Finite Element Method (FEM)

  1. Mathematical Foundations
    1. Review of Linear Algebra
      1. Vectors and Vector Spaces
        1. Vector Operations
          1. Linear Independence
            1. Basis and Dimension
            2. Matrices and Matrix Types
              1. Square Matrices
                1. Symmetric Matrices
                  1. Sparse Matrices
                    1. Positive Definite Matrices
                      1. Band Matrices
                      2. Matrix Operations
                        1. Addition and Subtraction
                          1. Multiplication
                            1. Transpose
                              1. Inverse
                                1. Pseudoinverse
                                2. Determinants and Rank
                                  1. Properties of Determinants
                                    1. Matrix Rank
                                      1. Singular Matrices
                                      2. Eigenvalues and Eigenvectors
                                        1. Characteristic Equation
                                          1. Eigenvalue Problems
                                            1. Generalized Eigenvalue Problems
                                            2. Solving Systems of Linear Equations
                                              1. Direct Methods
                                                1. Gaussian Elimination
                                                  1. LU Decomposition
                                                    1. Cholesky Decomposition
                                                    2. Iterative Methods
                                                      1. Jacobi Method
                                                        1. Gauss-Seidel Method
                                                          1. Conjugate Gradient Method
                                                      2. Review of Calculus
                                                        1. Functions of Several Variables
                                                          1. Continuity
                                                            1. Limits
                                                            2. Partial Derivatives
                                                              1. Chain Rule
                                                                1. Higher-Order Derivatives
                                                                2. Multiple Integrals
                                                                  1. Double Integrals
                                                                    1. Triple Integrals
                                                                      1. Change of Variables
                                                                      2. Integration by Parts
                                                                        1. Vector Calculus Theorems
                                                                          1. Green's Theorem
                                                                            1. Divergence Theorem
                                                                              1. Stokes' Theorem
                                                                              2. Vector Calculus Operations
                                                                                1. Gradient
                                                                                  1. Divergence
                                                                                    1. Curl
                                                                                      1. Laplacian
                                                                                    2. Governing Equations in Engineering
                                                                                      1. Partial Differential Equations
                                                                                        1. Classification of PDEs
                                                                                          1. Elliptic PDEs
                                                                                            1. Parabolic PDEs
                                                                                              1. Hyperbolic PDEs
                                                                                              2. Order of PDEs
                                                                                                1. Linear vs Nonlinear PDEs
                                                                                                2. Boundary Conditions
                                                                                                  1. Dirichlet Boundary Conditions
                                                                                                    1. Neumann Boundary Conditions
                                                                                                      1. Robin Boundary Conditions
                                                                                                        1. Mixed Boundary Conditions
                                                                                                        2. Initial Conditions
                                                                                                          1. Well-Posed Problems
                                                                                                          2. Variational Principles and Weak Formulations
                                                                                                            1. Calculus of Variations
                                                                                                              1. Functionals
                                                                                                                1. Euler-Lagrange Equations
                                                                                                                2. Principle of Minimum Potential Energy
                                                                                                                  1. Application to Structural Problems
                                                                                                                    1. Rayleigh-Ritz Method
                                                                                                                    2. Weighted Residual Methods
                                                                                                                      1. Concept of Residuals
                                                                                                                        1. Weight Functions
                                                                                                                          1. Galerkin Method
                                                                                                                            1. Petrov-Galerkin Method
                                                                                                                              1. Method of Least Squares
                                                                                                                                1. Collocation Method
                                                                                                                                2. Deriving Weak Forms
                                                                                                                                  1. Integration by Parts in Weak Formulation
                                                                                                                                    1. Boundary Terms
                                                                                                                                      1. Natural Boundary Conditions
                                                                                                                                        1. Essential Boundary Conditions
                                                                                                                                        2. Existence and Uniqueness of Solutions

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                                                                                                                                      1. Introduction to the Finite Element Method

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                                                                                                                                      3. Analysis of 1D Problems