Numerical solution techniques, advantages and suitability

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

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