Finite Element Method (FEM)

  1. Analysis of 2D and 3D Problems
    1. 2D Element Types and Formulations
      1. Triangular Elements
        1. Constant Strain Triangle
          1. Shape Functions
            1. Stiffness Matrix
              1. Applications and Limitations
              2. Linear Strain Triangle
                1. Six-Node Formulation
                  1. Shape Functions
                    1. Improved Accuracy
                    2. Higher-Order Triangular Elements
                    3. Quadrilateral Elements
                      1. Bilinear Quadrilateral
                        1. Four-Node Formulation
                          1. Shape Functions
                            1. Rectangular Elements
                            2. Quadratic Quadrilateral
                              1. Eight-Node Formulation
                                1. Nine-Node Formulation
                                2. Serendipity Elements
                                  1. Lagrangian Elements
                                  2. Element Performance Comparison
                                  3. 3D Element Types and Formulations
                                    1. Tetrahedral Elements
                                      1. Linear Tetrahedron
                                        1. Four-Node Formulation
                                          1. Shape Functions
                                            1. Volume Coordinates
                                            2. Quadratic Tetrahedron
                                              1. Ten-Node Formulation
                                                1. Improved Accuracy
                                              2. Hexahedral Elements
                                                1. Linear Hexahedron
                                                  1. Eight-Node Formulation
                                                    1. Trilinear Shape Functions
                                                    2. Quadratic Hexahedron
                                                      1. Twenty-Node Formulation
                                                        1. Twenty-Seven-Node Formulation
                                                      2. Wedge Elements
                                                        1. Six-Node Wedge
                                                          1. Fifteen-Node Wedge
                                                          2. Pyramid Elements
                                                            1. Five-Node Pyramid
                                                              1. Thirteen-Node Pyramid
                                                            2. Coordinate Systems and Shape Functions
                                                              1. Global Coordinate System
                                                                1. Local Coordinate Systems
                                                                  1. Natural Coordinate Systems
                                                                    1. Area Coordinates for Triangles
                                                                      1. Volume Coordinates for Tetrahedra
                                                                        1. Isoparametric Coordinates
                                                                        2. Shape Function Properties in 2D and 3D
                                                                          1. Coordinate Transformation
                                                                          2. Isoparametric Formulation
                                                                            1. Concept and Motivation
                                                                              1. Isoparametric Elements
                                                                                1. Subparametric Elements
                                                                                  1. Superparametric Elements
                                                                                    1. Mapping Between Coordinate Systems
                                                                                      1. Jacobian Matrix
                                                                                        1. Definition and Calculation
                                                                                          1. Determinant Interpretation
                                                                                            1. Invertibility Requirements
                                                                                              1. Distortion Effects
                                                                                              2. Strain-Displacement Relations
                                                                                                1. B-Matrix Derivation
                                                                                                  1. Geometric Interpretation
                                                                                                2. Numerical Integration
                                                                                                  1. Need for Numerical Integration
                                                                                                    1. Gauss Quadrature Rules
                                                                                                      1. One-Dimensional Gauss Quadrature
                                                                                                        1. Two-Dimensional Gauss Quadrature
                                                                                                          1. Quadrilateral Domains
                                                                                                            1. Triangular Domains
                                                                                                            2. Three-Dimensional Gauss Quadrature
                                                                                                            3. Integration Point Selection
                                                                                                              1. Full Integration
                                                                                                                1. Reduced Integration
                                                                                                                  1. Selective Integration
                                                                                                                  2. Integration Accuracy Requirements
                                                                                                                    1. Computational Efficiency Considerations
                                                                                                                      1. Locking Phenomena
                                                                                                                        1. Volumetric Locking
                                                                                                                          1. Shear Locking
                                                                                                                            1. Hourglass Modes

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                                                                                                                        3. Analysis of 1D Problems

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                                                                                                                        5. Application in Structural and Solid Mechanics