Mathematical Foundations I: Linear Algebra
Vectors, matrices, coordinate transformations, systems of equations, eigenvalues, and eigenvectors.
Mathematical Foundations II: Variational Calculus
Functionals, strong vs. weak formulations, boundary conditions, and the Euler-Lagrange equation.
Approximate Methods: Rayleigh-Ritz & Galerkin
Minimizing residuals of differential equations using trial functions, Galerkin residual weights, and energy minimization.
1D Spring & Stepped Bar Formulations
Shape functions, axial stiffness, local/global nodes, stepped bars, and global matrix assembly.
2D Pinned Truss Analysis
Coordinate transformations, direction cosines, element stiffness, and member force recovery.
Euler-Bernoulli Beam Elements
Transverse deflection, slopes, Hermite cubic shape functions, and flexural stiffness matrices.
2D Rigid Plane Frame Elements
Coupled axial, shear, and bending loads at rigid joints using a combined 6x6 element matrix.
Constant Strain Triangle (CST) Elements
Plane stress, plane strain, area coordinates, constant strain-displacement [B] matrices.
Linear Strain Triangle (LST) Elements
Higher-order quadratic triangular elements, shape functions, and solving high stress gradients.
Isoparametric Quadrilateral (Q4) Elements
Mapping skewed/curved quadrilaterals to a standard natural coordinates square, Jacobian matrix.
Numerical Integration: Gauss Quadrature
Integrating functions numerically using sampling points and weights in natural coordinate space.
Steady-State 1D Heat Conduction Elements
Conduction, thermal convection boundaries, heat generation vector, temperature profiles.
Dynamic & Vibration FEA: Mass Matrices
Consistent vs. lumped mass matrices, natural frequencies, and structural eigenvalues/eigenvectors.
Industry Best Practices & Commercial Solvers
Mesh convergence, warping, stress singularities, non-linearities, Abaqus/ANSYS workflow setup.