In this lecture, we'll cover the following:

• Introduction to numerical integration using Gauss–Legendre (Gauss quadrature)
• Evaluating integrals in finite element formulations that are difficult to solve analytically
• Core idea of approximating integrals via weighted sums of function values at selected sampling points
• Minimising integration error by choosing optimal weights and sample locations
• General formulation of Gauss quadrature and its role in stiffness matrix computation

In this lecture, we introduce Gauss quadrature as a practical method for numerical integration, particularly in the context of finite element analysis where analytical integration is often infeasible. We explore the fundamental idea of approximating an integral by evaluating a function at selected sampling points, applying appropriate weightings, and summing the results. This is framed through the need to integrate expressions such as those arising in stiffness matrix formulation.

We then focus on how Gauss quadrature achieves high accuracy by approximating the integrand as a polynomial and carefully selecting sampling points and weights to minimise error. Through a worked example with two sampling points, we see how these values are determined and how they relate to the degree of the approximating polynomial. We also outline the general form of the quadrature rule and briefly discuss how the number of sampling points affects accuracy. This method is foundational knowledge that will be applied to compute element stiffness matrices in subsequent lectures.

Next up

In the next lecture, we will apply this numerical integration technique directly to the plate element, transforming the stiffness matrix integral into a form that can be computed in code.