Computational Methods in Solid Mechanics / Edition 1

by A. Curnier

Overview

This volume presents an introduction to the three numerical methods most commonly used in the mechanical analysis of deformable solids, viz. the finite element method (FEM), the linear iteration method (LIM), and the finite difference method (FDM). The book has been written from the point of view of simplicity and unity; its originality lies in the comparable emphasis given to the spatial, temporal and nonlinear dimensions of problem solving. This leads to a neat global algorithm. Chapter 1 addresses the problem of a one-dimensional bar, with emphasis being given to the virtual work principle. Chapters 2—4 present the three numerical methods. Although the discussion relates to a one-dimensional model, the formalism used is extendable to two-dimensional situations. Chapter 5 is devoted to a detailed discussion of the compact combination of the three methods, and contains several sections concerning their computer implementation. Finally, Chapter 6 gives a generalization to two and three dimensions of both the mechanical and numerical aspects. For graduate students and researchers whose work involves the theory and application of computational solid mechanics.

Product Details

ISBN-13:	9780792327615
Publisher:	Springer Netherlands
Publication date:	05/31/1994
Series:	Solid Mechanics and Its Applications , #29
Edition description:	1994
Pages:	404
Product dimensions:	6.69(w) x 9.61(h) x 0.24(d)

Table of Contents

1 One-Dimensional Bar Model Problem (Principle of Virtual Work).- 1.1 Kinematics : material description.- 1.2 Dynamics: equilibrium of forces.- 1.3 Mechanics : principle of virtual work.- 1.4 Geometric and material non-linearities.- 1.5 Constitutive laws for solid materials.- 1.6 Discontinuities in space.- 1.7 Thermics : heat equation.- 1.8 Mathematics : functional analysis notions.- 1.9 Summary.- 2 Spatial Discretisation by the Finite Element Method.- 2.1 Global overview : Galerkin method.- 2.2 Nodal FEM : piecewise polynomial basis functions.- 2.3 Localisation of mesh nodal displacements.- 2.4 Interpolation of element nodal displacements.- 2.5 Integration of element nodal forces.- 2.6 Assembly of mesh nodal forces.- 2.7 Properties of force vectors.- 2.8 Automation: isoparametric maps and numerical integration.- 2.9 Boundary conditions condensation.- 2.10 Algorithm: element loop.- 2.11 Practice : heat equation discretisation.- 2.12 Accuracy : error norms and estimates.- 2.13 Summary.- 3 Solution of Non-Linearities by the Linear Iteration Method.- 3.1 Linearisation : classical and directional derivative.- 3.2 Nominal stress linearisation : nominal tangent modulus.- 3.3 Linearized equations of motion : mass and stiffness matrices.- 3.4 Finite element mass and stiffness matrices.- 3.5 Assembly of the mass and stiffness matrices.- 3.6 Properties of the mass and stiffness matrices.- 3.7 Linearized heat equation.- 3.8 Condensation of boundary conditions after linearisation.- 3.9 Linear iteration method : algorithm and variants.- 3.10 Standard and modified Newton methods.- 3.11 Secant or conjugate gradient methods.- 3.12 Gradient and Jacobi methods.- 3.13 Local and global convergence of iterative methods.- 3.14 Local convergence of the LIM : consistency and stability.- 3.15 Glocal convergence of the LIM : damping and continuation.- 3.16 Summary.- 4 Time Integration by the Finite Difference Method.- 4.1 Generalised trapezoidal rule or Euler scheme (applied to the linear heat equation).- 4.2 Modal analysis of the heat equation.- 4.3 General error analysis : summary and glossary.- 4.4 Stability of the heat-trapezoid algorithm.- 4.5 Consistency of the heat-trapezoid algorithm.- 4.6 Convergence of the heat-trapezoid algorithm.- 4.7 Generalized trapezoidal rale or Newmark scheme (applied to the linear wave equation).- 4.8 Modal analysis of the wave equation.- 4.9 Stability of the wave-trapezoid algorithm.- 4.10 Consistency of the wave-trapezoid algorithm.- 4.11 Convergence of the wave-trapezoid algorithm.- 4.12 Summary.- 5 Compact Combination of the Finite Element, Linear Iteration and Finite Difference Methods.- 5.1 Problem statement review.- 5.2 Galerkin-FE algorithm review.- 5.3 Newton-LI algorithm review.- 5.4 Newmark-FD algorithm review.- 5.5 Combining the FE, LI and FD algorithms.- 5.6 Nonlinear thermics algorithm.- 5.7 Nonlinear dynamics algorithm.- 5.8 Nonlinear thermodynamics synthesis.- 5.9 Convergence review.- 5.10 Programming guidelines (TACT example).- 5.11 FD and LI methods programming.- 5.12 FE and algebraic methods programming.- 5.13 Summary.- 6 Two- and Three-Dimensional Deformable Solids.- 6.1 Kinematics : material description.- 6.2 Dynamics: balance of forces.- 6.3 Mechanics : principle of virtual work.- 6.4 Objective constitutive laws.- 6.5 Nominal stress linearisation.- 6.6 Constitutive laws for solid materials.- 6.7 Spatial discretisation in three dimensions.- 6.8 Linearized discrete mechanics.- 6.9 Isoparametric solid finite elements.- 6.10 Finite difference time integration.- 6.11 Final algorithm.- 6.12 Summary.- Conclusion.- Appendix A : List of Symbols.- 1.- 2.- 3.- 4.- 5.- 6.- Appendix B : Exercises.- 1.- 2.- 3.- 4.- 5.- 6.