Computational techniques; why FEM predominates

Computational techniques

• Finite element method (FEM)
• Finite difference method (FDM)
• Boundary element method (BEM)
• Discrete element method (DEM)
• Finite volume method and mesh-free methods (SPH)

Why FEM predominates

• Handles any shape of boundary and any combination of loads and supports.
• Handles non-homogeneous, anisotropic and non-linear materials.
• Same formulation covers structures, soil, seepage, heat and dynamics.
• Mesh can be refined locally and element types can be mixed.
• Stiffness matrices are sparse and symmetric, so solution is efficient.
• Many mature commercial programs and a strong mathematical basis (energy principles) exist.

Steps in FEM

1. Discretisation: divide the body into finite elements joined at nodes (select element type and mesh).
2. Select interpolation (shape) functions: assume displacement within an element as $u = [N]\{d\}$.
3. Strain-displacement and stress-strain relations: $\{\varepsilon\} = [B]\{d\}$, $\{\sigma\} = [D]\{\varepsilon\}$.
4. Element stiffness matrix and load vector: $[k] = \int [B]^T[D][B]\,dV$ and equivalent nodal loads $\{f\}$.
5. Assembly: combine element matrices into the global equations $[K]\{U\} = \{F\}$.
6. Boundary conditions: apply supports (known displacements) and loads.
7. Solve for nodal displacements.
8. Post-processing: compute strains, stresses, reactions and plot results.