Finite Element Method (FEM)

  1. Analysis of 1D Problems
    1. Problem Setup and Domain Discretization
      1. 1D Domain Definition
        1. Division into Elements
          1. Element Types in 1D
            1. Node Placement and Numbering
              1. Element Connectivity
                1. Global and Local Numbering Systems
                2. Element Shape Functions
                  1. Definition and Purpose
                    1. Requirements for Shape Functions
                      1. Partition of Unity
                        1. Interpolatory Property
                          1. Continuity Requirements
                            1. Completeness
                            2. Linear Shape Functions
                              1. Derivation
                                1. Local Coordinates
                                2. Quadratic Shape Functions
                                  1. Derivation
                                    1. Three-Node Elements
                                    2. Higher-Order Shape Functions
                                      1. Cubic Elements
                                        1. Lagrangian Interpolation
                                        2. Hierarchical Shape Functions
                                        3. Element Formulation
                                          1. Direct Equilibrium Approach
                                            1. Force Balance
                                              1. Displacement-Force Relations
                                              2. Potential Energy Approach
                                                1. Element Potential Energy
                                                  1. Minimization Process
                                                  2. Galerkin Method Approach
                                                    1. Weak Form Development
                                                      1. Element Equations
                                                      2. Element Stiffness Matrix Derivation
                                                        1. Element Load Vector Derivation
                                                          1. Coordinate Transformations
                                                          2. Assembly Process
                                                            1. Global Stiffness Matrix Construction
                                                              1. Global Force Vector Construction
                                                                1. Direct Stiffness Method
                                                                  1. Assembly Algorithm
                                                                    1. Handling Element Connectivity
                                                                      1. Bandwidth Considerations
                                                                      2. Boundary Conditions Implementation
                                                                        1. Essential Boundary Conditions
                                                                          1. Elimination Method
                                                                            1. Penalty Method
                                                                              1. Lagrange Multiplier Method
                                                                              2. Natural Boundary Conditions
                                                                                1. Point Loads
                                                                                  1. Distributed Loads
                                                                                    1. Boundary Flux
                                                                                    2. Mixed Boundary Conditions
                                                                                    3. Solution of System Equations
                                                                                      1. Matrix Properties
                                                                                        1. Solution Techniques
                                                                                          1. Handling Ill-Conditioned Systems
                                                                                            1. Numerical Considerations
                                                                                            2. Post-Processing and Result Interpretation
                                                                                              1. Nodal Solution Recovery
                                                                                                1. Element-Level Calculations
                                                                                                  1. Derivative Calculations
                                                                                                    1. Reaction Force Calculations
                                                                                                      1. Error Assessment
                                                                                                        1. Result Visualization

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                                                                                                      2. Mathematical Foundations

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                                                                                                      4. Analysis of 2D and 3D Problems