---
title: "Mathematical modelling and applications of FEM"
subject: "Finite Element Method"
chapter: 1
key: "semester_8/finite_element_method:mathematical-modeling-fem-applications"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/mathematical-modeling-fem-applications
site: BCT Engineering
---

# Mathematical modelling and applications of FEM

Finite Element Method · Chapter 1: Overview

- Asked in: year not given, 5 marks
- Video: [Lect01: Finite Element Method](https://www.youtube.com/watch?v=RM-c-8xDwZk) — Dr. A. S. Sayyad
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/mathematical-modeling-fem-applications
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-1

## Question

Write short notes on: (a) mathematical modelling of a physical system, and (b) applications of the finite element method.

## Short answer

(a) Mathematical modelling idealizes a physical system, retaining key features, leading to governing differential equations with boundary conditions; the FEM solution is then validated. (b) Applications include structural stress analysis, heat transfer, fluid flow, vibration, and coupled electromagnetics/thermo-mechanical problems.

## Answer

#### (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.

```text
 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the aim when forming a mathematical model of a physical system?
   A. Keep the features that affect the answer and drop those that do not
   B. Include every detail of the real structure
   C. Remove the boundary conditions
   D. Replace the structure by a single number
   Answer: A. A model represents the important behaviour of the real system; irrelevant features are dropped. (Key point: Aim of modelling)

2. For an axially loaded bar the model is d/dx(AE du/dx) + f = 0 with boundary conditions. Which pair is correct?
   A. u(0) = P and AE du/dx at L = 0
   B. u(0) = 0 and AE du/dx at L = P
   C. u(L) = 0 and u(0) = P
   D. Only a force condition at x = 0
   Answer: B. The fixed end gives the displacement condition u(0) = 0, and the loaded end gives the force condition AE u'(L) = P. (Key point: Governing equation)

3. After solving the model, what is the correct next step?
   A. Stop, because the FEM answer is always correct
   B. Delete the assumptions
   C. Compare with the real system and refine the model if needed
   D. Increase the loads
   Answer: C. The loop ends with validation against the real system, refining the model if needed. (Key point: Validation)

4. Where can errors in a finite element solution come from?
   A. Only from the computer
   B. Only from the discretisation
   C. Neither, if the program is correct
   D. From both the idealisation and the discretisation
   Answer: D. The solution is only as good as the model, so errors arise from idealisation and from discretisation. (Key point: Sources of error)

## Related in Finite Element Method

- [Steps and advantages of finite element method](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Afem-steps-advantages) — Finite Element Method, chapter 1
- [Stepped bar under axial loads by FEM](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Astepped-bar-displacement-stress) — Finite Element Method, chapter 3
- [Composite wall conduction using finite elements](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Acomposite-wall-heat-conduction) — Finite Element Method, chapter 3
- [Derive bar element stiffness matrix](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Abar-element-stiffness-derivation) — Finite Element Method, chapter 3
- [Derive beam element stiffness matrix](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Abeam-element-stiffness-derivation) — Finite Element Method, chapter 3
- [Truss, beam and frame element differences](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Atruss-beam-frame-element-differences) — Finite Element Method, chapter 3

## Related in other subjects

- [FEM steps and geometric modelling](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Afem-steps-geometric-modelling) — Machine Design II, chapter 1
- [Physical vs mathematical models](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aphysical-vs-mathematical-model) — Simulation and Modeling, chapter 2
- [Basic steps in finite element analysis](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afem-basic-steps) — Computational Techniques in Civil Engineering, chapter 1
- [Analog filter applications and element scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aanalog-filter-applications-scaling) — Filter Design, chapter 1