Boundary knot method

Last updated January 11, 2020

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

Recent decades have witnessed a research boom on the meshfree numerical PDE techniques since the construction of a mesh in the standard finite element method and boundary element method is not trivial especially for moving boundary, and higher-dimensional problems. The boundary knot method is different from the other methods based on the fundamental solutions, such as boundary element method, method of fundamental solutions and singular boundary method in that the former does not require special techniques to cure the singularity. The BKM is truly meshfree, spectral convergent (numerical observations), symmetric (self-adjoint PDEs), integration-free, and easy to learn and implement. The method has successfully been tested to the Helmholtz, diffusion, convection-diffusion, and Possion equations with very irregular 2D and 3D domains.

Description

The BKM is basically a combination of the distance function, non-singular general solution, and dual reciprocity method (DRM). The distance function is employed in the BKM to approximate the inhomogeneous terms via the DRM, whereas the non-singular general solution of the partial differential equation leads to a boundary-only formulation for the homogeneous solution. Without the singular fundamental solution, the BKM removes the controversial artificial boundary in the method of fundamental solutions. Some preliminary numerical experiments show that the BKM can produce excellent results with relatively a small number of nodes for various linear and nonlinear problems.

Formulation

Consider the following problems,

(1)

Lu=f\left(x,y\right),\ \ \left(x,y\right)\in \Omega

(2)

u=g\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{D}

(3)

{\frac {\partial u}{\partial n}}=h\left(x,y\right),\ \ h\left(x,y\right)\in \partial \Omega _{N}

where $L$ is the differential operator, $\Omega$ represents the computational domain, $\partial \Omega _{D}$ and $\partial \Omega _{N}$ denote the Dirichlet and Neumann boundaries respectively, satisfied $\partial \Omega _{D}\cup \partial \Omega _{N}=\partial \Omega$ and $\partial \Omega _{D}\cap \partial \Omega _{N}=\varnothing$ . The BKM employs the non-singular general solution of the operator $L$ to approximate the numerical solution as follows,

(4)

u^{*}\left(x,y\right)=\sum \limits _{i=1}^{N}\alpha _{i}\phi \left(r_{i}\right)

where $r_{i}=\left\|\left(x,y\right)-\left(x_{i},y_{i}\right)\right\|_{2}$ denotes the Euclidean distance, $\phi \left(\cdot \right)$ is the general solution satisfied

(5)

L\phi =0

By employing the collocation technique to satisfy the boundary conditions (2) and (3),

(6)

{\begin{aligned}&g\left(x_{k},y_{k}\right)=\sum \limits _{i=1}^{N}\alpha _{i}\phi \left(r_{i}\right),\qquad k=1,\ldots ,m_{1}\\&h\left(x_{k},y_{k}\right)=\sum \limits _{i=1}^{N}\alpha _{i}{\frac {\partial \phi \left(r_{i}\right)}{\partial n}},\qquad k=m_{1}+1,\ldots ,m\\\end{aligned}}

where $\left(x_{k},y_{k}\right)|_{k=1}^{m_{1}}$ and $\left(x_{k},y_{k}\right)|_{k=m_{1}+1}^{m}$ denotes the collocation points located at Dirichlet boundary and Neumann boundary respectively. The unknown coefficients $\alpha _{i}$ can be uniquely determined by above Eq. (6). And then the BKM solution at any location of computational domain can be evaluated by the formulation (4).

History and recent developments

It has long been noted that boundary element method (BEM) is an alternative method to finite element method (FEM) and finite volume method (FVM) for infinite domain, thin-walled structures, and inverse problems, thanks to its dimensional reducibility. The major bottlenecks of BEM, however, are computationally expensive to evaluate integration of singular fundamental solution and to generate surface mesh or re-mesh. The method of fundamental solutions (MFS)^[1] has in recent decade emerged to alleviate these drawbacks and getting increasing attentions. The MFS is integration-free, spectral convergence and meshfree.

As its name implies, the fundamental solution of the governing equations is used as the basis function in the MFS. To avoid singularity of the fundamental solution, the artificial boundary outside the physical domain is required and has been a major bottleneck for the wide use of the MFS, since such fictitious boundary may cause computational instability. The BKM is classified as one kind of boundary-type meshfree methods without using mesh and artificial boundary.

The BKM has since been widely tested. In,^[2] the BKM is used to solve Laplace equation, Helmholtz Equation, and varying-parameter Helmholtz equations; in^[3] by analogy with Fasshauer’s Hermite RBF interpolation, a symmetric BKM scheme is proposed in the presence of mixed boundary conditions; in,^[4] numerical investigations are made on the convergence of BKM in the analysis of homogeneous Helmholtz, modified Helmholtz and convection-diffusion problems; in^[5] the BKM is employed to deal with complicated geometry of two and three dimension Helmholtz and convection-diffusion problems; in^[6] membrane vibration under mixed-type boundary conditions is investigated by symmetric boundary knot method; in^[7] the BKM is applied to some inverse Helmholtz problems; in^[8] the BKM solves Poisson equations; in^[9] the BKM calculates Cauchy inverse inhomogeneous Helmholtz equations; in^[10] the BKM simulates the anisotropic problems via the geodesic distance; in^[11]^[12] relationships among condition number, effective condition number, and regularizations are investigated; in^[13] heat conduction in nonlinear functionally graded material is examined by the BKM; in^[14] the BKM is also used to solve nonlinear Eikonal equation.

Related Research Articles

Wave equation Second-order linear differential equation important in physics

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Cutoff frequency frequency response boundary

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

Partial differential equation differential equation that contains unknown multivariable functions and their partial derivatives

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

In mathematics, the eigenvalue problem for the laplace operator is called Helmholtz equation. It corresponds to the linear partial differential equation:

Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. $. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.$

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field $E$ or the magnetic field $B$ , takes the form:

A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior.

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 neighbors. 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 parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments.

The finite element method (FEM) is the most widely used method for solving problems of engineering and mathematical models. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a particular numerical method for solving partial differential equations in two or three space variables. To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretisation in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology.

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.

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 scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a 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.

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.

The Kansa method is a computer method used to solve partial differential equations. Partial differential equations are mathematical models of things like stresses in a car's body, air flow around a wing, the shock wave in front of a supersonic airplane, quantum mechanical model of an atom, ocean waves, socio-economic models, digital image processing etc. The computer takes the known quantities such as pressure, temperature, air velocity, stress, and then uses the laws of physics to figure out what the rest of the quantities should be like a puzzle being fit together. Then, for example, the stresses in various parts of a car can be determined when that car hits a bump at 70 miles per hour.

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.

The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.

In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

References

↑ R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis, 638–650, 1977.
↑ W. Chen and M. Tanaka, A meshfree, exponential convergence, integration-free, and boundary-only RBF technique, Computers and Mathematics with Applications, 43, 379–391, 2002.
↑ W. Chen, Symmetric boundary knot method, Engineering Analysis with Boundary Elements, 26(6), 489–494, 2002.
↑ W. Chen and Y.C. Hon, Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz, and convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 192, 1859–1875, 2003.
↑ Y.C. Hon and W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems with complicated geometry, International Journal for Numerical Methods in Engineering, 1931-1948, 56(13), 2003.
↑ X.P. Chen, W.X. He and B.T. Jin, Symmetric boundary knot method for membrane vibrations under mixed-type boundary conditions, International Journal of Nonlinear Science and Numerical Simulation, 6, 421–424, 2005.
↑ B.T. Jing and Z. Yao, Boundary knot method for some inverse problems associated with the Helmholtz equation, International Journal for Numerical Methods in Engineering, 62, 1636–1651, 2005.
↑ W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for Poisson equations, Engineering Analysis with Boundary Elements, 29(8), 756–760, 2005.
↑ B.T. Jin, Y. Zheng, Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Engineering Analysis with Boundary Elements, 29, 925–935, 2005.
↑ B.T. Jin and W. Chen, Boundary knot method based on geodesic distance for anisotropic problems, Journal of Computational Physics, 215(2), 614–629, 2006.
↑ F.Z. Wang, W. Chen, X.R. Jiang, Investigation of regularized techniques for boundary knot method. International Journal for Numerical Methods in Biomedical Engineering, 26(12), 1868–1877, 2010
↑ F.Z. Wang, Leevan L, W. Chen, Effective condition number for boundary knot method. CMC: Computers, Materials, & Continua, 12(1), 57–70, 2009
↑ Z.J. Fu; W. Chen, Q.H Qin, Boundary knot method for heat conduction in nonlinear functionally graded material, Engineering Analysis with Boundary Elements, 35(5), 729–734, 2011.
↑ D. Mehdi and S. Rezvan, A boundary-only meshfree method for numerical solution of the Eikonal equation, Computational Mechanics, 47, 283–294, 2011.

Related website

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis, 638–650, 1977.

[2] W. Chen and M. Tanaka, A meshfree, exponential convergence, integration-free, and boundary-only RBF technique, Computers and Mathematics with Applications, 43, 379–391, 2002.

[3] W. Chen, Symmetric boundary knot method, Engineering Analysis with Boundary Elements, 26(6), 489–494, 2002.

[4] W. Chen and Y.C. Hon, Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz, and convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 192, 1859–1875, 2003.

[5] Y.C. Hon and W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems with complicated geometry, International Journal for Numerical Methods in Engineering, 1931-1948, 56(13), 2003.

[6] X.P. Chen, W.X. He and B.T. Jin, Symmetric boundary knot method for membrane vibrations under mixed-type boundary conditions, International Journal of Nonlinear Science and Numerical Simulation, 6, 421–424, 2005.

[7] B.T. Jing and Z. Yao, Boundary knot method for some inverse problems associated with the Helmholtz equation, International Journal for Numerical Methods in Engineering, 62, 1636–1651, 2005.

[8] W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for Poisson equations, Engineering Analysis with Boundary Elements, 29(8), 756–760, 2005.

[9] B.T. Jin, Y. Zheng, Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Engineering Analysis with Boundary Elements, 29, 925–935, 2005.

[10] B.T. Jin and W. Chen, Boundary knot method based on geodesic distance for anisotropic problems, Journal of Computational Physics, 215(2), 614–629, 2006.

[11] F.Z. Wang, W. Chen, X.R. Jiang, Investigation of regularized techniques for boundary knot method. International Journal for Numerical Methods in Biomedical Engineering, 26(12), 1868–1877, 2010

[12] F.Z. Wang, Leevan L, W. Chen, Effective condition number for boundary knot method. CMC: Computers, Materials, & Continua, 12(1), 57–70, 2009

[13] Z.J. Fu; W. Chen, Q.H Qin, Boundary knot method for heat conduction in nonlinear functionally graded material, Engineering Analysis with Boundary Elements, 35(5), 729–734, 2011.

[14] D. Mehdi and S. Rezvan, A boundary-only meshfree method for numerical solution of the Eikonal equation, Computational Mechanics, 47, 283–294, 2011.