---
title: "Techniques of solving problems; FEM and FDM"
subject: "Computational Techniques in Civil Engineering"
chapter: 1
key: "semester_8/computational_techniques_in_civil_engineering:fem-fdm-working-techniques"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/fem-fdm-working-techniques
site: BCT Engineering
---

# Techniques of solving problems; FEM and FDM

Computational Techniques in Civil Engineering · Chapter 1: Introduction

- Asked in: 2079 Jestha, 2+6 marks
- Video: [Understanding the Finite Element Method](https://www.youtube.com/watch?v=GHjopp47vvQ) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/fem-fdm-working-techniques
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-1

## Question

List the different techniques of solving civil engineering problems. Explain how the finite element method and finite difference method work to solve the problem.

## Short answer

FEM discretises the body into elements, assumes displacement via shape functions, forms element stiffness matrices, assembles global equations, applies boundary conditions and solves for displacements. FDM replaces derivatives in the governing differential equation by finite differences on a regular grid, giving algebraic equations at grid points, then solves them.

## Answer

#### Techniques of solving civil engineering problems
1. Analytical (exact, closed-form) methods
2. Numerical methods: finite element (FEM), finite difference (FDM), boundary element (BEM), discrete element (DEM), finite volume, mesh-free (SPH)
3. Experimental and model studies (laboratory and physical models)
4. Empirical and semi-empirical (code-based) methods

#### Finite element method
The body is divided into small finite elements connected at nodes. Inside each element the unknown (displacement $u$) is assumed as
$$
u = [N]\{d\}
$$
where $[N]$ are shape functions and $\{d\}$ are nodal values. Strain is $\{\varepsilon\}=[B]\{d\}$ and stress is $\{\sigma\}=[D]\{\varepsilon\}$. Using minimum potential energy or virtual work, the element stiffness is
$$
[k]=\int [B]^T[D][B]\,dV
$$
Element matrices are assembled into $[K]\{U\}=\{F\}$; supports are applied; the equations are solved for nodal displacements; strains and stresses then follow from $[B]$ and $[D]$.

#### Finite difference method
The domain is covered with a regular grid of points with spacing $\Delta x$, $\Delta y$. Derivatives in the governing differential equation are replaced by differences of nodal values, for example
$$
\frac{d^2u}{dx^2}\approx\frac{u_{i-1}-2u_i+u_{i+1}}{\Delta x^2}
$$
Writing this at every grid point gives one algebraic equation per point. Boundary values are inserted, and the simultaneous equations are solved (directly or iteratively). For time-dependent problems the time derivative is also replaced by a difference, using an explicit or implicit scheme.

```text
 FEM: elements + nodes      FDM: grid points
 o----o----o                 o  o  o  o
 | \  |  / |                 o  o  o  o
 o----o----o                 o  o  o  o
```

## Check yourself

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

1. In FEM, how is the unknown displacement inside an element expressed?
   A. u = [B]^T[D][B]
   B. u = [N]{d}, shape functions times nodal values
   C. As differences of grid values
   D. By a fundamental solution on the boundary
   Answer: B. Displacement within an element is interpolated from nodal values using shape functions [N]. (Key point: FEM interpolation)

2. Which expression gives the element stiffness matrix in FEM?
   A. [k] = integral of [B]^T[D][B] dV
   B. [k] = integral of [N]^T[D][N] dV
   C. [k] = [D][B]{d}
   D. [k] = (u(i-1) - 2u(i) + u(i+1))/dx^2
   Answer: A. Using minimum potential energy or virtual work, [k] = integral [B]^T [D] [B] dV. (Key point: Element stiffness)

3. How does FDM turn a differential equation into algebraic equations?
   A. Divide the body into elements and assemble stiffness
   B. Convert it to a boundary integral
   C. Integrate Newton's law for particles
   D. Replace derivatives by differences of values at grid points and write an equation at each point
   Answer: D. FDM replaces derivatives with differences on a grid, giving one algebraic equation per grid point. (Key point: FDM approach)

## Related in Computational Techniques in Civil Engineering

- [Concepts of FEM and FDM with advantages](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afem-fdm-concepts-advantages) — Computational Techniques in Civil Engineering, chapter 1
- [Software used for FEM and FDM](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asoftware-fem-fdm) — Computational Techniques in Civil Engineering, chapter 1
- [Necessity of computational techniques; FEM algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Anecessity-computational-fem-algorithm) — Computational Techniques in Civil Engineering, chapter 1
- [Types of problems solved by FEM](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afem-problem-types) — Computational Techniques in Civil Engineering, chapter 1
- [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
- [Finite difference method; explicit and implicit schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afdm-explicit-implicit) — Computational Techniques in Civil Engineering, chapter 5, asked 2×

## Related in other subjects

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