Time-space boundary elements for transient heat conduction problems

This paper is concerned with a generalized time-space boundary element formulation for transient heat conduction problems in anisotropic media. A weighted residual form of the governing equation is used to obtain the boundary integral equation in terms of the fundamental solution. The resulting boundary integral equation is discretized by means of a wide variety of boundary elements from constant-elements to higher-order isoparametric elements located both in time and space.     

A numerical method for heat conduction in shape reconstruction

This paper is concerned with the problem of the shape reconstruction of the inverse problem for heat conduction with two different boundary conditions in a multiple connected bounded domain. We derive the representation for domain derivative of the corresponding operator. This allows the investigation of the iterative regularization methods solving such ill-posed and nonlinear problem. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible.     

BEM-Fading regularization algorithm for Cauchy problems in 2D anisotropic heat conduction

Numerical Algorithms - We investigate the numerical reconstruction of the missing thermal boundary data on a part of the boundary for the steady-state heat conduction equation in anisotropic solids...     

Solving heat conduction problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method

As an improvement of the Meshless Local Petrov–Galerkin (MLPG), the Direct Meshless Local Petrov–Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low–degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.     

2D numerical manifold method based on quartic uniform B-spline interpolation and its application in thin plate bending

A new numerical manifold (NMM) method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and pol...     

A numerical conformal transformation method for harmonic mixed boundary value problems in polygonal domains

Summary A method is described for the numerical solution of harmonic mixed boundary value problems in simply-connected domains. The method consists of three successive conformal transformations of which the first is performed numerically.In particular the method is applied to problems, defined in polygonal regions, containing re-entrant cornes at which boundary singularities occur.

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Résumé Une méthode est décrite ici pour la solution numérique d'une classe de problèmes mixtes harmoniques aux valeurs aux limites dans des domaines simplement connexes. La méthode consiste en trois transformations conformes consécutives dont la première doit être accomplie numériauement.La méthode est plus particulièrement appliquée aux problèmes définis dans des domaines polygonaux possédant des angles rentrants où apparaissent des singularités.

     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

A direct general finite difference method for two-dimensional heat conduction problems

A numerical method, based on general finite difference forms, is introduced and applied for solving heat conduction problems in two-dimensional domains of any shape and compounded by subdomains with differing thermal properties.     

On the solution of heat conduction problems for thermosensitive bodies heated by convective heat exchange

We propose a method of solving heat conduction problems for thermosensitive bodies, in particular for a hollow cylinder from whose surfaces a convective heat exchange with the external environment occurs.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 83–86.     

A fast iterative method for solving stationary heat conduction problems on regions partitioned into rectangles

Summary The paper deals with some finite element approximation of stationary heat conduction problems on regions which can be partitioned into rectangular subregions. By a special superelement-technique employing fast elimination of the inner nodal parameters, the original finite element problem is reduced to a smaller problem, which is only connected with the nodes on the boundary of the superelements. To solve the reduced system of finite element equations, an efficient iterative algorithm is proposed. This algorithm is based either on the conjugate gradient method or the Tshebysheff method, using a special matrix by vector multiplication procedure. The explicit form of the matrix is not used. The presented numerical method is asymptotically optimal with respect to the memory requirement as well as to the operation count.     

Convergence of the alternating direction method for the numerical solution of a heat conduction equation with delay

Two-dimensional parabolic equations with delay effects in the time component are considered. An alternating direction scheme is constructed for the numerical solution of these equations. The question on the reduction of a problem with inhomogeneous boundary conditions to a problem with homogeneous boundary conditions is considered. The order of approximation error for the alternating direction scheme, stability, and convergence order are investigated.     

二维热传导型半导体瞬态问题的二次元特征有限体积元方法

将特征线方法与有限体积元方法相结合,采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间构造了二维热传导型半导体瞬态问题的全离散二次元特征有限体积元格式,并进行误差分析,得到了次优阶L^2模误差估计结果．     

Parameter selection by discrete mollification and the numerical solution of the inverse heat conduction problem

A procedure for the numerical solution of the one-dimensional inverse heat conduction problem, based on the computaion of the solution associated with a suitable filtered version of the noisy data by discrete mollification is presented and a parameter choice criterion, which automatically determines the radius of mollification as a function of the amount of noise in the data, is introduced. Several numerical examples of interest are also analyzed, showing the accuracy and stability properties of the method.     

A problem on heat conduction in an insulated wire solved by numerical inverse Laplace transform

A problem of transient heat conduction in an insulated wire is solved by use of Laplace transform and numerical inversion. The problem is solved for the radiation boundary condition and also for the boundary condition of no heat flux through the outer surface of the insulation. The results are presented both numerically with four significant figures and graphically. Asymptotic expansions are derived for small and large values of the time variable. The numerical inversion of the Laplace transform is checked by comparison with the asymptotic expansions and with the numerical results obtained by a numerical inversion formula utilizing one more abscissa than the previous one.     

The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.     

Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1289–1398, September, 1993.     

On transient heat conduction in the walls of a rectangular gas channel

In this paper a problem on transient heat conduction in the walls of a gas channel with a rectangular cross-section is solved. The temperature of the gas flow in the channel rises linearly while the temperature of the surrounding open air is constant.The differential equation and its auxiliary conditions are Laplace transformed, the subsidiary equations are solved by a method resembling the two-dimensional relaxation method for steady state heat conduction problems, and the resulting temperatures are obtained by numerical inversion.Numerical results are presented at the end of the paper.     

Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains

The formulation of certain classes of boundary value problems in terms of hypersingular integral equations is currently gaining increasing interest. In this paper we consider such type of equations on 2D polygonal domains, and assume we have to solve them by a collocation or a Galerkin BEM. In particular, given any (polynomial) local basis, we show how to compute efficiently, using a very low number of points, all integrals required by these methods. These integrals have kernels of the type log r, r⁻¹ and r⁻².The quadrature rules we propose to compute the above-mentioned integrals require the user to specify only the local polynomial degrees; therefore, they are quite suitable for the construction of a p or h−p version of the BEM.