Load vector 3D

Now we compute the load vector for the three-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 tetrahedron with vertices

The trial function on the tetrahedron has the form $\upsilon_{(i)(i+3)}(x, y, z)$ $= q_i \cdot \phi_i(x, y, z)$ $+ q_{i+1} \cdot \phi_{i+1}(x, y, z)$ $+ q_{i+2} \cdot \phi_{i+2}(x, y, z)$ $+ q_{i+3} \cdot \phi_{i+3}(x, y, z)$. To evaluate the integral, the source function $f(x, y, z)$ must be interpolated on the tetrahedron by a linear function. Let us represent $f(x, y, z)$ in the form $\widetilde{f}(x, y, z)$ $= e_1$ $+ e_2 \cdot x$ $+ e_3 \cdot y$ $+ e_4 \cdot z$. The coefficients $e_1, e_2, e_3, e_4$ 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 $f_i, f_{i+1}, f_{i+2}, f_{i+3}$ are the values of the source function at the vertices of the tetrahedron.

Substitute the interpolated source function and the trial function into (6.45$$\int_{\text{тет}} f(x, y, z) \cdot \upsilon \,dV.$$). We take into account (6.30$\displaystyle \phi_i(x, y, z)$ $\displaystyle = a_i$ $\displaystyle + b_i \cdot x$ $\displaystyle + c_i \cdot y$ $\displaystyle + d_i \cdot z$$\displaystyle \phi_{i+1}(x, y, z)$ $\displaystyle = a_{i+1}$ $\displaystyle + b_{i+1} \cdot x$ $\displaystyle + c_{i+1} \cdot y$ $\displaystyle + d_{i+1} \cdot z$$\displaystyle \phi_{i+2}(x, y, z)$ $\displaystyle = a_{i+2}$ $\displaystyle + b_{i+2} \cdot x$ $\displaystyle + c_{i+2} \cdot y$ $\displaystyle + d_{i+2} \cdot z$$\displaystyle \phi_{i+3}(x, y, z)$ $\displaystyle = a_{i+3}$ $\displaystyle + b_{i+3} \cdot x$ $\displaystyle + c_{i+3} \cdot y$ $\displaystyle + d_{i+3} \cdot z$) from the section on hat functions

where the indices $m$ and $n$ run over the vertices of the tetrahedron $\{i, i$ $+1, i$ $+2, i$ $+3\}$.

We use the relations (6.31$\displaystyle \int_{\text{тет}} \phi_m \cdot \phi_n \,dV$ $\displaystyle = \begin{cases} \frac{\displaystyle V_{\text{тет}}}{\displaystyle 10}, & m = n\\ \frac{\displaystyle V_{\text{тет}}}{\displaystyle 20}, & m \neq n \end{cases}$) for the integrals of products of hat functions, derived when computing the damping matrix, where $V_{\text{tet}}$ is the volume of the tetrahedron, which is computed by formula (6.15$\displaystyle V_{\text{тет}}$ $\displaystyle = \frac{\displaystyle |\Delta|}{\displaystyle 6},$). After collecting like terms we obtain

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

Thus, the local load vector for a three-dimensional tetrahedral element has the form

Note that the resulting load vector coincides with the product of the local damping matrix (6.33$$\begin{aligned}\mathbf{C}_{\text{тет}} = \begin{bmatrix} c_{(i)(i)} & c_{(i)(i+1)} & c_{(i)(i+2)} & c_{(i)(i+3)}\\ c_{(i+1)(i)} & c_{(i+1)(i+1)} & c_{(i+1)(i+2)} & c_{(i+1)(i+3)}\\ c_{(i+2)(i)} & c_{(i+2)(i+1)} & c_{(i+2)(i+2)} & c_{(i+2)(i+3)}\\ c_{(i+3)(i)} & c_{(i+3)(i+1)} & c_{(i+3)(i+2)} & c_{(i+3)(i+3)} \end{bmatrix} = \frac{\displaystyle V_{\text{тет}}}{\displaystyle 20} \begin{bmatrix} 2 & 1 & 1 & 1\\ 1 & 2 & 1 & 1\\ 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 2 \end{bmatrix}.\end{aligned}$$) and the vector of nodal source values, that is $\mathbf{R}_{\text{tet}}$ $= \mathbf{C}_{\text{tet}} \cdot \mathbf{f}$.

The global load vector $\mathbf{R}$ is obtained by summing the contributions from all tetrahedral 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 tetrahedral 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.