Regularized meshless method

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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.

In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function.

In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions and a number of points in the domain, and to select that solution which satisfies the given equation at the collocation points.

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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.

In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions. Thus a double layer potential u(x) is a scalar-valued function of x ∈ R3 given by

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.

Finite element method Numerical method for solving physical or engineering problems

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. The method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

Finite difference method In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations

In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives. FDMs are thus discretization methods. FDMs convert a linear (non-linear) ODE /PDE into a system of linear (non-linear) equations, which can then be solved by matrix algebra techniques. The reduction of the differential equation to a system of algebraic equations makes the problem of finding the solution to a given ODE ideally suited to modern computers, hence the widespread use of FDMs in modern numerical analysis.

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

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]

In numerical mathematics, the boundary knot method (BKM) is proposed as an alternative boundary-type meshfree distance function collocation scheme.

Singular boundary method

In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS), boundary knot method (BKM), regularized meshless method (RMM), boundary particle method (BPM), modified MFS, and so on. This family of strong-form collocation methods is designed to avoid singular numerical integration and mesh generation in the traditional boundary element method (BEM) in the numerical solution of boundary value problems with boundary nodes, in which a fundamental solution of the governing equation is explicitly known.

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

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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. 1 2 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 [ permanent dead link ]", 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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