Boundary particle method: Difference between revisions

Revision as of 16:42, 22 March 2013

Revision as of 06:28, 1 May 2013

Line 2:

In [[applied mathematics]], the '''boundary particle method (BPM)''' is a truly boundary-only [[meshfree method|meshless (meshfree)]] [[collocation method|collocation technique]], in the sense that none of inner nodes are required at all in the numerical solution of nonhomogeneous [[partial differential equations]]. Numerical experiments also show that the BPM has spectral [[convergence]]{{dn|date=June 2012}}. Its interpolation matrix can be symmetric and the method is easy-to-implement and free of integration and mesh. Thanks to its boundary-only merit, the BPM has clear edge over the other numerical schemes in the solution of [[optimization]] and [[inverse problems]], where only a part of boundary data is usually accessible.

== 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 the two most 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.

Line 13:

The boundary particle method (BPM) is then developed to a boundary-only discretization of inhomogeneous partial differential equation by combining the RC-MRM with a variety of the 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 since applied to a variety of problems such as nonhomogeneous [[Helmholtz]] and [[convection-diffusion equation]]. Numerical experiments are very encouraging. It is worthy of noting that the BPM interpolation representation is in fact 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 (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 [[Berger]], [[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> Thanks to its truly boundary-only merit, the BPM is more appealing in the solution of [[optimization]] and [[inverse problems]], where only a part of boundary data is usually accessible. The method has successfully 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–Taiwan 2009,17(3): 157–163</ref>

==Further comments==