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Beginner Guide

Introduction to Finite Element Method (FEM)

Mesh, stiffness matrix, boundary conditions — the core concepts of FEM explained clearly. Try free browser-based tools alongside this guide.

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.

Notes
Mesh quality tip: Keep aspect ratios below 5:1. Poor elements degrade accuracy.

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

ElementDimensionApplication
Beam / Bar1DFrames, trusses
CST / Q4 (plane stress)2DThin plates, plane problems
Shell2D curvedThin-walled structures
Tet / Hex (solid)3DGeneral 3D bodies

Try FEM Concepts in Your Browser

Beam Deflection & Stress
Real-time deflection curves, BMD, and SFD for beam problems
SIMP Topology Optimization
FEM-based topology optimization — find optimal material layout automatically
Euler Buckling Load
Critical load and buckling mode shapes for columns

How to Use

  1. Import or draw geometry (2D plate, beam, or 3D solid) in the CAD preprocessor
  2. Generate tetrahedral or hexahedral mesh; aim for 10,000–50,000 elements depending on feature size and accuracy requirements
  3. Define material properties (Young's modulus, Poisson's ratio, density) and assign element types (linear vs. quadratic)
  4. Apply boundary conditions: fixed supports, prescribed displacements, and external loads (point forces, pressure distributions)
  5. Assemble global stiffness matrix K and solve [K]{u} = {F} for nodal displacements
  6. Post-process results: visualize stress contours, strain fields, and deformation with color-coded legends

Worked Example

Aluminum cantilever beam: 500 mm length, 20 mm × 10 mm rectangular cross-section, E = 70 GPa, ν = 0.33. Fixed at left end, vertical point load 500 N at right end. Quadratic tetrahedral mesh: 8,500 elements. Computed maximum deflection: 3.2 mm at free end (theory: 3.21 mm). Maximum von Mises stress: 245 MPa at fixed support, well below 275 MPa yield strength. Convergence study shows less than 2% error with 12,000 elements.

Practical Notes

  • Mesh refinement near stress concentrations (fillets, holes) prevents artificial stress peaks; local element size ≤ 1/10 of feature radius
  • Quadratic elements (10-node tetrahedra, 20-node bricks) improve accuracy for curved geometries at modest computational cost versus linear elements
  • Always perform mesh independence study: run analyses at 50%, 100%, 150% element density; convergence below 5% confirms adequate mesh
  • Boundary condition errors are common: fully constrain rigid-body modes (all translational and rotational DOF at one point minimum)
  • For nonlinear problems (large deformation, plastic yield), enable Newton–Raphson iteration with load stepping over single-step analysis