2.11 The Finite Element Method for Two-Dimensional Diffusion

2.11.4 Construction of the Stiffness Matrix

Measurable Outcome 2.17, Measurable Outcome 2.20

The stiffness matrix arises in the calculation of \(\int _{\Omega } \nabla \phi _ i \cdot \left(k\nabla \tilde{T}\right)\, dA\). As in the one-dimensional case, the \(i\)-th row of the stiffness matrix \(K\) corresponds to the weighted residual of \(\phi _ i\). The \(j\)-th column in the \(i\)-th row corresponds to the dependence of the \(i\)-th weighted residual on \(a_ j\). Further drawing on the one-dimensional example, the weighted residuals are assembled by calculating the contribution to all of the residuals from within a single element. In the two-dimensional linear element situation, three weighted residuals are impacted by a given element, specifically, the weighted residuals corresponding to the nodal basis functions of the three nodes of the triangle. For example, in each element we must calculate

\[\int _{\Omega _ e} \nabla \phi _1 \cdot \left(k\nabla \tilde{T}\right)\, dA, \qquad \int _{\Omega _ e} \nabla \phi _2 \cdot \left(k\nabla \tilde{T}\right)\, dA, \qquad \int _{\Omega _ e} \nabla \phi _3 \cdot \left(k\nabla \tilde{T}\right)\, dA,\] (2.281)

where \(\Omega _ e\) is spatial domain for a specific element. As described in Section 2.11.3, the gradient of \(\tilde{T}\) can be written,

\[\nabla \tilde{T}(x,y)= \sum _{j=1}^{3} a_ j\nabla \phi _ j(x,y),\] (2.282)

thus the weighted residuals expand to,

\[\int _{\Omega } \nabla \phi _ i \cdot \left(k\nabla \tilde{T}\right)\, dA = \sum _{j=1}^{3} a_ j K_{i,j}, \qquad \mbox{where } \qquad K_{i,j} \equiv \int _{\Omega } \nabla \phi _ i \cdot \left(k\nabla \phi _ j\right)\, dA.\] (2.283)

For the situation in which \(k\) is constant and linear elements are used, then this reduces to

\[K_{i,j} \equiv k \nabla \phi _ i \cdot \nabla \phi _ j A_ e,\] (2.284)

where \(A_ e\) is the area of element \(e\).