Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state

The paper deals with the inverse determination of heat sources in steady 2-D heat conduction problem. The problem is described by Poisson equation in which the function of the right hand side is unknown. The identification of the strength of a heat source is given by using the boundary condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the method of fundamental solution with radial basis functions is proposed. The accurate results have been obtained for five test problems where the analytical solutions were available.     

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.     

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.     

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 technique for uncertainty analysis for inverse heat conduction problems

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the uncertainty analysis. An example calculation is presented. Parameter importance factors are defined and computed for the example problem; the volumetric heat capacity is the dominant parameter and an explanation is offered. Thoughts are presented on extending the analysis to include the non-linear problem of temperature dependent properties.     

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     

The method of fundamental solutions for solving free boundary saturated seepage problem

The problem of seepage flow through a dam is free boundary problem that is more conveniently solved by a meshless method than a mesh-based method such as finite element method (FEM) and finite difference method (FDM). This paper presents method of fundamental solutions, which is one kind of meshless methods, to solve a dam problem using the fundamental solution to the Laplace's equation. Solutions on free boundary are determined by iteration and cubic spline interpolation. The numerical solutions then are compared with the boundary element method (BEM), FDM and FEM to display the performance of present method.     

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

ＡＮｏｖｅｌＶａｒｉａｔｉｏｎａｌＦｏｒｍｕｌａｔｉｏｎｏｆＩｎｖｅｒｓｅＰｒｏｂｌｅｍｏｆＨｅａｔＣｏｎｄｕｃｔｉｏｎｗｉｔｈＦｒｅｅＢｏｕｎｄａｒｙｏｎａｎＩｍａｇｅＰｌａｎｅＧａｏ－ＬｉａｎＬｉｕ（ＳｈａｎｇｈａｉＩｎｓｔｉｔｕｔｅｏｆＭｅｃｈ...     

An inverse estimation of surface temperature using the maximum entropy method

The maximum entropy method (MEM) is applied to estimation of surface temperature from temperature readings. The inverse heat conduction problem is reformulated for MEM and a three-phase solution method utilizing the successive quadratic programming (SQP) is addressed. Computational results by the proposed MEM are presented and compared with results by the conventional methods.     

The application of the homotopy perturbation method to one-phase inverse Stefan problem

In this paper, the application of the homotopy perturbation method for solving the inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing temperature and heat flux on the boundary, when the position of the moving interface is known.     

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     

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.     

New Type of Basic Functions of FEM in Application to Solution of Inverse Heat Conduction Problem

The work presents the application of heat polynomials for solving an inverse problem. The heat polynomials form the Treffetz Method for non-stationary heat conduction problem. They have been used as base functions in Finite Element Method. Application of heat polynomials permits to reduce the order of numerical integration as compared to the classical Finite Element Method with formulation of the matrix of system of equations.     

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.     

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     

Thermodynamically consistent description of heat conduction with finite speed of heat propagation

In this work, governing equations for heat conduction with finite speed of heat propagation are derived directly from classical thermodynamics. For a one-dimensional flow of heat, the developed governing equation is linear and of parabolic type. In a three dimensional case, the system of nonlinear equations is formulated.Analytical solutions of the equations for one-dimensional flow of heat are obtained, and their analysis shows characteristic features of heat propagation with finite speed, being fully consistent with classical thermodynamics.     

A numerical method for natural convection and heat conduction around and in a horizontal circular pipe

Natural convection around a horizontal circular pipe coupled with heat conduction in the solid structure is numerically investigated using a preconditioning method for solving incompressible and compressible Navier–Stokes equations. In this method, fundamental equations are completely reduced to an equation of heat conduction when the flow field is static (zero velocity). Therefore, not only compressible flows but also very slow flows such as natural convection in a flow field and heat conduction in a static field can be simultaneously calculated using the same computational algorithm. In this study, we first calculated the compressible flow around a NACA0012 airfoil with conduction in the airfoil and then simulated natural convections around a horizontal circular pipe with a different heat conductivity. Finally, we numerically investigated the effect of heat conductivity of the pipe on natural convection.     

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.     

Training based, moving digital filter method for real time heat flux function estimation

In this paper the neural networks is utilized to estimate the “filter coefficients” needed to estimate heat flux in a particular system. In developing the training phase of the network inspiration is drawn from the Burgraff's exact solution of the IHCP as well as the filter method. Thus, the estimation phase neither requires any temperature field nor the sensitivity coefficients calculations. The neural network used in this work is a 2-layer perceptron. It is shown via classical triangular heat flux test cases that the method can yield very accurate, very efficient as well as stable estimations.