Boundary particle method: Difference between revisions

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TheIn Boundary[[applied Particlemathematics]], Methodthe '''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]]. 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)[1]<ref>Partridge PW, Brebbia CA, Wrobel LC, ''The dual reciprocity boundary element method''. Computational Mechanics Publications, 1992</ref> and [[multiple reciprocity method]] (MRM)[2]<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.

=== 滴答滴 ===

ItThe DRM has beenbecome claimeda incommon literaturesmethod thatto evaluate the MRMparticular hassolution. However, the strikingDRM requires inner nodes to guarantee the convergence and stability. The MRM has an advantage over the DRM in that it does not require using inner nodes at all for nonhomogeneous problems.{{cn|date=January However,2014}} the traditional MRM does also have disadvantages comparedCompared with the DRM. Firstly, the MRM is computationally much more expensive in the construction of the different interpolation matrices. Secondly, the methodand has limited applicability to general nonhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process.

In recent decades, the dual reciprocity method (DRM)[1] and multiple reciprocity method (MRM)[2] 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.

The DRM has become de facto the method of choice in the boundary methods to evaluate the particular solution, since it is easy to use, efficient, and flexible to handle a variety of problems. However, the DRM demands the inner nodes to guarantee the convergence and stability. Therefore, the method is not truly boundary-only.

AnThe improvedrecursive composite multiple reciprocity method (RC-MRM),<ref calledname="Chena">Chen theW, "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 methodtechnique".'' Engineering Analysis with Boundary Elements'' 2010,34(RC-MRM)[3,4],): is196–205</ref> was proposed to overcome the above-mentioned problems. The key idea of the RC-MRM is employingto employ high-order composite differential operators instead of high-order Laplacian operators to vanisheliminate a varietynumber of nonhomogeneous terms in the governing equation, which can not otherwise be handled by the traditional MRM. In addition, theThe RC-MRM takes advantage ofuses the recursive structures of the MRM interpolation matrix and significantlyto reducesreduce computational costs.

It has been claimed in literatures that the MRM has the striking advantage over the DRM in that it does not require using inner nodes at all for nonhomogeneous problems. However, the traditional MRM does also have disadvantages compared with the DRM. Firstly, the MRM is computationally much more expensive in the construction of the different interpolation matrices. Secondly, the method has limited applicability to general nonhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process.

An improved multiple reciprocity method, called the recursive composite multiple reciprocity method (RC-MRM)[3,4], 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.

The boundary particle method (BPM) is then developed to a boundary-only discretization of an 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 sincebeen applied to a variety of problems such as nonhomogeneous [[Helmholtz equation]] and convection-diffusion[[convection–diffusion equationsequation]]. Numerical experiments are very encouraging. It is worthy of noting that theThe BPM interpolation representation is in fact of a [[wavelet]] series.

For the application of the BPM to Helmholtz[3], Poisson[4] and plate bending problems[5], the high-order fundamental or general solutions, harmonic [6] or Trefftz functions (T-complete functions)[7] are often used, for instance, those of Berger, Winkler, and vibrational thin plate equations[8].

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 [[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''–Taiwan 2009,17(3): 157–163</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[9] and nonhomogeneous Helmholtz equations[10].

==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 inhomogeneousnonhomogeneous equation to a high-order homogeneous equationsequation, and the BPM can be simply implemented to solve these problems with boundary-only discretization.

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:

(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 inhomogeneous equation to a high-order homogeneous equations, 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.

==See also==

*  [[Meshfree method]]

* [[Radial basis function]]

* [[Boundary element method]]

* [[Trefftz method]]

* [[Method of fundamental solution]]

* [[Boundary knot method]]

* [[Singular boundary method]]

==References==

{{Reflist}}

==External links==

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

{{Numerical PDE}}

[[Category:Numerical analysis]]

*  Meshfree method

[[Category:Numerical differential equations]]