The method of fundamental solutions for oscillatory and porous buoyant flows

Accurate solutions of oscillatory Stokes flows in convection and convective flows in porous media are studied using the method of fundamental solutions (MFS). In the solution procedure, the flows are represented by a series of fundamental solutions where the intensities of these sources are determined by the collocation on the boundary data. The fundamental solutions are derived by transforming the governing equation into the product of harmonic and Helmholtz-type operators, which can be classified into three types depending on the oscillatory frequencies of temperature field. All the velocities, the pressure, and the stresses corresponding to the fundamental solutions are expressed explicitly in tensor forms for all the three cases. Three numerical examples were carried out to validate the proposed fundamental solutions and numerical schemes. Then, the method was also applied to study exterior flows around a sphere. In these studies, we derived the MFS formulas of drag forces. Numerical results were compared accurately with the analytical solutions, indicating the ability of the MFS for obtaining accurate solutions for problems with smooth boundary data. This study can also be treated as a preliminary research for nonlinear convective thermal flows if the particular solutions of the operators can be supplied, which are currently under investigations.     

Dipoles formulation for the method of fundamental solutions applied to potential problems

This paper generalizes and extends the traditional source formulation within the method of fundamental solutions (MFS) to use dipoles (or even series of multi-poles). In this paper, two-dimensional potential problems are considered. The present formulation is formulated by taking the limiting case of two adjacent sources. The necessary kernels are derived in explicit forms. Some numerical examples, covering different cases of boundary conditions and geometry, are solved. Several parametric studies are presented to demonstrate different configurations for placement of sources (monopoles or dipoles). The accuracy of the presented new formulation is verified by comparing its results to those obtained from other numerical and analytical methods. The main advantage of the present formulation lies in decreasing the oscillation in the sources intensity and reducing the number of sources required which is useful especially in concave and narrow geometries.     

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

The authors describe a meshless method for solving three-dimensional nonstationary heat conduction problems in anisotropic materials. A combination of dual reciprocity method using anisotropic radial basis function and method of fundamental solutions is used to solve the boundary-value problem. The method of fundamental solutions is used to obtain the homogenous part of the solution; the dual reciprocity method with the use of anisotropic radial basis functions allows obtaining a partial solution. The article shows the results of numerical solutions of two benchmark problems obtained by the developed numerical method; average relative, average absolute, and maximum errors are calculated.

     

Use of fundamental solutions in the collocation method in axisymmetric elastostatics

A method referred to as the fundamental collocation method is applied to problems of axisymmetric elastostatics. In the method the governing field equations are satisfied exactly using fundamental solutions corresponding to concentrated forces while the boundary conditions are satisfied approximately using an overdeterminate collocation technique. Numerical results are given for two stress concentration problems. The paper is concluded by a critical discussion of the merits of the method.     

Fundamental solutions for the collocation method in three-dimensional elastostatics

A method referred to as the fundamental collocation method is applied to traction problems of three-dimensional linear isotropic elastostatics. In the method the governing equations are satisfied exactly using fundamental solutions corresponding to concentrated forces, while the boundary conditions are satisfied approximately using an overdeterminate collocation technique. Numerical results are given for three sample problems. The paper is concluded with a critical discussion of the merits of the method.     

Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems

In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δ ^ℓ u=f, ℓ∈ℕ, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125–142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.     

A method of fundamental solutions for two-dimensional heat conduction

We investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend the MFS proposed in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem. 32 (2008), pp. 697–703] for one-dimensional heat conduction with the sources placed outside the space domain of interest, to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost.     

An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems

This paper presents an operator splitting-radial basis function (OS-RBF) method as a generic solution procedure for transient nonlinear Poisson problems by combining the concepts of operator splitting, radial basis function interpolation, particular solutions, and the method of fundamental solutions. The application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit Adams-Bashforth time marching for half the time step. This leaves a nonhomogeneous, modified Helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step. The resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions. Radial basis functions are used to construct approximate particular solutions, and a grid-free, dimension-independent method with high computational efficiency is obtained. This method is demonstrated for some prototypical nonlinear Poisson problems in heat and mass transfer and for a problem of transient convection with diffusion. The results obtained by the OS-RBF method compare very well with those obtained by other traditional techniques that are computationally more expensive. The new OS-RBF method is useful for both general (irregular) two- and three-dimensional geometry and provides a mesh-free technique with many mathematical flexibilities, and can be used in a variety of engineering applications.     

Particular solutions of products of Helmholtz-type equations using the Matern function

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.     

Boundary element solution for elastic orthotropic half-plane problems

This paper presents the boundary element solution for orthotropic half-plane problems. The complete fundamental solutions due to unit point loads within the half-plane are given. The boundary integral formulation using these fundamental solutions is presented. Expressions for stresses and displacements at internal points are also given. This formulation is applied to some classical problems. This solution procedure is highly accurate and computationally more efficient than the boundary element formulation using the Kelvin fundamental solution for orthotropic half-plane problems.     

Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem

The paper considers the determination of heat sources in unsteady 2-D heat conduction problem. The determination of the strength of a heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the θ-method with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem the Tikhonov regularization method based on SVD decomposition was used. In order to determine the optimum value of the regularization parameter the L-curve criterion was used. For testing purposes of the proposed algorithm the 2-D inverse boundary-initial-value problems in square region Ω with the known analytical solutions are considered. The numerical results show that the proposed method is easy to implement and pretty accurate. Moreover the accuracy of the results does not depend on the value of the θ parameter and is greater in the case of the identification of the temperature field than in the case of the identification of the heat sources function.     

Plate bending by a boundary point method

The problem of linear elastic plate bending is solved by a boundary point method. Four fundamental homogeneous solutions for each of a number of sources, which are situated outside the plate, are superimposed and combined with appropriate particular solutions. Each source point is associated with a boundary point, which may be clamped or simply supported. At each boundary point, four edge conditions are enforced which allow the scalar coefficients, introduced in the superposition process, to be determined, and, hence, the plate displacement and stress solution to be obtained. Four non-rectangular example plates are considered, under uniform and hydrostatic loading, for which no internal sub-division is required, and it is demonstrated that accurate solutions may be obtained with 10–14 boundary points.     

Meshless method to solve nonstationary heat conduction problems using atomic radial basis functions

The authors consider a meshless method to solve 3D nonstationary boundary-value heat conduction problems. It is implemented through an iterative scheme based on a combination of the double substitution method and the method of fundamental solutions with the use of atomic radial basis functions. The approaches to the visualization of the desired solution are considered.     

Efficient MFS Algorithms for Problems in Thermoelasticity

We propose efficient fast Fourier transform (FFT)-based algorithms using the method of fundamental solutions (MFS) for the numerical solution of certain problems in planar thermoelasticity. In particular, we consider problems in domains possessing radial symmetry, namely disks and annuli and it is shown that the MFS matrices arising in such problems possess circulant or block-circulant structures. The solution of the resulting systems is facilitated by appropriately diagonalizing these matrices using FFTs. Numerical experiments demonstrating the applicability of these algorithms are also presented.     

Boundary element analysis of linear viscoelastic problems using realistic relaxation functions

This paper presents an efficient method for the stress analysis of realistic viscoelastic solids by the time-domain boundary element method. The fundamental solutions and stress kernels are obtained using the elastic-viscoelastic correspondence principle. Since it is inconvenient to obtain the Laplace transform of the relaxation functions of realistic viscoelastic solids, the method of collocation has been employed and the relaxation function has been expanded in a sum of exponentials. Numerical results of example problems show the effectiveness and applicability of the proposed method.     

Beam and plate stability by boundary elements

The direct boundary element method is used for the linear elastic stability analysis of Bernoulli-Euler beams and Kirchhoff thin plates. The formulation is based on the reciprocal work theorem of Betti and utilizes either fundamental solutions which incorporate the effect of axial and in-plane forces on bending, or fundamental solutions which correspond to pure flexure. In the former case. only a boundary discretization of the structure is required, while in the latter case discretization of the boundary as well as of the interior is necessary. However, the fundamental solutions in the latter case are less complicated than the ones in the former case. Numerical examples are subsequently presented to illustrate the methodology. The basic conclusion is that the simpler fundamental solutions are adequate and, by virtue of being more general, greatly expand the versatility of the boundary element method.     

Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition

Applications of the multidomain Local Fourier Basis method [1], for the solution of PDEs on parallel computers are described. The present approach utilizes, in an explicit way, the rapid (exponential) decay of the fundamental solutions of elliptic operators resulting from semi-implicit discretizations of parabolic time-dependent problems. As a result, the global matching relations for the elemental solutions are decoupled into local interactions between pairs of solutions in neighboring domains. Such interactions require only local communications between processors with short communication links. Thus the present algorithm overcomes the global coupling, inherent both in the use of the spectral Fourier method and implicit time discretization scheme.This research is supported partly by a grant from the French-Israeli Binational Foundation for 1991–1992.     

Application of the finite method in micromechanical analyses of creep fracture problems

The finite element method is applied to determine the generalized stress intensity factors for two problems fundamental in the micromechanical study of intergranular creep fracture. Special techniques to simulate multipoint constraints arising from symmetry and periodicity, and a singularity element have been developed and proved effective. The good agreements of finite element results with local field solutions are discussed.     

对极几何估计的鲁棒性新算法

以８参数模型的Ｆ矩阵为基础,深入研究的Ｆ阵的稳定性求解中的若干问题,将求解对极点的问题归结为求解二元六次非线性方程的最优解问题,并将所求最优解作为进上步计算的最佳修造对极点,最后以余差最小准则获得Ｆ阵的全局最优解,实验结果表明,该方法不但使对极点的稳定性有较大提高,也能高精度地估计祟Ｆ阵。