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.     

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...     

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.     

Numerical solution of transient heat conduction equation with variable thermophysical properties by the Tau method

In this article, we proposed the operational approach to the Tau method for solving linear and nonlinear one‐dimensional transient heat conduction equations with variable thermophysical properties which can involve heat generation term. To solve heat conduction equation, first we recall the Tau method to obtain a matrix form of the governing differential equation. Then boundary and initial conditions are transformed into a matrix form. Finally the resulting systems of linear or nonlinear algebraic equations are given. Afterwards, efficient error estimation is also introduced for this method. Some numerical examples are given to illustrate the efficiency and high accuracy of the proposed method and also results are compared with solutions obtained by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 964–977, 2014     

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.     

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 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.

     

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 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.     

Hydraulic fracturing modeling using the enriched numerical manifold method

Recent attempts to solve rock mechanics problems using the numerical manifold method (NMM) have been regarded as fruitful. In this paper, a coupled hydro-mechanical (HM) model is incorporated into the enriched NMM to simulate fluid driven fracturing in rocks. In this HM model, a “cubic law” is employed to model fluid flow through fractures. Several benchmark problems are investigated to verify the coupled HM model. The simulation results agree well with the analytical and experimental results, indicating that the coupled HM model is able to simulate the hydraulic fracturing process reliably and correctly.     

A weighted algorithm based on the homotopy analysis method: Application to inverse heat conduction problems

In this paper, a weighted algorithm based on the homotopy analysis method (HAM), for solving inverse heat conduction problems, is introduced . Using the HAM, it is possible to find the exact solution or an approximate solution of the problem in the form of a series.     

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.     

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

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.     

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.     

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．     

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.