Boundary particle method: Difference between revisions

Revision as of 00:28, 18 January 2014

Revision as of 00:50, 18 January 2014

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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. Tthe 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.

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==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 solution of such problems involves:{{cn|date=January 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 reduce the nonhomogeneous equation to a high-order homogeneous equation, and the BPM can be implemented to solve these problems with boundary-only discretization.