Singular boundary method

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Fig. 1. Problem sketch and nodes distribution using the MFS: (a) interior problems, (b) exterior problems (please click to see big pictures) SBM and MFS 01.PNG
Fig. 1. Problem sketch and nodes distribution using the MFS: (a) interior problems, (b) exterior problems (please click to see big pictures)
Fig. 2. Problem sketch and nodes distribution using the SBM: (c) interior problems, (d) exterior problems (please click to see big pictures) MFS and SBM 02.PNG
Fig. 2. Problem sketch and nodes distribution using the SBM: (c) interior problems, (d) exterior problems (please click to see big pictures)

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), [1] [2] [3] boundary knot method (BKM), [4] regularized meshless method (RMM), [5] boundary particle method (BPM), [6] modified MFS, [7] 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.

Numerical analysis study of algorithms that use numerical approximation for the problems of mathematical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

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.

In scientific computation and simulation, the method of fundamental solutions (MFS) is getting a growing attention. The method is essentially based on the fundamental solution of a partial differential equation of interest as its basis function. The MFS was developed to overcome the major drawbacks in the boundary element method (BEM) which also uses the fundamental solution to satisfy the governing equation. Consequently, both the MFS and the BEM are of a boundary discretization numerical technique and reduce the computational complexity by one dimensionality and have particular edge over the domain-type numerical techniques such as the finite element and finite volume methods on the solution of infinite domain, thin-walled structures, and inverse problems.

Contents

The salient feature of the SBM is to overcome the fictitious boundary in the method of fundamental solution, while keeping all merits of the latter. The method offers several advantages over the classical domain or boundary discretization methods, among which are:

The SBM provides a significant and promising alternative to popular boundary-type methods such as the BEM and MFS, in particular, for infinite domain, wave, thin-walled structures, and inverse problems.

History of the singular boundary method

The methodology of the SBM was firstly proposed by Chen and his collaborators in 2009. [8] [9] The basic idea is to introduce a concept of the origin intensity factor to isolate the singularity of the fundamental solutions so that the source points can be placed directly on the real boundary. In comparison, the method of fundamental solutions requires a fictitious boundary for placing the source points to avoid the singularity of fundamental solution. The SBM has since been successfully applied to a variety of physical problems, such as potential problems, [10] [11] infinite domain problem, [12] Helmholtz problem, [13] and plane elasticity problem. [14]

There are the two techniques to evaluate the origin intensity factor. The first approach is to place a cluster of sample nodes inside the problem domain and to calculate the algebraic equations. The strategy leads to extra computational costs and makes the method is not as efficient as expected compared to the MFS. The second approach [15] [16] is to employ a regularization technique to cancel the singularities of the fundamental solution and its derivatives. Consequently, the origin intensity factors can be determined directly without using any sample nodes. This scheme makes the method more stable, accurate, efficient, and extends its applicability.

Recent developments

Boundary layer effect problems

Like all the other boundary-type numerical methods, also it is observed that the SBM encounters a dramatic drop of solution accuracy at the region nearby boundary. Unlike singularity at origin, the fundamental solution at near-boundary regions remains finite. However, instead of being a flat function, the interpolation function develops a sharp peak as the field point approaches the boundary. Consequently, the kernels become “nearly singular” and can not accurately be calculated. This is similar to the so-called boundary layer effect encountered in the BEM-based methods.

A nonlinear transformation, based on the sinh function, can be employed to remove or damp out the rapid variations of the nearly singular kernels. [17] As a result, the troublesome boundary layer effect in the SBM has been successfully remedied. The implementation of this transformation is straightforward and can easily be embedded in existing SBM programs. For the test problems studied, very promising results are obtained even when the distance between the field point and the boundary is as small as 1×10−10.

Large-scale problems

Like the MFS and BEM, the SBM will produce dense coefficient matrices, whose operation count and the memory requirements for matrix equation buildup are of the order of O(N2) which is computationally too expensive to simulate large-scale problems.

The fast multipole method (FMM) can reduce both CPU time and memory requirement from O(N2) to O(N) or O(NlogN). With the help of FMM, the SBM can be fully capable of solving a large scale problem of several million unknowns on a desktop. This fast algorithm dramatically expands the applicable territory of the SBM to far greater problems than were previously possible.

The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.

See also

In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbours. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field.

A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that ; or alternatively on the distance from some other point , called a center, so that . Any function that satisfies the property is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible.

In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz(de) (1888–1937). It falls within the class of finite element methods.

Related Research Articles

In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory. In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ which is the fundamental solution of the Laplace equation. It is named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the fundamental gravitational potential in Newton's law of universal gravitation. In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential.

Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).

The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations. including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and contact mechanics.

Trajectory optimization is the process of designing a trajectory that minimizes some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop solution is either impossible or impractical.

The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. It was initially developed by O.D.L. Strack at the University of Minnesota. It is similar in nature to the boundary element method (BEM), as it does not rely upon discretization of volumes or areas in the modeled system; only internal and external boundaries are discretized. One of the primary distinctions between AEM and BEMs is that the boundary integrals are calculated analytically.

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.

hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size (h) and polynomial degree (p). The origins of hp-FEM date back to the pioneering work of Ivo Babuska et al. who discovered that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and p-refinements. The exponential convergence makes the method a very attractive choice compared to most other finite element methods which only converge with an algebraic rate. The exponential convergence of the hp-FEM was not only predicted theoretically but also observed by numerous independent researchers.

Weakened weak form is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems.

In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of nonhomogeneous partial differential equations. Numerical experiments show that the BPM has spectral convergence. Its interpolation matrix can be symmetric.

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

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.

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The discrete least squares meshless (DLSM) method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. While most of the existing meshless methods need background cells for numerical integration, DLSM did not require a numerical integration procedure due to the use of the discrete least squares method to discretize the governing differential equation. A Moving least squares (MLS) approximation method is used to construct the shape function, making the approach a fully least squares-based approach.

Alexander G. Ramm is an American mathematician. His research focuses on differential and integral equations, operator theory, ill-posed and inverse problems, scattering theory, functional analysis, spectral theory, numerical analysis, theoretical electrical engineering, signal estimation, and tomography.

The finite point method (FPM) is a meshfree method for solving partial differential equations (PDEs) on scattered distributions of points. The FPM was proposed in the mid-nineties in, and with the purpose to facilitate the solution of problems involving complex geometries, free surfaces, moving boundaries and adaptive refinement. Since then, the FPM has evolved considerably, showing satisfactory accuracy and capabilities to deal with different fluid and solid mechanics problems.

The generalized-strain mesh-free (GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form collocation. This method was first presented by Oliveira and Portela (2016), in order to further improve the computational efficiency of meshfree methods in numerical analysis. Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this formulation results in a local form of the work theorem that is reduced to regular boundary terms only, integration-free and free of volumetric locking.

References

  1. ↑ method of fundamental solutions (MFS)
  2. ↑ Golberg MA, Chen CS, Ganesh M, "Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions", Eng Anal Bound Elem 2000;24(7–8): 539–47.
  3. ↑ Fairweather G, Karageorghis A, "The method of fundamental solutions for elliptic boundary value problems", Adv Comput Math 1998;9(1): 69–95.
  4. ↑ Chen W, Tanaka M, "A meshless, integration-free, and boundary-only RBF technique Archived 2016-03-04 at the Wayback Machine .", Comput Math Appl 2002;43(3–5): 379–91.
  5. ↑ D.L. Young, K.H. Chen, C.W. Lee, "Novel meshless method for solving the potential problems with arbitrary domain", J Comput Phys 2005;209(1): 290–321.
  6. ↑ boundary particle method (BPM)
  7. ↑ Sarler B, "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.
  8. ↑ Chen W, "Singular boundary method: A novel, simple, meshfree, boundary collocation numerical method", Chin J Solid Mech 2009;30(6): 592–9.
  9. ↑ Chen W, Wang FZ, "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.
  10. ↑ Wei X, Chen W, Fu ZJ, "Solving inhomogeneous problems by singular boundary method", J Mar SCI Tech 2012; 20(5).
  11. ↑ Chen W, Fu ZJ, Wei X, "Potential Problems by Singular Boundary Method Satisfying Moment Condition", Comput Model Eng Sci 2009;54(1): 65–85.
  12. ↑ Chen W, Fu Z, "A novel numerical method for infinite domain potential problems", Chin Sci Bull 2010;55(16): 1598–603.
  13. ↑ Fu ZJ, Chen W, "A novel boundary meshless method for radiation and scattering problems", Advances in Boundary Element Techniques XI, Proceedings of the 11th international Conference, 12–14 July 2010, 83–90, Published by EC Ltd, United Kingdom (ISBN   978-0-9547783-7-8)
  14. ↑ Gu Y, Chen W, Zhang CZ., "Singular boundary method for solving plane strain elastostatic problems", Int J Solids Struct 2011;48(18): 2549–56.
  15. ↑ Chen W, Gu Y, "Recent advances on singular boundary method", Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II, Taiwan 2011.
  16. ↑ Gu Y, Chen, W, "Improved singular boundary method for three dimensional potential problems", Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(2): 351-360 (in Chinese)
  17. ↑ Gu Y, Chen W, Zhang J, "Investigation on near-boundary solutions by singular boundary method", Eng Anal Bound Elem 2012;36(8): 117–82.
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