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.     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

Application of Wavelet in Highly Ill-Posed Inverse Heat Conduction Problem

In this work, the prefiltering of the sensor data is taken into consideration when solving an inverse heat conduction problem. The temperature data obtained from each sensor is considered as a discrete signal, and discrete wavelet transform in a multi-resolution filter bank structure is utilized for the signal analysis, after which wavelet denoising algorithm is applied to remove noise from data signal. Subsequently, noisy and denoised temperatures are separately used as input data to an inverse heat conduction problem for comparison. The inverse heat conduction problem considered in this article is an inverse volumetric heat source problem, and it is solved using the conjugate gradient method along with the associated adjoint problem used to obtain the gradient of the objective function. Three sets of results in two case studies are compared (i.e., the result obtained from non-noisy data, noisy data, and denoised data). In the case of noisy data, iterative regularization is used to regularize the solution. The root mean square error of the estimated heat source from denoised data is reduced approximately by a factor of seven to nine as compared to those obtained from noisy data.     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

On estimating thermal diffusivity using analytical inverse solution for unsteady one-dimensional heat conduction

A modified procedure for calculating the thermal diffusivity of solids based on temperature measurements at two points and the semi-infinite boundary condition is presented. The method makes use of a solution to the unsteady one-dimensional inverse heat conduction problem for the semi-infinite solid. The procedure gives accurate results based on temperature changes produced by an arbitrary fluctuating heat flux source at the boundary.     

Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

The present work addresses the direct and inverse problems for convective heat transfer with incompressible laminar gas flow in micro-channels, within the range of validity of the slip-flow regime. The direct problem analysis combines the classical integral transform method and the generalized integral transform technique (GITT), by analytically solving the two-dimensional steady-state convection problem and finding a hybrid numerical-analytical solution for the required eigenvalue problem. The inverse problem analysis makes use of the accuracy and robustness of the direct problem solution and focus on the simultaneous identification of the momentum and thermal accommodation coefficients, related to gas flow and heat transfer within micro-channels, besides the usually unknown boundary condition parameters, here represented by the external Biot number. The inverse analysis is based on the availability solely of temperature measurements at the channel external wall, along its length, as obtained for instance via infrared camera thermography. A Bayesian inference approach is adopted in the solution of the identification problem based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings sampling algorithm. A typical example of slip flow in parallel-plates micro-channel is selected to illustrate both the direct and inverse problems solution approaches.     

A Sequential One-Dimensional Solution for the Sideways Heat Equation in a Semi-Infinite,Finite, and Multilayer Slab

The sideways heat equation concerns an inverse problem where an unknown and inaccessible surface temperature is estimated using sensor data gathered from an interior point of a slab. The article presents how the ill-posed problem is solved using a finite-difference technique, where the space variable is continuous and the time variable is discrete. The inverse procedure is based on a recursive deconvolution algorithm, derived from the direct solution. Regularization is completed with digital filters. The test runs give promising results. However, due to the time derivative approximation and disturbances, abrupt changes in the estimated signal cannot be perfectly reproduced.     

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.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

A parameter estimation approach to solve the inverse problem of point heat sources identification

This paper deals with an inverse problem that consists of the identification of multiple line heat sources placed in a homogeneous domain. In the inverse problem under investigation the location and strength of the line heat sources are unknown. The estimation procedure is based on the boundary element method. As the discrete problem is non-linear if the location of the line heat sources is unknown, an iterative procedure has to be applied to find out the location of the sources. The proposed approach has been tested for steady and transient experiments. In the present study we propose an original approach to solve the steady problem. As in the steady heat conduction case we have a limited number of unknown for each source, a “parameter estimation” approach can be applied to estimate the sources. Using the techniques of parameter estimation, we can also estimate the confidence interval of the estimated locations, which permits to design an optimal experiment. We intend to present some numerical and experimental 2D results.     

Determination of thermal contact resistance from transient temperature measurements

The temperature distribution in combustion engine components is highly influenced by thermal contact resistance. For the prediction and optimisation of the thermal behaviour of modern combustion engines knowledge about the contact heat transfer is crucial.Available correlations to predict the contact resistance are simplifications of the real geometric conditions and only tested for moderate pressures up to 7 MPa. Typical combustion engine applications include contact pressures up to 250 MPa.The experimental approach presented here to derive the thermal contact resistance in terms of contact heat transfer coefficients for high temperature and high pressure conditions is based on transient infrared temperature measurements. Two bodies initially at two different temperatures are brought in contact and the surface temperature histories are recorded with a high-speed infrared camera. The contact heat flux is calculated by solving the related inverse problem. From the contact heat flux and from the measured temperature jump at the interface the contact heat transfer coefficient is calculated.The inverse method used for the calculation of the heat flux is based on the analytical solution for a semi-infinite body and a step response to a Neumann boundary condition. This method provides an algorithm that is used in a sequential manner. The use of “future” temperature data greatly improve the stability of the governing equations and reduce the sensitivity to measurement errors.     

Inverse alloy solidification problem including the material shrinkage phenomenon solved by using the bee algorithm

In the paper we solve the one-phase inverse problem of alloy solidifying within the casting mould, including the shrinkage of metal which results from the difference between densities of the liquid and solid phases. The process is modeled by means of the solidification in the temperature interval basing on the heat conduction equation with the source element enclosed, whereas the shrinkage of metal is modeled by the proper application of the mass balance equation. The investigated inverse problem consists in reconstruction of the heat transfer coefficient on the boundary of the casting mould on the basis of measurements of temperature read from the sensor placed in the middle of the mould. Functional expressing the error of approximate solution is minimized with the aid of Artificial Bee Colony Optimization algorithm.     

The BEM for point heat source estimation: application to multiple static sources and moving sources

This paper deals with an inverse problem, which consists of the identification of point heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The two-dimensional and three-dimensional linear heat conduction problems are considered here. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a quadratic norm. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. A numerical example is presented for the 3D application.     

An inverse problem in estimating the base heat flux of an annular fin based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent base heat flux of an annular fin from the knowledge of temperature measurements taken within the fin. The inverse solutions will be justified based on the numerical experiments in which two specific cases to determine the unknown base heat flux are examined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux can be obtained for the test cases considered in this study.     

Multiple transient point heat sources identification in heat diffusion: application to experimental 2D problems

This paper deals with an inverse problem, which consists of the experimental identification of line heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a cost function comparing the modelled heat source term and the measurements. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. In some particular examples, internal thermocouples can be used. A time regularization procedure associated to future time-steps is used to correctly solve the ill-posed problem.     

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.     

The BFGS method for estimating the interface temperature and convection coefficient in ultrasonic welding

An inverse heat transfer problem is investigated in the present study by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to predict the unknown time-dependent heat generation at the weld interface and convection heat transfer coefficient during an ultrasonic metal welding process based on the knowledge of temperature measurements taken on the horn. With known temperature data at some locations on the horn, the inverse solution was rapidly obtained by solving nonlinear direct problem, Central Finite Difference and Simple Step Method. The proposed method which did not need solving adjoint and sensitivity problem revealed the characteristics of high efficiency, lower iterations for a computational algorithm and high accuracy for estimating values even when measurement error was considered. Besides, a comparison of the BFGS method with some previous methods (i.e. CGM, SCGM) was established. These results show that an excellent estimation on interfacial heat generation (or temperature), as well as a convection heat transfer coefficient, can be simultaneously obtained in this study. The current methodology will provide a useful tool to optimize welding conditions in ultrasonic welding.     

Inverse identification of temperature-dependent thermal conductivity via genetic algorithm with cost function-based rearrangement of genes

A new application of the cost function-based rearrangement of genes (proposed by Liu (2008) [1]) is presented in this paper through the genetic algorithm-based solution of the inverse heat conduction problem of identifying the temperature dependent thermal conductivity of a solid material using transient temperature histories. The inverse problem was defined according to the evaluation of the BICOND thermophysical property measurement method. Through the solution of the inverse problem (using simulated measurements), different approaches of the application of the rearrangement of genes were studied and compared. Application of the rearrangement significantly improved the convergence performance and accuracy of the inverse solution compared to a real-valued genetic algorithm, which was adapted to the problem by the authors. In the algorithm that performed best, the rearrangement was applied in an approach different from Liu’s. The effect of random noise added to the temperature history and the effect of regularization was also studied. With significant improvement in computational efficiency, the proposed algorithm is likely to be very effective in evaluation of real measured temperature histories to determine thermophysical properties.