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Techniques of solving problems; FEM and FDM

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.

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

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 uu) is assumed as

u=[N]{d}u = [N]\{d\}

where [N][N] are shape functions and {d}\{d\} are nodal values. Strain is {ε}=[B]{d}\{\varepsilon\}=[B]\{d\} and stress is {σ}=[D]{ε}\{\sigma\}=[D]\{\varepsilon\}. Using minimum potential energy or virtual work, the element stiffness is

[k]=∫[B]T[D][B] dV[k]=\int [B]^T[D][B]\,dV

Element matrices are assembled into [K]{U}={F}[K]\{U\}=\{F\}; supports are applied; the equations are solved for nodal displacements; strains and stresses then follow from [B][B] and [D][D].

Finite difference method

The domain is covered with a regular grid of points with spacing Δx\Delta x, Δy\Delta y. Derivatives in the governing differential equation are replaced by differences of nodal values, for example

d2udx2≈ui−1−2ui+ui+1Δx2\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.

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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