Solution of the Stationary 2D Inverse Heat Conduction Problem by Treffetz Method

The paper presents analysis of a solution of Laplace equation with the use of FEM harmonic basic functions. The essence of the problem is aimed at presenting an approximate solution based on possibly large finite element. Introduction of harmonic functions allows to reduce the order of numerical integration as compared to a classical Finite Element Method. Numerical calculations conform good efficiency of the use of basic harmonic functions for resolving direct and inverse problems of stationary heat conduction.Further part of the paper shows the use of basic harmonic functions for solving Poisson's equation and for drawing up a complete system of biharmonic and polyharmonic basic functions     

A solution to the two‐dimensional inverse heat conduction problem

A simple method is developed in this paper to solve two‐dimensional nonlinear steady inverse heat conduction problems. The unknown boundary conditions can be numerically obtained by using the iteration and modification method. The effect of measurement errors of the wall temperature on the algorithm is numerically tested. The results prove that this method has the advantages of fast convergence, high precision, and good stability. The method is successfully applied to estimate the convective heat transfer coefficient in the case of a fluid flowing in an electrically heated helically coiled tube. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 113–119, 2000     

Analytical solution of the inverse unsteady wall heat conduction problem and experimental application

Based on the analytical solution of the unsteady heat conduction differential equation, a solution procedure is presented for the inverse unsteady wall heat conduction problem, i.e. for the calculation of the thermal properties of structural elements of existing buildings under real transient conditions, using on-site temperature measurements. Previous procedures, which were based on the finite-difference method, required a considerable number of temperature measurements in space and time within the wall. The advantage of the present analytical procedure is that it requires only two temperature measurements, apart from some information on the outdoor and indoor temperature variations. The two temperature measurements may be taken on the outdoor and indoor wall surfaces at the same time level, or on one of these surfaces at two different time levels. The proposed analytical procedure provides the values of the thermal conductivity and heat capacity of structural elements, and therefore it may be used in practice for ex post checking of the materials used by the constructor, or for load calculation when heating or cooling systems are to be installed in old buildings of unknown wall properties. Experimental examples are presented which show that the proposed analytical procedure may be applied in practice with very good accuracy.     

Solution of inverse heat conduction problem using the lattice Boltzmann method

In this paper the D2Q9 lattice Boltzmann method (LBM) was utilized for the solution of a two-dimensional inverse heat conduction (IHCP) problem. The accuracy of the LBM results was validated against those obtained from prevalent numerical methods using a common benchmark problem. The conjugate gradient method was used in order to estimate the heat flux test case. A complete error analysis was performed. As the LBM is attuned to parallel computations, its use is recommended in conjugation with IHCP solution methods.     

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

The paper deals with the non-iterative inverse determination of the temperature-dependent thermal conductivity in 2-D steady-state heat conduction problem. The thermal conductivity is modeled as a polynomial function of temperature with the unknown coefficients. The identification of the thermal conductivity is obtained by using the boundary data and additionally from the knowledge of temperature inside the domain. The method of fundamental solutions is used to solve the 2-D heat conduction problem. The golden section search is used to find the optimal place for pseudo-boundary on which are placed the singularities in the frame of method of fundamental solutions.     

A Novel Variational Formulation of Inverse Problem of Heat Conduction with Free Boundary on an Image

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Calculation Error of Numerical Solution for a Boundary—Value Inverse Heat Conduction Problem

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Analytical approach with Laplace transform to the inverse problem of one‐dimensional heat conduction transfer: Application to second and third boundary conditions

An analytical method using Laplace transformation has been developed for one‐dimensional heat conduction. This method succeeded in explicitly deriving the analytical solution by which the surface temperature for the first kind of boundary condition can be well predicted. The analytical solutions for the surface temperature and heat flux are applied to the second and third of the boundary conditions. These solutions are also found to estimate the corresponding surface conditions with a high degree of accuracy when the surface conditions smoothly change. On the other hand, when these conditions erratically change such as the first derivative of temperature with time, the accuracy of the estimation becomes slightly less than that for a smooth condition. This trend in the estimation is similar irrespective of any kind of boundary condition. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 29–41, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10069     

Numerical Methods for Inverse Problem of Heat Conduction with Unknown Boundary Based on Variational Principles with Variable Domain

In this article a variable-domain variational approach to the entitled problem is presented.A pair of comple-mentary variational principles with a variable domain in terms of temperature and heat-streamfunction are firstestablished.Based on them,two methods of solution—generalized Ritz method and variable-domain FEM—both capable of handling problems with unknown boundaries,are suggested.Then,three sample numericalexamples have been tested.The computational process is quite stable,and the results are encouraging.Thisvariational approach can be extended straightforwardly to 3-D inverse problems as well as to other problems inmathematical physics.     

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric inverse heat conduction problem (IHCP). In the radially symmetric IHCP data on an inner fixed boundary is determined from Cauchy data given on an outer boundary. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach of Johansson et al. (2008) for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

Application of the homotopy perturbation method for the solution of inverse heat conduction problem

This paper deals with an application of the homotopy perturbation method for the solution of inverse heat conduction problem. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of functions describing the temperature and heat flux on the boundary, when the temperature measurements in the domain are known. Examples illustrating discussed application and confirming utility of this method in such a type of problem was also presented.     

Application of inverse solution in two‐dimensional heat conduction problem using Laplace transformation

An inverse solution has been explicitly derived for two‐dimensional heat conduction in cylindrical coordinates using the Laplace transformation. The applicability of the inverse solution is checked using the numerical temperatures with a normal random error calculated from the corresponding direct solution. For a gradual temperature change in a solid, the surface heat flux and temperature can be satisfactorily predicted, while for a rapid change in the temperature this method needs the help of a time partition method, in which the entire measurement time is split into several partitions. The solution with the time partitions is found to make an improvement in the prediction of the surface heat flux and temperature. It is found that the solution can be applied to experimental data, leading to good prediction. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 602–617, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10115     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

Any solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10117     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

An inverse analysis of two-phase Stefan problems using imaginary heat sources

This paper presents a new methodology for the inverse analysis of time-dependent two-phase Stefan problems. The problem considered here is that of determining the time dependence of a phase-change interface at several observed temperatures. In our method, imaginary heat sources are arranged in an imaginary domain and then the phase-change interface is identified as the isothermal surface at the melting temperature by controlling the imaginary heat source intensities. Using delta-function imaginary heat sources and their corresponding Green functions, which are pre-calculated numerically, it is shown that the phase-change interface is determined non-iteratively at each time step. We offer numerical examples to demonstrate the capability of the proposed method. © 1998 Scripta Technica, Heat Trans Jpn Res, 27(3): 179–191, 1998     

Integral transform solution for hyperbolic heat conduction in a finite slab

An analytical integral transformation of the thermal wave propagation problem in a finite slab is obtained through the generalized integral transform technique (GITT). The use of the GITT approach in the analysis of the hyperbolic heat conduction equation leads to a coupled system of second order ordinary differential equations in the time variable. The resulting transformed ODE system is then numerically solved by Gear's method for stiff initial value problems. Numerical results are presented for the local and average temperatures with different Biot numbers and dimensionless thermal relaxation times, permitting a critical evaluation of the technique performance. A comparison is also performed with previously reported results in the literature for special cases and with those produced through the application of the Laplace transform method (LTM), and the finite volume-Gear method (FVGM).     

Thermal diffusivity estimation using numerical inverse solution for 1D heat conduction

This work presents an improved apparatus and a numerical approach to obtain the estimate of thermal diffusivity of complex materials. Transient thermal response at the axis of cylindrical sample is measured when boundary temperature is suddenly changed. Instead of assuming an ideal step temperature excitement, a measured temperature of a material boundary was employed. An iterative procedure, based on minimizing a sum of squares function with the Levenberg–Marquardt method, is used to solve the inverse problem. A graphical user interface is built to enable easy use of the inverse thermal diffusivity estimation method. The reference materials used to evaluate the method are Agar water gel, glycerol and Ottawa quartz sand.