FEM steps and geometric modelling

(a) Steps of the finite element method

The finite element method (FEM) divides a body into small elements joined at nodes, solves a simple equation on each element and assembles them to get the behaviour of the whole body.

1. Preprocessing: create the geometry, choose element type (bar, beam, triangle, quadrilateral, tetrahedron, brick), assign material properties ($E$, $\nu$) and section data.
2. Discretisation (meshing): divide the body into elements. Use a finer mesh where stress changes quickly (holes, fillets).
3. Element equations: for each element write $[k]^e\{u\}^e = \{f\}^e$ using shape functions. For a bar element: $[k]=\dfrac{AE}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$.
4. Assembly: combine the element matrices into the global equation $[K]\{U\}=\{F\}$.
5. Boundary conditions and loads: fix the supports and apply forces.
6. Solution: solve for nodal displacements $\{U\}$.
7. Postprocessing: calculate strains, stresses and reactions, plot contours, and check convergence by refining the mesh.

  Geometry -> Mesh -> Element k -> Assemble K
     -> Apply loads/supports -> Solve U
     -> Stress, strain -> Check, refine mesh

(b) Wireframe, surface and solid models