Computational Methods in Solid Mechanics

Inhaltsverzeichnis

• 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.