Solution of a stationary inverse heat conduction problem by means of Trefftz non-continuous method

In the paper, the non-continuous FEM with Trefftz base functions (FEMT) applied to direct and inverse problem of heat conduction equation has been presented. For the finite number of base functions in each finite element the temperature field becomes non-continuous on the border between elements. This non-continuity has been decreased with the penalty function added to optimised functional. The numerical entropy distribution and energy dissipation function have been analysed on the common boundaries of elements. Increasing the number of base functions in the finite element substantially decreases the inaccuracies of direct and inverse problem solution.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

On the accuracy of the collocation Trefftz method for solving two- and three-dimensional heat equations

ABSTRACT

In this article, the accuracy of the collocation Trefftz method (CTM) for solving two- and three-dimensional heat equations is investigated. The numerical solutions are approximated by superpositioning T-complete functions formulated using cylindrical harmonics. To avoid the ill-conditioning of the CTM, the characteristic lengths and the multiple-scale Trefftz method are adopted. The results reveal that for two-dimensional problems, the CTM can provide highly accurate numerical solutions, with the accuracy increasing with the order of the terms. For three-dimensional problems, highly accurate numerical solutions can be obtained using a certain order of terms, where the order is determined by performing an accuracy assessment.     

Analysis and solution of the ill-posed inverse heat conduction problem

The inverse conduction problem arises when experimental measurements are taken in the interior of a body, and it is desired to calculate temperature and heat flux values on the surface. The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. A special solution procedure is developed for the one-dimensional case which replaces the heat conduction equation with an approximating hyperbolic equation. If viewed from a new perspective, where the roles of the spatial and time variables are interchanged, then an initial value problem for the damped wave equation is obtained. Since the formulation is well-posed, both analytic and numerical solution procedures are readily available. Sample calculations confirm that this approach produces consistent, reliable results for both linear and nonlinear problems.     

Numerical method for hyperbolic inverse heat conduction problems

The Laplace transform technique and control volume method in conjunction with the hyperbolic shape function and least-squares scheme are applied to estimate the unknown surface conditions of one-dimensional hyperbolic inverse heat conduction problems. In the present study, the expression of the unknown surface conditions is not given a priori. To obtain the more accurate estimates, the whole time domain is divided into several analysis sub-time intervals. Afterward, the unknown surface conditions in each analysis interval are estimated. To evidence the accuracy of the present method, a comparison between the present estimations and exact results is made. Results show that good estimations on the unknown surface conditions can be obtained from the transient temperature recordings only at one selected location even for the cases with measurement errors.     

On the iterative regularization of inverse heat conduction problems by conjugate gradient method

The convergence and regularization properties of the conjugate gradient algorithm applied to the inverse heat conduction problem are considered for a time-dependent boundary heat flux. An analysis based on both numerical and analytical results clearly shows that the convergence process of the algorithm is strongly frequency-dependent and provides in this way a very efficient regularization mechanism against the destabilizing effect of random errors in the input data.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

Nonlinear inverse heat conduction problem of surface temperature estimation by calibration integral equation method

A calibration integral equation method is proposed for estimating the surface temperature in the context of a nonlinear inverse heat conduction problem. The temperature-dependent thermophysical properties and probe positioning are implicitly accounted in the integral equation formulation through calibration tests. A first kind Chebyshev expansion is applied to represent the temperature-dependent property transform function. The undetermined expansion coefficients associated with the Chebyshev expansion are then estimated through two calibration tests. Regularization of the ill-posed problem is achieved by the future-time method. The optimal regularization parameter is estimated using a phase plane and cross-correlation phase plane analyses. Numerical simulation for stainless steel yields highly favorable surface temperature prediction.     

Solution of inverse heat conduction problems using maximum entropy method

A solution scheme based on the maximum entropy method (MEM) for the solution of one-dimensional inverse heat conduction problem is proposed. The present work introduces MEM in order to build a robust formulation of the inverse problem. MEM finds the solution which maximizes the entropy functional under the given temperature measurements. In order to seek the most likely inverse solution, the present method converts the inverse problem to a non-linear constrained optimization problem. The constraint of the problem is the statistical consistency between the measured temperature and the estimated temperature. Successive quadratic programming (SQP) facilitates the maximum entropy estimation. The characteristic feature of the method is discussed with the sample numerical results. The presented results show considerable enhancement in the resolution of the inverse problem and bias reduction in comparison with the conventional methods.     

A regularized integral equation method for the inverse geometry heat conduction problem

This paper addresses a new technique for solving the inverse geometry heat conduction problem of the Laplace equation in a two-dimensional rectangle, which, named regularized integral equation method (RIEM), consists of three parts. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, y). Second, we consider a Lavrentiev regularization by adding a term αg(x) to obtain a second kind Fredholm integral equation. The termwise separable property of the kernel function allows us to transform the inverse geometry heat conduction problem into a two-point boundary value problem and therefore, an analytical regularized solution is derived in the final part by using orthogonality. Principally, the RIEM possesses the following advantages: it does not need any guess of the initial profile, it does not need any iteration and a regularized closed-form solution can be obtained. The uniform convergence and error estimate of the regularized solution u^α(x, y) are proved and a boundary geometry p(x) is solved by half-interval method. Several numerical examples present the effectiveness of our novel approach in providing excellent estimates of unknown boundary shapes from given data.     

Fourier analysis of conjugate gradient method applied to inverse heat conduction problems

The convergence and regularization mechanism of the conjugate gradient algorithm applied to inverse heat conduction problems are studied within the context of a Fourier analysis, for a square enclosure subjected to an unknown time-varying heat flux on one side, and to known boundary conditions on the remaining sides. Analytic solutions are derived for the Fourier components of the unknown flux over a given time interval. The convergence rate of the algorithm is thereby shown to depend essentially on the time frequency of the data. Numerical solutions are also presented to describe in details the convergence process and solution regularization power of the conjugate gradient method, when the unknown heat flux contains many frequency components and the measurement data are noisy. It is found that an unknown time-dependent heat flux may be satisfactorily recovered using a single sensor even when the temperature field becomes two-dimensional, and that the sensor should be placed in a symmetric manner for better results.     

Intelligent fuzzy weighted input estimation method applied to inverse heat conduction problems

The innovative intelligent fuzzy weighted input estimation method which efficiently and robustly estimates the unknown time-varying heat flux in real-time is presented in this paper. The algorithm includes the Kalman Filter (KF) and the recursive least square estimator (RLSE), which is weighted by the fuzzy weighting factor proposed based on the fuzzy logic inference system. To directly synthesize the Kalman filter with the estimator, this work presents an efficient robust forgetting zone, which is capable of providing a reasonable compromise between the tracking capability and the flexibility against noises. The capability of this inverse method are demonstrated in one- and two-dimensional time-varying estimation cases and the proposed algorithm is compared by alternating between the constant and adaptive weighting factors. The results show that this method has the properties of faster convergence in the initial response, better target tracking capability and more effective noise reduction.     

An inversion approach for the inverse heat conduction problems

The inverse heat conduction problems (IHCP) analysis method provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. Successful applications of the IHCP method depend mainly on the efficiency of the inversion algorithms. In this paper, a generalized objective functional, which has been developed using a generalized stabilizing functional and a combinational estimation that integrates the advantages of the least trimmed squares (LTS) estimation and the M-estimation, is proposed. The objective functional unifies the regularized M-estimation, the regularized least squares (LS) estimation, the regularized LTS estimation, the regularized combinational estimation of the LTS estimation and the M-estimation, and the regularized combinational estimation of the LS estimation and the M-estimation into a concise formula. The filled function method, which is coupled with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, is developed for searching a possible global optimal solution. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. Favorable numerical performances and satisfactory results are observed, which indicates that the proposed algorithm is successful in solving the IHCP.     

A decentralized fuzzy inference method for solving the two-dimensional steady inverse heat conduction problem of estimating boundary condition

This paper addresses a new technique for solving the two-dimensional steady inverse heat conduction problem, which named decentralized fuzzy inference (DFI) method. First of all, a group of decentralized fuzzy inference units are designed, and the fuzzy inference for each fuzzy inference unit is conducted which bases on the difference between the measured and the computed temperature at each measuring location. The computed temperatures are obtained by solving the direct heat conduction problem with the finite difference method. And then, inference results of fuzzy inference units are weighted to yield compensation values of the unknown boundary temperatures. The unknown boundary temperatures are estimated by updating guess temperatures continuously with compensation values. Numerical experiments are carried out with different initial guesses, the number of measuring points and measurement errors. Comparing results of DFI method and Levenberg–Marquardt (L–M) method, we can conclude that DFI method is valid.     

A general method for the solution of inverse heat conduction problems with partially unknown system geometries

An auxiliary problem is introduced in the solution of inverse heat conduction problems with geometries not fully specified. Resolving the position of the unknown boundary subject to a Dirichlet condition leads to the solution of a nonlinear algebraic equation. Imposing Neumann or Robin conditions at the unknown boundary requires the solution of a first-order nonlinear, ordinary differential equation. The method yields accurate results for exact data, while measurement errors render the Neumann problem insoluble. The Dirichlet and Robin problems are still solvable, and for these problems, the errors in the investigated boundaries increase with their depth, a nature of the problem being investigated.     

A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source

This work considers a new approach for solving the inverse heat conduction problem of estimating unknown plan heat source. It is shown that the physical heat transfer problem can be formulated as an optimization problem with differential equation constraints. A modified genetic algorithm is developed for solving the resulting optimization problem. The proposed algorithm provides a global optimum instead of a local optimum of the inverse heat transfer problem with highly-improved convergence performance. Some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.     

The meshless analog equation method for solving heat transfer to molten polymer flow in tubes

In this paper, we proposed a meshless analog equation method (MAEM) to solve a heat transfer problem of molten polymer flow, which is considered to be a generalized Newtonian viscous flow. The MAEM, free from mesh generation and numerical integration, is a powerful meshless method. The numerical solutions are expressed by a linear combination of the derived radial basis functions (RBFs). This paper considers two different viscosity models for the molten polymer; one is temperature-independent power-law model and the other is temperature-dependent power-law model. The viscous dissipation term is included in the energy equation to capture the relevant physical phenomena. From the comparisons of numerical simulation, the meshless solutions are in good agreement with some analytical solutions and other finite element solutions. Moreover, the MAEM uses much less CPU-time and computer memory to simulate molten polymer flows. Therefore, it is believed that the RBF-based meshless method of the MAEM is a promising and flexible numerical scheme for molten polymer flow simulation.     

Shape identification for inverse geometry heat conduction problems by FEM without iteration

The boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.     

A data method using the inner temperature difference to improve the stability of the inverse heat conduction problems

Abstract

This article proposes a method to construct two series systems for improving the stability of the inverse heat conduction problems (IHCP) in a finite slab. The transfer function between the surface heat flux or temperature and the inner temperature difference is respectively obtained by Laplace transform technique firstly. Then the series systems which can solve IHCP based on the inner temperature difference are constructed by replacing the unsuitable zero and pole points of the transfer function approximated by è approximation. Finally the effects of the series systems are evaluated by a typical example. The results of the evaluation show that this method can obtain the surface heat flux and temperature by the inner temperature difference, and enhance the response speed of the measurement system at the same time. In addition this method can also improve the signal to noise ratio (SNR) of the inverse solutions by selectively amplifying the high SNR parts of the inner temperature difference. The present work provides an effective method to improve the stability of IHCP.