Boundary element method

Boundary Element Method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e., in boundary integral form). It is applicable to a variety of engineering and physical problems, notably in the fields of fluid mechanics, acoustics, electromagnetics, and fracture mechanics. The key advantage of BEM over other numerical methods such as the Finite Element Method (FEM) is its ability to discretize only the boundary of the domain rather than the entire volume, leading to a reduction in the dimensionality of the problem by one. This can result in significant computational savings, especially for problems involving large domains or infinite domains.

Overview

The Boundary Element Method transforms a domain problem to a boundary problem using the principle of boundary reduction. This is achieved by expressing the unknown function over the domain in terms of its values on the boundary, using a fundamental solution of the differential equation. The method involves three main steps: discretization of the boundary into elements, formulation of the boundary integral equation, and solution of the resulting system of equations.

Mathematical Formulation

The mathematical foundation of BEM is based on Green's identities, which relate the values of a function within a domain to values on the boundary of the domain. For a given differential operator L, a fundamental solution G can be found such that L(G) = δ, where δ is the Dirac delta function. The unknown field variable in the domain can then be represented as an integral over the boundary of the domain, involving the fundamental solution and the boundary values of the function and its normal derivative.

Applications

BEM is widely used in engineering and physics for solving problems such as:

• Acoustic scattering
• Electromagnetic field computation
• Heat transfer
• Elasticity and fracture mechanics
• Fluid flow in porous media

Advantages and Limitations

Advantages:

• Requires discretization of only the boundary, reducing the problem size.
• Well-suited for problems involving infinite or semi-infinite domains.
• Provides high accuracy for boundary quantities.

Limitations:

• Primarily applicable to linear problems.
• Formulation of the integral equation can be complex for certain problems.
• Handling of internal discontinuities or singularities can be challenging.

Software and Tools

Several software packages and tools have been developed for implementing the Boundary Element Method, including both commercial and open-source options. Examples include:

• BEM++: An open-source C++ library for solving boundary integral equations.
• ANSYS: A commercial software offering BEM capabilities for certain types of problems.

See Also

• Finite Element Method
• Mesh generation
• Numerical analysis
• Partial differential equation