Finite Element Method (FEM)

  1. Advanced Topics in FEM
    1. Dynamic Analysis
      1. Fundamentals of Structural Dynamics
        1. Equation of Motion
          1. Degrees of Freedom
          2. Mass Matrix Formulation
            1. Consistent Mass Matrix
              1. Derivation
              2. Lumped Mass Matrix
                1. Diagonal Form
                  1. Row Sum Technique
                  2. Comparison and Selection
                  3. Damping Matrix Formulation
                    1. Rayleigh Damping
                      1. Viscous Damping Models
                      2. Free Vibration Analysis
                        1. Eigenvalue Problem Formulation
                          1. Natural Frequencies
                            1. Mode Shapes
                              1. Solution Techniques
                                1. Subspace Iteration
                                  1. Lanczos Method
                                2. Forced Vibration Analysis
                                  1. Harmonic Response
                                    1. Random Vibration
                                    2. Transient Dynamic Analysis
                                      1. Direct Integration Methods
                                        1. Implicit Methods
                                          1. Newmark Method
                                            1. Wilson-Theta Method
                                              1. HHT-Alpha Method
                                              2. Explicit Methods
                                                1. Central Difference Method
                                                  1. Stability Limits
                                                  2. Time Step Selection
                                                    1. Stability and Accuracy
                                                  3. Nonlinear Analysis
                                                    1. Sources of Nonlinearity
                                                      1. Geometric Nonlinearity
                                                        1. Large Displacements
                                                          1. Large Rotations
                                                            1. Large Strains
                                                              1. Updated Lagrangian Formulation
                                                                1. Total Lagrangian Formulation
                                                                2. Material Nonlinearity
                                                                  1. Elastoplasticity
                                                                    1. Hyperelasticity
                                                                      1. Viscoelasticity
                                                                        1. Damage Models
                                                                        2. Contact Nonlinearity
                                                                          1. Contact Detection
                                                                            1. Contact Forces
                                                                              1. Friction Models
                                                                            2. Solution Techniques
                                                                              1. Incremental Methods
                                                                                1. Iterative Methods
                                                                                  1. Newton-Raphson Method
                                                                                    1. Tangent Stiffness Matrix
                                                                                      1. Convergence Criteria
                                                                                      2. Modified Newton Methods
                                                                                        1. Quasi-Newton Methods
                                                                                          1. Arc-Length Methods
                                                                                            1. Riks Method
                                                                                              1. Crisfield Method
                                                                                            2. Load Control vs Displacement Control
                                                                                              1. Convergence and Stability Issues
                                                                                              2. Error Analysis and Convergence
                                                                                                1. Sources of Error
                                                                                                  1. Modeling Error
                                                                                                    1. Discretization Error
                                                                                                      1. Numerical Error
                                                                                                        1. Round-off Error
                                                                                                        2. Convergence Requirements
                                                                                                          1. h-Convergence
                                                                                                            1. p-Convergence
                                                                                                              1. Monotonic Convergence
                                                                                                              2. Error Measures
                                                                                                                1. Energy Norm
                                                                                                                  1. L2 Norm
                                                                                                                    1. Maximum Norm
                                                                                                                    2. A Priori Error Estimation
                                                                                                                      1. Theoretical Bounds
                                                                                                                        1. Convergence Rates
                                                                                                                        2. A Posteriori Error Estimation
                                                                                                                          1. Residual-Based Estimators
                                                                                                                            1. Recovery-Based Estimators
                                                                                                                              1. Equilibrium-Based Estimators
                                                                                                                              2. Adaptive Strategies
                                                                                                                                1. Error Indicators
                                                                                                                                  1. Refinement Criteria
                                                                                                                                2. Mesh Generation and Adaptivity
                                                                                                                                  1. Mesh Generation Fundamentals
                                                                                                                                    1. Mesh Quality Requirements
                                                                                                                                      1. Element Shape Quality
                                                                                                                                      2. Meshing Strategies
                                                                                                                                        1. Structured Meshing
                                                                                                                                          1. Regular Grids
                                                                                                                                            1. Mapped Meshing
                                                                                                                                            2. Unstructured Meshing
                                                                                                                                              1. Delaunay Triangulation
                                                                                                                                                1. Advancing Front Method
                                                                                                                                                2. Hybrid Meshing
                                                                                                                                                3. Mesh Quality Metrics
                                                                                                                                                  1. Aspect Ratio
                                                                                                                                                    1. Skewness
                                                                                                                                                      1. Jacobian Ratio
                                                                                                                                                        1. Element Distortion
                                                                                                                                                          1. Warpage
                                                                                                                                                          2. Adaptive Mesh Refinement
                                                                                                                                                            1. h-Refinement
                                                                                                                                                              1. Element Subdivision
                                                                                                                                                                1. Hanging Nodes
                                                                                                                                                                  1. Transition Elements
                                                                                                                                                                  2. p-Refinement
                                                                                                                                                                    1. Polynomial Order Increase
                                                                                                                                                                      1. Hierarchical Basis Functions
                                                                                                                                                                      2. r-Refinement
                                                                                                                                                                        1. Node Relocation
                                                                                                                                                                          1. Mesh Smoothing
                                                                                                                                                                          2. hp-Refinement
                                                                                                                                                                          3. Automatic Mesh Generation
                                                                                                                                                                            1. CAD Integration
                                                                                                                                                                              1. Mesh Size Control
                                                                                                                                                                                1. Boundary Layer Meshing

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                                                                                                                                                                            6. Application in Heat Transfer and Other Field Problems

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                                                                                                                                                                            8. Practical Implementation and Software