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.     

The dual reciprocity boundary element formulation for nonlinear diffusion problems

This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.     

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.     

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 method of fundamental solutions for three-dimensional elastostatics problems

We consider the application of the method of fundamental solutions to isotropic elastostatics problems in three space dimensions. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity, which are expressed in terms of sources placed outside the domain of the problem under consideration. The final positions of the sources and the coefficients of the fundamental solutions are determined by enforcing the satisfaction of the boundary conditions in a least squares sense. The applicability of the method is demonstrated on two test problems. The numerical experiments indicate that accurate results can be obtained with relatively few degrees of freedom.     

Analytical solutions of some steady-state electrical problems in the rectangular domain

Analytical solutions for the electrical potential are developed within a rectangular domain under dirichlet and Neumann boundary conditions. Examples enable us to examine currently used numerical techniques. All solutions are applicable to the boundary element method as typical fundamental solutions.     

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.

     

Edge function analysis of anisotropic materials with holes and cracks

The edge function method for anisotropic elasticity is based on the complex variable formulation and on the superposition of analytical solutions to the field equations. Herein the method is used in the study of anisotropic materials with holes and cracks. Several examples are studied in order to provide confidence in the model through literature comparisons. The effects of anisotropy and ellipticity of the cavities are examined. The potential of the method for assessing complex stress states is demonstrated through a numerical investigation of a sheet with several cracks and cavities.     

Applications of the complete multiple reciprocity method for solving the 1D Helmholtz equation of a semi-infinite domain

In this paper, the complete multiple reciprocity method is adopted to solve the one-dimensional (1D) Helmholtz equation for the semi-infinite domain. In order to recover the information that is missing when the conventional multiple reciprocity method is used, an appropriate complex number in the zeroth order fundamental solution is added such that the kernels derived using this proposed method are fully equivalent to those derived using the complex-valued formulation. Two examples including the Dirichlet and Neumann boundary conditions are investigated to show the validity of the proposed method analytically and numerically. The numerical results show good agreement with the analytical solutions.     

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.     

Application of the extended Kantorovich method to the bending of variable thickness plates

The governing equations of the classical plate theory for a uniform or a unidirectional variable thickness rectangular plate under transverse applied loading are solved by means of the extended Kantorovich method. The plate may be either simply supported or clamped along the edges. The solution procedure is iterative and must be carried out numerically. This necessitates the calculation of the two missing pieces of boundary data along the edges of the plate. The missing boundary data are determined utilizing the method of adjoints of the shooting method. The numerical values of the deflection and bending moments for uniform and variable thickness plates are compared with those from the exact solutions and finite element analysis, respectively.     

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 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.     

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.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

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.     

Solution of Polynomial Systems Derived from Differential Equations

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.     

A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations

This paper presents a new family of fourth- compact finite difference schemes for the numerical solution of three-dimensional nonlinear biharmonic equations using coupled approach. The numerical solutions of unknown variable and its first- derivatives as well as v(=Δ u) are obtained not only in the interior but also at the boundary. A prominent contribution of this work is that the boundary conditions for the variable v are approximated more accurately, which plays an important role for the efficiency of calculation. Finally, numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these schemes, including the steady Navier–Stokes equation in terms of vorticity-stream function formulation.