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.     

The method of fundamental solutions for heat conduction in layered materials

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two‐dimensional problems of steady‐state heat conduction in isotropic and anisotropic bimaterials. Two approaches are used: a domain decomposition technique and a single‐domain approach in which modified fundamental solutions are employed. The modified fundamental solutions satisfy the interface continuity conditions automatically for planar interfaces. The two approaches are tested and compared on several test problems and their relative merits and disadvantages discussed. Finally, we use the domain decomposition approach to investigate bimaterial problems where the interface is non‐planar and the modified fundamental solutions cannot be used. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Methods of fundamental solutions for time‐dependent heat conduction problems

A time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

MFS with time-dependent fundamental solutions for unsteady Stokes equations

This paper describes the applications of the method of fundamental solutions (MFS) for 2D and 3D unsteady Stokes equations. The desired solutions are represented by a series of unsteady Stokeslets, which are the time-dependent fundamental solutions of the unsteady Stokes equations. To obtain the unknown intensities of the fundamental solutions, the source points are properly located in the time–space domain and then the initial and boundary conditions at the time–space field points are collocated. In the time-marching process, the prescribed collocation procedure is applied in a time–space box with suitable time increment, thus the solutions are advanced in time. Numerical experiments of unsteady Stokes problems in 2D and 3D peanut-shaped domains with unsteady analytical solutions are carried out and the effects of time increments and source points on the solution accuracy are studied. The time evolution of history of numerical results shows good agreement with the analytical solutions, so it demonstrates that the proposed meshless numerical method with the concept of space–time unification is a promising meshless numerical scheme to solve the unsteady Stokes equations. In the spirit of the method of fundamental solutions, the present meshless method is free from numerical integrations as well as singularities in the spatial variables.     

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.     

The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media

In this paper, the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady‐state heat conduction in an anisotropic medium. Since the resulting matrix equation is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, while the optimal regularization parameter is chosen according to the L‐curve criterion. Numerical results are presented for both two‐ and three‐dimensional problems, as well as exact and noisy data. The convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries, and decreasing the amount of noise added into the input data, respectively, are analysed. A sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed. The numerical results obtained show that the proposed numerical method is accurate, convergent, stable and computationally efficient, and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady‐state heat conduction. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Three-dimensional fundamental solutions for transient heat transfer by conduction in an unbounded medium,half-space,slab and layered media

This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers.This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, António J, Simões N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.     

The method of fundamental solutions for axisymmetric potential problems

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright © 1999 John Wiley & Sons, Ltd.     

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.