Finite Element Mathematics

Weak forms to stiffness matrices. The mathematics behind FEM software: strong and weak forms, essential and natural boundary conditions, Rayleigh–Ritz and Galerkin methods, shape functions, element stiffness, global assembly, solution and convergence.

• Weak Form, Variational Methods & Galerkin — The finite element method rests on rewriting a differential equation in its weak (integral) form and seeking the best approximation within a finite set of trial functions. Learn strong vs weak form, integration by parts, essential vs natural boundary conditions, the Rayleigh–Ritz energy method, the Galerkin method, and why the approximation is optimal — with a SymPy Galerkin solution.
• 1-D Finite Elements: Stiffness Assembly — Build a complete finite element solver for an axial bar: linear shape functions, element stiffness matrix and load vector, global assembly, applying boundary conditions, solving, and post-processing stresses and reactions. Understand why linear elements give exact nodal values in 1-D, how convergence works, and how the same steps scale to 2-D/3-D FEM software.