Steps and advantages of finite element method

The finite element method (FEM) is a numerical technique in which a continuous body is divided into a finite number of small pieces called elements, connected at nodes. The unknown field (displacement, temperature, etc.) is approximated over each element, and the element equations are assembled to give the equations of the whole body.

Steps of FEM

1. Discretisation. Divide the body into elements (bar, beam, triangle, quadrilateral, brick) and number the nodes and elements. Finer mesh is used where gradients are high, such as near holes and corners.
2. Select the displacement (interpolation) function. Assume a polynomial such as $u = a_1 + a_2 x$ for each element and express it in terms of nodal values using shape functions: $u = [N]\{d\}$.
3. Strain-displacement and stress-strain relations. Obtain $\{\varepsilon\} = [B]\{d\}$ and $\{\sigma\} = [D]\{\varepsilon\}$.
4. Derive the element stiffness matrix and load vector. Using minimum potential energy or the Galerkin method,

$$[k] = \int_V [B]^T [D] [B]\, dV$$

5. Assemble the global equations. Add the element matrices according to node connectivity to get $[K]\{D\} = \{F\}$.
6. Apply boundary conditions. Fix the supported degrees of freedom (remove rows and columns, or use the penalty method).
7. Solve the equations for nodal displacements.
8. Post-processing. Compute strains, stresses, reactions and principal values; check and plot results.

 Body -> Mesh -> Element eq. -> Assemble -> Apply BC
                                              |
 Stresses <- Strains <- Displacements <- Solve

Advantages

• Can handle complex geometry, irregular boundaries and different materials in one model.
• Can take any type of loading and boundary condition (point, distributed, thermal, dynamic).
• Mesh can be refined where needed, and the results improve as the mesh is refined; the same program can solve structural, thermal and fluid problems.