---
title: "Numerical solution techniques, advantages and suitability"
subject: "Computational Techniques in Civil Engineering"
chapter: 1
key: "semester_8/computational_techniques_in_civil_engineering:numerical-solution-techniques-compare"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/numerical-solution-techniques-compare
site: BCT Engineering
---

# Numerical solution techniques, advantages and suitability

Computational Techniques in Civil Engineering · Chapter 1: Introduction

- Asked in: 2079 Shrawan, 4 marks; 2073 Magh, 4+4 marks; 2072 Asoj, 8 marks; 2070 Bhadra, 8 marks (asked 4 times)
- 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/numerical-solution-techniques-compare
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-1

## Question

Describe the different numerical solution techniques used in solving civil engineering problems (with the help of mechanics), giving their advantages, disadvantages and suitability.

## Short answer

FEM divides the whole domain into elements and suits complex geometry and materials but struggles with large deformation. FDM uses regular grids, is simple, but poor for irregular boundaries. BEM discretises only the boundary, good for infinite domains, weak for nonlinearity. DEM models particles or blocks for discontinuities but is computationally costly.

## Answer

Civil engineering problems (stress analysis, seepage, flow in channels, soil-structure interaction) are governed by differential equations that rarely have closed-form solutions for real geometry and loading. Numerical techniques replace the continuous problem by a set of algebraic equations. The main techniques are FEM, FDM, BEM, DEM and mesh-free methods (SPH).

#### 1. Finite Element Method (FEM)
The body is divided into elements joined at nodes; displacement inside an element is interpolated by shape functions and the equations $[K]\{U\}=\{F\}$ are formed from energy or virtual work (e.g. a beam with $[k]$ from Hermite functions).
- **Advantages:** handles irregular geometry, mixed materials, non-linearity, any boundary condition; systematic and general; large software base.
- **Disadvantages:** whole domain must be meshed; large equations; poor for infinite domains, cracks and large deformation (mesh distortion).
- **Suitability:** structures (frames, trusses, plates, shells), foundations, dams, soil stress analysis.

#### 2. Finite Difference Method (FDM)
Derivatives are replaced by differences between grid-point values, e.g. $\dfrac{\partial u}{\partial x}\approx\dfrac{u_{i+1}-u_{i-1}}{2\Delta x}$.
- **Advantages:** simple concept and programming; efficient on regular grids; explicit and implicit time stepping.
- **Disadvantages:** difficult for curved or irregular boundaries; grid refinement is global; stability limits on $\Delta t$.
- **Suitability:** groundwater flow, seepage, open-channel and pipe flow, heat flow, flow nets.

#### 3. Boundary Element Method (BEM)
The governing PDE is converted into an integral equation on the boundary only; only the boundary is discretised.
- **Advantages:** reduces dimension by one; small data; good for infinite or semi-infinite domains and stress concentration.
- **Disadvantages:** full, unsymmetric matrices; needs a fundamental solution; poor for non-linear and heterogeneous media.
- **Suitability:** soil-structure interaction, tunnels, fracture, potential flow.

#### 4. Discrete Element Method (DEM)
Material is modelled as an assembly of separate blocks or particles; contact forces are found from force-displacement laws and Newton's second law is integrated in time.
- **Advantages:** models large displacement, sliding and separation naturally.
- **Disadvantages:** costly; contact parameters hard to calibrate.
- **Suitability:** granular soils, jointed rock, masonry, rockfall.

#### Comparison

| Method | Discretises | Strength | Weakness |
|---|---|---|---|
| FEM | Whole domain (elements) | Complex geometry, materials | Large deformation |
| FDM | Whole domain (grid) | Simplicity | Irregular boundary |
| BEM | Boundary only | Infinite domains | Non-linear, non-homogeneous |
| DEM | Particles/blocks | Discontinua | Computational cost |

FEM is the most widely used because it suits both solids and fluids and complex geometry.

## Check yourself

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

1. Which part of the domain is discretised in the boundary element method?
   A. The whole domain with a grid
   B. Only the boundary
   C. Only the interior nodes
   D. Individual particles or blocks
   Answer: B. BEM converts the PDE into an integral equation on the boundary, so only the boundary is divided into elements. (Key point: BEM discretisation)

2. Why is the finite difference method difficult to apply to some problems?
   A. Irregular or curved boundaries do not fit a regular grid
   B. It cannot handle time-dependent problems
   C. It needs a fundamental solution
   D. It cannot model flow problems
   Answer: A. FDM works best on regular grids; curved or irregular boundaries are hard, and grid refinement is global. (Key point: FDM weakness)

3. Which problem is best suited to the discrete element method?
   A. Stress in a plate with a hole
   B. Seepage under a weir on a grid
   C. Potential flow in an infinite domain
   D. Granular soil or jointed rock with sliding and separation
   Answer: D. DEM models separate blocks or particles with contact laws, so it handles discontinua, sliding and separation. (Key point: DEM suitability)

4. What is a main disadvantage of FEM?
   A. It cannot handle mixed materials
   B. It works only on regular grids
   C. Mesh distortion under large deformation and meshing of the whole domain
   D. It produces full unsymmetric matrices from boundary integrals
   Answer: C. FEM meshes the whole domain and its elements distort under large deformation, though it handles materials and geometry well. (Key point: FEM limitation)

## Related in Computational Techniques in Civil Engineering

- [Techniques of solving problems; FEM and FDM](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afem-fdm-working-techniques) — Computational Techniques in Civil Engineering, chapter 1
- [Linear solution techniques; CG two iterations](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asolution-techniques-cg-numerical-two-iter) — Computational Techniques in Civil Engineering, chapter 2
- [Linear solution techniques; CG algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asolution-techniques-cg-algorithm) — Computational Techniques in Civil Engineering, chapter 2
- [Solution methods of linear equations with examples](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asolution-methods-linear-examples) — Computational Techniques in Civil Engineering, chapter 2
- [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
- [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

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