Key words: meshfree, Green second identity, boundary particle method, dual reciprocity BEM, multiple reciprocity BEM, recursive multiple reciprocity, general solution, fundamental solution, composite multiple reciprocity, differentiation elimination, wavelets.
In last decade, the dual reciprocity BEM (DR-BEM) and multiple reciprocity BEM (MR-BEM) have been emerging as two most promising BEM techniques to handle inhomogeneous problems. The advantages and disadvantages of the MR-BEM relative to the DR-BEM are that it does not use inner nodes at all for inhomogeneous problems but that it requires more computing effort. The MR-BEM uses high-order fundamental solutions to approximate high-order homogeneous solutions and then get the particular solutions.
1. Original boundary particle method (BPM)---multiple reciprocity method
By analogy with the MR-BEM, we introduced the meshfree boundary particle method (BPM), which combines the distance function and multiple reciprocity principle to formulate a simple and efficient truly boundary-only meshfree collocation method. The method is also spectral convergent (numerical observations), symmetric (self-adjoint PDE), integration-free, and easy to learn and implement. In particular, the BPM formulation is a wavelet series.
Numerical experiments of the BPM are very encouraging. It is worth stressing that our multiple reciprocity scheme is recursive and only requires producing one system matrix and is thereofre called the recursive multiple reciprocity method, which dramatically reduces computing cost of the multiple reciprocity techniques.
We also discovered the high-order fundamental and general solutions of convection-diffusion, Bergers, Winkler, vibration thin plate equations, and high-order T-complete functions of Laplacian and Helmholtz operators.
2. Improved boundary particle method (BPM)---composite multiple reciprocity method
However, the multiple reciprocity method (MRM) does also have another disadvantage compared with the dual reciprocity method (DRM) in that the standard MRM has limited feasibility for general inhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process. We recently proposed an improved MRM, called composite multiple reciprocity (CMRM) technique to efficiently handle various inhomogeneous terms in the governing equation. The key point of CMRM is employing high-order composite differential operators instead of high-order Laplacian operators to vanish the inhomogeneous terms in the governing equation.
The BPM coupled with CMRM are applied to Poisson equation and inhomogeneous Helmholtz equation, plate bending analysis.
Due to its truly boundary-only merit, The BPM coupled with CMRM is far more appealing in the solution of inverse Cauchy problems, where only a part of boundary data are usually accessible. The BPM-CMRM is applied to inverse Cauchy problem associated with Poisson equation and inhomogeneous Helmholtz equation. Numerical demonstration show the BPM-CMRM performs well with inverse Cauchy problems under often-encountered types of inner source term f(x) (polynomial, exponential and trigonometric functions or the combination of these functions).
3. Further development
One may argue that the present BPM may not be suitable for complex functions (e.g., non-smooth functions), large-gradient functions or a set of diescrete measured data in the inner source term f(x). In our opinion, it can express the complex functions or a set of diescrete measured data by a sum of polynomial or trigonometric function series, and then the present BPM can simply be implemented to solve these problems with the boundary-only discretization. For the cases with large-gradient functions, it may combine the present BPM with domain decomposition method to deal with. These issues are still under study.
W. Chen, Several new domain-type and boundary-type numerical discretization schemes with radial basis function, CoRR preprint, 2001.
W. Chen, New RBF collocation schemes and kernel RBFs with applications, Lecture Notes in Computational Science and Engineering, Vol 26, 75-86, 2002.
W. Chen, RBF-based meshless boundary knot method and boundary particle method. Proc. of the China Congress on Computational Mechanics's 2001, pp. 319-326, Guangzhou, China, Dec. 2001.
W. Chen, High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method, Engng. Anal. Bound. Elem., 26(7), 571-575, 2002.
W. Chen, Meshfree boundary particle method applied to Helmholtz problems, Engng. Anal. Bound. Elem., 26(7), 577-581, 2002.
W. Chen, Distance function wavelets – Part III: "Exotic" transforms and series, Research report of Simula Research Laboratory, CoRR preprint, June, 2002.
W. Chen, Some recent advances on the RBF. Proceedings of BEM 24, pp. 125-134, Portugal, June, 2002.
W. Chen, Z.J. Fu, B.T. Jin, A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique. Engineering Analysis with Boundary Elements, 34, 196–205, 2010
Z.J. Fu, W. Chen, A truly boundary-only meshfree method applied to Kirchhoff plate bending problem. Advances in Applied Mathematics and Mechanics, 1(3), 341-352, 2009.
Z.J. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method. Computational Mechanics, 44(6), 757-763, 2009. (matlab program)
W. Chen, Z.J. Fu, Q.H. Qin, Boundary particle method with high-order Trefftz functions. CMC: Computers, Materials, & Continua, 13(3), 201-218, 2009.
W. Chen, Z.J. Fu, Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations. Journal of Marine Science and Technology, 17(3), 157-163, 2009
Z.J. Fu, W. Chen, C.Z. Zhang, Boundary Particle Method for Cauchy inhomgeneous potential problems. Inverse Problem in Science and Engineering, 20(2), 189-207, 2012 (matlab toolbox)
Z.J. Fu, W. Chen, H.T. Yang, Boundary Particle Method For Laplace Transformed Time Fractional Diffusion Equations. Journal of Computational Physic, 235, 52-66, 2013