# Optimization of Composite Structures with Thin Rigid Fibers Using Bioinspired Algorithms
## 4 Boundary Element Method for the Composite Including Stiff, Thin Fibers  
The external boundaries of the plate and the fibers are split into boundary elements in order to obtain the numerical solution by BEM. Nodal values and quadratic shape functions are used to interpolate the displacements, traction, and coordinates of external boundaries and the traction operating on the fibers. Because the fibers are straight and stiff, the displacements and coordinates are not interpolated. The relations between nodal displacements and traction are established by substituting the interpolated displacements, traction, and coordinates into the boundary integral [47]. Nodes along the outside edges of the plate and the fibers are known as collocation points.  
Fiber node displacements are computed analytically [47]. The following formulas for every fiber node can be expressed in matrix form:  
\[\mathbf{u}=\mathbf{I}\mathbf{u_{f}} \tag{10}\]  
where the matrix \(\mathbf{I}\) depends on the coordinates of nodes, the matrix \(\mathbf{u}\) contains the displacements of fiber nodes, and the matrix \(\mathbf{u_{f}}\) contains the displacements of fiber tips and the angles of fiber rotations.  
The following matrix form can be used to express the equilibrium equations for traction acting on the fibers:  
\[\mathbf{E_{f}}=0 \tag{11}\]  
where the nodal values of the traction operating on the fibers are contained in the matrix \(\mathbf{t}_{f}\), and the matrix \(\mathbf{E}\) is dependent on the coordinates of the fiber nodes. Analytical integration of the formulas given in the paper [47] yields the matrix \(\mathbf{E}\).  
The conventional BEM matrix equation is produced by numerically computing the boundary integrals using the Gauss quadrature. Equation (10) represents the collocation fiber node displacements. When the equilibrium is included in the system of equations (11), the matrix equation that results is as follows:  
\[\left[\begin{array}{cc}\mathrm{H_{ee}}&0\\ \mathrm{H_{fe}}&\mathrm{I}\\ 0&0\end{array}\right]\left[\begin{array}{cc}\mathrm{u_{e}}\\ \mathrm{u_{f}}\end{array}\right]=\left[\begin{array}{cc}\mathrm{G_{ee}}& \mathrm{G_{ef}}\\ \mathrm{G_{fe}}&\mathrm{G_{ff}}\\ 0&\mathrm{E}\end{array}\right]\left[\begin{array}{cc}\mathrm{t_{e}}\\ \mathrm{t_{f}}\end{array}\right] \tag{12}\]  
where the submatrices with the index \(\mathbf{e}\) denote the external boundaries and the submatrices with the index \(\mathbf{f}\) denote the fibers. The submatrices \(\mathbf{H}\) and \(\mathbf{G}\) are the standard BEM matrices, which depend on the boundary integrals of fundamental solutions, shape functions, and Jacobians [47]. There is rearrangement of the matrix Equation (12). The equation's left side should contain the unknown variables, while the right side should contain the known boundary conditions. The appropriate column and the unknown fiber traction \(\mathbf{t_{f}}\) are rearranged as follows:  
\[\left[\begin{array}{cc}\mathrm{H_{ee}}&-\mathrm{G_{ef}}&0\\ \mathrm{H_{fe}}&-\mathrm{G_{ee}}&\mathrm{I}\\ 0&\mathrm{E}&0\end{array}\right]\left[\begin{array}{cc}\mathrm{u_{e}}\\ \mathrm{t_{f}}\\ \mathrm{u_{f}}\end{array}\right]=\left[\begin{array}{cc}\mathrm{G_{ee}}\\ \mathrm{G_{fe}}\\ 0\end{array}\right]\left[\begin{array}{cc}\mathrm{t_{e}}\\ \mathrm{t_{e}}\end{array}\right] \tag{13}\]  
In the case of the mixed boundary conditions on the external boundaries, the matrices \(\mathbf{u_{e}}\) and \(\mathbf{t_{e}}\) and the corresponding columns of matrices \(\mathbf{H}\) and \(\mathbf{G}\) are additionally rearranged, as in standard BEM [47]. The fiber displacements and traction, as well as the unknown nodal displacements and traction at the external boundaries, are the direct solutions of the matrix equation. With the current method, displacements, strains, and stresses may be calculated at any location in the matrix.