Mathematical modelling and applications of FEM

(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.