Boundary Meshless Method Research Articles

Formulation of the method of fundamental solutions for two-phase Stokes flow

The method of fundamental solutions with a subdomain technique is used for the solution of the free boundary problem associated with a two-phase Stokes flow in a 2D geometry. The solution procedure is based on the collocation of the boundary conditions with the Stokeslets. It is formulated for the flow of unmixing fluids in contact, where the velocity, pressure field, and position of the free boundary between the fluids must be determined. The standard formulation of the method of fundamental solutions is, for the first time, upgraded for the case with mixed velocity and pressure boundary conditions and verified on a T-splitter single-phase flow with unsymmetric pressure boundary conditions. The standard control volume method is used for the reference solution. The accurate evaluation of the velocity derivatives, which are required to calculate the balance of forces at the free boundary between the fluids, is achieved in a closed form in contrast to previous numerical attempts. An algorithm for iteratively calculating the position of the free boundary that involves displacement, smoothing and repositioning of the nodes is elaborated. The procedure is verified for a concurrent flow of two fluids in a channel. The velocity and velocity derivatives show fast convergence to the analytical solution. The developed boundary meshless method is easy to code, accurate and computationally efficient since only collocation at the fixed and free boundaries is needed.

The enhanced boundary knot method with fictitious sources for solving Helmholtz-type equations

The boundary knot method (BKM) is a boundary meshless method for solving homogeneous boundary value problems. The fact that the BKM uses non-singular general solution without the need of source points outside the domain is at the expense of the accuracy when compared to the well-known method of fundamental solutions. In this paper, a new approach called the enhanced BKM (EBKM) is proposed to greatly improve the accuracy of the BKM by introducing fictitious sources which can be placed inside and/or outside the domain quite freely. We also adopt various techniques to place these fictitious sources in an optimal way.

Numerical simulation of anti-plane wave propagation in heterogeneous media

This paper formulates a semi-analytical boundary meshless method, the singular boundary method (SBM), to simulate the anti-plane wave motion in the heterogeneous media. Two different continuous inhomogeneity variations are studied, and the corresponding fundamental solutions are derived by means of a simple variable transformation. The derivation reveals that the fundamental solutions are determined by the wavenumbers of the resulted governing equations after the variable transformation. The source singularities of these fundamental solutions are overcome by the novel origin intensity factors (OIFs). The fundamental solutions and OIFs make the SBM applicable for the simulation of the anti-plane wave scattering and diffraction in nonhomogeneous media. Numerical experiments, including the finite and semi-infinite cases with different variations and boundary conditions, are presented to illustrate the convergence, accuracy and effectiveness of the proposed SBM.

A boundary meshless method for dynamic coupled thermoelasticity problems

This work develops the boundary knot method (BKM) for two-dimensional (2D) coupled thermoelasticity problems in the frequency domain. By taking the non-singular general solution satisfying the governing equations as the basis function, the BKM does not require domain discretization. Nevertheless, the non-singular solution for the considered problems is absent. The Helmholtz decomposition and eigen-analysis are introduced to firstly derive the non-singular general solution, and thus the BKM can be formulated in terms of the discretized boundary points. In the final, the exponential window method (EWM) in conjunction with the BKM is employed to perform the transient analysis. The accuracy and feasibility of the proposed method are presented via numerical examples.

A meshless singular boundary method for elastic wave propagation in 2D partially saturated poroelastic media

This paper presents a boundary meshless method, singular boundary method (SBM), in conjunction with the exponential window method (EWM) to simulate the dynamic response of the 2D partially saturated poroelastic media. The problem is firstly solved in the frequency-domain resulted from the time-domain governing equations via the Fourier transformation. Then the SBM approximates the solutions with a linear combination of fundamental solutions with respect to the source points on the physical boundary. To successfully apply the SBM, two issues are addressed. Firstly, the kernel functions, namely the fundamental solutions of frequency-domain governing equations, are derived based on the eigen-analysis. Secondly, the source singularities of the fundamental solutions are desingularized with analytical formulas. After obtaining the SBM formulation, the time-domain solutions can be obtained by the EWM, where the frequency domain windowing technique is introduced to stabilize the oscillations caused by damping parameters. Four numerical examples are carried out to show the validity and accuracy of the proposed method.

Regularized singular boundary method for groundwater flow in a cofferdam

The boundary integral methods form one group of methods for solving differential equations. The boundary element method (BEM) is the basic method of this group. However, it requires the boundary mesh of elements and the evaluation of improper singular integrals, that arise due to fundamental solution singularity. Therefore, boundary meshless methods have recently have come into focus to remove these shortcomings. One of the most promising boundary collocation numerical schemes is the singular boundary method (SBM). To tackle the singularity of the fundamental solution, this method adopts the concept of original intensity factors (OIFs). The application of the proposed SBM scheme to groundwater flow in the cofferdam structure is presented and compared to the finite element method (FEM) solution.

The MFS and MAFS for solving Laplace and biharmonic equations

The method of fundamental solutions (MFS) has been known as an effective boundary meshless method for solving homogeneous differential equations with smooth boundary conditions and boundary shapes. Despite many attractive features of the MFS, the determination of the source location and the boundaries with sharp corners still pose a certain degree of challenges. In this paper, we revisit another powerful boundary method, the method of approximate fundamental solutions (MAFS), which approximates the fundamental solution using trigonometric functions. In the MAFS, the fundamental solutions for various governed equations can be easily constructed. The placement of the source points is also simple. In this paper, we will apply the MAFS for solving the Laplace equation with non-harmonic boundary conditions and the biharmonic equation with non-biharmonic boundary conditions with highly irregular or non-smooth domains. We will compare the performance of the MAFS and the MFS in these types of problems.

Non-Singular Method of Fundamental Solutions for Two-dimensional Thermoelasticity

A framework for simulation of thermomechanical processing of microstructures by the method of fundamental solutions, free of artificial boundary, is shown. The formulation of the method for two-dimensional thermal and mechanical models is presented. In particular, the formulation for elastic and thermo-elastic problems is discussed. In the thermal and mechanical models, the concentrated point sources are replaced by the distributed sources over the sphere around the singularity to regularize the singularities. The balance of heat fluxes and forces is used to calculate some of the otherwise singular diagonal coefficients. This procedure enables also to solve a broad spectra of thermomechanical problems. The novel boundary meshless method has been assessed by comparison with the method of fundamental solutions, and analytical solution. The method is easy to code, accurate, and efficient.

Complex variable moving Kriging interpolation for boundary meshless method

In this paper, we proposed the complex variable moving Kriging interpolation (CVMKI) to approximate functions on two-dimensional (2D) boundaries. The CVMKI is based on complex variable theory and the moving Kriging interpolation (MKI). It requires no curvilinear coordinate, and can construct shape functions possessing Kronecker delta function property and partition of unity property. Further, the complex variable boundary node method (CVBNM) is proposed for potential problems based on CVMKI and boundary integration equation (BIE). CVBNM is an efficient and accurate method that can directly impose the boundary conditions. Three 2D example problems are presented to verify the accuracy and efficiency of CVBNM.

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.

Calculation of the Scattering Coefficient of Large Scale Periodic Structure

The periodic structure is common used in engineering to improve the sound diffusion in room acoustics. It usually has large dimensions and it is difficult to calculate the scattering coefficient based on its original scale. In order to represent the scattering coefficients of a large scale periodic structure by those of a smaller one, the relations of the scattering coefficients of the periodic structures with different dimensions are analyzed in this paper. At first, a BMM (Boundary Meshless Method) for calculating the scattering coefficient is derived. The scattering coefficients of the periodic structures which have different numbers of sub-structures are calculated and compared. The computation results of different sub-structure numbers show that the periodic structure with 15 sub-structures can represent the samples which have more sub-structures. In addition, it can be proved that the square samples can be represented by those of the small and rectangular ones which have the same numbers of sub-periods. These conclusions not only greatly enhance the computational efficiency, but also show good prospect for fast evaluation of the periodic structures in engineering applications.

Thermal Properties of Short Fibre Composites Modeled by Meshless Method

Computational model using continuous source functions along the fibre axis is presented for simulation of temperature/heat flux in composites reinforced by short fibres with large aspect ratio. The aspect ratio of short fibres reinforcing composite material is often as large as 103 : 1–106 : 1, or even larger. 1D continuous source functions enable simulating the interaction of each fibre with the matrix and also with other fibres. The developed method of continuous source functions is a boundary meshless method reducing the problem considerably comparing to other methods like FEM, BEM, meshless methods, or fast multipole BEM formulation.

The Galerkin boundary node method for magneto-hydrodynamic (MHD) equation

In this research, a variational reformulation for magneto-hydrodynamic (MHD) problem is derived. Existence and uniqueness of the weak solution are discussed. The Galerkin boundary node method is a boundary meshless method which uses MLS basis functions to approximate solution of problems. This paper tries to apply this method for a variational form of the magneto-hydrodynamic (MHD) problem. Numerical experiments reveal that the method is effective and convenient for this problem.

Coupling of polynomial approximations with application to a boundary meshless method

SUMMARYNew bridging techniques are introduced to match high degree polynomials. This permits piecewise resolutions of elliptic partial differential equations (PDEs) in the framework of a boundary meshless method introduced recently. This new meshless method relies on the computation of Taylor series approximations deduced from the PDE, the shape functions being high degree polynomials. In this way, the PDE is solved quasi‐exactly inside the subdomains so that only discretization of the boundary and the interfaces are needed, which leads to small size matricial problems. The bridging techniques are based on the introduction of Lagrange multipliers and a set of collocation points on the boundary and the interfaces. Several numerical applications establish that the method is robust and permits an exponential convergence with the degree. Copyright © 2013 John Wiley & Sons, Ltd.

A FAST METHOD OF FUNDAMENTAL SOLUTIONS FOR SOLVING HELMHOLTZ-TYPE EQUATIONS

The method of fundamental solution (MFS) has been known as a simple and effective boundary meshless method. However, the MFS generates dense square coefficient matrix and thus requires a large amount of computational time and memory storage for solving large-scale problems. The fast multipole method (FMM) is a technique that reduces computational operations and storage requirements for solving such dense matrix equations. This study makes the first attempt to apply the FMM to the MFS calculation of acoustic problems, where the operations are reduced to O(N log N) while O(N2) operations are required for the traditional MFS using the standard iterative methods. Numerical examples with up to 100,000 points are successfully tested on a desktop personal computer. Our results clearly demonstrate efficiency and accuracy of the fast multipole MFS for solving large-scale Helmholtz-type problems.

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

AbstractThis paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.

Convergence analysis and detection of singularities within a boundary meshless method based on Taylor series

We study a meshless method based on Taylor series approximation. This method solves quasi-exactly the Partial Differential Equation (PDE) in the domain. The boundary conditions are applied by using a least square method as proposed by Zhang et al. (2001 [3]), for stabilising collocation method. After showing accuracy and efficiency of the technique, we analyse the convergence of the series. In this aim, we introduce techniques as Domb Sykes plot, Hadamard criterion or Darboux criterion. This analysis leads to the radius of convergence and to the location and the nature of the singularities of the sought solution.

Computational Modelling of Composite Materials Reinforced by Glass Fibers

Abstract Composite materials reinforced by micro particles are one of the topics of interest of researchers. Properties of fiber composites reinforced by long fibers significantly depend on the selection of fiber and matrix. By increasing length of fibers, the reinforcement is more effective at load carrying. In this paper the Method of Continuous Source Functions (MCSF) and Trefftz Radial Basis Functions (TRBF) will be presented. They are boundary meshless methods which do not need any mesh. The TRBF are source functions having their source points outside the domain. Special attention will be given to the application of the TRBF in the form of dipoles to the simulation of composites reinforced by fibres of finite length with large aspect ratio. In linear problems, only nodes on the domain boundaries and a set of source functions in points outside the domain are necessary to satisfy boundary conditions. Finally we will show MCSF to modelling of reinforced composites with glass fiber and epoxy matrix. For the sake of simplicity we consider only a patch of non-overlaying rows of fibers.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.