A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

Helmholtz方程的楔形基区域分解法

基于楔形基函数和无网格配点法,提出了一种求解Helmholtz型方程区域分解法。该方法克服了在求解大规模问题时用一般的全域配点法所带来的配置矩阵为非对称满阵,且高度病态的问题。通过数值结果表明,该算法在求解Helmholtz型方程降低系数矩阵条件数的同时,也能够降低误差,并达到满意的收敛效果。     

Boundary value identification of inverse Cauchy problems in arbitrary plane domain through meshless radial point Hermite interpolation

This article presents an efficient method to solve elliptic partial differential equations which are the nucleus of several physical problems, especially in the electromagnetic and mechanics, such as the Poisson and Laplace equations, while the subject is to recover a harmonic data from the knowledge of Cauchy data on some part of the boundary of the arbitrary plane domain. This method is a local nodal meshless Hermite-type collocation technique. In this method, we use the radial-based functions to call out the shape functions that form the local base in the vicinity of the nodal points. We also take into account the Hermit interpolation technique for imposing the derivative conditions directly. The proposed technique called pseudospectral meshless radial point Hermit interpolation is applied on some illustrative examples by adding random noises on source function and reliable results are observed.

     

A BEM-based domain meshless method for the analysis of Mindlin plates with general boundary conditions

In this paper, a BEM-based domain meshless method is developed for the analysis of moderately thick plates modeled by Mindlin’s theory which permits the satisfaction of three physical conditions along the plate boundary. The presented method is achieved using the concept of the analog equation of Katsikadelis. According to this concept, the original governing differential equations are replaced by three uncoupled Poisson’s equations with fictitious sources under the same boundary conditions. The fictitious sources are established using a technique based on BEM and approximated by radial basis functions series. The solution of the actual problem is obtained from the known integral representation of the potential problem. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The presented method has the advantages of the BEM in the sense that the discretization and integration are performed only on the boundary, and consequently Mindlin plates with general boundary conditions can be analyzed without difficulty. To illustrate the effectiveness, applicability as well as accuracy of the method, numerical results of various example problems are presented.     

A meshless average source boundary node method for steady-state heat conduction in general anisotropic media

The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.     

Global and local Trefftz boundary integral formulations for sound vibration

The sound-pressure field harmonically varying in time is governed by the Helmholtz equation. The Trefftz boundary integral equation method is presented to solve two-dimensional boundary value problems. Both direct and indirect BIE formulations are given. Non-singular Trefftz formulations lead to regular integrals counterpart to the conventional BIE with the singular fundamental solution. The paper presents also the local boundary integral equations with Trefftz functions as a test function. Physical fields are approximated by the moving least-square in the meshless implementation. Numerical results are given for a square patch test and a circular disc.     

A generalized finite-difference method based on minimizing global residual

An improved generalized finite-difference method is proposed in this paper, as an alternative meshless method to solve differential equations. The method establishes discrete equations by minimizing a global residual. A general frame for constructing difference schemes is first described. As one choice the moving least square method is used in this paper. Compared with other generalized finite-difference methods, the improved method yields a set of discrete equations having the favorable properties such as symmetric, positive definite and well conditioned. Compared with meshless methods based on a variational principle or a weak form, the method described in this paper does not need a numerical integration and thus provides an alternative way to avoid the difficulties in implementing a numerical integration. In the proposed method there is no such inconvenience in applying essential boundary conditions as commonly encountered in other meshless methods. Numerical examples show that the improved method has a high convergence rate and can produce accurate results even with a coarse mesh.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

The discrete collocation method for Fredholm–Hammerstein integral equations based on moving least squares method

The purpose of this paper is to investigate the discrete collocation method based on moving least squares (MLS) approximation for Fredholm–Hammerstein integral equations. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.     

On a Galerkin boundary node method for potential problems

A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.     

Moving-boundary problems solved by adaptive radial basis functions

The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach.     

Pricing American options under jump-diffusion models using local weak form meshless techniques

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.     

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.     

Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.     

Data discovering of inverse Robin boundary conditions problem in arbitrary connected domain through meshless radial point Hermite interpolation

In this paper, a suitable method is presented to treat the partial derivative equations, especially the Laplace equation having the Robin boundary conditions. These equations come from classical physics, especially the branch of thermodynamics, and have an efficient role in the field of heat and temperature. Our motivation is to reset a harmonic data obtained from Robin’s conditions in the arbitrary plane domain particularly on its boundaries. The applied method is a nodal Hermite meshless collocation technique at which it is formed of radial basis functions to get out the shape functions which is the key to construct the local bases in the neighborhoods of the nodal points. Moreover, by taking into consideration the Hermite interpolation technique, we can impose the boundary conditions directly, the named technique is called “MRPHI,” meshless radial point Hermite interpolation, and it is done on some examples so that trustworthy results are obtained.

     

Radial basis function meshless method for the steady incompressible Navier–Stokes equations

A meshless collocation method based on radial basis functions is proposed for solving the steady incompressible Navier–Stokes equations. This method has the capability of solving the governing equations using scattered nodes in the domain. We use the streamfunction formulation, and a trust-region method for solving the nonlinear problem. The no-slip boundary conditions are satisfied using a ghost node strategy. The efficiency of this method is demonstrated by solving three model problems: the driven cavity flows in square and rectangular domains and flow over a backward-facing step. The results obtained are in good agreement with benchmark solutions.     

More accurate results for two-dimensional heat equation with Neumann’s and non-classical boundary conditions

In this article, recently proposed spectral meshless radial point interpolation (SMRPI) method is applied to the two-dimensional diffusion equation with a mixed group of Dirichlet’s and Neumann’s and non-classical boundary conditions. The present method is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high-order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach. A comparison study of the efficiency and accuracy of the present method and other meshless methods is given by applying on mentioned diffusion equation. Stability and convergence of this meshless approach are discussed and theoretically proven. Convergence studies in the numerical examples show that SMRPI method possesses excellent rates of convergence.     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.     

Boundary point method for linear elasticity using constant and quadratic moving elements

Based on the boundary integral equations and stimulated by the work of Young et al. [J Comput Phys 2005;209:290–321], the boundary point method (BPM) is a newly developed boundary-type meshless method enjoying the favorable features of both the method of fundamental solution (MFS) and the boundary element method (BEM). The present paper extends the BPM to the numerical analysis of linear elasticity. In addition to the constant moving elements, the quadratic moving elements are introduced to improve the accuracy of the stresses near the boundaries in the post processing and to enhance the analysis for thin-wall structures. Numerical tests of the BPM are carried out by benchmark examples in the two- and three-dimensional elasticity. Good agreement is observed between the numerical and the exact solutions.     

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.