Load vector 2D

Now we compute the load vector for the two-dimensional case. The load vector is related to the integral of the product of the source function and the trial function. Consider the integral for a single triangle with vertices $(x_i, y_i), (x_{i+1}, y_{i+1}), (x_{i+2}, y_{i+2})$

The trial function on the triangle has the form $\upsilon_{(i)(i+2)}(x, y)$ $= q_i \cdot \phi_i(x, y)$ $+ q_{i+1} \cdot \phi_{i+1}(x, y)$ $+ q_{i+2} \cdot \phi_{i+2}(x, y)$. To evaluate the integral, the source function $f(x, y)$ must be interpolated on the triangle $\triangle$ by a linear function. We represent $f(x, y)$ in the form $\widetilde{f}(x, y)$ $= e_1$ $+ e_2 \cdot x$ $+ e_3 \cdot y$. The coefficients $e_1, e_2, e_3$ are determined from the interpolation conditions at the nodes

Since the hat functions $\phi_n$ equal one at their own node and zero at the others, the linear interpolant of the source function, expressed through them, takes the form

where $f_i, f_{i+1}, f_{i+2}$ are the values of the source function at the vertices of the triangle.

Substitute the interpolated source function and the trial function into (6.40$$\int_{\triangle} f(x, y) \cdot \upsilon \,dS.$$). We take into account (6.25$\displaystyle \phi_i(x, y)$ $\displaystyle = a_i$ $\displaystyle + b_i \cdot x$ $\displaystyle + c_i \cdot y$$\displaystyle \phi_{i+1}(x, y)$ $\displaystyle = a_{i+1}$ $\displaystyle + b_{i+1} \cdot x$ $\displaystyle + c_{i+1} \cdot y$$\displaystyle \phi_{i+2}(x, y)$ $\displaystyle = a_{i+2}$ $\displaystyle + b_{i+2} \cdot x$ $\displaystyle + c_{i+2} \cdot y$) from the section on hat functions

where the indices $m$ and $n$ run over the vertices of the triangle $\{i, i$ $+1, i$ $+2\}$.

We use the relations (6.26$\displaystyle \int_{\triangle} \phi_m \cdot \phi_n \,dS$ $\displaystyle = \begin{cases} \frac{\displaystyle S_{\triangle}}{\displaystyle 6}, & m = n\\ \frac{\displaystyle S_{\triangle}}{\displaystyle 12}, & m \neq n \end{cases}$) for the integrals of products of hat functions, derived when computing the damping matrix, where $S_{\triangle}$ is the area of the triangle, which is computed by formula (6.9$\displaystyle S_{\triangle}$ $\displaystyle = \frac{\displaystyle |d|}{\displaystyle 2},$). After collecting like terms we obtain

We introduce the notation for the elements of the local load vector of the triangle

Thus, the local load vector for a two-dimensional triangular element has the form

Note that the resulting load vector coincides with the product of the local damping matrix (6.28$$\begin{aligned}\mathbf{C}_{\triangle} = \begin{bmatrix} c_{(i)(i)} & c_{(i)(i+1)} & c_{(i)(i+2)}\\ c_{(i+1)(i)} & c_{(i+1)(i+1)} & c_{(i+1)(i+2)}\\ c_{(i+2)(i)} & c_{(i+2)(i+1)} & c_{(i+2)(i+2)} \end{bmatrix} = \frac{\displaystyle S_{\triangle}}{\displaystyle 12} \begin{bmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{bmatrix}.\end{aligned}$$) and the vector of nodal source values, that is $\mathbf{R}_{\triangle}$ $= \mathbf{C}_{\triangle} \cdot \mathbf{f}$.

The global load vector $\mathbf{R}$ is obtained by summing the contributions from all triangular elements of the mesh using the assembly method: the elements of the local vectors are added to the corresponding elements of the global vector according to the global node numbering. The dimension of the global load vector equals $N$, where $N$ is the total number of nodes in the mesh. It is important to note that for interior nodes the contributions from all adjacent triangular elements containing the given node are summed. For boundary nodes the element contributions are assembled in the same way, but the corresponding components of the system may be modified when boundary conditions are imposed.