The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem

We investigate an application of the method of fundamental solutions (MFS) to the time-dependent two-dimensional Cauchy heat conduction problem, which is an inverse ill-posed problem. Data in the form of the solution and its normal derivative is given on a part of the boundary and no data is prescribed on the remaining part of the boundary of the solution domain. To generate a numerical approximation we generalize the work for the stationary case in Marin (2011) [23] to the time-dependent setting building on the MFS proposed in Johansson and Lesnic (2008) [15], for the one-dimensional heat conduction problem. We incorporate Tikhonov regularization to obtain stable results. The proposed approach is flexible and can be adjusted rather easily to various solution domains and data. An additional advantage is that the initial data does not need to be known a priori, but can be reconstructed as well.     

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.     

The method of fundamental solutions for inhomogeneous elliptic problems

We investigate the use of the Method of Fundamental Solutions (MFS) for solving inhomogeneous harmonic and biharmonic problems. These are transformed to homogeneous problems by subtracting a particular solution of the governing equation. This particular solution is taken to be a Newton potential and the resulting homogeneous problem is solved using the MFS. The numerical calculations indicate that accurate results can be obtained with relatively few degrees of freedom. Two methods for the special case where the inhomogeneous term is harmonic are also examined.     

A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

Recently, Hon and Wei proposed a method of fundamental solutions for solving isotropic inverse heat conduction problems (IHCP). It provides an efficient global approximation scheme in both spatial and time domains. In this paper, we try to extend the inherently meshless and integration-free method to solve 2D IHCP in an anisotropic medium. First, we acquire the fundamental solution of the governing equation through variables transformation. Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned. Results for several numerical examples are presented to demonstrate the efficiency of the method proposed. The relationship between the accuracy of the numerical solutions and the value of the parameter T is also investigated.     

Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions

A new method for the boundary element analysis of unsteady heat conduction problems involving non-homogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented. Nonlinear terms are converted to a fictitious heat source and implemented in the present formulation. The domain integrals are efficiently treated by the recently introduced Cartesian transformation method. Similar to the dual reciprocity method, some internal grid points are considered for the treatment of the domain integrals. In the present method, unlike the dual reciprocity method, there is no need to find particular solution for the shape functions in the interpolation computations and the form of the shape functions can be arbitrary and sufficiently complicated. In the present method, at each time step the temperature at boundary nodes and some internal grid points is computed and used as pseudo-initial values for the next time step. Most of the generated matrices are constant at all time steps and computations can be carried out fast. An example with different forms of heat sources is presented to show the efficiency and accuracy of the proposed method.     

The method of fundamental solutions for the inverse space-dependent heat source problem

In this study, the inverse heat source problem in which the heat source is space-dependent is treated. The method proposed in Yan et al. [The method of fundamental solutions for the inverse heat source problem. Eng Anal Boundary Elem 2008;32:216–22] where the heat source is considered to be only time-dependent, is modified so that it can be applied to only space-dependent problems. We have used a new transformation to simplify the problem.     

An element-by-element method for heat conduction CAE including composite problems

Mechanical computer aided engineering (CAE) implies large finite element problems due to the geometric complexity of the ‘true’ 3D designs. Application of the standard finite element technique is not practical for such problems because the direct solution of the global matrix equations is too costly. This paper considers the element-by-element implicit algorithm for the CAE application of transient heat conduction. The direct solution is avoided by an operator splitting or approximate factorization technique. This results in both the execution time and storage requirements for each time step being linearly proportional to the number of elements while retaining unconditional stability. However, the approximate factorization introduces additional truncation error and incorrect jump conditions at material interfaces. Detailed analyses and numerical experiments are carried out in one dimension to assess the nature and mechanism of these inaccuracies. A three dimensional implementation is then compared with one dimensional results. The need for an additional predictor–corrector element-by-element algorithm for 3D composite problems is also presented.     

A boundary element method for the solution of a class of heat conduction problems for inhomogeneous media

A boundary element method is derived for solving the two-dimensional heat equation for an inhomogeneous body subject to suitably prescribed temperature and/or heat flux on the boundary of the solution domain. Numerical results for a specific test problem is given.     

Some heat conduction problems in an inhomogeneous body

Solutions are obtained for the heat conduction equation in the case when the thermal conductivity is a homogeneous function of the coordinates.     

Certain exact solutions to steady-state problems in the theory of heat conduction applied to inhomogeneous bodies

A method is shown of constructing exact analytical solutions to steady-state problems in the theory of heat conduction where the thermal conductivity is a special kind of function of the space coordinates.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 3, pp. 554–556, September, 1972.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

An improved meshless method with almost interpolation property for isotropic heat conduction problems

In the paper an improved element free Galerkin method is presented for heat conduction problems with heat generation and spatially varying conductivity. In order to improve computational efficiency of meshless method based on Galerkin weak form, the nodal influence domain of meshless method is extended to have arbitrary polygon shape. When the dimensionless size of the nodal influence domain approaches 1, the Gauss quadrature point only contributes to those nodes in whose background cell the Gauss quadrature point is located. Thus, the bandwidth of global stiff matrix decreases obviously and the node search procedure is also avoided. Moreover, the shape functions almost possess the Kronecker delta function property, and essential boundary conditions can be implemented without any difficulties. Numerical results show that arbitrary polygon shape nodal influence domain not only has high computational accuracy, but also enhances computational efficiency of meshless method greatly.     

An explicit numerical method for solving transient combined heat conduction and convection problems

This paper describes an explicit numerical method for solving transient combined heat conduction and convection problems. Applications include the cooling of many types of engineering plant; for example, stator and rotor windings of turbogenerators and high voltage underground cables. The Du Fort–Frankel and the fully implicit finite difference schemes have been used to solve the conduction and convection equations, respectively. It is shown that, with a suitable order of calculation, the overall method becomes explicit. Computational procedures are outlined and stability, accuracy and convergence are considered. Numerical examples are given to illustrate the use of the method and to validate some of the theoretical points. Results have also been obtained using existing numerical methods and have been compared with those from the proposed method. For certain problems, it is shown that the proposed method uses less overall computing time than other methods such as that devised by Crank and Nicolson.     

Solution of heat conduction problems for a body of revolution made of inhomogeneous material

Solutions of the heat conduction equation are obtained which permit methods of solving plane boundary problems for analytic functions to be applied when the boundary conditions on a surface of revolution are satisfied.     

Iterative approximation method for solving heat conduction equations

Convergence analysis is given for solving equations of heat conduction in the threedimensional space on an electronic computer under involved boundary conditions; an iterative approximation method is used in which the sought function is determined by successive approximations with a subsequent approximation by Chebyshev polynomial of the same degree.     

Optimality of the method of fundamental solutions

The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is also right-hand side vector dependent. The first part of this work, in [EABE 33(5): 637-43], revealed the close connection between the ECN and the accuracy of the Method of Fundamental Solutions (MFS) for each given problem. In this paper, we show how the ECN can help achieve the problem-dependent quasi-optimal settings for MFS calculations—that is, determining the position and density of the source points. A series of examples on Dirichlet and mixed boundary conditions shows the reliability of the proposed scheme; whenever the MFS fails, the corresponding value of the ECN strongly indicates to the user to switch to other numerical methods.     

Discontinuous solutions in problems of nonlinear heat conduction

We propose a method of constructing discontinuous solutions for nonlinear problems in the theory of heat conduction. The method is a modification of the pivot methods for the solution of linear boundary-value problems. As an illustration we present numerical solutions of two problems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 1, pp. 111–117, July, 1969.     

Applicability of the method of fundamental solutions

The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number.Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.