Two-dimensional Inverse Heat Conduction Problem Research Articles

Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques

A two-dimensional transient inverse heat-conduction problem (2DIHCP) was established to determine the mold heat flux using observed temperatures. The sequential regularization method (SRM) was used with zeroth-, first-, and second-order spatial regularization to solve the 2DIHCP. The accuracy of the 2DIHCP was investigated under two strict test conditions (Case 1: heat flux with time-spatial periodically varying, and Case 2: that with sharp variations). The effects of the number of future time steps, regularization parameters, order of regularization, discrete grids, and time step size on the accuracy of the 2DIHCP were analyzed. The results showed that the minimum relative error (epred) of the predicted Case 1 heat flux is 5.05%, 5.39%, and 5.88% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 6.31%, 6.30%, and 6.36%. Notably, zeroth- and first-order spatial regularization had higher accuracy than second-order spatial regularization, while zeroth-order spatial regularization was comparable to first-order. Additionally, first-order spatial regularization was more accurate in reconstructing heat flux containing sharp spatial variations. The CPU time of the predicted Case 2 heat flux is 1.71, 1.71, and 1.70 s for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 1 heat flux are 6.18, 6.15, and 6.17 s. It is noteworthy that the choice of spatial regularization order does not significantly impact the required computing time. Lastly, the minimum epred of Case 2 heat flux with zeroth-order spatial regularization is 7.96%, 6.42%, and 7.87% for time step sizes of 1/fs, 1/2fs, and 1/5fs, respectively. The accuracy of the inverse analysis displays an initial improvement followed by degradation as the time step size decreases. A recommended time step size is 1/2fs, where fs denotes the temperature-sampling rate.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

A quasi-reversibility method for solving a two-dimensional time-fractional inverse heat conduction problem

In this paper, we consider a two-dimensional time-fractional inverse heat conduction problem, which is severely ill-posed. A quasi-reversibility method is proposed to solve this problem with disturbed boundary value. We give the selection of regularization parameters of quasi-reversibility method both under a priori and a posteriori rules, and give the proof of error estimates between the exact solution and its regularization approximation. Further more, numerical results are included to verify the effectiveness of the proposed method.

Influence of the Slag Rim on the Heat Transfer Behavior of a Mold

To explore the influence of the slag rim on the heat transfer behavior of a mold, a slag rim was extracted from a stainless steel production factory for high-temperature continuous casting simulation experiments. The 2DIHCP (Two-Dimensional Inverse Heat Conduction Problem) two-dimensional inverse problem calculation model was used to analyze the heat transfer characteristics in the mold, and the temperature and heat flux distribution of the hot surface of the mold copper wall under different working conditions were investigated. The results showed that under a casting speed of 0.5 Â m/min, vibration frequency of 2 Hz, and amplitude of 10 mm, the seepage flow of the mold flux was 9.314 and 6.326 g with and without the slag rim, respectively, the thickness of the slag film was 2.1 and 1.3 mm, respectively, the temperature of the mold hot surface was 118.24 and 134.83°C, respectively, and the maximum heat flux density of the mold was 0.98 and 1.95 MW/m2, respectively. Through data comparison, when there is a slag rim, the amount of slag infiltration decreased by 32%, the thickness of the slag film decreased by 38%, the temperature of the hot surface of the mold increased by 14%, the heat flux density increased by 99%, and the internal crystal morphology of the slag film was complete. This indicates that the slag rim affects the heat transfer behavior in the mold by restricting the inflow of the mold flux and affecting the formation of the slag film.

A posteriori regularization method for the two-dimensional inverse heat conduction problem

Abstract In this article, we consider a two-dimensional inverse heat conduction problem that determines the surface temperature distribution from measured data at the fixed location. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A quasi-boundary value regularization method in conjunction with the a posteriori parameter choice strategy is proposed to solve the problem. A Hölder-type error estimate between the approximate solution and its exact solution is also given. The error estimate shows that the regularized solution is dependent continuously on the data.

Size determination of interior defects by reconstruction of subsurface virtual heat flux for step heating thermography

Step heating thermography is a non-destructive testing (NDT) technique that inspects interior defects by observing surface temperature rises during a long pulse heat stimulation. Due to the heat diffusion induced blurring of defect shapes, it is a challenging task to accurately determine the sizes of deep and small defects. This work focuses on estimating the size of a subsurface defect by an inversion of temperature images for the step heating thermography, to overcome the information loss during the propagation of thermal waves. We assume a space- and time-dependent virtual heat flux on the defect surface, which is reconstructed from surface temperature measurements by solving a two-dimensional inverse heat conduction problem. The local future time concept is combined with the Tikhonov's regularization to stabilize the inverse solution. A prudent choice of the regularization parameter is carried out through the L-curve. The feasibility of the proposed approach is assessed by numerical simulations and experiments. It is shown that the inversion brings an improvement in defect sizing accuracy for defects with low width-to-depth ratios and an efficient suppression of measurement noises.

LBM combined with LM algorithm to estimate the unknown heat flux – A new inverse approach

The objective of the present work is the application of the Levenberg–Marquardt method as an inverse method for the estimation of the heat flux. In this paper inverse estimation of heat flux for a two-dimensional heat conduction problem is carried out. As a direct method, in the first attempt the solution of two-dimensional inverse heat conduction problem is formulated by using Lattice Boltzmann Method as a forward model. Later the solution to the problem is also obtained by using Finite Difference Method (FDM) as the forward model for the purpose of validation. Once the forward model is established, Levenberg-Marquardt Method is used as an inverse model to estimate the input parameter i.e. heat flux which is reported. A complete error analysis of inverse model with known values is performed. As the Lattice Boltzmann Method (LBM) is acclimatizing to parallel computation, its use is recommended in Levenberg-Marquardt method for the solution of inverse heat conduction problem which is evident from the results.

Explicit Sensitivity Coefficients for Estimation of Temperature-Dependent Thermophysical Properties in Inverse Transient Heat Conduction Problems

Explicit expressions are obtained for sensitivity coefficients to separately estimate temperature-dependent thermophysical properties, such as specific heat and thermal conductivity, in two-dimensional inverse transient heat conduction problems for bodies with irregular shape from temperature measurement readings of a single sensor inside the body. The proposed sensitivity analysis scheme allows for the computation of all sensitivity coefficients in only one direct problem solution at each iteration with no need to solve the sensitivity and adjoint problems. In this method, a boundary-fitted grid generation (elliptic) method is used to mesh the irregular shape of the heat conducting body. Explicit expressions are obtained to calculate the sensitivity coefficients efficiently and the conjugate gradient method as an iterative gradient-based optimization method is used to minimize the objective function and reach the solution. A test case with different initial guesses and sensor locations is presented to investigate the proposed inverse analysis.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Two-dimensional inverse heat conduction problem in a quarter plane: integral approach

We consider a two-dimensional inverse heat conduction problem in the region $$\lbrace x>0, y >0 \rbrace $$ with infinite boundary which consists to reconstruct the boundary condition $$f(y,t)=u(0,y,t)$$ on one side from the measured temperature $$g(y,t)=u(1,y,t)$$ on accessible interior region. The numerical solution of the direct problem is computed by a boundary integral equation method. The inverse problem is equivalent to an ill-posed integral equation. For its approximation we use the regularization of Tikhonov after the mollification of the noised data $$g_\delta $$ of exact data g. We show some numerical examples to illustrate the validity of the method.

A modified quasi-boundary value method for a two-dimensional inverse heat conduction problem

In this study, we consider a two-dimensional inverse heat conduction problem, which determines the surface temperature and heat flux distribution from measured data at the fixed location. The problem is seriously ill posed in the Hadamard sense and a conditional stability is given for it. We propose a modified quasi-boundary value regularization method to deal with the ill-posed problem. By choosing suitable regularization parameters and introducing some technical inequalities, we obtain quite sharp error estimates between the approximate solution and its exact solution.

On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by using many source points through the addition theorem such that the ill-posedness is greatly reduced. The MSMM has the same superiorities as the CTM, such as the boundary discretization only, and is advantageous for solving inverse problems. Several numerical examples are conducted to validate the accuracy of solving inverse heat conduction problems using boundary conditions with different levels of noise. Moreover, the domain decomposition method is adopted for problems in the doubly-connected domain. The results demonstrate that the proposed method may recover the unknown data with remarkably high accuracy, even though the over-specified boundary data are assigned a portion that is less than 1/10 of the overall domain boundary.

Minimax Optimization Method in the Two-Dimensional Boundary-Value Inverse Heat Conduction Problem

A method is proposed for the two-dimensional inverse heat conduction problem via reconstruction of the spatial and temporal density of a boundary heat flux. It is based on the optimal control theory for objects with distributed parameters. The method limits the set of desired solutions to the class of physically realized functions, which makes it possible to represent the desired-effect structure as a product of two one-variable functions. The problem of semi-infinite optimization, which minimizes temperature residuals in the uniform estimation metric, is formulated based on the parameterization of the desired characteristic (considered a control action). Analytical solution of the problem with the alternance properties of the desired optimal temperature deviations makes it possible to obtain the optimal values of the parameter vector.

Solving of Two-Dimensional Unsteady Inverse Heat Conduction Problems Based on Boundary Element Method and Sequential Function Specification Method

The boundary element method (BEM) and sequential function specification method (SFSM) are used to research the inverse problem of boundary heat flux identification in the two-dimensional heat conduction system. The future time step in the SFSM is optimized by introducing the residual error principles to get the more accurate inversion results. For the forward problems, the BEM is used to calculate the required temperature value of discrete point; for the inverse problems, the impacts of different future time steps, measuring point position, and measuring error on the inversion results are discussed. Furthermore, the comparison is made for the optimal future time step obtained by introducing the residual error principle and the inherent future time step. The example analysis shows that the method proposed still has higher accuracy when the measuring error exists or the measuring point position is far away from the boundary heat flux.

An extended approach of a Kalman smoothing technique applied to a transient nonlinear two-dimensional inverse heat conduction problem

An extended Kalman smoother (EKS) introducing the concept of future measurement information is developed to handle nonlinear inverse heat conduction problems. In the present study, the formulation of the EKS algorithm is generalized in the case of a two-dimensional problem in order to reconstruct a non-homogeneous heating condition on the front surface of a cylindrical sample. This leads to apply the new method in a multiple sensor case for the simultaneous reconstruction of several parameters at each time. The analyzed inverse problem is nonlinear due to the variability of the thermophysical properties with temperature and to the presence of radiation boundary conditions. Inverse estimation is successfully performed where temporal and spatial variations of the front surface heat flux are recovered based on non intrusive transient temperature measurements made on the back surface. Numerical experiments show that the use of an optimal number of future data greatly improves the solution of the EKS compared to the extended Kalman filter (EKF) estimates, where a noticeable reduction of time lag and sensitivity to measurement errors is observed. Inversion results show that the optimal number of future data, chosen on the basis of a minimum measure of the heat flux bias, depends on the modeling error, the measurement time step, the distance between the sensors and the front heated surface and the measurement noise level. The proposed algorithm is robust in recovering different heat flux profiles and provides for all the examined patterns symmetric and stable solutions that superposes well with the exact functions. The EKS formulation based on an augmented state vector allows to efficiently recover the front surface temperature simultaneously with the heat flux.

An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.

Using multiple graphics accelerators to solve the two-dimensional inverse heat conduction problem

In this paper, we present a novel parallel algorithm implemented on graphics accelerators to solve the two-dimensional inverse heat conduction problem (estimation of the temporo-spatial heat transfer coefficients without any prior knowledge of the thermal boundary conditions) based on Particle Swarm Optimisation method. In the absence of analytical solutions, there are several heuristic methods to solve the problem, but the unacceptable high runtime (several days) makes these unsuitable for practical use. This paper presents the methods on how to adapt the original sequential algorithm to an efficient data-parallel one, keeping in mind the main features of graphics processing units (launching multiple threads on all multiprocessors, storing data in fast on-chip memory, eliminating warp divergence and memory transfer latency, using the host and device together, etc.). The achieved ∼45× speed-up (without any accuracy degradation) makes the heuristic methods suitable for practical use. Some of the proposed ideas are generally useable; therefore, this paper can be considered a step-by-step guide for researchers of other fields to speed-up general purpose calculations and evaluate the results.

Identification of space- and temperature-dependent heat transfer coefficient

The aim of this paper is to present a very efficient and accurate numerical algorithm to identify a variable (space- and temperature-dependent) heat transfer coefficient in two-dimensional inverse steady-state heat conduction problems involving irregular heat-conducting body shapes in the presence of Dirichlet, Neumann, and Robin boundary conditions. In this numerical method, a boundary-fitted grid generation technique (elliptic) is used to discretize the physical domain (heat-conducting body) and solve for the steady-state heat conduction equation by approximating the derivatives of the field variable (temperature) by algebraic ones. This paper describes a very accurate and efficient sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the variable heat transfer coefficient. The main advantage of the sensitivity analysis scheme is that it does not require the solution of adjoint equation. The conjugate gradient method (CGM) is used to reduce the mismatch between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results confirm that the proposed algorithm is very accurate, efficient, and robust.

Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets

In this study, a two-dimensional linear transition inverse heat conduction problem (IHCP) was solved using the Generalized Minimal Residual Method (GMRES) in quenching process by water jets. The inverse solution method was validated by set of artificial data and solution sensitivity analysis was done on data noise level, regularization parameter, cell size, etc. An experimental study has been carried out on quenching a rotary hollow cylinder by one row of subcooled water jets. The inverse solution approach enabled prediction of surface temperature and heat flux distribution of test specimen in the quenching experiments by using measured internal specimen temperature. Three different boiling curves were defined in the quenching process of a rotary cylinder. Result obtained by the inverse solution showed clear footprint of rotation in surface temperature and heat flux on each revolution of cylinder and temperature variation damping from quenching surface toward interior of specimen.

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

ABSTRACTThe 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.