Boundary element method

Source: Wikipedia, the free encyclopedia.

The boundary element method (BEM) is a numerical computational method of solving linear

electromagnetics (where the technique is known as method of moments or abbreviated as MoM),[1] fracture mechanics,[2] and contact mechanics.[3][4]

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 conditions
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

discretised before solution can be attempted, removing one of the most often cited advantages of BEM[citation needed]. 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 functions
(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 "

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 functions, or fundamental solutions, 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.[5][6][7][8] This method is known as discrete complex image method.[7][8]

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.

natural frequencies of liquid sloshing in tanks.[10][11][12] Boundary element method is one of the most effective methods for numerical simulation of contact problems,[13] in particular for simulation of adhesive contacts.[14]

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

References

  1. ^ In electromagnetics, the more traditional term "method of moments" is often used, though not always, as a synonymous of "boundary element method": see (Gibson 2008) for further information on the subject.
  2. ^ The boundary element method is well suited for analyzing cracks in solids. There are several boundary element approaches for crack problems. One such approach is to formulate the conditions on the cracks in terms of hypersingular boundary integral equations, see (Ang 2013).
  3. S2CID 137494525
    .
  4. ^ "BEM Based Contact Pressure Calculation Tutorial". www.tribonet.org. 9 November 2017.
  5. doi:10.1109/22.75309
    .
  6. hdl:11693/10779
    .
  7. ^
    doi:10.1109/75.877225
    .
  8. ^
    doi:10.1109/TMTT.2002.807834
    .
  9. ^ See (Katsikadelis 2002).
  10. doi:10.1016/j.euromechflu.2015.06.001
    .
  11. doi:10.1016/j.compfluid.2014.10.013
    .
  12. ISBN 978-0-7918-4647-6
    .
  13. ISBN 9783662530801
    .
  14. ^ Pohrt, Roman; Popov, Valentin L. (2015-04-09). "Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method". Facta Universitatis, Series: Mechanical Engineering. 13 (1): 3–10.
  15. ^ Najarzadeh, L., Movahedian, B. and Azhari, M., 2022. Numerical solution of water wave propagation problems over variable bathymetries using the modified radial integration boundary element method. Ocean Engineering, 257, p.111613.
  16. ^ 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.

Bibliography

Further reading

External links

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,
    ISBN 978-3-211-71574-1
    (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).
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