Finite Element Method (FEM)

The Finite Element Method (FEM) is a numerical technique for solving boundary value problems governed by partial differential equations (PDEs). It works by discretizing a continuous domain into smaller subdomains called finite elements, connected at nodes. Within each element, unknown field variables (such as displacement, temperature, or pressure) are approximated using interpolation functions known as shape functions. The governing PDE is transformed into a system of algebraic equations using variational or weighted residual methods, most commonly the Galerkin method. The assembled global system takes the form [K]{u} = {F}, where [K] is the global stiffness matrix, {u} is the vector of nodal unknowns (e.g., displacements), and {F} is the global force vector. Key steps in FEM include preprocessing (geometry definition, meshing, material properties, boundary conditions), formulation of element stiffness matrices, assembly into the global system, application of boundary conditions, solution of the algebraic system, and post-processing of results. FEM is widely used in structural mechanics, heat transfer, fluid dynamics, and multiphysics problems due to its ability to handle complex geometries, heterogeneous materials, and non-standard boundary conditions. Limitations include computational cost, dependence on mesh quality, and the need for careful interpretation of numerical results.