
About
In the Finite Element Method (FEM), we often need to compute integrals over each element, for example when assembling the stiffness matrix or computing internal forces. However, these integrals usually can’t be solved analytically, especially when the shape functions are complex or the material properties vary.
That’s where Gauss–Legendre integration comes in. It’s a numerical integration method that lets us approximate these integrals efficiently and accurately. Instead of integrating over the entire element, we evaluate the function at a few special points (called Gauss points) and multiply by weights.
These Gauss points and weights are carefully chosen to make the approximation very accurate, especially for polynomial functions, which are common in FEM.
Gauss–Legendre Integration
Gauss–Legendre integration approximates the integral of a function f(x) over the interval [-1, 1] using a weighted sum of function values at specific points (Gauss points):
Where:
are the Gauss points are the corresponding weights n is the number of integration points (e.g. 2-point, 3-point quadrature)
This method is exact for polynomials of degree or less.
In actual computation in FEA, the Jacobean J is multiplied to above formula since coordinate conversion is applied.
Example for Stiffness Matrix
In linear elasticity, the element stiffness matrix \mathbf{K}^e is computed by integrating over the element volume:
Where:
• is the strain–displacement matrix
• is the material stiffness matrix
• is the domain of the element
After mapping to the reference element and using Gauss integration, the formula becomes: