Two-dimensional non-linear inverse heat conduction problem based on the singular value decomposition

In this paper an efficient sequential method is developed in order to estimate the unknown boundary condition on the surface of a body from transient temperature measurements inside the solid. This numerical approach for solving an inverse heat conduction problem (IHCP) takes into account two-dimensional problems, planar or axisymmetric cylindrical, composite materials with irregular boundaries and temperature-dependent thermal properties. The unknown surface condition is assumed to have abrupt changes at unknown times. The regularization procedure used for the solution of the IHCP is based on the singular value decomposition technique. An overall estimate of error is defined in order to find the optimal estimation in the 2D IHCP (linear and non-linear). The stability and accuracy of the scheme presented is evaluated by comparison with the Function Specification Method. This comparative study has been carried out using numerically simulated data, and the parameters considered include shape of input, noise level of measurement, size of time step and temperature-dependent thermal properties. A good agreement was found between both methods. Beside this, the slight differences on estimations and number of future temperatures are discussed in this paper.     

Reduction of the computational time and noise filtration in the IHCP by using the proper orthogonal decomposition (POD) method

In this paper a comprehensive investigation is performed on the application of the proper orthogonal decomposition (POD) in solving the inverse heat conduction problems (IHCP's). Inverse heat conduction problems are those in which the boundary or the initial condition, or the physical properties are unknown; instead temperature measurements at some locations within the body are available. Two approaches may be taken to solve these mathematically ill-posed problems. These approaches are classified according to whether the problem is modeled using a PDE or a matrix form. In this paper a new approach, namely the proper orthogonal decomposition (POD) method is used for solving the IHCP. In this approach, based on the dynamics of the problem on hand, two courses of action may be taken: In the first course the governing PDE is reduced to a system of ODE's which reduces the computational time considerably. In the second approach the ill-conditioned matrix is modified using the singular value decomposition (SVD) which reduces the destructive effects of the random noise of the temperature data. Although both courses of action introduce a bias error into the inverse solution, the considerable reduction in the variance error in the solution makes the method to be quite attractive. In this paper these points are demonstrated through solutions obtained for two standard IHCP's. The results obtained by the methods suggested in this paper are compared with those obtained by the well-known conjugate gradient and Tikhonov regularization methods.     

NUMERICAL SOLUTION OF 2-D NONLINEAR INVERSE HEAT CONDUCTION PROBLEMS USING FINITE-ELEMENT TECHNIQUES

A general method is presented for solving different classes of nonlinear inverse heat conduction problems (IHCP) for two-dimensional, arbitrarily shaped bodies. It is based on the systematic use of a finite-element library. It is shown that, following this approach, the conjugate gradient method can be easily implemented. The method offers a very wide field of practical applications in inverse thermal analysis, while reducing very significantly the amount of work which remains specific for each particular IHPC. Two numerical experiments illustrate the influence of data errors and the iterative regularization principle.     

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

The estimation of temporal dependent heat source in transient heat conduction problem is investigated. A stochastic method known as quantum-behaved particle swarm optimization (QPSO) is used to estimate the heat source without a priori information on its functional form, which is classified as the function estimation by inverse calculation. Because of the ill-posedness of this kind of inverse problems, Tikhonov regularization method is applied in this paper to stabilize the solution. Numerical experiments indicate the validity and stability of the QPSO method. Comparison with the conjugate gradient method (CGM) is also presented in this paper.     

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 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.     

Solution of inverse heat conduction problem using the Tikhonov regularization method

It is hard to solve ill-posed problems, as calculated temperatures are very sensitive to errors made while calculating “measured” temperatures or performing real-time measurements. The errors can create temperature oscillation, which can be the cause of an unstable solution. In order to overcome such difficulties, a variety of techniques have been proposed in literature, including regularization, future time steps and smoothing digital filters. In this paper, the Tikhonov regularization is applied to stabilize the solution of the inverse heat conduction problem. The impact on the inverse solution stability and accuracy is demonstrated.     

A regularization method for the inverse design of solidification processes with natural convection

An inverse algorithm is developed for the design of the solidification processing systems. The algorithm entails the use of the Tikhonov regularization method, along with the L-curve method to select an optimal regularization parameter. Both the direct solution of moving boundary problems and the inverse design formulation are presented. The design algorithm is applied to determine the optimal boundary heat flux distribution to obtain a unidirectional solid–liquid interface in a 2-D cavity. The inverse calculation is also performed with a prescribed sinuous solid–liquid interface. To this end, a whole time-domain method and a sequential method are used and evaluated. The L-curve based regularization method is found to be reasonably accurate for the purpose of designing solidification processing systems. We also found that the sequential method with appropriately selected time domains is comparative to the whole time-domain method.     

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.     

Estimation of boundary conditions in the presence of unknown moving boundary caused by ablation

Ablative materials can sustain very high temperatures in which surface thermochemical processes are significant enough to cause surface recession. Existence of moving boundary over a wide range of temperatures, temperature-dependent thermophysical properties of ablators, and no prior knowledge about the location of the moving surface augment the difficulty for predicting the exposed heat flux at the receding surface of ablators. In this paper, the conjugate gradient method is proposed to estimate the unknown surface recession and time-varying net surface heat flux for these kinds of problems. The first order Tikhonov regularization is employed to stabilize the inverse solution. Considering the complicated phenomena that are taking place, it is shown via simulated experiment that unknown quantities can be obtained with reasonable accuracy using this method despite existing noises in the measurement data.     

A spectral method for the estimation of a thermomechanical heat source from infrared temperature measurements

The inverse problem of 2D time-dependent heat source reconstruction is solved. The scientific objectives are the quantification of thermal effects associated to the mechanical deformation of materials during tensile tests. The experiment provides infrared measurements of the specimen’s surface temperature and the inverse algorithm aims at providing a volumic heat source that is free of errors due to heat diffusion. This algorithm is based on an analytical solution of the direct problem in the Laplace-Fourier domain. The solution proposed here is compared to a previously used method [1] based on an adjoint formulation and a regularization of Tikhonov type. This allows to check the validity of the results.     

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 two-dimensional inverse heat conduction problem in estimating the fluid temperature in a pipeline

An inverse heat conduction problem (IHCP) was investigated in the two-dimensional section of a pipe elbow with thermal stratification to estimate the unknown transient fluid temperatures near the inner wall of the pipeline. An inverse algorithm based on the conjugate gradient method (CGM) was proposed to solve the IHCP using temperature measurements on the outer wall. In order to examine the accuracy of estimations, some comparisons have been made in this case. The temperatures obtained from the solution of the direct heat conduction problem (DHCP) using the finite element method (FEM) were pseudo-experimental input data on the outer wall for the IHCP. Comparisons of the estimated fluid temperatures with experimental fluid temperatures near the inner wall showed that the IHCP could accurately capture the actual temperature in form of the frequency of the temperature fluctuations. The analysis also showed that the IHCP needed at least 13 measurement points for the average absolute error to be dramatically reduced for the present IHCP with 37 nodes on each half of the pipe wall.     

Boundary reconstruction in two-dimensional steady state anisotropic heat conduction using a regularized meshless method

We study the stable numerical identification of an unknown portion of the boundary on which either a Dirichlet or a Robin boundary condition is provided, while additional Cauchy data are given on the remaining known part of the boundary of a two-dimensional domain, in the case of steady state anisotropic heat conduction problems. This inverse geometric problem is solved using the method of fundamental solutions (MFS) in conjunction with the Tikhonov regularization method [53]. The optimal value for the regularization parameter is chosen according to Hansen’s L-curve criterion [17]. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples in both smooth and piecewise smooth geometries.     

Two dimensional steady-state inverse heat conduction problems

In this work we estimate the surface temperature in two dimensional steady-state in a rectangular region by two different methods, the singular value decomposition (SVD) with boundary element method (BEM) and the least-squares approach with integral transform method (ITM). The BEM method is efficient for solving inverse heat conduction problems (IHCP) because only the boundary of the region needs to be discretized. Furthermore, both temperature and heat flux at the unknown boundary are estimated at the same time. The least-squares technique involves solving the equations constructed from the measured temperature and the exact solution. The measured data are simulated by adding random errors to the exact solution of the direct problem. The effects of random errors on the accuracy of the predictions are examined. The sensitivity coefficients are also presented to illustrate the effect of sensor location on the estimated surface conditions. Numerical experiments are given to demonstrate the accuracy of the present approaches.     

MULTIPLE TRANSIENT POINT HEAT SOURCES IDENTIFICATION IN HEAT DIFFUSION: APPLICATION TO NUMERICAL TWO- AND THREE-DIMENSIONAL PROBLEMS

This article deals with an inverse problem, which consists of the location and strength identification of multiple-point heat sources in transient heat conduction. The identification procedure is based on a boundary integral formulation using space and time Green functions. The discretized problem is nonlinear if the location of the point heat sources is unknown. In order to reduce the sensitivity of the solution to errors, we use the future time step procedure associated to a Tikhonov regularization procedure. The proposed numerical approach is applied to numerical two- and three-dimensional examples.     

Practical application of inverse heat conduction for wall condition estimation on a rotating cylinder

The solution of the linear, inverse, transient heat conduction problem (IHCP) in a cylindrical geometry is analysed. The rotating cylinder under investigation is experiencing boiling convection induced by the impingement of a water jet. The initial temperature is known, additional temperature measurements in time are taken with sensors positioned at a constant radius within the solid material, and the estimation of the wall heat flux at the external radius is sought. First, simulated temperature measurements inside the cylinder are processed in order to be used to estimate the wall heat flux. When noise is present in the data, some of the simulated results obtained using the least squares method exhibit oscillatory behavior, but these large oscillations are substantially reduced by the implementation of a regularization technique. Real experimental data are also used for the wall condition estimation and for the subsequent building of local boiling curves are plotted and discussed. The question of the possible effect of a temperature dependent conductivity on the reconstructed wall condition is also considered.     

Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem

In the paper, an example is presented of the application of a genetic algorithm to a design inverse Stefan problem. The problem consists in the reconstruction of the function which describes the heat transfer coefficient, where the positions of phase change moving interfaces are well-known. In numerical calculations, the Tikhonov regularization, a genetic algorithm and a generalized alternating phase truncation method were used. The featured examples of calculations show a very good approximation of the exact solution.     

Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting method

Abstract

In this article, an explicit Lie-group shooting method (LGSM) is developed to solve the time-dependent heat source and the initial data for backward heat conduction problems. To recover both unknown data simultaneously, it is very difficult to obtain a stable solution by explicit or implicit schemes. To solve these problems by using conventional numerical schemes, numerical iterative regularization techniques and numerical integration techniques are necessary. To avoid these numerical techniques and to increase the computational efficiency, an explicit LGSM is developed. According to the solution of the quadratic equation of the LGSM, the initial condition can be directly obtained by using the final condition and boundary conditions at the initial time and final time. Using the reciprocal relationship of the solutions for the initial condition and the final condition, the proposed algorithm can avoid numerical integration and numerical iteration. Additionally, a closed-form formula from a two-point Lie-group equation can be directly used to calculate the heat source term. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and stable solution, even with noisy measurement data, by comparing the estimation results with the existing literature.     

A Hybrid Method for Reconstructing Temperature Distributions in a Radiant Enclosure

A hybrid method combining Tikhonov regularization and generalized singular value decomposition (TR–GSVD) was proposed to reconstruct temperature distributions in radiant enclosures. The regularization parameter, which is crucial to accurately inverse temperature distributions, could be fast updated in TR–GSVD. Numerical reconstruction of three-dimensional temperature fields in a 10 m × 10 m × 20 m furnace proved TR–GSVD possesses the same computational accuracy but higher efficiency compared with TR. Experimental reconstruction results from practical flames in the furnace also showed the regularization parameter changed significantly in some combustion conditions. It is necessary to update the regularization parameter during the reconstruction.