Classical Boundary Element Method Algorithms

The Boundary Element Method (BEM) is an efficient numerical computation technique widely applied to various engineering and physics problems, including electric potential distribution, heat conduction, and elasticity mechanics. By transforming problems into boundary integral equations, BEM significantly reduces computational complexity, making it particularly suitable for infinite or semi-infinite domain problems.

Implementing BEM in MATLAB typically involves these critical steps:

Boundary Discretization: The problem domain's boundary is divided into small elements, typically using linear or quadratic elements. Each element is defined by nodes where unknown quantities (such as potential or displacement) become the solution targets. In MATLAB, this can be implemented using geometric decomposition functions and node coordinate management.

Green's Function Selection: Appropriate Green's functions are chosen based on the physical nature of the problem. For instance, electric potential problems typically use Laplace equation Green's functions, while elasticity mechanics employs Navier equation Green's functions. MATLAB's symbolic toolbox can assist in deriving and verifying these functions.

Integral Equation Formulation: Based on Green's theorem, differential equations are converted into boundary integral equations. For potential problems, this generally involves single-layer and double-layer potential integrals. MATLAB's integral functions can handle these computations with proper singularity treatment.

Coefficient Matrix Assembly: The discretized integral equations form linear systems. MATLAB's matrix operations efficiently handle this process through vectorized computations and sparse matrix optimization.

Equation System Solution: MATLAB's built-in linear algebra solvers (e.g., backslash operator or iterative methods) process the resulting algebraic systems. For large systems, preconditioned iterative methods like GMRES are often employed.

Result Post-processing: Solutions at domain interior points (such as potential values) are computed and visualized. MATLAB's advanced plotting capabilities enable intuitive result presentation through contour plots and 3D visualizations.

BEM's primary advantage lies in its dimension-reduction characteristic: 3D problems transform into 2D boundary integrals, while 2D problems become 1D integrals. This makes it more efficient than Finite Element Method for large spatial domains. However, the method has limitations when handling nonlinear or heterogeneous material problems.

Key implementation challenges in MATLAB include singular integral treatment and coefficient matrix storage optimization. Established BEM libraries like BEMLIB provide excellent references, but MATLAB's flexibility allows customized development for specific requirements through careful algorithm design and memory management techniques.