Regularized meshless method

In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose fundamental solution is explicitly known. The RMM is a strong-form collocation method with merits being meshless, integration-free, easy-to-implement, and high stability. Until now this method has been successfully applied to some typical problems, such as potential, acoustics, water wave, and inverse problems of bounded and unbounded domains.

Description

The RMM employs the double layer potentials from the potential theory as its basis/kernel functions. Like the method of fundamental solutions (MFS),[1][2] the numerical solution is approximated by a linear combination of double layer kernel functions with respect to different source points. Unlike the MFS, the collocation and source points of the RMM, however, are coincident and placed on the physical boundary without the need of a fictitious boundary in the MFS. Thus, the RMM overcomes the major bottleneck in the MFS applications to the real world problems.

Upon the coincidence of the collocation and source points, the double layer kernel functions will present various orders of singularity. Thus, a subtracting and adding-back regularizing technique [3] is introduced and, hence, removes or cancels such singularities.

History and recent development

These days the finite element method (FEM), finite difference method (FDM), finite volume method (FVM), and boundary element method (BEM) are dominant numerical techniques in numerical modelings of many fields of engineering and sciences. Mesh generation is tedious and even very challenging problems in their solution of high-dimensional moving or complex-shaped boundary problems and is computationally costly and often mathematically troublesome.

The BEM has long been claimed to alleviate such drawbacks thanks to the boundary-only discretizations and its semi-analytical nature. Despite these merits, the BEM, however, involves quite sophisticated mathematics and some tricky singular integrals. Moreover, surface meshing in a three-dimensional domain remains to be a nontrivial task. Over the past decades, considerable efforts have been devoted to alleviating or eliminating these difficulties, leading to the development of meshless/meshfree boundary collocation methods which require neither domain nor boundary meshing. Among these methods, the MFS is the most popular with the merit of easy programming, mathematical simplicity, high accuracy, and fast convergence.

In the MFS, a fictitious boundary outside the problem domain is required in order to avoid the singularity of the fundamental solution. However, determining the optimal location of the fictitious boundary is a nontrivial task to be studied. Dramatic efforts have ever since been made to remove this long perplexing issue. Recent advances include, for example, boundary knot method (BKM),[4][5] regularized meshless method (RMM),[3] modified MFS (MMFS),[6] and singular boundary method (SBM) [7]

The methodology of the RMM was firstly proposed by Young and his collaborators in 2005. The key idea is to introduce a subtracting and adding-back regularizing technique to remove the singularity of the double layer kernel function at the origin, so that the source points can be placed directly on the real boundary. Up to now, the RMM has successfully been applied to a variety of physical problems, such as potential,[3] exterior acoustics [8] antiplane piezo-electricity,[9] acoustic eigenproblem with multiply-connected domain,[10] inverse problem,[11] possion’ equation [12] and water wave problems.[13] Furthermore, some improved formulations have been made aiming to further improve the feasibility and efficiency of this method, see, for example, the weighted RMM for irregular domain problems [14] and analytical RMM for 2D Laplace problems.[15]

See also

References
  1. A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95.
  2. M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61.
  3. D.L. Young, K.H. Chen, C.W. Lee. Novel meshless method for solving the potential problems with arbitrary domains. Journal of Computational Physics 2005; 209(1): 290–321.
  4. W. Chen and M. Tanaka, "A meshfree, exponential convergence, integration-free, and boundary-only RBF technique Archived 2016-03-04 at the Wayback Machine", Computers and Mathematics with Applications, 43, 379–391, 2002.
  5. W. Chen and Y.C. Hon, "Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz, and convection-diffusion problems Archived 2015-06-20 at the Wayback Machine", Computer Methods in Applied Mechanics and Engineering, 192, 1859–1875, 2003.
  6. B. Sarler, "Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions", Eng Anal Bound Elem 2009;33(12): 1374–82.
  7. W. Chen, F.Z. Wang, "A method of fundamental solutions without fictitious boundary Archived 2015-06-06 at the Wayback Machine", Eng Anal Bound Elem 2010;34(5): 530–32.
  8. D.L. Young, K.H. Chen, C.W. Lee. Singular meshless method using double layer potentials for exterior acoustics.Journal of the Acoustical Society of America 2006;119(1):96–107.
  9. K.H. Chen, J.H. Kao, J.T. Chen. Regularized meshless method for antiplane piezo- electricity problems with multiple inclusions. Computers, Materials, & Con- tinua 2009;9(3):253–79.
  10. K.H. Chen, J.T. Chen, J.H. Kao. Regularized meshless method for solving acoustic eigenproblem with multiply-connected domain. Computer Modeling in Engineering & Sciences 2006;16(1):27–39.
  11. K.H. Chen, J.H. Kao, J.T. Chen, K.L. Wu. Desingularized meshless method for solving Laplace equation with over-specified boundary conditions using regularization techniques. Computational Mechanics 2009;43:827–37
  12. W. Chen, J. Lin, F.Z. Wang, "Regularized meshless method for nonhomogeneous problems Archived 2015-06-06 at the Wayback Machine", Eng. Anal. Bound. Elem. 35 (2011) 253–257.
  13. K.H. Chen, M.C. Lu, H.M. Hsu, Regularized meshless method analysis of the problem of obliquely incident water wave, Eng. Anal. Bound. Elem. 35 (2011) 355–362.
  14. R.C. Song, W. Chen,"An investigation on the regularized meshless method for irregular domain problems", CMES-Comput. Model. Eng. Sci. 42 (2009) 59–70.
  15. W. Chen, R.C. Song, Analytical diagonal elements of regularized meshless method for regular domains of 2D Dirichlet Laplace problems, Eng. Anal. Bound. Elem. 34 (2010) 2–8.
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