Programming the finite element method

Presents compact and portable FORTRAN programs for solution of problems using the finite element method. All code is presented in full, and the user is encouraged to look into the subroutines to see how they actually work. Covers the main areas of engineering analysis: equilibrium, transient, and eigenvalue problems. In the structures/solid mechanics area, programs solve problems ranging from simple structural analysis, through linear elastic analyses of two-dimensional solids with a variety of finite elements, to complex three-dimensional analyses involving nonlinear material properties. Dynamics applications are covered, including both time-domain and eigenvalue problems. Programs for solving transient problems of fluid/heat flow are presented, including novel, element-by-element algorithms.

Preface

Acknowledgement

1 Preliminaries: Computer Strategies

1.1 Introduction

1.2 Hardware

1.3 Memory management

1.4 Vector processors

1.5 Parallel processors

1.6 BLAS libraries

1.7 MPI libraries

1.8 Applications software

1.8.1 Arithmetic

1.8.2 Conditions

1.8.3 Loops

1.9 Array features

1.9.1 Dynamic arrays

1.9.2 Broadcasting

1.9.3 Constructors

1.9.4 Vector subscripts

1.9.5 Array sections

1.9.6 Whole-array manipulations

1.9.7 Intrinsic procedures for arrays

1.9.8 Additional Fortran 95 features

1.9.9 Subprogram libraries

1.9.10 Structured programming

1.10 Conclusions

References

2 Spatial Discretisation by Finite Elements

2.1 Introduction

2.2 Rod element

2.2.1 Rod stiffness matrix

2.2.2 Rod mass element

2.3 The eigenvalue equation

2.4 Beam element

2.4.1 Beam element stiffness matrix

2.4.2 Beam element mass matrix

2.5 Beam with an axial force

2.6 Beam on an elastic foundation

2.7 General remarks on the discretisation process

2.8 Alternative derivation of element stiffness

2.9 Two-dimensional elements: plane strain and plane stress

2.10 Energy approach

2.11 Plane element mass matrix

2.12 Axisymmetric stress and strain

2.13 Three-dimensional stress and strain

2.14 Plate-bending element

2.15 Summary of element equations for solids

2.16 Flow of fluids: Navier-Stokes equations

2.17 Simplified flow equations

2.17.1 Steady state

2.17.2 Transient state

2.17.3 Advection

2.18 Further coupled equations: Biot consolidation

2.19 Conclusions

References

3 Programming Finite Element Computations

3.1 Introduction

3.2 Local coordinates for quadrilateral elements

3.2.1 Numerical integration for quadrilaterals

3.2.2 Analytical integration for quadrilaterals

3.3 Local coordinates for triangular elements

3.3.1 Numerical integration for triangles

3.4 Multi-element assemblies

3.5 "Element-by-element" or "Mesh-free" techniques

3.5.1 Conjugate gradient method

3.5.2 Preconditioning

3.5.3 Unsymmetric systems

3.5.4 Symmetric non-positive definite equations

3.5.5 Symmetric eigenvalue systems

3.6 Incorporation of boundary conditions

3.7 Programming using building blocks

3.7.1 Black box routines

3.7.2 Special purpose routines

3.7.3 Plane elastic analysis using quadrilateral elements

3.7.4 Plane elastic analysis using triangular elements

3.7.5 Axisymmetric strain of elastic solids

3.7.6 Plane steady laminar fluid flow

3.7.7 Mass matrix formation

3.7.8 Higher-order 2D elements

3.7.9 Three-dimensional elements

3.7.10 Assembly of elements

3.8 Solution of equilibrium equations

3.9 Evaluation of eigenvalues and eigenvectors

3.9.1 Jacobi algorithm

3.9.2 Lanczos algorithm

3.10 Solution of first order time dependent problems

3.11 Solution of coupled Navier-Stokes problems

3.12 Solution of coupled transient problems

3.12.1 Absolute load version

3.12.2 Incremental load version

3.13 Solution of second order time dependent problems

3.13.1 Modal superposition

3.13.2 Newmark or Crank-Nicolson method

3.13.3 Wilson's method

3.13.4 Explicit methods and other storage-saving strategies

References

4 Static Equilibrium of Structures

4.1 Introduction

Program 4.1 One-dimensional analysis of axially loaded elastic rods using 2-node rod elements

Program 4.2 Analysis of elastic pin-jointed frames using 2-node rod elements in two or three dimensions

Program 4.3 Analysis of elastic beams using 2-node beam elements (elastic foundation optional)

Program 4.4 Analysis of elastic rigid-jointed frames using 2-node beam/ro

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Summary

Summary

Table of Contents