Boundary particle method: Difference between revisions

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{{expert-subject|1=Mathematics|date=April 2012}}

In [[applied mathematics]], the '''boundary particle method (BPM)''' is a boundary-only [[meshfree method|meshless (meshfree)]] [[collocation method|collocation technique]], in the sense that none of inner nodes are required in the numerical solution of nonhomogeneous [[partial differential equations]]. Numerical experiments show that the BPM has [[spectral convergence]]. Its interpolation matrix can be symmetric.

== History and recent developments ==

In recent decades, the [[dual reciprocity method]] (DRM)<ref>Partridge PW, Brebbia CA, Wrobel LC, ''The dual reciprocity boundary element method''. Computational Mechanics Publications, 1992</ref> and [[multiple reciprocity method]] (MRM)<ref>Nowak AJ, Neves AC, ''The multiple reciprocity boundary element method''. Computational Mechanics Publication, 1994</ref> have been emerging as promising techniques to evaluate the particular solution of nonhomogeneous [[partial differential equations]] in conjunction with the boundary discretization techniques, such as [[boundary element method]] (BEM). For instance, the so-called DR-BEM and MR-BEM are popular BEM techniques in the numerical solution of nonhomogeneous problems.

The DRM has become a common method to evaluate the particular solution. However, the DRM requires inner nodes to guarantee the convergence and stability. TtheThe MRM has an advantage over the DRM in that it does not require using inner nodes for nonhomogeneous problems.{{cn|date=January 2014}} Compared with the DRM, the MRM is computationally more expensive in the construction of the interpolation matrices and has limited applicability to general nonhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process.

The recursive composite multiple reciprocity method (RC-MRM),<ref name="Chena">Chen W, "Meshfree boundary particle method applied to Helmholtz problems". ''Engineering Analysis with Boundary Elements'' 2002,26(7): 577–581</ref><ref name="Chenb">Chen W, Fu ZJ, Jin BT, "A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique".'' Engineering Analysis with Boundary Elements'' 2010,34(3): 196–205</ref> was proposed to overcome the above-mentioned problems. The key idea of the RC-MRM is to employ high-order composite differential operators instead of high-order Laplacian operators to eliminate a number of nonhomogeneous terms in the governing equation. The RC-MRM uses the recursive structures of the MRM interpolation matrix to reduce computational costs.

The boundary particle method (BPM) is a boundary-only discretization of an inhomogeneous partial differential equation by combining the RC-MRM with strong-form meshless boundary collocation discretization schemes, such as the [[method of fundamental solution]] (MFS), [[boundary knot method]] (BKM), [[regularized meshless method]] (RMM), [[singular boundary method]] (SBM), and [[Trefftz method]] (TM). The BPM has been applied to problems such as nonhomogeneous [[Helmholtz equation]] and [[convection-diffusionconvection–diffusion equation]]. The BPM interpolation representation is of a [[wavelet]] series.

For the application of the BPM to [[Helmholtz]],<ref name="Chena" /> [[Siméon Denis Poisson|Poisson]]<ref name="Chenb" /> and [[plate]] bending]] problems,<ref>Fu ZJ, Chen W, Yang W, Winkler plate bending problems by a truly boundary-only boundary particle method. Computational Mechanics 2009,44(6): 757–563</ref> the high-order [[fundamental solution]] or general solution, harmonic function<ref>Hon YC, Wu ZM, "A numerical computation for inverse boundary determination problem." ''Engineering Analysis with Boundary Elements'' 2000,24(7–8): 599–606</ref> or [[Trefftz function]] function (T-complete functions)<ref>Chen W, Fu ZJ, Qin QH, "Boundary particle method with high-order Trefftz functions". ''CMC: Computers, Materials & Continua'' 2010,13(3): 201–217</ref> are often used, for instance, those of [[Marcel Berger|Berger]], [[Peter Winkler|Winkler]], and vibrational thin plate equations.<ref>Chen W, Shen ZJ, Shen LJ, Yuan GW, "General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates." ''Engineering Analysis with Boundary Elements'' 2005,29(7): 699–702</ref> The method has been applied to inverse Cauchy problem associated with [[Siméon Denis Poisson|Poisson]]<ref>Fu ZJ, Chen W, Zhang CZ, "Boundary particle method for Cauchy inhomogeneous potential problems". ''Inverse Problems in Science and Engineering'' 2012,20(2): 189–207</ref> and nonhomogeneous [[Helmholtz]] equations.<ref>Chen W, Fu ZJ, "Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations". ''Journal of Marine Science and Technology–TaiwanTechnology''–Taiwan 2009,17(3): 157–163</ref>

==Further comments==

The BPM may encounter difficulty in the solution of problems having complex source functions, such as non-smooth, large-gradient functions, or a set of discrete measured data. The road map for the BPM solution of such problems isinvolves:{{cn|date=January briefly outlined below:2014}}

(1) The complex functions or a set of discrete measured data can be interpolated by a sum of [[polynomial]] or [[trigonometric]] function series. Then, the RC-MRM can easily reduce the nonhomogeneous equation to a high-order homogeneous equationsequation, and the BPM can be simply implemented to solve these problems with boundary-only discretization.

(2) The [[___domain decomposition]] may be used to in the BPM boundary-only solution of large-gradient source functions problems.

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==External links==

* [https://web.archive.org/web/20160303222653/http://www.ccms.ac.cn/fuzj/Boundary%20Particle%20Method.htm Boundary Particle Method]

==Free software and Matlab codes==

* [http://www.ccms.ac.cn/fuzj/download/BPM-Winkler%20Plate.rar Winker plate bending analysis]

* [http://www.ccms.ac.cn/shijh/bpmtoolbox.zip BPM toolbox for inverse Cauchy problems]

{{Numerical PDE}}

{{DEFAULTSORT:Meshfree Methods}}

[[Category:Numerical analysis]]

[[Category:Numerical differential equations]]