MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas

This paper investigates the solitary wave solutions of the two-dimensional regularized long-wave equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas. The main idea behind the numerical solution is to use a combination of boundary knot method and the analog equation method. The boundary knot method is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution, the boundary knot method uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to method of fundamental solution, the radial basis function is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method. According to the analog equation method, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Furthermore, in order to show the efficiency and accuracy of the proposed method, the present work is compared with finite difference scheme. The new method is analyzed for the local truncation error and the conservation properties. The results of several numerical experiments are given for both the single and double-soliton waves.     

Variably saturated flow described with the anisotropic Lattice Boltzmann methods

This paper addresses the numerical solution of highly nonlinear parabolic equations with Lattice Boltzmann techniques. They are first developed for generic advection and anisotropic dispersion equations (AADE). Collision configurations handle the anisotropic diffusion forms by using either anisotropic eigenvalue sets or anisotropic equilibrium functions. The coordinate transformation from the orthorhombic (rectangular) discretization grid to the cuboid computational grid is equivalent for the AADE to the anisotropic rescaling of the convection/diffusion terms. The collision components (eigenvalues and/or equilibrium functions) become discontinuous on the boundaries of the computational sub-domains which have different space scaling factors. We focus on the analysis of the boundary continuity conditions by using anisotropic LB techniques. The developed schemes are applied to Richards’ equation for variably saturated flow. The anisotropy of the Richard’s equation originates from distinct soil conductivity values in both the vertical and horizontal directions. The method should on the interface between the heterogeneous layers maintain the continuity of the normal component of the Darcy’s velocity (total flux). Also the method should accommodate steep jumps of the moisture content variable (conserved quantity) resulting from the continuity of the pressure variable, a given non-linear function of the moisture content. The coupling between heterogeneity and the anisotropy is examined by using the distinct space steps in neighboring layers and tested against uniform grid solutions. Different formulations of the Richard’s equation illustrate the construction of distinct diffusion forms and their integral transforms via specification of the equilibrium components. Integral transforms are used to overcome the difficulties coming from the rapid change of the main variables on sharp fronts. The numerical assessment of the stability criteria and the interface boundary conditions extend the analysis of the Lattice Boltzmann schemes to nonlinear problems with discontinuous coefficients.     

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.     

Recovering temperature-dependent heat conductivity in 2D and 3D domains with homogenization functions as the bases

The paper solves the parameters identification problem in a nonlinear heat equation with homogenization functions as the bases, which are constructed from the boundary data of the temperature in the 2D and 3D space-time domains. To satisfy the over-specified Neumann boundary condition, a linear equations system is derived and then used to determine the expansion coefficients of the solution. Then, after back substituting the solution and collocating points to satisfy the governing equations, the space-time-dependent and temperature-dependent heat conductivity functions in 2D and 3D nonlinear heat equations are identified by solving other linear systems. The novel methods do not need iteration and solving nonlinear equations, since the unknown heat conductivities are retrieved from the solutions of linear systems. The solutions and the heat conductivity functions recovered are quite accurate in the entire space-time domain. We find that even for the inverse problems of nonlinear heat equations, the homogenization functions method is easily used to recover 2D and 3D space-time-dependent and temperature-dependent heat conductivity functions. It is interesting that the present paper makes a significant contribution to the engineering and science in the field of inverse problems of heat conductivity, merely solving linear equations and without employing iteration and solving nonlinear equations to solve nonlinear inverse problems.

     

Application of the method of moments to acoustic scattering from fluid-filled cylinders using three different formulations

In this work, we present three different formulations, namely the pressure field integral equation formulation (PFIE), the velocity field integral equation formulation (VFIE), and the combined field integral equation formulation (CFIE) for solving acoustic scattering problems associated with two-dimensional fluid-filled bodies of arbitrary cross-section. In particular using the boundary conditions on the surface of the body, two equivalent problems, each valid for the outside and inside regions of the scatterer, are derived. By properly selecting the associated equations for these equivalent problems, the three different formulations are derived. The PFIE, VFIE, and CFIE are then solved by approximating the cylindrical cross-section by linear segments and employing the method of moments. Further, it is shown that the moment matrices generated by the PFIE and VFIE are ill-conditioned at resonant frequencies of the cylinder, whereas the CFIE generates a well-conditioned matrix at all frequencies. The solution techniques presented in this work are simple, efficient and applicable to truly arbitrary geometries. Numerical results are presented for certain canonical shapes and compared with other available data.     

DRM multidomain mass conservative interpolation approach for the BEM solution of the two-dimensional navier-stokes equations

The dual reciprocity method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach, the nonlinear terms are usually approximated by an interpolation applied to the convective terms of the Navier-Stokes equations. In this paper, we introduce a radial basis function interpolation scheme for the velocity field, that satisfies the continuity equation (mass conservative). The proposed method performs better than the classical interpolation used in the DRM approach to represent such a field. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the nonlinear terms in the Navier-Stokes equations.     

Rigid body mode and spurious mode in the dual boundary element formulation for the Laplace problems

In this paper, the general formulation for the static stiffness is analytically derived using the dual integral formulations. It is found that the same stiffness matrix is derived by using the integral equation no matter what the rigid body mode and the complementary solutions are superimposed in the fundamental solution. For the Laplace problem with a circular domain, the circulant was employed to derive the stiffness analytically in the discrete system. In deriving the static stiffness, the degenerate scale problem occurs when the singular influence matrix can not be inverted. The Fredholm alternative theorem and the SVD updating technique are employed to study the degenerate scale problem mathematically and numerically. The direct treatment in the matrix level is achieved to deal with the degenerate scale problems instead of using a modified fundamental solution. The addition of a rigid body term in the fundamental solution is found to shift the zero singular value for the singular matrix without disturbing the stiffness.     

A new complex-valued formulation and eigenvalue analysis of the Helmholtz equation by boundary element method

By considering the close relationship between the multiple reciprocity boundary element formulation and that of the fundamental solution of the Helmholtz differential operator, we present a new complex-valued integral equation formulation for the eigenvalue analysis of the scalar-valued Helmholtz equation. Eigenvalues are determined as local minima of the determinant of the coefficient matrix of the discretized equation iteratively by the Newton scheme. The necessary recurrence formula is derived and computed with high efficiency, due to polynomial representation of the matrix components. Some example computations demonstrate the utility of the proposed formulation and eigenvalue determination scheme, and construction of adaptive boundary elements for the eigenvalue determination is attempted.     

基于结合扩展精度技术的基本解方法的非线性功能梯度材料热传导问题求解

采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量,将时域问题转化为频域问题求解;采用基本解方法计算得到高精度的频域数值解,再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例,分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点,能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.     

Boundary element analysis of nonlinear transient heat conduction problems

This paper is concerned with a boundary element formulation and its numerical implementation for the nonlinear transient heat conduction problems with temperature-dependent material properties. By using the Kirchhoff transformation for the material properties a set of pseudo-linear integral equations is obtained in space and time for the fully three-dimensional nonlinear problems under consideration. The resulting boundary integral equations are solved by means of the usual boundary element method. Emphasis is placed on the numerical solution procedure employing constant elements with respect to time. It is shown that in this case there is no need to evaluate the domain integrals resulting from the nonlinearity of the problem. Finally, the powerful usefulness of the proposed method is demonstrated through the numerical computation of several sample problems.     

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.

     

Dynamic inelastic structural analysis by boundary element methods

Summary Boundary element methodologies for the determination of the response of inelastic two-and three-dimensional solids and structures as well as beams and flexural plates to dynamic loads are briefly presented and critically discussed. Elastoplastic and viscoplastic material behaviour in the framework of small deformation theories are considered. These methodologies can be separated into four main categories: those which employ the elastodynamic fundamental solution in their formulation, those which employ the elastostatic fundamental solution in their formulation, those which combine boundary and finite elements for the creation of an efficient hybrid scheme and those representing special boundary element techniques. The first category, in addition to the boundary discretization, requires a discretization of those parts of the interior domain expected to become inelastic, while the second category a discretization of the whole interior domain, unless the inertial domain integrals are transformed by the dual reciprocity technique into boundary ones, in which case only the inelastic parts of the domain have to be discretized. The third category employs finite elements for one part of the structure and boundary elements for its remaining part in an effort to combine the advantages of both methods. Finally, the fourth category includes special boundary element techniques for inelastic beams and plates and symmetric boundary element formulations. The discretized equations of motion in all the above methodologies are solved by efficient step-by-step time integration algorithms. Numerical examples involving two-and three-dimensional solids and structures and flexural plates are presented to illustrate all these methodologies and demonstrate their advantages. Finally, directions for future research in the area are suggested.     

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.     

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.     

Nonlinearity problem in the numerical solution of superstiff Cauchy problems

For the numerical solution of Cauchy stiff initial problems, many schemes have been proposed for ordinary differential equation systems. They work well on linear and weakly nonlinear problems. The article presents a study of a number of well-known schemes on nonlinear problems (which include, for example, the problem of chemical kinetics). It is shown that on these problems, the known numerical methods are unreliable. They require a sufficient step reducing at some critical moments, and to determine these moments, sufficiently reliable algorithms have not been developed. It is shown that in the choice of time as an argument, the difficulty is associated with the boundary layer. If the length of the integral curve arc is taken as an argument, difficulties are caused by the transition zone between the boundary layer and regular solution.     

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.     

Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems

For a given mixed type potential problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions. Numerical results show that the conventional boundary integral formulation yields incorrect potential and flux results when the distance scale in the fundamental solution approaches its degenerate value. Such a kind of non-equivalence of the conventional boundary integral equation can be eliminated by the use of the necessary and sufficient boundary integral formulation which always ensures the equivalence of solutions.