Computational Methods in Solid Mechanics [electronic resource] /

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.