In this lecture, we'll cover the following:

• Transforming the element stiffness matrix integral from physical to natural coordinates using the Jacobian matrix.
• Applying Gauss quadrature to evaluate the double integral.
• Understanding sampling points and weighting in two-dimensional integration.
• Extending the same numerical integration approach to compute equivalent nodal force vectors.

In this lecture, we focus on how to adapt the element stiffness matrix formulation so it can be evaluated efficiently in natural coordinates. We begin by recognising that while the strain–displacement matrix depends on the natural coordinates $(r, s)$, the area element $\mathrm{d}A$ is defined in physical space. To resolve this mismatch, we derive an expression for $\mathrm{d}A$ in terms of $(r, s)$ using the determinant of the Jacobian matrix. This allows us to rewrite the entire stiffness matrix integral consistently in the natural coordinate system.

We then apply Gauss quadrature to approximate the resulting double integral. By sampling the integrand at specific points in the $(r, s)$ domain and applying appropriate weights, we convert the integration into a double summation. For a two-point Gauss scheme in each direction, this leads to four sampling points across the element. We evaluate the strain–displacement matrix, constitutive matrix, and Jacobian determinant at each of these points and sum their contributions. The same procedure is also extended to compute equivalent nodal force vectors for distributed loads.

Overall, we see that what initially appears to be a complex integral becomes a structured and computationally manageable process through coordinate transformation and numerical integration. This sets the stage for implementing these ideas numerically and explicitly constructing the element stiffness matrix.

Next up

With the integration framework in place, in the next lecture we begin the hands-on implementation by setting up our Python environment and defining the inputs for a stiffness matrix calculation.