Finite Element Method (FEM)

7.

Advanced Topics in FEM

7.1.

Dynamic Analysis

7.1.1.

Fundamentals of Structural Dynamics

7.1.1.1.

Equation of Motion

7.1.1.2.

Degrees of Freedom

7.1.1.3.

Modal Analysis Concepts

7.1.2.

Mass Matrix Formulation

7.1.2.1.

Consistent Mass Matrix

7.1.2.1.1.

7.1.2.2.

Lumped Mass Matrix

7.1.2.2.1.

7.1.2.2.2.

Row Sum Technique

7.1.2.3.

Comparison and Selection

7.1.3.

Damping Matrix Formulation

7.1.3.1.

Rayleigh Damping

7.1.3.2.

7.1.3.3.

Viscous Damping Models

7.1.4.

Free Vibration Analysis

7.1.4.1.

Eigenvalue Problem Formulation

7.1.4.2.

Natural Frequencies

7.1.4.3.

7.1.4.4.

Modal Orthogonality

7.1.4.5.

Solution Techniques

7.1.4.5.1.

Subspace Iteration

7.1.4.5.2.

7.1.5.

Forced Vibration Analysis

7.1.5.1.

Harmonic Response

7.1.5.2.

Random Vibration

7.1.5.3.

Modal Superposition

7.1.6.

Transient Dynamic Analysis

7.1.6.1.

Direct Integration Methods

7.1.6.2.

Implicit Methods

7.1.6.2.1.

7.1.6.2.2.

Wilson-Theta Method

7.1.6.2.3.

HHT-Alpha Method

7.1.6.3.

Explicit Methods

7.1.6.3.1.

Central Difference Method

7.1.6.3.2.

Stability Limits

7.1.6.4.

Time Step Selection

7.1.6.5.

Stability and Accuracy

7.2.

Nonlinear Analysis

7.2.1.

Sources of Nonlinearity

7.2.1.1.

Geometric Nonlinearity

7.2.1.1.1.

Large Displacements

7.2.1.1.2.

Large Rotations

7.2.1.1.3.

7.2.1.1.4.

Updated Lagrangian Formulation

7.2.1.1.5.

Total Lagrangian Formulation

7.2.1.2.

Material Nonlinearity

7.2.1.2.1.

Elastoplasticity

7.2.1.2.2.

Hyperelasticity

7.2.1.2.3.

Viscoelasticity

7.2.1.2.4.

7.2.1.3.

Contact Nonlinearity

7.2.1.3.1.

Contact Detection

7.2.1.3.2.

7.2.1.3.3.

Friction Models

7.2.2.

Solution Techniques

7.2.2.1.

Incremental Methods

7.2.2.2.

Iterative Methods

7.2.2.3.

Newton-Raphson Method

7.2.2.3.1.

Tangent Stiffness Matrix

7.2.2.3.2.

Convergence Criteria

7.2.2.4.

Modified Newton Methods

7.2.2.5.

Quasi-Newton Methods

7.2.2.6.

Arc-Length Methods

7.2.2.6.1.

7.2.2.6.2.

Crisfield Method

7.2.3.

Load Control vs Displacement Control

7.2.4.

Convergence and Stability Issues

7.3.

Error Analysis and Convergence

7.3.1.

Sources of Error

7.3.1.1.

7.3.1.2.

Discretization Error

7.3.1.3.

Numerical Error

7.3.1.4.

Round-off Error

7.3.2.

Convergence Requirements

7.3.2.1.

7.3.2.2.

7.3.2.3.

Monotonic Convergence

7.3.3.

7.3.3.1.

7.3.3.2.

7.3.3.3.

7.3.4.

A Priori Error Estimation

7.3.4.1.

Theoretical Bounds

7.3.4.2.

Convergence Rates

7.3.5.

A Posteriori Error Estimation

7.3.5.1.

Residual-Based Estimators

7.3.5.2.

Recovery-Based Estimators

7.3.5.3.

Equilibrium-Based Estimators

7.3.6.

Adaptive Strategies

7.3.6.1.

Error Indicators

7.3.6.2.

Refinement Criteria

7.4.

Mesh Generation and Adaptivity

7.4.1.

Mesh Generation Fundamentals

7.4.1.1.

Mesh Quality Requirements

7.4.1.2.

Element Shape Quality

7.4.2.

Meshing Strategies

7.4.2.1.

Structured Meshing

7.4.2.1.1.

7.4.2.1.2.

7.4.2.2.

Unstructured Meshing

7.4.2.2.1.

Delaunay Triangulation

7.4.2.2.2.

Advancing Front Method

7.4.2.3.

7.4.3.

Mesh Quality Metrics

7.4.3.1.

7.4.3.2.

7.4.3.3.

7.4.3.4.

Element Distortion

7.4.3.5.

7.4.4.

Adaptive Mesh Refinement

7.4.4.1.

7.4.4.1.1.

Element Subdivision

7.4.4.1.2.

7.4.4.1.3.

Transition Elements

7.4.4.2.

7.4.4.2.1.

Polynomial Order Increase

7.4.4.2.2.

Hierarchical Basis Functions

7.4.4.3.

7.4.4.3.1.

Node Relocation

7.4.4.3.2.

7.4.4.4.

7.4.5.

Automatic Mesh Generation

7.4.5.1.

CAD Integration

7.4.5.2.

Mesh Size Control

7.4.5.3.

Boundary Layer Meshing

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8. Practical Implementation and Software