Boundary particle method: Difference between revisions

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The '''Boundary Particle Method''' (BPM) is a truly boundary-only meshless (meshfree) 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]]. 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==

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An improved multiple reciprocity method, called 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>, is proposed to overcome the above-mentioned problems. The key idea of the RC-MRM is employing high-order composite differential operators instead of high-order Laplacian operators to vanish a variety of nonhomogeneous terms in the governing equation, which can not otherwise be handled by the traditional MRM. In addition, the RC-MRM takes advantage of the recursive structures of the MRM interpolation matrix and significantly reduces computational costs.<br />

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]] equations. Numerical experiments are very encouraging. It is worthy of noting that the BPM interpolation representation is in fact of a [[wavelet]] series.<br />

For the application of the BPM to [[Helmholtz]]<ref name="Chena" />, [[Poisson]]<ref name="Chenb" /> and [[plate bending]] 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 solutionssolution]] or general solutionssolution, harmonic functionsfunction<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 functions]] 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 [[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==

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 is briefly outlined below:<br />