augmented primary surface, and the triangulating legs in 3D. This step

is to help with mathematic validation and visual aesthetics. Figure 6.20 displays an example of

the final configuration after optimization.

Figure 6.20: Plot example using Axis3d

6.5.10. Results

The goal for this section is to formulate an efficient workflow that creates a doubly-curved,

deployed primary surface. To produce a constructible deployed layer, optimization is used to

connect all units in the primary surface by incrementally changing set parameters derived in

Chapter 5.

The SciPy library comes equipped with several algorithms for optimization, and to produce the

best results, or models with the smallest objective value, these different methods are tested and

modeled. More images of these results are located in Appendix D.


76

The preliminary results investigate a 3x14 strip at the pavilion’s right-most boundary condition as

seen in Figure 6.3. It is observed that optimization at this smaller scale is relatively proportional

to the method’s performance at the full pavilion scale. Scaling for optimization allows for more

iterations to be completed more efficiently. The strip for analysis is depicted in Figure 6.21

Figure 6.21: Sample 3x14 assembly for optimization analysis

Algorithms to be explored in this section are Nelder-Mead, Powell, TNC, and SLSQP.

6.5.10.1. Nelder-Mead

Nelder-Mead is a heuristic optimization technique, similar to Genetic Algorithms. This method is

primarily employed because of the complexity of the configuration-space. With every parameter

introduced, the configuration-space becomes increasingly non-linear, from adding new

dimensions that the optimizer needs to navigate. For this example, the optimization explores a 2N

configuration-space, so for a 3x14 Assembly, that is 84-dimensions of complexity.

Nelder-Mead locates a local minimum by searching spaces within randomly generated,

multidimensional polygons known as simplexes. At each iteration, these polygons shift one vertex

at a time towards an optimal region in the configuration-space, until converging to a local

minimum.



77

Figure 6.22: Nelder-Mead optimization results

Table 4: Nelder-Mead Optimization Results

Case theta phi exp # of iterations initial distance final distance % reduction

Nelder-Mead 0.5 0.01 0.075 5000 2.628 1.683 35.95%

The heuristic approach has a reduction of 35.95%. Moreover, the final result displayed in Figure

6.22 has a relatively smooth deployed surface, barring a couple of inconsistencies especially at the

boundaries. Downsides to using the Nelder-Mead method are its long computational time and its

exploration of a relatively small configuration-space, which is reflected by its small percent

reduction.

6.5.10.2. Powell

Powell method is a gradient descent optimizer. Gradient descent optimizers take small steps in the

configuration-space in the direction of the current gradient, or the direction of steepest descent. It

will continue taking steps in that direction unless the objective value increases or converges. By

taking the gradient, the optimization converges faster to a solution than the traditional heuristic

approach taken by the Nelder-Mead method.


78

To combat the complexity of the configuration-space, the Python script includes a penalty function

if the optimizer explores a space that does not properly create a deployed surface. If the objective

value reads NaN or False, the optimizer automatically assigns the objective value to be 100,000.

This influences the configuration-space to always explore spaces where a coherent surface is

formed.

Figure 6.23: Powell optimization results

Table 5: Powell Optimization Results

Case theta phi exp # of

iterations

initial

distance final distance

%

reduction

Powell 0.5 0.01 0.075 5000 2.628 0.538 79.54%

The Powell method has a reduction of 79.54%. While yielding the lowest objective value, the final

surface created is composed of discrete, jagged units that do not coalesce into a single surface.

This most likely occurred because of the non-linear nature of the configuration-space. Although

taking the path of steepest descent, its initial position and the added non-linearity of the discrete

penalty function caused the optimizer to find a sub-optimal local minimum.

To achieve the accuracy of the Nelder-Mead method but the efficiency of the Powell method,

bounded steepest descent optimization is investigated in the following sections. This allows for

efficient locating of the local minimum, but within a set region in the configuration-space, similar



79

to the heuristic method. Two methods in SciPy that allow for bounded, unconstrained optimization

without a Jacobian or Hessian function are TNC and SLSQP.

6.5.10.3. TNC (Truncated Newton Conjugate-Gradient)

The Truncated Newton Conjugate-Gradient, or TNC, optimizer uses the Newton method for

finding local minimums. Newton method is an extended version of the gradient descent optimizer

used in the Powell method. It includes a more refined way of reaching a minimum that molds its

steps better to more naturally navigate the configuration-space. Figure 6.24 depicts this

mathematic phenomenon.

Figure 6.24: Steepest descent (blue) vs. Newton method (black) for locating local minima

Moreover, this method is preferred because it is “truncated.” SciPy’s TNC algorithm allows users

to incorporate assigned bounds for its parameters, limiting the explored configuration-space to

create a more constructible surface.

Therefore, three different bounds cases are explored, adjusting 𝜃 and 𝜑’s navigable space. These

cases alongside their objective values are shown in Figure 6.25 and Table 6.


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Figure 6.25: TNC 1: Theta [.301,.800], Phi [.001,.5] (left),

TNC 2: Theta [.301,.900], Phi [.001,.6] (middle),

TNC 3: Theta [.401,.900], Phi [.001,.5] (right)

Table 6: TNC Optimization Results


iterations

initial

distance

final

distance

%

reduction

TNC 1 0.301 - 0.8 0.001 - 0.5 0.075 100 2.724 1.457 46.48%

TNC 2 0.301 - 0.9 0.001 - 0.6 0.075 100 2.724 1.432 47.40%

TNC 3 0.401 - 0.9 0.001 - 0.5 0.075 100 2.724 1.210 53.92%

As observed by Table 6, the highest percent reduction came from the bounds with the highest theta

values, TNC Case 3 being the largest with Case 2 as second. Moreover, referring to Figure 6.25,

Case 3 has the smoothest primary surface created with Case 1 following. These results depict that

the configuration-space rewards higher theta values with a greater percent reduction, but punishes

having a wider range, creating non-uniform meshes in the primary surface.

6.5.10.4. SLSQP (Sequential Least Squares Programming)

The same case study for bounds is investigated with the Sequential Least Squares Programming,

or SLSQP, optimization method in SciPy. SLSQP is a nonlinear optimizer that incorporates the

Newton method logic from Section 6.5.10.3 and the Broyden-Fletcher-Goldfarb-Shanno (BFGS)

algorithm. BFGS is a numerical Hessian function update strategy that helps with optimization. The

findings for the bounded SLSQP study are represented in Figure 6.26 and Table 7.



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Figure 6.26: SLSQP 1: Theta [.301,.800], Phi [.001,.5] (left), SLSQP 2: Theta [.301,.900], Phi [.001,.6] (middle),

SLSQP 3: Theta [.401,.900], Phi [.001,.5] (right)

Table 7: SLSQP Optimization Results


iterations

initial

distance

final

distance

%

reduction

SLSQP 1 0.301 - 0.8 0.001 - 0.5 0.075 100 2.627 1.000 61.89%

SLSQP 2 0.301 - 0.9 0.001 - 0.6 0.075 100 2.627 0.724 72.42%

SLSQP 3 0.401 - 0.9 0.001 - 0.5 0.075 100 2.627 0.696 73.51%

As observed by Table 7, the greatest percent reduction came from the bounds with the largest theta

values, SLSQP Case 3 being the largest with Case 2 as second. This is a similar observation to the

TNC cases. Figure 6.26 illustrates remotely consistent smoothness throughout all the cases. A

singularity occurs at one corner of the 3x14 assembly for every case however. In comparison to

the TNC cases in regards to percent reduction, SLSQP reliably reduces the objective value by at

least 15% more.


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6.5.10.5. Summary of 3x14 Assembly Optimization Case Study

Table 8: Summary of 3x14 Assembly Optimization Results


iterations

initial

distance

final

distance

%

reduction

Nelder-

Mead 0.5 0.01 0.075 5000 2.628 1.683 35.95%

Powell 0.5 0.01 0.075 5000 2.628 0.538 79.54%

TNC 1 0.301 - 0.8 0.001 - 0.5 0.075 100 2.724 1.457 46.48%

TNC 2 0.301 - 0.9 0.001 - 0.6 0.075 100 2.724 1.432 47.40%

TNC 3 0.401 - 0.9 0.001 - 0.5 0.075 100 2.724 1.210 53.92%

SLSQP 1 0.301 - 0.8 0.001 - 0.5 0.075 100 2.627 1.000 61.89%

SLSQP 2 0.301 - 0.9 0.001 - 0.6 0.075 100 2.627 0.724 72.42%

SLSQP 3 0.401 - 0.9 0.001 - 0.5 0.075 100 2.627 0.696 73.51%

6.5.10.6. Final Pavilion

The final pavilion is modeled using both the TNC and SLSQP methods. TNC noticeably gives a

cleaner, smoother primary surface while SLSQP yields a consistently lower objective value. This

experiment will be conducted using Case 3 in both methods. Figure 6.27 shows the non-optimized

primary surface before running the Python script.

Figure 6.27: Pavilion design prior to optimization



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Figures 6.28 and 6.29 display the optimized pavilion using the TNC and SLSQP methods

respectively. Table 9 shows the final values for both optimization methods.

Figure 6.28: TNC optimization for proposed pavilion design

Figure 6.29: SLSQP optimization for proposed pavilion design

Table 9: Proposed Pavilion Optimization Results


iterations

initial

distance

final

distance

%

reduction

TNC 0.401 - 0.9 0.001 - 0.5 0.075 100 25.351 15.219 39.97%

SLSQP 0.401 - 0.9 0.001 - 0.5 0.075 100 25.351 9.847 61.17%

Table 9 reflects the observations found Section 6.5.10.4, where the SLSQP method has a percent

reduction almost 20% greater than the TNC method. In other words, deployed units are closer

together, creating a more idealized primary surface. Figures 6.28 and 6.29 both display relatively


84

smooth optimized surfaces, with no one method seemingly more advantageous than the other.

Therefore, for future optimization of the finalized pavilion design, the SLSQP algorithm from the

SciPy library will be utilized.

6.5.11. Improvements for the Future

Two main modifications should be looked into for future installments of the Python script in

Section 6.5. The optimizer currently used in SciPy is non-modifiable, so having a built-in optimizer

within the Python code allows for more variability depending on the current conditions of the

problem. Moreover, equating a Jacobian or Hessian function for Objective is another step that

allows for more efficient optimization. Currently, the SciPy packages calculate approximations for

both values; therefore, having an analytical way to describe the gradient will enhance the

performance of the system.



85

Chapter 7

Finite Element Analysis of Proposed Pavilion

Design

This chapter discusses performing finite element analysis on a computationally formed doubly-

curved pavilion to investigate its structural behavior. Computational analysis is required due to

compounding the already complex Spin-Valence geometry with double-curvature.

Recent computational analysis of the kirigami frame by Sherrow-Groves and Baker (2019) has

been continued for the proposed pavilion design as noted in Chapter 6. Computational analysis is

conducted through a collaborative effort between Arup and MIT. Sherrow-Groves is investigating

the global effects of loading on the structure, specifically looking at the global stress distributions

and behavior related to construction and support conditions.

The thesis author has investigated the original computational model (Baker and Sherrow-Groves,

2019) in order to numerically analyze, using Abaqus, the local effects of a particular section of the

pavilion. Upon validating the model created using Abaqus based on results of prior analyses

(Sherrow-Groves and Baker, 2019) (Sahuc et al., 2019), the goal of the present study is to create a

detailed 3x3 unit assembly of the pavilion section shown in Figure 7.1. The aim of this study is to

understand the effect of stress distributions on connection behavior at the most highly stressed

section of the pavilion.

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Testing Spin-Valence with different configurations through computation enables a greater

understanding of how the system performs. This allows designers to have a better intuition of the

kirigami frame’s strengths and weaknesses as a spanning structure. Moreover, it provides a level

of forethought for later design modifications, such as adding stronger material or shifting the

magnitude of curvature in certain places.

Details for this ongoing work are expected to be published.

Figure 7.1: 3x3 unit assembly for detailed finite element analysis

CHAPTER 7. FINITE ELEMENT ANALYSIS OF PROPOSED PAVILION

DESIGN

87

Chapter 8

Conclusion

This chapter summarizes the work presented in this MEng thesis and discusses the next steps for

deployable Spin-Valence spatial frames. It also reiterates the potential impacts this ongoing

research may achieve.

8.1. Summary of Contributions

The derivation for the deployment of an individual Spin-Valence unit has been found. It solves the

system as a 2D configuration-space.

A preliminary workflow has been proposed to translate input surfaces from Rhino to fully realized,

doubly-curved kirigami spatial frames.

A modified algorithm for forming the secondary surface in Grasshopper from input surfaces has

been achieved through collaborations between Harriss and Ramirez.

Using the mathematic deployment algorithm, a cohesive primary surface can be formulated in

Python using the Sequential Least Squares Programming, or SLSQP, optimization method in

SciPy.

88

A model of the flat 3x3 unit assembly has been set up in Abaqus for validation based on the prior

numerical study (Sherrow-Groves and Baker, 2019). It is used to supplement the analysis of

connection behavior in a 3x3 unit section extracted from the pavilion structure.

8.2. Future Work

In terms of geometry, the algorithm derived in this thesis grants a better mathematic understanding

of the Spin-Valence system. It can be used to extend the kirigami configurations beyond

rhombuses to parallelograms or other polygons. Current projects that are being discussed are

triangular or hexagonal space frames or geodesic domes created using the Spin-Valence pattern

logic.

As mentioned in Section 6.5.11, optimization can be improved by programming a built-in

optimizer within the Python script that does not rely on the SciPy library or configuring Jacobian

and Hessian equations for the objective function. These modifications would enable greater system

variability and computational speed.

The final results of the verification model along with the local 3x3 subassembly of the pavilion are

expected to be published in the paper entitled “DCSV Pavilion: Double-Curved Spin-Valence

Space Frame,” and presented at the IASS 2021 Symposium. Investigation of structural behavior

of doubly-curved Spin-Valence frames is important for developing new configurations for the

frame. Understanding global and local stress distributions, buckling modes, and behavior will

provide better intuition for creating more efficient Spin-Valence frames. Improvements in the

computational and structural analysis domains of kirigami spatial frames can make Spin-Valence

a more viable concept in the field of deployable mechanisms.

CHAPTER 8. CONCLUSION

89

8.3. Concluding Remarks and Potential Impacts

This thesis derives a new way of extending the Spin-Valence pattern logic to doubly-curved input

surfaces. Introducing curvature to the originally flat design enables the kirigami frame to

counterbalance internal element forces with better material efficiency and architecturally

variability. By creating a simplistic workflow that generates fully computational Spin-Valence

models, users have greater freedom to express their creativity whether they are producing

structural art installments or solving interesting boundary conditions using the kirigami

deployment method.

With Spin-Valence being a deployable mechanism, these creations can be easily transported and

deployed to a variety of locations. Hopefully, the Spin-Valence kirigami spatial frames will be

used in practice to create beautiful and economic forms in the built environment.

8.3. CONCLUDING REMARKS AND POTENTIAL IMPACTS

90


CHAPTER 8. CONCLUSION

91

Chapter 9

References

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Addis, B. (2006, Mar). The Crystal Palace and its place in stuctural history. International

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Baker, E. (2014). Search for a Rooted Aesthetic: Study in Spin-Valence. In F. Gramazio, M.

Kohler, & S. Langenberg (Ed.), Fabricate: Negotiating Design & Making, (pp. 128-135).

Zurich, Switzerland.

Baker, E. (2018). Spin-Valence: Kirigami Space Frame. In C. Mueller, & S. Adriaenssens (Ed.),

IASS Symposium 2018, Creativity in Structural Design, (pp. 1-7(7)). International

Association for Shell and Spatial Structures.

Baker, E., & Harriss, E. (n.d.). Computational Method for Mapping Kirigami Space Frames onto

Input Surfaces. (manuscript in preparation)

Belongie, S. (n.d.). Rodrigues' Rotation Formula. MathWorld - A Wolfram Web Resource.

WolfRam Research, Inc.

Beorkem, C. (2013). Flatform. Material Strategies in Digital Fabrication, 85-89.

Eureka. (2014). Rapid Deployable System (RDS).

Gelfand, I., & Silverman, R. (2000). Calculus of Variations. Chelmsford, MA: Courier

Corporation.

Gore, N. (2004). Craft and Innovation: Serious Play and the Direct Experience of the Real.

Journal of Architectural Education, 58(1), 39-44.

Hanaor, A., & Levy, R. (2001). Evaluation of deployable structures for space enclosures.

International Journal of Space Structures, 16, 221-229.

Hiroshima University. (2015). PreAssemble type of Bridge: Mobile Bridge. Hiroshima.

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International Association of Shell and Spatial Structures. (2020). IASS ANNUAL SYMPOSIUM

AND SPATIAL STRUCTURES CONFERENCE 2020. Retrieved from

https://www.surrey.ac.uk/iass2020

Lin, S., & Huang, Z. (2016). Comparative Design of Structures: Concepts and Methodologies.

Shanghai, China: Shanghai Jiao Tong University Press.

Livermore Software Technology Corporation. (1978, March). LS-DYNA.

Miyamoto, Y. (2015). Rotational Erection System (RES): Origami Extended with Cuts. (K.

Miura, T. Kawasaki, T. Tachi, R. Uehara, R. Land, & P. Wang-Iverson, Eds.) Origami9

II: Technology, Art, Education.

Rawn, E. (2019). Spotlight: Frei Otto. Arch Daily.

Robert McNeel & Associates. (1980). Rhinoceros.

Robert NcNeel & Associates. (2007, September). Grasshopper 3D.

Sahuc, J., Herning, G., & Baker, E. (2019). Experimental and Analytical Investigation of the

Spin-Valence Kirigami Space Frame. Proceedings of the International Association for

Shell and Spatial Structures (IASS) Symposium. Barcelona, Spain.

Saiidi, U. (2019). Long layover in Singapore? 'Jewel' makes the world's greatest airport even

better. CNBC.

Sherrow-Groves, N., & Baker, E. (2019). Structural finite-element analysis of steel kirigami

space frame to characterize global behavior. IASS 2019 Annual Symposium (International

Association of Shell and Spatial Structures).

SOM. (2000). San Francisco Airport Lenticular Truss Roofing. San Francisco, California.

Song, P., Fu, C., Goswami, P., Zheng, J., Mitra, N., & Cohen-Or, D. (2014). An interactive

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Wolfram Research. (1988, June). Wolfram Mathematica.

CHAPTER 9. REFERENCES

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Appendix A: Individual Deployment Logic in

Mathematica

This appendix contains the Mathematica code for solving the 2D configuration-space for the

deployment of an individual Spin-Valence unit. The algorithm was developed by Edmund

Harriss from the University of Arkansas and Michael Ramirez from MIT.

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99

Appendix B: Sample Calculation of

Deployment in Mathematica

This appendix contains a Mathematica script for a sample calculation of the algorithm derived in

Chapter 5.

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Appendix C: Computational Primary Surface

Python Script

This appendix contains the Python script for the creation of a computationally produced primary

surface for the kirigami space frame. This code is broken into five parts: Objective, File_Read,

Kirigami, Distance, and Plot.

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Objective

109

File_Read

110

Kirigami

114

Distance

118

Plot

120


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Appendix D: Additional Images from

Chapter 6 Results

This appendix contains the additional images for the optimization results from Section 6.5.10.

122

3x14 Assembly

123

Nelder-Mead Optimization

Powell Optimization

124

TNC Optimization

125

SLSQP Optimization

126

Non-Optimized Pavilion

127

TNC Optimized Pavilion

128

SLSQP Optimized Pavilion

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