---
title: "Steps and advantages of finite element method"
subject: "Finite Element Method"
chapter: 1
key: "semester_8/finite_element_method:fem-steps-advantages"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/fem-steps-advantages
site: BCT Engineering
---

# Steps and advantages of finite element method

Finite Element Method · Chapter 1: Overview

- Asked in: year not given, 5+3 marks
- Video: [Basic Steps in FEA | Finite Element Analysis - 8 Steps | E3](https://www.youtube.com/watch?v=_ZBSdLZ53T0) — Msquare Analysis Projects
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/fem-steps-advantages
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-1

## Question

Explain the general steps involved in the finite element method for solving a structural problem. State any three advantages of the method.

## Short answer

The method involves discretization, selection of interpolation functions, derivation of element stiffness matrices, assembly of global equations, application of boundary conditions, solution for displacements, and post-processing. Three advantages: it handles complex geometry and varied materials, accepts any loading/boundary condition, and refines mesh where needed for improved accuracy, with a single code for multiple physics.

## Answer

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.

```text
 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.

## Check yourself

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

1. In the first FEM step, how should the mesh be chosen?
   A. Finer where gradients are high, such as near holes and corners
   B. Uniformly fine everywhere, regardless of the stress pattern
   C. Coarser near holes and corners to save time
   D. Only one element for the whole body
   Answer: A. The body is divided into elements joined at nodes, and the mesh is made finer where stresses or fields change quickly, such as near holes and corners. (Key point: Discretisation)

2. Which expression gives the element stiffness matrix in terms of [B] and [D]?
   A. [k] = integral of [B][D]^T dV
   B. [k] = integral of [B]^T[D][B] dV
   C. [k] = integral of [D]^T[B] dV
   D. [k] = integral of [B][D][B]^T dV
   Answer: B. Using minimum potential energy or Galerkin, the element stiffness is [k] = integral of [B]^T [D] [B] dV. (Key point: Element stiffness matrix)

3. After the global equations [K]{D} = {F} are assembled, what must be done before solving?
   A. Compute the stresses
   B. Select the displacement function
   C. Apply boundary conditions to the supported degrees of freedom
   D. Refine the mesh
   Answer: C. Supported degrees of freedom are fixed (rows and columns removed, or the penalty method) so the system is not singular. Stresses come only after solving. (Key point: Order of steps)

4. What is the role of the shape functions in the second step?
   A. They give the loads at the nodes
   B. They combine the element matrices into the global matrix
   C. They apply the support conditions
   D. They express the assumed field over an element in terms of nodal values, u = [N]{d}
   Answer: D. A polynomial is assumed for each element and written in terms of nodal values using shape functions, u = [N]{d}. (Key point: Interpolation function)

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