The boundary element method (BEM) is a numerical computational method of solving linear
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
which have been formulated as
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s (i.e. in ''boundary integral'' form), including
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
,
acoustics,
electromagnetics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
(where the technique is known as
method of moments or abbreviated as MoM),
fracture mechanics, and
contact mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points.Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical P ...
.
Mathematical basis
The integral equation may be regarded as an exact solution of the governing partial differential equation. The boundary element method attempts to use the given
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post-processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain.
BEM is applicable to problems for which
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s can be calculated. These usually involve fields in
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
homogeneous media. This places considerable restrictions on the range and generality of problems to which boundary elements can usefully be applied. Nonlinearities can be included in the formulation, although they will generally introduce volume integrals which then require the volume to be
discretised before solution can be attempted, removing one of the most often cited advantages of BEM. A useful technique for treating the volume integral without discretising the volume is the
dual-reciprocity method. The technique approximates part of the integrand using
radial basis function A radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed ...
s (local interpolating functions) and converts the volume integral into boundary integral after collocating at selected points distributed throughout the volume domain (including the boundary). In the dual-reciprocity BEM, although there is no need to discretize the volume into meshes, unknowns at chosen points inside the solution domain are involved in the linear algebraic equations approximating the problem being considered.
The Green's function elements connecting pairs of source and field patches defined by the mesh form a matrix, which is solved numerically. Unless the Green's function is well behaved, at least for pairs of patches near each other, the Green's function must be integrated over either or both the source patch and the field patch. The form of the method in which the integrals over the source and field patches are the same is called "
Galerkin's method". Galerkin's method is the obvious approach for problems which are symmetrical with respect to exchanging the source and field points. In frequency domain electromagnetics, this is assured by
electromagnetic reciprocity. The cost of computation involved in naive Galerkin implementations is typically quite severe. One must loop over each pair of elements (so we get n
2 interactions) and for each pair of elements we loop through
Gauss points in the elements producing a multiplicative factor proportional to the number of Gauss-points squared. Also, the function evaluations required are typically quite expensive, involving trigonometric/hyperbolic function calls. Nonetheless, the principal source of the computational cost is this double-loop over elements producing a fully populated matrix.
The
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s, or
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
s, are often problematic to integrate as they are based on a solution of the system equations subject to a singularity load (e.g. the electrical field arising from a point charge). Integrating such singular fields is not easy. For simple element geometries (e.g. planar triangles) analytical integration can be used. For more general elements, it is possible to design purely numerical schemes that adapt to the singularity, but at great computational cost. Of course, when source point and target element (where the integration is done) are far-apart, the local gradient surrounding the point need not be quantified exactly and it becomes possible to integrate easily due to the smooth decay of the fundamental solution. It is this feature that is typically employed in schemes designed to accelerate boundary element problem calculations.
Derivation of closed-form Green's functions is of particular interest in boundary element method, especially in electromagnetics. Specifically in the analysis of layered media, derivation of spatial-domain Green's function necessitates the inversion of analytically-derivable spectral-domain Green's function through Sommerfeld path integral. This integral can not be evaluated analytically and its numerical integration is costly due to its oscillatory and slowly-converging behaviour. For a robust analysis, spatial Green's functions are approximated as complex exponentials with methods such as
Prony's method
Prony analysis (Prony's method) was developed by Gaspard Riche de Prony in 1795. However, practical use of the method awaited the digital computer. Similar to the Fourier transform, Prony's method extracts valuable information from a uniformly samp ...
or
generalized pencil of function, and the integral is evaluated with
Sommerfeld identity.
This method is known as discrete complex image method.
Comparison to other methods
The boundary element method is often more efficient than other methods, including finite elements, in terms of computational resources for problems where there is a small surface/volume ratio. Conceptually, it works by constructing a "
mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, exp ...
" over the modelled surface. However, for many problems boundary element methods are significantly less efficient than volume-discretisation methods (
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
,
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
,
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
). A good example of application of the boundary element method is efficient calculation of
natural frequencies
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In ...
of
liquid sloshing in tanks. Boundary element method is one of the most effective methods for numerical simulation of contact problems, in particular for simulation of adhesive contacts.
Boundary element formulations typically give rise to fully populated matrices. This means that the storage requirements and computational time will tend to grow according to the square of the problem size. By contrast, finite element matrices are typically banded (elements are only locally connected) and the storage requirements for the system matrices typically grow quite linearly with the problem size. Compression techniques (e.g. multipole expansions or adaptive cross approximation/
hierarchical matrices) can be used to ameliorate these problems, though at the cost of added complexity and with a success-rate that depends heavily on the nature of the problem being solved and the geometry involved.
See also
*
Analytic element method The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. It was initially developed by O.D.L. Strack at the University of Minnesota. It is similar in nature to the boundary element method (BE ...
*
Computational electromagnetics
*
Meshfree methods
*
Immersed boundary method
In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the flui ...
*
Stretched grid method
*
Modified radial integration method[Najarzadeh, L., Movahedian, B. and Azhari, M., 2019. Numerical solution of scalar wave equation by the modified radial integration boundary element method. Engineering Analysis with Boundary Elements, 105, pp.267-278.]
References
Bibliography
*.
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*, available als
here
*.
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* (in two volumes).
Further reading
*
External links
An Online Resource for Boundary ElementsWhat lies beneath the surface? A guide to the Boundary Element Method and Green's functions for the students and professionals*
ttp://personal.ntu.edu.sg/mwtang/anghyperbook.html Boundary elements for plane crack problemsbr>
Electromagnetic Modeling web site at Clemson University(includes list of currently available software)
Concept Analyst Boundary Element Analysis softwareKlimpke, Bruce ''A Hybrid Magnetic Field Solver Using a Combined Finite Element/Boundary Element Field Solver'', U.K. Magnetics Society Conference, 2003which compares FEM and BEM methods as well as hybrid approaches
Free software
BembelA 3D, isogeometric, higher-order, open-source BEM software for Laplace, Helmholtz and Maxwell problems utilizing a fast multipole method for compression and reduction of computational cost
boundary-element-method.comAn open-source BEM software for solving acoustics / Helmholtz and Laplace problems
Puma-EMAn open-source and high-performance Method of Moments / Multilevel Fast Multipole Method parallel program
AcouSTOAcoustics Simulation TOol, a free and open-source parallel BEM solver for the Kirchhoff-Helmholtz Integral Equation (KHIE)
FastBEMFree fast multipole boundary element programs for solving 2D/3D potential, elasticity, Stokes flow and acoustic problems
ParaFEMIncludes the free and open-source parallel BEM solver for elasticity problems described in Gernot Beer, Ian Smith, Christian Duenser, ''The Boundary Element Method with Programming: For Engineers and Scientists'', Springer, (2008)
Boundary Element Template Library (BETL)A general purpose C++ software library for the discretisation of boundary integral operators
NemohAn open source hydrodynamics BEM software dedicated to the computation of first-order wave loads on offshore structures (added mass, radiation damping, diffraction forces)
Bempp An open-source BEM software for 3D Laplace, Helmholtz and Maxwell problems
MNPBEM An open-source Matlab toolbox to solve Maxwell's equations for arbitrarily shaped nanostructures
Contact Mechanics and Tribology Simulator Free, BEM based software
{{DEFAULTSORT:Boundary Element Method
Numerical differential equations
Computational fluid dynamics
Computational electromagnetics