Introduction to Finite Element Method (FEM)

What is FEM?

The Finite Element Method (FEM) is a numerical technique for solving engineering problems — structural deformation, thermal distribution, electromagnetic fields — that cannot be solved analytically for complex geometries. Originating in NASA's aerospace programs in the 1960s, FEM is now the backbone of all major CAE software.

The key idea: divide a continuous body into a finite number of elements, approximate the solution (e.g. displacement) as a polynomial within each element, then solve the resulting system of equations.

FEM Workflow

1. Geometry and Meshing

Discretize the domain into triangles/quads (2D) or tetrahedra/hexahedra (3D). Refine the mesh near stress concentrations; coarser elsewhere.

2. Element Stiffness Matrix

For each element: [k] = ∫[B]ᵀ[D][B]dV, where [B] is the strain-displacement matrix and [D] is the material constitutive matrix.

3. Global Assembly

Assemble element stiffness matrices into the global stiffness matrix [K] using node numbering.

4. Apply Boundary Conditions

Set displacements (fixed supports) and forces/pressures to make the system [K]{u} = {F} solvable.

5. Solve and Post-process

Solve for nodal displacements {u}, then compute strains and stresses at integration points.

Element Types

Try FEM Concepts in Your Browser