Sequential Function Specification Research Articles

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

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.

Trend analysis in the 1D solution of the inverse heat conduction problem by the sequential function specification technique

Inverse heat conduction problems are categorized by the solution technique or algorithm (Function Specification, Regularization, Laplace Transform, Conjugate Gradient, Mollification), by the solution method (Duhamel's Theorem, Difference Method Finite, Finite Element Method), and by the time domain (Stoltz Method, Sequential Method, and Complete Domain). In the direct approach, the unique solution of an effect due to a cause is obtained. From the cause-effect view, inverse problems are characterized by not meeting the existing criteria and the uniqueness of determining the cause when analyzing an effect. However, this has promoted the implementation of robust methods to optimize the stability of a solution. In this work, the study system consists of a long solid cylinder at an elevated temperature that is cooled. The implemented methodology allowed the creation of data trends through linear extrapolation to improve estimation accuracy for abrupt changes in the function (boundary condition). The results show an acceptable increase in punctual precision in the estimation, and it is a consequence of the solution that the calculated thermal histories already contain an implicit degree of error.

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.

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.

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.

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

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.

Estimation of the Time-Varying High-Intensity Heat Flux for a Two-Layer Hollow Cylinder

Gun barrels are subjected to time-varying high-intensity heat flux under multiple firing, which may damage the material and limit the overall performance of the gun. In order to monitor the thermal state of a gun barrel, an inverse method coupling the finite difference method with the sequential function specification method was developed to estimate the unknown time-varying heat flux imposed on the inner wall of a gun barrel. A two-layer hollow cylindrical tube was assumed with the convection heat transfer boundary condition on the outer wall of the tube. A direct heat transfer model was developed, and was used to estimate the temporal distribution of boundary heat flux in approximately real time based on the measured transient temperature at some positions on the outer wall of the gun barrel. Numerical tests were performed to verify the effectiveness and reliability of this method by investigating the influence of temperature measurement noises and future time step selection. The results show that the proposed method has high precision and efficiency in extracting the time-varying heat flux under one-shot and three-shot firing conditions. When there is a measurement noise, this method has good anti-illness characteristics and can achieve better results by appropriately selecting the value of a future time step.

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.

Time Traveling Regularization for Inverse Heat Transfer Problems

This work presents a technique called Time Traveling Regularization (TTR) applied to an optimization technique in order to solve ill-posed problems. This new methodology does not interfere in the minimization technique process. The Golden Section method together with TTR are applied only to the objective function which will be minimized. It consists of finding an ideal timeline that minimizes an objective function in a defined future time step. In order to apply the proposed methodology, inverse heat conduction problems were studied. Controlled experiments were performed on 5052 aluminum and AISI 304 stainless steel samples to validate the proposed technique. One-dimensional and three-dimensional heat input experiments were carried out for the 5052 aluminum and AISI 304 stainless steel samples, respectively. The Sequential Function Specification Method (SFSM) was also used to be compared with the results of heat flux obtained by TTR. The estimated heat flux presented a good agreement when compared with experimental values and those estimated by SFSM. Moreover, TTR presented lower residuals than the SFSM.

Evaluation of generalized polynomial function specification methods

The sequential function specification method (SFSM) for solution of the inverse heat conduction problem was introduced by Beck (1968) [1], and described in more detail [2] in 1985. This method assumes a functional form for the unknown heat flux in order to add more (future) data to regularize the problem and facilitate estimation of the next heat flux component. Two basic forms for the unknown heat flux have been well-explored: a constant function (zeroth order polynomial), and a linear function of time (a first order polynomial). Until now, only these two “traditional” forms have been used and higher order polynomial function approximations have not been considered.A generalized polynomial sequential function specification method (SFSM) is developed and evaluated in this paper. For higher order polynomials, the proper balance of past and future time step information is not obvious; the optimal balance is investigated herein. A generalized algorithm is used to define polynomials of any degree and estimate heat fluxes using SFSM with any number of past and future data. Seven heat flux test cases, including three non-polynomial heat flux variations, are used to evaluate the traditional and new methods, and the “optimal” number of future time steps is chosen for each case. The filter coefficient concept [3] provides additional insights into the new SFSM approaches. Filter coefficients are calculated with the optimal number of future time steps and the results are discussed.

Boundary Condition Estimation of Ablating Material Using Modified Sequential Function Specification Method

An inverse analysis is applied in a one-dimensional ablation problem for simultaneous estimation of unknown time-dependent boundary conditions including the applied heat flux and the Biot number at the two end surfaces of the ablating material. The sequential function specification method is employed to minimize the least-square objective function. Due to the ablation process, a moving boundary is created at the heated surface. The estimated boundary conditions along with the direct equations are used to calculate the front position of the moving surface. To deal with the high nonlinearity caused by the moving boundary and to treat the ill posedness associated with deeply embedded sensors of the current inverse problem, a small number of future time steps are used as the regularizing parameters to stabilize the inverse problem. Furthermore, to fasten the solution of the inverse problem, a new adaptive overrelaxation factor is developed that increases the speed of convergence significantly. The simulated temperature measurements at specific sensor positions inside the ablating materials are used to evaluate the objective function. The accuracy of an applied inverse analysis is examined by the step and triangular boundary condition profiles containing discontinuities and sharp corners, respectively. In spite of high nonlinearity, an appropriate consistency between the estimated values and the exact ones are obtained.

A multiple model adaptive inverse method for nonlinear heat transfer system with temperature-dependent thermophysical properties

A multiple model adaptive inverse (MMAI) method is proposed to estimate the boundary heat flux distribution of nonlinear heat transfer system with temperature-dependent thermophysical properties. In the temperature space of system, the nonlinear heat transfer system is divided into several linear subspaces, and for each subspace, the linear prediction submodel of the temperature at the measurement point is established. Furthermore, according to the instantaneous matching degree between each prediction submodel and the actual heat transfer system, the prediction submodels then are weighted and synthesized to gain the global prediction model of the nonlinear heat transfer system. Finally, based on the global prediction model, the boundary heat fluxes are simultaneously estimated through rolling optimization. Numerical experiments are performed to study the effects of system nonlinearity and measurement errors on the inversion results. Comparisons with the existing dynamic matrix control inverse method and the adaptive sequential function specification method are also conducted, and they all show the validity of the inverse method established in this paper.