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  • Aalborg Universitet

    Object-Oriented Structuring of Finite Elements

    Hededal, O.

    Publication date:1994

    Document VersionPublisher's PDF, also known as Version of record

    Link to publication from Aalborg University

    Citation for published version (APA):Hededal, O. (1994). Object-Oriented Structuring of Finite Elements. Aalborg: Aalborg Universitetsforlag. R :Institut for Bygningsteknik, Aalborg Universitet, No. Paper no. 1

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  • OBJECT-ORIENTED STRUCTURING

    OF FINITE ELEMENTS

    OLE HEDEDAL

    AALBORG UNIVERSITY

    September 1994

  • Acknowledgements

    This thesis, Object-oriented Structuring of Finite Elements, has been prepared in connec-

    tion with a Ph.D. study carried out in the period September 1991 to April 1994 at the

    Department of Building Technology and Structural Engineering, University of Aalborg,

    Denmark.

    A visit at the Department of Structural Engineering, Chalmers Technical University,

    Sweden, in the period March to May 1993 was used for writing a major part of the program

    code. I thank Professor N.E. Wiberg and my colleagues for making the visit a professionalas well as social success.

    I greatly acknowledge the inspiration and guidance given to me by my supervisor Pro-fessor, Dr. Techn. Steen Krenk. Furthermore, I thank Norma Hornung for assistance inthe preparation of the illustrations in the thesis.

    The work has been nanced by a grant from the Technical Faculty at the University ofAalborg. The visit to Chalmers Technical University was supported by Nordisk Forskerut-danningsakademi (NorFA).

    Aalborg, September 1994 Ole Hededal

    i

  • ii

  • Contents

    1 Introduction 1

    1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Numerical requirements . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.2 Requirements to the program structure . . . . . . . . . . . . . . . . 3

    1.1.3 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The structure of nite elements . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2.1 Sample program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.3 Object-oriented programming . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Objects and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    1.3.3 Polymorphism and dynamic binding . . . . . . . . . . . . . . . . . 111.3.4 Objects and C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.4 Review of literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.1 Matrix and vector classes . . . . . . . . . . . . . . . . . . . . . . . . 14

    1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2 Concepts in nite elements 17

    2.1 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Finite element approximation . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Elasticity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    3 Classes in nite elements 29

    3.1 The class structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.1.1 Requirements to ObjectFEM . . . . . . . . . . . . . . . . . . . . . 31

    3.2 The FEM classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 The Node class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    3.2.2 The Element class . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    3.2.3 The Material class . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.4 The Property class . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.3 Model storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 The List class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.3.2 Customizing the lists . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3.4 Algebraic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    iii

  • iv CONTENTS

    3.5 The application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    3.5.1 Model denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    3.5.2 Model generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    3.5.3 Forming the global equation system . . . . . . . . . . . . . . . . . . 48

    3.5.4 Solving the global equation system . . . . . . . . . . . . . . . . . . 48

    3.5.5 Postprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    3.5.6 Linear and non-linear applications . . . . . . . . . . . . . . . . . . . 50

    4 Customizing the FEM classes 53

    4.1 Potential element with linear materials . . . . . . . . . . . . . . . . . . . . 53

    4.2 Isoparametric elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    4.2.1 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.3 The isoparametric element class . . . . . . . . . . . . . . . . . . . . . . . . 59

    4.3.1 The Gausspoint class . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    4.3.2 Potential elements in 2D and 3D . . . . . . . . . . . . . . . . . . . 65

    4.4 Customizing Material and Property . . . . . . . . . . . . . . . . . . . . . . 67

    5 Solid elements 69

    5.1 Solid element for linear elasticity . . . . . . . . . . . . . . . . . . . . . . . 695.2 Isoparametric solid element . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    5.3 Elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    6 Non-linear nite elements 75

    6.1 Solution of non-linear nite element equations . . . . . . . . . . . . . . . . 786.2 The orthogonal residual method . . . . . . . . . . . . . . . . . . . . . . . . 83

    6.2.1 Dual orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.2 Implementation of the orthogonal residual method . . . . . . . . . . 88

    6.3 Extensions to the Element class . . . . . . . . . . . . . . . . . . . . . . . . 94

    7 Bar elements 95

    7.1 Elastic bar element with nite deformations . . . . . . . . . . . . . . . . . 957.1.1 Tangent stiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    7.1.2 Total and updated Lagrangian formulation . . . . . . . . . . . . . . 99

    7.2 Linear bar element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    7.3 Geometrically non-linear bar elements . . . . . . . . . . . . . . . . . . . . . 101

    7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

    7.4.1 Example 1: Two-bar truss . . . . . . . . . . . . . . . . . . . . . . . 104

    7.4.2 Example 2: 12-bar truss . . . . . . . . . . . . . . . . . . . . . . . . 107

    7.4.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    8 Elasto-plastic materials 113

    8.1 Hardening plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    8.1.1 Hardening rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2 Integration of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    8.2.1 Explicit integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

  • CONTENTS v

    8.2.2 Return mapping algorithms . . . . . . . . . . . . . . . . . . . . . . 120

    8.3 Classes in elasto-plastic analysis . . . . . . . . . . . . . . . . . . . . . . . . 121

    8.3.1 Extension to the Element class . . . . . . . . . . . . . . . . . . . . 122

    8.3.2 The Gausspoint class . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    8.3.3 The Plastic material class . . . . . . . . . . . . . . . . . . . . . . . 125

    8.4 von Mises plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    8.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

    8.5.1 Example: Plate with hole . . . . . . . . . . . . . . . . . . . . . . . 132

    9 Conclusion 137

    9.1 Algebraic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

    9.2 FEM classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    9.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

    9.4 An open, expandable framework . . . . . . . . . . . . . . . . . . . . . . . . 140

    10 References 143

    A Algebraic classes 147

    A.1 Class de