Inverse Hyperbolic Heat Conduction Research Articles

Convergence analysis of a variational quasi-reversibility approach for an inverse hyperbolic heat conduction problem

Abstract We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell–Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational quasi-reversibility method for the classical time-reversed heat conduction problem, which obeys the Fourier or Fick law, can be adapted to cope with this hyperbolic scenario. We establish a generic regularization scheme in the sense that we perturb both spatial operators involved in the PDE. Driven by a Carleman weight function, we exploit the natural energy method to prove the well-posedness of this regularized scheme. Moreover, we prove the Hölder rate of convergence in the mixed L 2 {L^{2}} – H 1 {H^{1}} spaces.

An inverse hyperbolic heat conduction problem in estimating pulse heat flux with a dual-phase-lag model

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 with the dual-phase-lag heat transfer model in estimating the unknown boundary pulse heat flux in an infinitely long solid cylinder from the temperature measurements taken within the medium. An efficient numerical scheme involving the hybrid application of the Laplace transform and control volume methods in conjunction with hyperbolic shape functions is used to solve the hyperbolic direct problem. The inverse solutions will be justified based on the numerical experiments in which two different heat flux distributions are to be determined. 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 pulse heat flux can be obtained for the test cases considered in this study.

Estimation of surface heat flux and temperature distributions in a multilayer tissue 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 surface heat flux in a skin tissue, which is stratified into epidermis, dermis, and subcutaneous layers, from the temperature measurements taken within the medium. Subsequently, the temperature distributions in the tissue can be calculated as well. The concept of finite heat propagation velocity is applied to the modeling of the bioheat transfer problem. The inverse solutions will be justified based on the numerical experiments in which two different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors on the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.

Interval inverse analysis of hyperbolic heat conduction problem

A general numerical model was presented to analyze the interval inverse hyperbolic heat conduction problem, with Bregman distances and weighted Bregman distances as regularization terms. By using the interval finite element method and interval extension theory, the direct and inverse models were established for uncertainties. The eight-point isoparametric elements were applied for the discretization in the space domain, and the Precise algorithm in time domain was empolyed. The inverse problems were implicitly formulated as optimization problems, using squared residues between the calculated and measured quantities as the objective function of the inverse identification. Results show that the proposed numerical models can identify single and combined interval thermal parameters and boundary conditions for hyperbolic heat transfer problems accurately and efficiently.

Estimation of energy absorption rate and temperature distributions in short-pulse laser heating of metals with a dual-phase-lag model

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 with a dual-phase-lag model to estimate the unknown space- and time-dependent energy absorption rate in a thin metal film exposed to short-pulse laser heating from the temperature measurements taken within the medium. Subsequently, the temperature distributions in the metal film can be determined as well. The inverse solutions are justified based on the numerical experiments in which two different energy absorption rates are estimated. 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 laser energy absorption rate can be obtained for the test cases considered in this study.

An inverse hyperbolic heat conduction problem in estimating base heat flux of two-dimensional cylindrical pin fins

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 space- and time-dependent base heat flux of a cylindrical pin fin from the knowledge of temperature measurements taken within the medium. The inverse solutions have been justified based on the numerical experiments in which three 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 space- and time-dependent base heat flux can be obtained for the test cases considered in this study.

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.

Inverse hyperbolic thermoelastic analysis of a functionally graded hollow circular cylinder in estimating surface heat flux and thermal stresses

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 inner-wall heat flux of a hollow cylinder from the knowledge of temperature measurements taken within the medium. The inverse solutions have been justified through the numerical experiments in two specific cases to determine the unknown heat flux. 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 excellent estimation on the time-dependent heat flux can be obtained for the test cases considered in this study. Once heat flux variation is accurately estimated, the evolution of temperature, displacement, and stress distributions can be calculated in great precision.

Real valued genetic algorithm for solving an inverse hyperbolic problem: multi-core parallelisation approach

In this paper, a numerical approach combining the use of the least squares method and the genetic algorithm (sequential and multi-core parallelisation approach) is proposed for the determination of temperature in an inverse hyperbolic heat conduction problem (IHHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.4 GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.4 GHz for each core.

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

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 surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. 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 surface heat flux can be obtained for the test cases considered in this study.

Inverse Hyperbolic Heat Conduction in Fins with Arbitrary Profiles

This study aims to estimate unknown base temperature distribution in different non-Fourier fins. The Cattaneo–Vernotte (CV) heat model is used to predict the heat conduction behavior in these fins. This inverse problem is solved by the function-estimation version of the Adjoint conjugate gradient method (ACGM) based on boundary temperature measurements. The ACGM includes direct, sensitivity, and adjoint problems. For each of these problems, a one-dimensional general formulation of the non-Fourier model for longitudinal fins with arbitrary profile is driven and solved by an implicit finite difference method. In this study, three different profiles are considered: triangular, convex parabolic, and concave parabolic. For each of them, two different base temperature distributions are estimated using an inverse method. Moreover, the effects of sensor positions at the fin tip and a specific place in-between are considered on the base temperature estimation. A close agreement between the exact values and the estimated results is found, confirming the validity and accuracy of the proposed method. The results show that the ACGM is an accurate and stable method to determine the thermal boundary conditions in different non-Fourier fin problems.

Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.

Inverse Hyperbolic Conduction Problem in Estimating Two Unknown Surface Heat Fluxes Simultaneously

An inverse hyperbolic heat conduction problem is solved in the present study by an iterative regularization method, that is, the conjugate gradient method, to estimate simultaneously two unknown boundary heat fluxes based on the interior temperature measurements. The inverse solutions will be justified based on the numerical experiments in which two specific cases in determining the unknown boundary heat flux distributions are examined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat fluxes and that the position of each sensor should be as close to each boundary as possible to obtain accurate estimations. Finally, it is concluded that accurate boundary heat fluxes can be estimated in the present study when large measurement errors are considered.

The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.

An inverse hyperbolic heat conduction problem in estimating surface heat flux by the conjugate gradient method

In the present study an inverse hyperbolic heat conduction problem is solved by the conjugate gradient method (CGM) in estimating the unknown boundary heat flux based on the boundary temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where three different heat flux distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat flux. Moreover, the drawbacks of the previous study for this similar inverse problem, such as (1) the inverse solution has phase error and (2) the inverse solution is sensitive to measurement error, can be avoided in the present algorithm. Finally, it is concluded that accurate boundary heat flux can be estimated in this study.

Estimating the boundary condition in a 3D inverse hyperbolic heat conduction problem

This study is intended to provide an inverse method for estimating the unknown boundary condition T( x, 0, z, t) in a 3D non-Fourier heat conduction problem. In this study, finite-difference methods are employed to discretize the problem domain, and then a linear inverse model is constructed to identify the unknown boundary condition. The matrix forms of the original differential governing equations are rearranged in order that the unknown conditions can be represented explicitly, and the linear least-squares method is then adopted to determine their solution. The results show that the proposed method is capable of generating precise solutions using just a small number of measuring points. Furthermore, it is indicated that the method is capable of providing good numerical approximations even where measurement errors are present. The phenomenon of complicated reflection and interaction of thermal waves reflects the fact that the inverse non-Fourier heat conduction problem is different from the inverse Fourier heat conduction problem. In contrast to traditional approaches, the proposed inverse analysis method: requires no prior information regarding the functional form of the unknown quantities. In addition, only one iteration is necessary for calculation, and no initial guess is required. Furthermore, the existence and uniqueness of the solutions can be easily identified.

Direct and Inverse Solutions of the Hyperbolic Heat Conduction Problems

A sequential method for estimating boundary conditions in the field of hyperbolic heat-conduction problems is proposed. An inverse solution is deduced from a finite-difference method, the concept of future time, and a modified Newton‐Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of nonlinear equations, which are formulated from the measured temperature and the calculated temperature. Then an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Two examples are used to demonstrate the characteristics of the proposed method. In the first example, a well-known problem is used to demonstrate the validity of the proposed direct method and then an inverse solution is evaluated. In the second example, a larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is used to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method for determining the boundary conditions in inverse hyperbolic heat-conduction problems.