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Key Laboratory of Advanced Technology for Aerospace Vehicles of Liaoning Province，State Key Laboratory of Structural Analysis for Industrial Equipment，Dalian University of Technology，Dalian 116024，P.R.China

Abstract:Accurately solving transient nonlinear inverse heat conduction problems in complex structures is of great importance to provide key parameters for modeling coupled heat transfer process and the structure’s optimization design.The finite element method in ABAQUS is employed to solve the direct transient nonlinear heat conduction problem.Improved particle swarm optimization（PSO）method is developed and used to solve the transient nonlinear inverse problem.To investigate the inverse performances，some numerical tests are provided.Boundary conditions at inaccessible surfaces of a scramjet combustor with the regenerative cooling system are inversely identified.The results show that the new methodology can accurately and efficiently determine the boundary conditions in the scramjet combustor with the regenerative cooling system.By solving the transient nonlinear inverse problem，the improved particle swarm optimization for solving the transient nonlinear inverse heat conduction problem in a complex structure is verified.

Key words：improved particle swarm optimization；transient nonlinear heat conduction problem；inverse identification；finite element method；complex structure

0 Introduction

For the traditional heat conduction problem，the temperature field of an object is calculated by giving the initial condition and boundary condition of the material property，which is called the direct heat conduction problem［1］.Many different innovation methods were proposed by scholars to solve the direct heat conduction problems.However，limited by various factors，solving the inverse problems of heat conduction under some circumstances is difficult.A large number of inverse methods were proposed and studied，which could help estimate unknown values or unmeasurable quantities by some known and easily measured information［2］.So，solving the inverse problems of heat conduction is of great importance.

In order to solve the inverse problems of heat conduction with excellent performances，numerous algorithms have been proposed and developed［3］.An enormous amount of research has been put into solving inverse problems of heat conduction［4］.Based on conjugate gradient method and discrepancy principle，Yang et al.［5］proposed a new anti-identification algorithm for the heat flow of disc brake system with time and space changes.Liu et al.［6］solved the problem with heat source and boundary condition recovery by multiple-scale polynomial Trefftz method.Mohasseb et al.［7］presented an innovative hybrid optimization algorithm to solve inverse heat conduction problems，which used genetic algorithm as the leading optimizer and used sequential quadratic programming to reduce the calculation time.In addition，a number of other methods were innovatively proposed，such as the Levenberg-Marquardt method［8-10］，the krill herd algorithm［11］，the Fourier transform［12］，the fundamental solutions［13］，a domain decomposition method［14］，the bee colony algorithm［15］，the particle swarm optimization［16-17］，and the genetic algorithm［18］，which were introduced in Ref.［4］.

Those algorithms could be divided into two groups，the gradient-based algorithm and stochastic algorithm［3］.For most inverse problems，the gradient-based methods could usually converge rapidly，and had high inverse identification accuracy.However，the properties of these methods were easily affected by some factors，e.g.cost function of the inverse problem，which made the optimization method fall into the local optimum and missed the global optimal solution.Moreover，sensitivity coefficients must be evaluated in this way.As for the stochastic methods，they have excellent global searching ability because of the randomness of the initial population and the calculation based on lots of possible solutions instead of an initial solution.However，stochastic methods are generally with a slow convergence rate，and a mass of iterations are required inevitably.

For linear inverse heat conduction problems，satisfactory results could be obtained by using gradient-based methods.But for nonlinear heat conduction problems（e.g.the present work），the sensitivity coefficient matrix is difficult to be precisely determined，and then sometimes the inversion results obtained by the gradient-based method are unreasonable.For example，with the same random number，the error of inverse identification would become smaller with the increase of measuring error，when the algorithm［4］was employed to solve the corresponding nonlinear inverse problem.However，compared with gradient-based methods，the particle swarm optimization（PSO）has some advantages in solving nonlinear heat conduction problems.The stronger ability of global search prevents the identification from falling into local optimum.Only a few parameters need to be input before calculation，so the operation is very convenient and estimation results are very stable.In addition，PSO algorithm combined interpolation method can quickly solve the nonlinear heat conduction problems without repeated calculation of direct problems.Therefore，to accurately and robustly identify the boundary conditions of a scramjet combustor with the regenerative cooling system in Ref.［4］，this paper tries to propose a new stochastic algorithm，namely the improved PSO，to avoid the determination of the sensitivity coefficient matrix and ensure the accuracy，efficiency and robustness of inverse identification at the same time.

1 Principle of Traditional PSO

PSO was first proposed by Eberhart and Kennedy［19］.The idea of PSO was derived from the predation behavior of birds，which generated swarm intelligence and guided the inverse identification by consociation and competition of particles［20-21］.Recently，as a high efficiency，excellent stability and strong ability of global searching algorithm，the PSO algorithm has attracted wide extension of researchers.In the PSO algorithm，every particle in the searching space represented a possible solution，which adjusted the next site according to the flight experiences of individual and group［22-23］.There areMindividual particles inD-dimensional searching space with velocity vectorVand position coordinateX.The fitness values of individual particles are estimated by objective function according to their position coordinates.The biggest or smallest fitness values of individual particle and particle swarm encountered so far are named as local best positionPiand global best positionPg，respectively，which are regarded as individual flying experience and group particle’s experience，and affect the positions of nextgeneration particle swarm.The basic PSO algorithm formula can be described as

whereidenotes theith individual particle，i=1，2，…，N；ωthe inertia weight factor；V i（t）the velocity of theith particle intth generation；V i（t+1）the velocity of theith particle in generation（t+1）；Xi（t）the location of theith particle at generationt；Pi（t）the individual particle best location at generationt；andPg（t）the global best location of particle swarm at generationt.C1，C2denote the acceleration coefficients；andR1，R2the random numbers which are uniformly distributed in［0，1］.

2 Improved PSO

2.1 Coefficients of improved PSO

For the inversion identification algorithm，its performance is not only affected by the mechanism of the algorithm itself，but also by the values of coefficients.As we know，an excellent inversion identification algorithm usually has powerful global exploration at the initial stage and outstanding local exploration at the last stage.However，the inertia weight factorωand acceleration coefficientsC1andC2in PSO method are constant，which limits the property of PSO method.Some studies show that the larger coefficients of inertia weight and accelerationC1and the smaller coefficient of accelerationC2can enhance the global searching ability.In contrast，the smaller coefficients of inertia weight and accelerationC1，as well as the larger coefficient of accelerationC2，maintain the local searching capability of the algorithm.Therefore，inertia weight factor and acceleration coefficients in improved PSO method have been redefined，which are shown as

whereωinidenotes the initial value ofωandωfinthe final value；Mis an integer representing the biggest value of the generation.C1，iniandC1，findenote the initial and final value ofC1，respectively；C2.iniand andC2，finthe initial and final value ofC2，respectively.

2.2 Iteration equation

The velocity of particles in the traditional PSO algorithm is only affected by the individual best position，group best position and the speed of the previous generation，in which iteration strategy cannot be adjusted in time according to the speed of convergence and population distribution.Therefore，the PSO method is easy to fall into the local optimum for solving the complicated inverse problem.To overcome this drawback of the traditional PSO algorithm，we propose the improved PSO method with a new evolution strategy in this paper.For the new improved PSO method，the randomness of the particles’evolution is increased while the fitness of global best position remains unchanged for a certain number of iterations and the value of particle diversity is very low，which enables the particles to escape from the region nearby the local optimum and to search for the global optimum.The iteration equation of the（t+1）th generation can be expressed as

whereR1，R2，R3，R4andR5denote random numbers uniformly distributed between 0 and 1，andξ，nthe restrictions of diversity（t）andNquit，respectively.Nquitis the number of the algorithm convergence stagnation andαa parameter used to control the probability of particle initialization.

2.3 Par ticle diversity

The particle diversity is a parameter used to evaluate the dispersion degree of particle distribution in the whole searching space.In the process of identification，the particle diversity decreases slowly with the enhancement of particle tracking ability to the optimal position of group.Particle diversity is related to the ability of particles to jump out of the area near optimal local location.In some cases，the optimal position of the group is not the global optimal solution，and particles of traditional PSO algorithm cannot escape from the area near the local optimal solution at the last stage，which inevitably causes the algorithm to fail to identify.Particles diversity is defined as diversity(t)=

whereLmax（j）andLmin（j）denote the maximum and minimum values of thejth dimension in the searching space，respectively；diversity（t）denotes particle diversity at generationtandDthe total number of identified parameters.

3 Direct Problem：Three-Dimensional Transient Nonlinear Heat Conduction Problem

The direct problem is a three-dimensional transient nonlinear heat conduction problem，which is solved by the finite element method（FEM）in ABAQUS.Initial conditions，boundary conditions and physical property parameters of the model are certain，then the temperature field for inverse identification can be obtained.The energy equations of the direct problem can be described as

with the initial condition

and boundary conditions

4 Inverse Problem

For the inverse heat conduction problem，temperatures at some positions in the scramjet combustor are known，but some important unknown parameters of boundary conditions，e.g.heat flux in boundary，need to be inversely identified.In the process of inverse identification，the possible solutions are assessed by the minimization objective function，which are determined by the temperatures of estimated and measured.The objective function of the inverse problem is described as

Iteration stops until the generationMor the objective function value of global best position less than a specified valuef.The procedure for inverse identification is described as follows，and the flow chart is drawn in Fig.1.

Fig.1 Flow chart of identification

Step 1Prepare necessary data of the inverse problem and control parameters of the algorithm：Input the number of particlesN，the maximum number of iterationM，number of identified parametersDand searching space［Lmin（j），Lmax（j），j=1，2，…，D］. Input acceleration coefficientsC1，ini，C1，fin，C2，ini，C2，fin，initial and final inertia weight factorsωiniandωfin，restrictionsξandn，control parametersαand tolerance value for objective functionf.

Step 2Initialize the positions of particle swarm inD-dimensional searching space and the velocities of particle swarm，and turn to Step 4.

Step 3Calculate the velocities of particle swarm by Eq.（6）and the positions of particle swarm inD-dimensional search space by Eq.（7）.

Step 4Compute the values of temperatures at measuring positions by solving the direct problem，then the objective function of each particle can be obtained.Calculate the individual best positionPi（t），global best positionPg（t），the objective function value objiofPi（t）and the objective function value objgofPg（t）.

Step 5Calculate the diversity（t）of the particle swarm andNquitat generationt.Generate a random numberR3，and compare with the criterion of running the iteration equation：If the conditions diversity(t) ≤ξ，Nquit≥nandR3≤Nquit/αare achieved，turn to Step 6；Otherwise，turn to Step 7.

Step 6Generate the random numbersR4andR5，and initialize the positions and velocities of particles at generation（t+1）by Eq.（8）and Eq.（9），then turn to Step 8.

Step 7Generate the random numbersR1andR2，and calculate the new position of particle according to Eq.（6）and Eq.（7）.If the position is outside of the searching space at thejth dimension，make it to be equal toLmin（j）orLmax（j）.Turn to Step 8.

Step 8Calculate the objective function value of the new particles；Update the individual best position and the global best position.Check the stop criteria：If the value of objgis smaller than the specified valuefor the number of current iteration is equalM，turn to Step 9；Otherwise，turn to Step 3.

Step 9Output the computation，results and stop the program.

5 Examples of Improved PSO Algorithm for Solving Transient Nonlinear Inverse Problems

In this section，a model of outer wall structure［4］with steel mental material is used，and it is a part of the scramjet combustor in Ref.［4］.When the inversion methodology in Ref.［4］is applied to solve the corresponding nonlinear inverse problem，the results are unacceptable，due to the fact that the sensitivity coefficients cannot be evaluated accurately.Therefore，the improved PSO algorithm is used to solve the transient nonlinear inverse problem，in which sensitivity coefficients are unnecessary.The computer that was used to produce results of estimation has an Intel Core i7-4900 3.60 GHz CPU，16.0 GB of memory.

In the physical model，the bottom surface is a wall of the combustor，and 14 regenerative cooling channels are designed inside.But only one of them is considered in this work because the heat transfer mechanism in each one is the same.The length，width and height of selected structure are 0.2，0.005 and 0.019 m，respectively.The diameter of the channel is 0.001 5 m，and the density is 8 240 kg/m3.

In order to investigate the property of improved PSO to solve the transient nonlinear heat conduction problem in complex structures，some physical property parameters，such as conductivity and specific heat of the selected structure，are not constant and change nonlinearly with temperature.The curves of temperature-dependent conductivity and specific heat are shown in Fig.2.

Fig.2 Temperature-dependent thermal properties

Theinitial temperaturein theexampleis 1 200 K.The convective heat transfer boundary condition is imposed on the inner channel，and the temperature of the coolant is 300 K，which is a constant.Heat flux on the bottom surface is known.On the left，right，front，back and upper surfaces，the heat flux is zero.Boundary conditions of the selected structure are shown in Fig.3［4］.

Fig.3 Boundary conditions

The unstructured grids are used in the FEM model for ABAQUS，which contains 23 188 nodes and 109 992 elements.The temperature field of the three-dimensional transient nonlinear heat conduction problem is shown in Fig.4.It can be seen that temperature in the region near the middle of bottom surface increases with the increase of time，even the convective heat transfer has been imposed on the inner channel to cool the selected structure.This is because the heat fluxes on the bottom surface are so high that coolant in the channel cannot fully absorb the heat.In addition，the conductivity of the nonlinear problem increases with the increase of temperature，which accelerates the heat transfer rate in the selected structure and makes the temperature distribution in Fig.4 more uniform，that is，the temperature difference between different positions in Fig.4 is smaller than that in Ref.［4］.

Fig.4 Nonlinear transient temperature fields of the selected structure

Solutions to the direct problem will provide temperature information for the inverse identification of the boundary conditions.

5.1 Impact of measuring points’position

In the direct problem，each ABAQUS calculation needs about 3 min.Therefore，it is time-consuming to use the improved PSO algorithm to identify boundary conditions in the scramjet combustor with a regenerative cooling system，which needs multiple times to call ABAQUS.Long identification time is unfavorable to practice application.To overcome this deficiency，the database is used in this paper.We divide each dimension of searching space intoQequal parts，and the temperature measurements with eachP（i，j，k）（i，j，k=1，2，…，Q+1）are obtained by solving the direct problem.All the temperature measurements are stored in the database.During the estimation using the improved PSO algorithm，there is no need to call ABAQUS to solve the direct problem.Instead，temperature measurements are obtained from the database by interpolation.Using the database in inverse analysis instead of calling ABAQUS can be reused in inversion processes.This methodology can significantly reduce the inversion time and improve the efficiency，which is favorable for practical applications，especially in real-time occasions.

By solving the inverse transient nonlinear heat conduction problem in a complex configuration，the capability of the improved PSO algorithm will be demonstrated in this part.Boundary conditions at three points，on the bottom surface of the model，are selected to be identified.The exact values of the identified heat fluxes are 460，480，and 500 k W/m2，respectively.To examine the accuracy and efficiency of the improved PSO algorithm，measuring temperatures at six points are numbered 1，2，3，4，5，and 6，and all the results summarized in the following part are averages of the best five estimation selected from several calculations for clear conclusions.Temperatures of selected points are measured every 10 s until 200 s.Some parameters of the improved PSO algorithm are defined：D=3，C1，ini=C2，fin=2.5，C1，fin=C2，ini=0.5，ωmax=0.9，ωmin=0.4，ξ=20，n=10，α=100.It is necessary to point out that the values of the coefficients are based on results of multiple calculation，which can guarantee that the results are stable and accurate.Also，comparing with other kinds of interpolation functions，the cubic spline interpolation usually has high accuracy of results，so it is used in the inverse identification.

Generally speaking，the performances of the improved PSO algorithm are not only related to the algorithm itself，but also related to specific inverse problems，such as the number of measuring points，the position of measuring points，the number of particles，the measuring errors and so on.

To analyze the impact of positions of measuring points and numbers of measuring points on estimation，measuring points are divided into three groups：Group 1 contains points 1—3，group 2 contains points 4—6，and group 3 contains all the six points.Estimation results of the improved PSO algorithm with noiseless measurements are depicted in Tables 1—3，and the corresponding objective function value curves with different number of particles are shown in Fig.5.The program stops running if the iteration number reaches 50.The “Error” stands for the absolute error，which is an absolute value of the difference between the exact and identified value of heat fluxes，and the relative error is the ratio of absolute error to the exact value times 100%.” It is observed that all the relative errors of identified heat fluxes are smaller than 0.56%.Comparing the estimation of group 1 and group 2 can find that the results of group 1 obviously have bigger relative errors than group 2.This is because the position of point 2 in group 1 is on the right side of the upper surface，and temperature in that position only varies slightly with different boundary conditions on the bottom surface during the cooling process，so the inverse identification becomes more difficult.It is indicated that the accuracy of results can be affected by the positions of measuring points.Therefore，in order to improve the accuracy of the inversion，the measuring points whose temperatures change obviously with the boundary conditions should be selected.

Table 1 Estimation results of the improved PSO with different particles number in group 1

Table 2 Estimation r esults of the improved PSO with different par ticles number in group 2

Table 3 Estimation results of the improved PSO with different particles number in group 3

5.2 Effects of measuring points’number

In addition to the position of measuring points，the influence of measuring points’number should be also analyzed.Results using measuring points in group 1 have been introduced before，and their accuracy is the worst among the results of the three groups of measuring points.Compared with group 1，group 3 has three more measuring points 4—6.Estimation results for group 3 are shown in Table 3.It is found that all the best fitness is less than 1.5×10-8in group 3.However，the relative errors are zero when the particle numbers are 100，1 000 and 10 000，which reveals the results in group 3 are very accurate.In addition，it indicates that more measuring points can improve the accuracy of estimation.

5.3 Estimation of PSO method

In order to examine the performances of the improved PSO method for solving the inverse nonlinear transient heat conduct problem in a complex structure，the PSO method is also used to retrieve the identified parameters with measuring points of group 1，and the particle numbers are 100，1 000 and 10 000，respectively.The program stops running when the iteration number equals 50.Fig.6 shows the objective function value curves with three different number of particles.It is observed that the results of the first and the third parameters with different particle number are accurate in Table 4，whereas the relative errors of the second parameter are unsatisfied.Comparing the results of the PSO method with the improved PSO method，it is easy to see that relative errors of the second parameter of the PSO method are many times as much as that of the improved method.Hence，for inverse identification the boundary conditions in scramjet combustor，the improved PSO method shows higher accuracy than the PSO method.Also，the time shown in Table 1 and Table 4 is CPU time，which is used to compare the efficiency of the PSO method and the improved PSO method.The results show that both of the two methods with the same particle number take about the same time.

Fig.6 Objective function value curves with different number of particles

Table 4 Estimated results of PSO with different particles number

5.4 Efficiency and accuracy of the impr oved PSO method

In this section，the program stops running until the iteration number reaches 100 or the best fitness is less than 1×10-6.Measuring points of group 1 are used in this analysis.In Table 5，it can be seen that the convergence speed for 20 000 particles is the most fast and only three iterations are needed，while 23 iterations are needed when the particle number is 5 000.It is observed that the iterations decrease with the increase of particle number，which is as expected.Also，results of three selected representative estimations with three different numbers of particles are chosen as observations，which are shown in Figs.7—8 and Table 5.From Fig.8，it is obvious that the objective function value of the improved PSO with 20 000 particles rapidly decreases.While the methods with 10 000 particles and 5 000 particles experience a period of stability in process of estimation，which cause the convergence speed slowly.So it inevitably needs more time for identification.Convergence curves of three heat fluxes with different particle numbers are shown in Fig.7.It can be concluded that an increasing in particle number can speed up the estimation.Computation time shown in Table 5 is only about 1 s.In addition，all the relative errors shown in Table 6 are less than 0.5%.It indicates that the improved PSO method can predict the boundary conditions of nonlinear transient heat conduction problems in a complex structure efficiently and accurately.

Fig.7 Convergence curves of three heat flux with different numbers of particles

Fig.8 Objective function value curves with different numbers of particles

Table 5 Inversion time with differ ent par ticle number s

Table 6 Estimated r esults of the impr oved PSO method with different particles numbers

5.5 Effects of measur ing er rors

In above analysis，the performances of the improved PSO method to estimate the inverse nonlinear heat conduction problem with noiseless temperature measurement are examined.However，noiseless data is unavailable in practice.Hence，it is necessary to investigate the property of the improved PSO method with certain measuring errors.A random error term is added to measuring temperature to simulate the measuring error，which is described as

whereζis a random number follow normally distributed with zero mean；ηis a certain number used to control the range of measurement error，hereηare set as 1%，3%and 5%.

The ratio of the noise to the exact solution（NSR）of measuring temperature is depicted as Eq.（17）.The NSR distributions with the measuring errors of 1%，3%and 5%are shown in Fig.9，which are turbulent.

Fig.9 NSR distributions with different measuring errors

To assess the results of inverse identification，relative error is used in this work.The definition of relative error is described as

Measuring errors of 1%，3%and 5%are used to examine the performance of the improved PSO method.The particle number is 25，and measuring points are No.1，No.2 and No.6.The program of the algorithm will stop running until iteration reaches 25.Fig.10 shows the convergence curves of three heat flux with different measuring errors.Fig.11 shows the convergence curves with different measuring errors of temperature，and results and corresponding relative errors of identification are listed in Table 7.It is observed that all the relative errors of identified parameters are smaller than 1%，even the biggest measuring error is 5%.In addition，relative errors of identification are smaller than measuring errors.It is validated that the improved PSO method has ability to overcome the inevitable measuring errors for identifying the boundary conditions in scramjet combustor with a regenerative cooling system accurately.

Fig.11 Objective function values with different random temperature measuring errors

Table 7 Estimated results of impr oved PSO with different measuring errors

6 Conclusions

An improved PSO is proposed and used to solve three-dimensional transient nonlinear inverse heat conduction problems with complex structures，then the boundary conditions in the scramjet combustor with a regenerative cooling system are identified.By analyzing the performances of the improved PSO method，several conclusions can be obtained as follows：

（1）Comparing with the traditional PSO method，the improved PSO method can solve the transient nonlinear inverse heat conduct problem more accurately with the same efficiency.

（2）The methodology of obtaining temperature measurements by using database can obviously reduce the estimation time，i.e.，improve the efficiency.

（3）The improved PSO method can identify the boundary conditions in the scramjet combustor accurately and efficiently，for different measuring points and positions，which effectively solves the problem encountered by using the method in Ref.［4］.

（4）The improved PSO method can predict the boundary conditions in the scramjet combustor accurately，even in the case of containing certain measuring errors.

In general，it is verified the improved PSO method can solve the transient nonlinear inverse heat conduction problems in complex structures with good performances.