Course Topics

NONLINEAR FORMULATIONS AND SOLUTION STRATEGIES

• Nonlinear FEM in Engineering Historical Perspective Linear Benchmark Problems; Patch Tests Nonlinear Benchmark Problems, Test Problems Nonlinear FEM Analysis Geometric and Material Nonlinearities Stress and Strain Measures: Piola-Kirchhoff stresses, Green Strain, Rate-of-deformation Examples of Material Models: Hypoelastic, Hyperelastic, Elastic-plastic, Damage Models Conservation Equations

SEMIDISCRETIZATION AND SOLUTION METHODS

• FE Methods of Nonlinear Mechanics Semidiscretization of Continuum Equations Static and Dynamic Discrete Equations Lagrangian, Eulerian, and Arbitrary Lagrangian Eulerian (ALE) meshes Frame Invariant Stress Rates Total and Updated Lagrangian Formulations Material and Geometric Stiffness Solution Algorithms for Nonlinear Problems Newton and Modified Newton Methods Consistent Linearization Line Search Quasi-Newton Updates ("BFGS", etc.) Arc-Length Strategies Time Integration Procedures Stability, Consistency, and Convergence Survey of Algorithms Formulation of Algorithms for Nonlinear Problems Implicit-Explicit Element Partitions Space-Time Finite ElementsExplicit Dynamic Integration Implicit and Explicit Methods Element Eigenvalue Inequalities; Time Step Selection Accuracy and Stability; Mass Lumping Time Step Partitions; Subcycling Symplectic and Energy Conserving Integrators Direct and Iterative Equation Solvers Direct Solvers : Band, Profile and Sparse Anatomy of an Iterative Equation Solver: Driver Algorithms, Preconditioners Residual CalculationsCAD and Finite Elements: Isogeometric Analysis Computational Geometry NURBS, T-splines Comparison of p- and k-refinement Methods Structural, Vibration and Wave Propagation Analysis Fluid-Structure Interaction: ALE method Cardiovascular Modeling

ELEMENT TECHNOLOGY

• Element Technology - I : Incompressible and Slightly Compressible Media Mixed and Displacement Methods Volumetric Locking Babuska-Brezzi (BB) Condition Survey of Effective Elements Reduced and Selective Integration Techniques Pressure Oscillations Strain Projection Methods: B-bar; Linear and Nonlinear Cases Element Technology - II : Underintegrated Elements Element accuracy and rates of convergence Stiffness Matrix Rank and Rank Deficiency Spurious Singular Modes (Hourglassing) Mixed Variational Principles : Hu-Washizu Stabilization by Perturbation, Assumed Strain, and Variational Methods Physical Hourglass Control Element Technology - III : Plates and Shells C0 and C1 Flexural Theories; Discrete Kirchhoff Theory Continuum Based (Degenerated )Elements Shear Locking and Elimination of Locking by Assumed Strain Membrane Locking and Inextensional Modes Hourglass Modes and Control Shear Instabilities; Physical Hourglass Control Survey and Comparison of Elements

CONSTITUTIVE MODELS

• Rate-Independent Deviatoric Plasticity Small and Finite Deformation Formulations Radial Return Methods Algorithms for the Finite Deformation Case Material Instabilities, Fracture and Failure Material Instabilities : Strain-Softening, Nonassociated Plasticity Loss of Ellipticity (Hyperbolicity); Localization Regularization: Viscous, Gradient, Nonlocal Explicit and Smeared Crack Models Failure Modeling : Static and Dynamic Crack Propagation Discontinuous Elemennts (XFEM, others) Level Sets for Evovling Discontinuities Coarse graining of failure processes Hierarchical and concurrent multiscale methods Quantum/Molecular/Continuum Multiscale Methods Return Mapping Algorithms for General Classes of Inelastic Materials Cutting Plane Algorithm Closest Point Projection Algorithm Elastic Damage and Viscoplastic Models Operator Splitting

OTHER TOPICS

• Contact-Impact Penalty and Lagrange Multiplier Methods Augmented Lagrangian Methods Regularization of Impact and Friction; Pinball Algorithm Crashworthiness Analysis Adaptivity and Meshfree Methods p-, h-, and r-Adaptivity Error Indicators: Residual, Global and Local Projection Smooth Particle Hydrodynamics Element-free Galerkin Method RKPM, Petrov-Galerkin meshfree methods Moving least square and radial basis functions Visibility method for discontinuities, smoothing Integration methods: background mesh; stress-points Partitions of Unity Fluid-Structure Interaction: Immersed FEM, level sets Fluids - I and II Scalar Advection-Diffusion Equation SUPG and Galerkin/Least-Squares Method Space-Time Generalizations Discontinuous Galerkin Method Advective-Diffusive Systems Incompressible Euler and Navier-Stokes Equations Stokes Equations; Methods which Circumvent the BB-Condition Compressible Euler and Navier-Stokes Equations Entropy Variables Conservation and Physical Variables Shock-Capturing Operators Domain Decomposition Iterative Procedures; GMRES Matrix Free Algorithms Turbulence, LES Variational Multiscale Methods Isogeometric Fluid Analysis Acoustic Wave Propagation Phase Field Modeling

Copyright 2008 Hughes - Belytschko