Finite Element Method

The FEM process begins with the ‘discretization’ of the problem domain into a mesh of finite elements (e.g., triangles or quadrilaterals in 2D, tetrahedra or hexahedra in 3D). Within each element, the unknown field variable (e.g., displacement) is approximated by simple polynomial functions, known as shape functions. The values of the field at the element nodes become the new unknowns of the problem.

A system of algebraic equations for the entire domain is derived, typically using a variational principle like the principle of minimum potential energy or a weighted residual الطريقة like the Galerkin method. This process generates an ‘element stiffness matrix’ [latex][k_e][/latex] for each element, which relates the nodal forces [latex]\{f_e\}[/latex] to the nodal displacements [latex]\{u_e\}[/latex] via [latex][k_e] \{u_e\} = \{f_e\}[/latex]. These individual element matrices are then systematically combined (‘assembled’) into a single global stiffness matrix [latex][K][/latex] for the entire structure. After applying known boundary conditions (forces and constraints), the resulting large system of linear equations, [latex][K] \{U\} = \{F\}[/latex], is solved numerically for the unknown global displacement vector [latex]\{U\}[/latex]. Once the nodal displacements are known, other quantities like strains and stresses can be calculated for each element.