Overview

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.

(a) Mathematical modelling

A mathematical model is a set of equations that represents the important behaviour of a real physical system. The aim is to keep the features that affect the answer and to drop those that do not.

Steps:

1. Idealise the real structure: choose geometry, supports, material (linear elastic, isotropic) and loads (point, distributed, thermal).
2. Make assumptions, such as small deformation, plane stress, or a bar carrying only axial force.
3. Write the governing differential equation with boundary conditions. For an axially loaded bar:

$$\frac{d}{dx}\left(AE\frac{du}{dx}\right) + f(x) = 0,\quad u(0)=0,\ AE\frac{du}{dx}\Big|_{L} = P$$

4. Solve this model by an exact or numerical (FEM) method, then compare with the real system and refine the model if needed.

 Real system -> Idealisation -> Math model (ODE/PDE + BC)
        ^                              |
        |                              v
   Validation  <------  FEM solution (numerical)

The finite element solution is only as good as the model: errors can come from the idealisation and from the discretisation.

(b) Applications of FEM

• Structural and solid mechanics: stress analysis of beams, frames, trusses, plates, shells, pressure vessels, machine parts, bridges and buildings.
• Heat transfer: steady and transient conduction, fins, heat exchangers, thermal stresses.
• Fluid flow: seepage through dams, flow in pipes and ducts, potential flow.
• Dynamics: vibration, natural frequencies, buckling, crash analysis.
• Electromagnetics and coupled problems: motors, transformers, piezoelectric and thermo-mechanical problems.