Implementation Details¶

This section describes the software architecture and numerical implementation of the BEM solver in Panl.

Core Architecture¶

The solver is divided into four main components:

Kernels (panelyze.analysis.kernels.BEMKernels): Implements the fundamental solutions from NASA CR-1934.
Geometry (panelyze.analysis.geometry): Simple boundary discretization into constant linear elements.
Solver (panelyze.analysis.solver.BEMSolver): Assembles the global matrices $H$ and $G$ and solves the system.

Matrix Assembly¶

The boundary is discretized into $M$ elements. For each element $i$, we collocate the boundary integral equation at its center. This leads to a system of equations:

\[[H] \{u\} = [G] \{t\}\]

where: - $\{u\}$ is the vector of boundary displacements. - $\{t\}$ is the vector of boundary tractions.

Singular Integrals¶

When the collocation point $i$ lies on the element $j$ being integrated, the kernels become singular ($1/r$ for $T$ and $\ln(r)$ for $U$). - The singular $G$ matrix components (integrals of $\ln z$) are handled analytically using the identity $\int \ln z dz = z(\ln z - 1)$. - The singular $H$ matrix components involve a “jump term” (typically 0.5) and the Cauchy Principal Value (CPV). Panelyze uses the Rigid-Body Motion trick to determine the diagonal terms of $H$:

\[H_{ii} = - \sum_{j \neq i} H_{ij}\]

This ensures that a rigid body translation results in zero tractions, which is a physical requirement of the system.

Numerical Integration¶

For non-singular elements, the kernels are integrated analytically using the endpoint complex coordinates $z_1$ and $z_2$. This approach is more robust than Gauss quadrature for elements very close to each other (e.g., near sharp cutouts).

Internal Point Evaluation¶

Displacements and stresses at interior points are computed using the Somigliana identity after the boundary values are fully solved.

\[u_i(\xi) = \sum_{j=1}^M \left( \int_{\Gamma_j} U_{ki} d\Gamma \right) t_k(x) - \sum_{j=1}^M \left( \int_{\Gamma_j} T_{ki} d\Gamma \right) u_k(x)\]

Note that in the implementation, Reciprocity is accounted for. The fundamental solution $U_{ki}(\xi, x)$ represents the displacement in direction $i$ at $x$ due to a load at $xi$. To get the displacement at $xi$, the kernel results are transposed.

Stress Calculation¶

Stresses at interior points are calculated by taking the derivative of the displacement identity. Direct differentiation of the kernels is used to avoid numerical differentiation errors.

\[\sigma_{ab}(\xi) = C_{abcd} \frac{\partial u_c(\xi)}{\partial x_d}\]