Implementing Boundary Element Method in MATLAB

Boundary Element Method (BEM) is a numerical technique for solving potential field problems, particularly suitable for infinite or semi-infinite domains. MATLAB serves as a powerful numerical computation tool commonly employed to implement BEM-related algorithms.

### Core Concept of Boundary Element Method BEM transforms governing equations into boundary integral equations, reducing the computational domain from entire regions to boundaries only. For potential field problems, integral equations are typically constructed using Green's functions or fundamental solutions, with boundary unknowns obtained through boundary discretization and linear system solutions.

### Key Implementation Steps in MATLAB Boundary Discretization: Divide boundaries into elements (linear/quadratic) described by nodes and shape functions using functions like `pdegeom` for geometry definition. Integral Equation Formulation: Establish integral equations using Green's functions, handling singular integrals through techniques like `quadgk` for adaptive quadrature. Matrix Assembly: Construct coefficient matrices from discretized equations, computing influence coefficients via vectorized operations like `meshgrid` for efficient node pairing. Boundary Condition Application: Adjust equations for Dirichlet/Neumann conditions using logical indexing (`A(boundaryNodes,:)=0`) and constraint enforcement. Equation Solving: Utilize MATLAB solvers (`\` operator for direct methods, `gmres` for iterative solutions) with preconditioning for dense matrices. Post-processing: Compute internal field values using `scatteredInterpolant` for potential reconstruction and `gradient` for field derivatives.

### Code Optimization Considerations Singular integrals require special handling through polar coordinate transformations (`pol2cart`) or analytical integration techniques. Dense matrices benefit from Fast Multipole Method (FMM) implementations (`fmm3d` libraries) for O(n) complexity. Vectorization with `arrayfun` or `bsxfun` avoids loops, while `sparse` storage manages memory for large-scale problems.

For practical implementation, reference established resources like BEMLIB, adapting discretization strategies (`delaunayTriangulation` for mesh generation) and integration methods (`integral2` for surface integrals) to specific problem requirements.