Function Specification Method Research Articles

A review of regularization strategies and solution techniques for ill-posed inverse problems, with application to inverse heat transfer problems

With the presence of a large number of inversion algorithms for inverse heat transfer problems (IHTPs) and non-IHTPs, a need for review to have a holistic view is seen. An exhaustive literature review, with the motivation of selecting the inversion technique best fit for a given problem, was made for a general inverse problem. For ill-posedness, a classification of available regularization algorithms namely Tikhonov’s regularization, Bayesian regularization, mollification method, Beck’s sequential approach and Alifanov’s iterative approach, has been provided. Inversion methods like singular value decomposition, truncated singular value decomposition, Tikhonov regularization and total variation regularization are explained. Optimization methods namely steepest descent method, conjugate gradient method, Newton method, Levenberg–Marquardt method, Lagrange method, adjoint method, function specification method, genetic algorithm, differential evolution and particle swarm optimization (PSO) are reviewed. Further, a technique based on neural networks is studied, and wavelet methods like shrinking and wavelet vaguelette decomposition are reviewed. Associated literature has also been listed, highlighting the gaps. The usability of various algorithms in IHTP, starting from the golden section search method, for retrieval of a single parameter, to the regularized versions of the inversion technique, for retrieval of multiple parameters with uncertainty, demonstrates real-life applications to fins in IHTP. An inversion algorithm capable to handle every kind of nonlinearity is sought in literature, whose absence raises the research question, “Is there a technique that works globally for every inverse problem?”, is asked prior to, “What if the available techniques were not utilized to an extent that they should?” is posed. In lieu of this gap, a general comparative framework is developed, such that an efficient technique is selected, based on the total minimum error, which can be used in any field of interest.

Estimation of thermal input in thermite reaction for innovative wellbore plugging & abandonment techniques

In this work, a methodology is presented to simulate heat transfer in wellbores for plugging & abandonment operations, where the thermal input is provided by a thermite reaction. The operation of burning a thermite column inside the well's production tubing to form a metal plug is studied. The objective is to eliminate the need to simulate the thermite domain and chemical reaction during the process, simplifying the physical model and reducing computational cost. In this model, it is assumed that the thermal input from the thermite reaction is provided to the model through multiple heat flux boundary conditions along the inner wall of the production tubing. The unknown heat flux from the thermite to the inner wall of the tube is estimated by solving an inverse heat conduction problem (IHCP). The Adaptive Function Specification Method is used to estimate multiple heat flux functions at the boundary through information from multiple temperature sensors located on the outer surface along the tube's height. A characteristic behavior of the heat flux curve was verified in all segments of the tube, and the average heat flux curve was used to simulate heat transfer during a 15 m thermite burning process inside a well.

Inverse Estimation of Transient Heat Flux Using Sequential Function Specification Method

Knowledge of surface heat flux is of great importance to numerous engineering fields. In this study, the sequential function specification method (SFSM) was employed to estimate the transient heat flux on the surface of semitransparent medium. The coupled radiation and conduction heat transfer was solved by discrete ordinate method combined with finite volume method. The temperature was used as input in estimation task which was solved by SFSM. Different types of heat fluxes were taken into account to test performance of the present technique. Retrieval results illustrated that the transient heat flux can be accurately determined using the SFSM. It was proved to be effective for different semitransparent media. For the heat flux with drastic changes, the SFSM showed some limitations. Moreover, the influences of future time step and measurement error on determination results were investigated. The SFSM was applicable even when the random deviation was increased to 1.5. The future time step of SFSM is suggested to be 3 for the estimation of transient heat flux on semitransparent medium surface.

Real-time identification of severe heat loads over external interface of lightweight thermal protection system

The lightweight thermal protection system (LTPS) is typically employed on the exterior surface of high-speed aircraft to prevent the penetration of high-temperature heat flux. The sequential function specification method (SFSM) is exploited in this study to solve the inverse issue in anticipation of the external-surface heating conditions of a lightweight thermal protection system by entering temperature data at a single measurement station. The results show that the inversion accuracy is significantly dependent on the temperature acquisition position as well as the forecast time step. Adding noise to the temperature data reveals that temperature noise aggravates the volatility of the inverse heat flux, necessitating an increase in the forecast time step to enhance the inverse accuracy. The experiment of a light thermal protection system demonstrates that the approach can efficiently and correctly recover the boundary heat flux by employing the inner wall temperature of the thermal protection structure.

Estimating the absorption efficiency in a laser welding process using a nonlinear inverse problem method

A numerical-experimental methodology is presented in this study to estimate the absorption efficiency in a laser welding process by estimating the rate of energy transferred to a metal plate. The iterative Function Specification Method was modified to account for moving temperature sensor thermal sensitivity as a function of time and position relative to the welding bead. Thus, highly nonlinear problems can be solved by using a high-temperature gradient in the measurement sensor region. Three experiments on an AISI 1020 steel sheet were carried out using a 3 kW fiber laser and a 3 m/min welding speed. A thermo-fluid model was used with solid–liquid phase changes, buoyancy forces, and the Marangoni effect in the welding pool to model the physical phenomena. A code in Matlab was developed to solve the inverse problem. The direct problem was solved using COMSOL Multiphysics through the Livelink for Matlab feature. The average absorption efficiency was 79.5% for the welding process. A comparison was made between the geometry of the welding bead obtained in experiments with the numerically calculated welding bead to validate the model. The results obtained in this article are intended to assist simulations in laser welding processes and are in agreement with the literature data.

An improved sequential function specification coupled with Broyden combined method for determination of transient temperature field of the steel billet

Accurate knowledge of the steel billet temperature field in the reheating furnace is of crucial importance to improvement of product quality and energy conservation. For this purpose, the determination of the transient temperature field of the steel billet during reheating is investigated based on the inverse heat conduction problem. The transient two-dimensional heat transfer model of the billet in the furnace is established and used as the direct problem. The sequential function specification coupled with the Broyden combined method (BC-SFSM) is developed to obtain the temperature field of steel billet. A real measurement experiment is conducted for validation of the inverse model, which indicates that the BC-SFSM can accurately determine the billet temperature field from the temperatures measured at three points inside the billet. A series of numerical experiments are implemented to analyze the performance of the BC-SFSM and study the effects of the measurement point locations and of the number of future time steps on inverse results. The results illustrate that the BC-SFSM can take only 49.0% of the computing time of the sequential function specification method to estimate the billet temperatures, and the relative root-mean-square error of estimated temperatures is generally less than 1.5% when the reasonable number of future time steps and suitable measurement point locations are adopted. Therefore, the BC-SFSM in this work is a promising algorithm to determine the billet temperature field in the reheating furnace with high accuracy and efficiency.

Sequential Inverse Heat Conduction Problem in OpenFOAM

The solution of the inverse heat conduction problem (IHCP) is commonly found with the sequential algorithm known as the function specification method with explicit updating formulas and sensitivity coefficients of heat flux. This paper presents a different approach namely a direct mathematical optimization of minimizing the least squares norm between experimental data and simulation. A CFD open-source code OpenFOAM is used together with NLOPT and DLIB optimization libraries. To guarantee credibility of the simulation tool developed herein, real experimental data is used from spray cooling of a fast-moving hot steel plate. As the IHCP is inherently an ill-posed problem, the proposed sequential algorithm is stabilized using future time stepping and thereof the optimal number is explained. An assumption about the profile of thermal boundary condition during future steps must be made. It is shown that assuming a linear change of the heat transfer coefficient during each sequence of future time steps yields more accurate results than setting a constant value. For the problem size considered with less than 10k cells, the preconditioned conjugate gradient (FDIC) linear solver converges faster than the multigrid solver (GAMG). However, the latter performs better as the accuracy is concerned. Concerning the best choice of minimizer, the BOBYQA algorithm (quadratic approximation) is found superior to other methods. The proposed IHCP solver is compared with the well-established one.

A modified sequential gradient-based method for the inverse estimation of transient heat transfer coefficients in non-linear one-dimensional heat conduction problems

In this communication, we introduce an online sequential implementation of a gradient-based method for reconstructing transient heat transfer coefficients in the context of non-linear one-dimensional heat conduction problems. Such a method employs a quasi-Newton updating strategy for computing the descent direction, in contrast with the traditional approach based on the conjugate gradient method. We denote the resulting procedure as the sequential quasi-Newton method (SQNM). The performance of the proposed algorithm was tested in the reconstruction of triangle-, sine-, and square-wave functions that models different transient heat transfer coefficients and compared with the results obtained using a standard sequential function specification method. The SQNM was capable of properly reconstruct the aforementioned exact functions independently of the location of the temperature sensors within the body. The proposed strategy is fast, robust, and reliable, which demonstrates the suitability of employing the sequential gradient-based implementation, together with the quasi-Newton updating strategy, for reconstructing transient heat transfer coefficients in the context of one- and two-dimensional non-linear heat conduction problems. Thus, the proposed method is a novel alternative strategy to other online inverse estimation procedures.

Inverse ARX (IARX) method for boundary specification in heat conduction problems

Heat conduction problems are usually solved either with analytical or numerical simulations, or with a reduced model using system identification. The use of polynomial models, often used in automation theory, gained the attention of the thermal community in the last decades to elaborate these reduced models because of their simplicity and performance to characterize an invariable system. They allow, for example, estimating a local temperature with a known input heat source. However, using polynomial models (or identified systems) in inverse conduction problems is not straightforward, usually requiring either a second inversion step. In this paper, we present a novel inverse technique based on the polynomial model ARX (autoregressive with exogenous input) that allows the estimation of an unknown input (like an imposed heat flux on a boundary) using a known output (temperature measurement). This new method, named inverse ARX or IARX, only requires a calibration step as a regular polynomial model and, then, it can estimate the input by a direct calculation with the identified parameters. The difference between IARX and ARX is the presence of future exogenous parameters, which were deduced using the initial discrete form of the ARX model. We present herein a numerical example using IARX of a 1D heat conduction simulation and IARX succeeded to estimate the input heat flux, even with high discontinuities and high measurement noises. Finally, we compare the proposed method with the classical Beck’s function specification method. IARX presented advantages like having no restriction for the number of future terms used in the model and performing the calculation 45% faster and with much less memory space consumption than with Beck’s method.

An application of conjugate gradient technique for determination of thermal conductivity as an inverse engineering problem

Inverse problems frequently occur when boundary conditions/thermal properties of a material are unknown, and are determined from known internal temperature measurements. A direct problem is the opposite of an inverse problem. For example, it would be the task of determining the temperature field within a material body, given boundary conditions and thermal properties. In an ill-posed, inverse problem, small variations in internal temperature measurement cause large variations in the estimated quantity such as boundary heat flux. Measured temperature variations may be due to noise, measurement error and instrumentation error, among other factors. There are various methods for solving inverse problems in materials engineering. In the paper, an inverse problem of estimating boundary heat flux for one-dimensional heat conduction is solved using the classical method of Beck and temperature dependent thermal conductivity is estimated using the more recent and advanced Conjugate Gradient method (CGM). The temperature values from the finite difference application to a direct heat transfer problem are used along with noise added to the data, in order to simulate real world, experimental conditions. The data is applied to the inverse problem of finding boundary heat flux and thermal conductivity. It is seen that Conjugate Gradient Method (CGM) along with sequential function specification method recovers the boundary heat flux and thermal conductivity quite accurately for noise levels of 1% and 2%. It is further observed that addition of future time steps stabilize the algorithm and increase accuracy of the desired results. It is also observed that CGM can recover the temperature dependent thermal conductivity quickly within few iterations, for different values of thermal conductivity function coefficients, k0 and k1. CGM is found to be fast and robust when applied to one-dimensional case and can be extended to complex geometries.

Modified nonlinear function specification and temperature moving sensor to estimate the heat supply in a welding process

The estimation of the imposed heat rate in welding processes has always been an obstacle to improve the thermal efficiency of these processes. Thus, the use of inverse problem techniques is an alternative procedure to estimate the heat rate. In this study, the problem is highly nonlinear; therefore, the heat flux provided by the welding process is estimated by the iterative Function Specification method. This method was modified to estimate the heat supply based on the heat rate sensitivity coefficient, which represents the influence of the welding power in the temperatures of 304 stainless steel plates. In order to avoid the problem of low thermal sensitivity due to the movement of the welding source on the upper face of the plate, this methodology is based on the concept of temperature moving sensor. The software COMSOL was used to solve the three-dimensional heat diffusion equation with enthalpy function to model the phase change problem. 10 type K thermocouples, equally spaced, were attached by capacitive discharge on the opposite heated surface to measure the temperature. The efficiency of the process was calculated with the estimated net heat rate which ranged from 63% to 87%. The average efficiency obtained was 75% for this welding process. A relation between the energy rate given by the power supply and the estimated efficiency was also observed. The thermal efficiency of the arc decreases as the power increases. Thus, an analysis using the Taguchi method was performed in order to better understand the influence of the welding parameters on the efficiency of the process. It was concluded that with a significance level of 10%, the welding current and arc length have a strong correlation with thermal efficiency. These results are discussed and compared with results obtained experimentally by other researchers. Besides, an uncertainty analysis in the estimated heat rate is also presented.

Simultaneous estimation of fluid temperature and convective heat transfer coefficient by sequential function specification method

In pressurized water reactor (PWR) with multiple loops, abnormal working conditions occur when coolant pumps do not work in some loops. In these closed-loop pipelines, thermal stratification is formed under the effect of natural convection cooling and this poses a threat to the safe operation of the reactor. In this paper, the sequential function specification method (SFSM) is implemented to simultaneously predict the spatially and temporally varying internal fluid temperature and convective heat transfer coefficient of two-dimensional pipe under thermal stratification. In the direct problem, the finite volume method (FVM) is adopted to solve the governing equations subjected to initial and boundary conditions. In the inverse problem, the least-square method is employed to obtain the heat flux of the inner wall, and then Newton's law of cooling and the natural-convection correlation are adopted to obtain the fluid temperature and convective heat transfer coefficient. Numerical experiments of natural convection cooling are carried out under different outer wall boundary conditions to demonstrate the effectiveness of the proposed inversion method. The results of temperature estimation are consistent with the results of natural cooling numerical experiment and the relative error is below 4%, whereas the convective heat transfer coefficient is slightly worse due to the difference in the extraction positions. The effects of the number of involved future time steps and the measurement noise on the accuracy of the inverse analysis are also investigated in detail. Our results show that the fluid temperature and convective heat transfer coefficient of natural cooling may be effectively estimated using the proposed inverse method.

Heat flux estimation in journal bearings using inverse heat transfer method

Journal bearings are used in industrial equipment at high rotational speed. Normal and also fault operations of bearing increase the temperature of the bearing. Estimation of heat flux on the inner surface of the bearing and prediction of temperature of the bearing can prevent the damage to the bearing. In this research, these quantities are estimated in normal operation by an inverse method. The sequential function specification method is applied and the best number of temperature sensors and their positions are obtained. The temperature sensors are embedded in the bearing. The results show that the accuracy of heat flux estimation has a district relation with the number and the position of the sensors and also with the measurement frequency.

A sequential conjugate gradient method to estimate heat flux for nonlinear inverse heat conduction problem

In this paper, a sequential conjugate gradient method is presented to reconstruct the undetermined surface heat flux for nonlinear inverse heat conduction problem (IHCP). This modified method combines the merits of sequential function specification method (SFSM) and conjugate gradient method (CGM). The proposed method can effectively eliminate the leading error which is inevitably generated in SFSM due to the existence of temporary assumption of the unknown parameters and reduce the computing time of nonlinear IHCP. The square wave and sine wave heat flux are used to evaluate the proposed method and the influence of the number of future time steps and measurement errors on the predicted results is studied. The numerical experiments verify the effectiveness and timeliness of the proposed nonlinear IHCP algorism in reconstructing the heat flux. The comparisons with the typical SFSM and CGM are conducted to demonstrate the superiority of the proposed method. The results show that proposed method is more accurate compared with that of the SFSM under the condition of large measurement errors and the computing time is much faster than that of CGM.

Contact resistance analysis applied to simultaneous estimation of thermal properties of metals

This work aims to present a method for simultaneously estimating the thermal conductivity, k, and specific heat, cp, in samples of stainless steel 304 and cemented carbide by accounting the imperfect thermal contact at the heater–sample interface. A transient one-dimensional heat conduction model with constant thermal properties is used. The metallic sample is placed in the middle of a polyimide heater and a thermal insulator. A constant heat flux is applied on the specimen upside surface and a thermal insulation condition is maintained on the opposing surface, where a type T thermocouple measures the temperature response. Contact resistance is determined and accounted for as a reducing factor on heat flux. Instead of considering a lumped model, a microscopic approach of the contacting regions is used to describe both the surface roughness and the fluid gap. With the intention of guaranteeing simultaneous and reliable estimates for both thermal properties, sensitivity analysis is used to establish heat flux intensity, experiment duration, time step for data acquisition, and other characteristics of the experiments. To simultaneously achieve both thermal properties, the optimization method BFGS (Broyden–Fletcher–Goldfarb–Shanno) is employed to minimize a least-squares objective function comparing experientially measured and numerically calculated temperatures. The numerical solution for the transient conduction problem is determined with COMSOL, which solves the transient heat diffusion problem by discretizing it applying the finite element method (FEM). The linear system of equations is solved using the PARDISO solver and backward differentiation formula (BDF) method. Heat flux is estimated employing the sequential function specification method (SFSM) as a means of verifying the robustness of parameter estimation and experimental procedure. Moreover, uncertainty analysis is performed to confirm the quality of the results obtained. Finally, the uncertainty analysis outcomes and thermal properties estimated are compared with literature.

Experimental study on transient heat transfer characteristics of intermittent spray cooling

ABSTRACT Experiments were performed to investigate the transient cooling characteristics of intermittent spray cooling (ISC) in single-phase and nucleate-boiling regimes. The consumption of residual liquid in the non-injection period was confirmed via observations of the dynamic behavior through visualization experiments. The time variation of the surface heat transfer is relatively small in the single-phase regime and minimal in the initial nucleate-boiling regime. Further, ISC was considered as a process where the transient mass flow rate was specified by the characteristic time. This idea was successfully evaluated in the prediction of the critical heat flux in cryogen spray cooling. Abbreviations: CHF: critical heat flux; CSC: cryogen spray cooling; DC: duty cycle; ISC: intermittent spray cooling; SFSM: sequential function specification method

3D-transient identification of surface heat sources through infrared thermography measurements on the rear face

A study of estimation of space and time distribution of heat flux at the front face of a thin plate is presented in this paper. This identification is carried out from the temperature data collected on the rear face by an infrared camera. The inversion procedure is based on a sequential estimation method: the state representation of the 3D parabolic model of heat conduction in solids. Of course, the parameterized heat flux distribution is discrete in both space (2D) and time. The high number of unknowns to be estimated simultaneously, as well as the measurement noise, makes the ill-posed character of this multi-dimensional inverse problem quite high. Regularization is implemented by using two techniques: the stabilization by the function specification method and the Tikhonov penalization.

Inverse estimation of model parameters for newborn brain cooling process simulations

In this work, a three-dimensional simplified computational model was built to simulate the passive thermophysiological response of part of a newborn's head for neonate’s selective brain cooling. Both metabolic heat generation and blood perfusion were considered. The set of model parameters was selected and a sensitivity study was carried out. Analysis of dimensionless sensitivity coefficients showed that the most important are: the contact thermal resistance between the cool-cap and skin, the thermal resistance of the plastic wall material, and deep (arterial) blood temperature. The function specification method was applied to estimate the value of the contact resistance. Two, four and six computationally simulated measurements with different uncertainties were used to adjust random contact resistance value to the assumed one. Results showed that when using only two measurements having 2% of the uncertainty, the error of estimation does not exceed 9.8%. However, when using six measurements having 1% of uncertainty, the resulting estimation is burdened with an error of 0.3% only.

Verification of sequential function specification method with intermittent spray cooling

Intermittent spray cooling (ISC) is an active flow control technique that is used to provide rapid response and increase the spray efficiency. The transient feature of surface heat flux and temperature is important for the understanding of ISC performance. It is also a sensitive parameter in the implementation of the control strategy of ISC. In this study, a sequential-function-specification method (SFSM) was used to determine the time-varying surface heat flux from an internal temperature measurement in ISC. The simulated heat flux input was used to evaluate the accuracy of the method while considering regularization, random noise, size of the time step and pulse period in the simulation. Experiments were conducted and the results were compared with the simulation results to examine a detailed characteristic of surface heat flux calculation. Regularization dampens random noise but induces deterministic bias. Increasing the time step and decreasing the pulse period result in high deterministic bias. The standard deviation of these estimates decreases as the time step increases. The present study gives a concise insight into the parameter effect of SFSM in ISC, offering guidance to select the appropriate parameter of SFSM and proper experimental settings.

Estimation of the moving heat source intensity using the multiple model adaptive inverse method

This study proposes a multiple model adaptive inverse (MMAI) method to estimate the heat source intensity of nonlinear heat transfer system with a moving heat source. In the whole motion space of heat source, the nonlinear heat transfer system is divided into several linearized subspaces, and a corresponding linear prediction sub-model for the temperature at the measurement point is established for each subspace. Based on the linear prediction sub-models, the components of the heat source intensity compensation corresponding to each linearized subspace are obtained through rolling optimization. Then, according to the instantaneous matching degree between each prediction sub-model and the actual heat transfer system, the components are weighted and synthesized to gain the compensation of the guessed value. Finally, the estimation of instantaneous moving heat source intensity can be realized. Numerical experiments are performed to study the effects of the number of prediction sub-models, the moving velocity of heat source, the temperature measurement errors and the number of future time steps on the inversion results. Comparison with the existing adaptive sequential function specification method is also conducted, and it shows the validity of the inverse method developed in this paper.