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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 involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
which have been formulated as
integral equation In mathematical analysis, 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,\ldots,x_n ; u(x_1,x_2 ...
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 plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
electromagnetics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
(where the technique is known as method of moments or abbreviated as MoM),
fracture mechanics Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics t ...
, and
contact mechanics Contact mechanics is the study of the Deformation (mechanics), deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between Stress (mechanics), stresses acting perpendicular to the cont ...
.


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 the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
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 (or Green 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 L is a linear dif ...
s can be calculated. These usually involve fields in
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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 In mathematics 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\, ), o ...
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 n2 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 (or Green 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 L is a linear dif ...
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 ...
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 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 Medical Subject Headings (MeSH) is a comprehensive controlled vocabulary for the purpose of indexing journal articles and books in the life sciences. It serves as a thesaurus of index terms that facilitates searching. Created and updated by th ...
" over the modelled surface. However, for many problems boundary element methods are significantly less efficient than volume-discretisation methods (
finite element method 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 tran ...
,
finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial doma ...
,
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 divergen ...
). A good example of application of the boundary element method is efficient calculation of natural frequencies 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 * Computational electromagnetics * Meshfree methods * Immersed boundary method * Stretched grid method * Modified radial integration methodNajarzadeh, 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

*. *. *. * *, available als
here
*. *. * (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, 2003
which compares FEM and BEM methods as well as hybrid approaches


Free software


Bembel
A 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.com
An open-source BEM software for solving acoustics / Helmholtz and Laplace problems
Puma-EM
An open-source and high-performance Method of Moments / Multilevel Fast Multipole Method parallel program
AcouSTO
Acoustics Simulation TOol, a free and open-source parallel BEM solver for the Kirchhoff-Helmholtz Integral Equation (KHIE)
FastBEM
Free fast multipole boundary element programs for solving 2D/3D potential, elasticity, Stokes flow and acoustic problems
ParaFEM
Includes 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
Nemoh
An 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
MultiFEBE
BEM-FEM solver for computational mechanics, allowing coupling of 2D and 3D viscoelastic or poroelastic media with beam and shell structural elements (for dynamic soil-structure interaction problems, for instance).
BE-STATIK
Free BE-Programs for 2D potential, elasticity and plate bending problems (Kirchhoff). {{DEFAULTSORT:Boundary Element Method Numerical differential equations Computational fluid dynamics Computational electromagnetics