Structural optimization for multiple structure cases and multiple payload cases with a two-level multipoint approximation method

An Haichao,Chen Shenyan,Huang Hai

School of Astronautics,Beihang University,Beijing 100083,China

Structural optimization for multiple structure cases and multiple payload cases with a two-level multipoint approximation method

An Haichao,Chen Shenyan*,Huang Hai

School of Astronautics,Beihang University,Beijing 100083,China

This paper is to address structural optimization problems where multiple structure cases or multiple payload cases can be considered simultaneously.Both types of optimization problems involve multiple finite element models at each iteration step,which draws high demands in optimization methods.Considering the common characteristic for these two types of problems,which is that the design domain keeps the same no matter what the structure cases or payload cases are,both problems can be formulated into the unified expressions.A two-level multipoint approximation (TMA) method is firstly improved with the use of analytical sensitivity analysis for structural mass,and the n this improved method is utilized to tackle these two types of problems.Based on the commercial finite element software MSC.Patran/Nastran,an optimization system for multiple structure cases and multiple payload cases is developed.Numerical examples are conducted to verify its feasibility and efficiency,and the necessity for the simultaneous optimizations of multiple structure cases and multiple payload cases are illustrated as well.

1.Introduction

Since the concept was proposed in the 1960s,structural optimization has experienced its significant progress and now itis a practical design tool in the field of structural engineering like aircraft and aerospace systems.1Three main optimization problems are considered and investigated which include sizing,shape and topology optimization,2,3and each kind of optimization problem owns its characteristics and difficulties.Whichever type the optimization problem belongs to,it always consists of three parts,i.e.design variables,objectives and constraints.When taking practical conditions into consideration,the complexities have been increased in structural optimization problems by involving multiple variables,objectives,constraints,etc.4–6For these problems,when finite element(FE)methods are used for structural analysis,it can be found that there is only a single FE model involved in general cases.

In reality,it is also required to conduct structural optimizations in multiple structure cases,7in which more than one FE model should be considered during the optimization process.Here,the so-called multiple structure cases refer to a structural system with different working modes or states.For instance,when a variable-sweep aircraft takes off or flies at high speed,it corresponds to two working modes,i.e.low and high sweep angles,for the wings;for another,the flexible attachments in a spacecraft,like the solar array panels and antenna,have compacted and deployed states,which correspond to launch stage and orbital status.The mechanical requirements such as deformations and strengths under the involved structure cases are different,indicating different design constraints,while the structure system is still composed of the same components,implying the same design variables.Accordingly,structural optimization under multiple structure cases is to design the shared structure components and to minimize the structural mass meanwhile simultaneously considering all constraints under each structure case.Therefore,at each iteration step,structural analysis should be carried out for all the FE models when conducting optimizations for multiple structure cases.So it can be seen that optimization in multiple structure cases is quite different from the usual optimization considering multiple load cases and it has drawn high demands in optimization schemes.

In terms of multiple payload cases,it refers to a structure acting as a platform to be equipped with multiple payload systems.For instance,in spacecraft designs,to decrease the design period and costs,a same satellite platform8,9could be adopted with different payloads.Under this condition,it is required that the optimal design to this platform should be carried out by simultaneously considering various known payload cases.Similar to the multiple structure cases,the structural response demands for different payload cases are various,while the structure systems have the same platform.In addition,it also involves different FE models and each model is related to one payload case.Correspondingly,more than one structural analysis is needed at each iteration step.From this view point,structural optimizations for multiple structure cases and multiple payload cases can be classified into the same category of optimization problems.However,the related publications on multiple structure cases are limited and more research work is needed to be developed and explored.

Based on a two-level multipoint approximation(TMA)method,Huang et al.7,8,10developed an optimization system which considers single structure case with high efficiency even for large-scale engineering problems;afterwards,this method was improved and utilized11,12in an optimization design of a satellite platform by taking multiple structure cases into account.However,as a backward difference method was used for the sensitivity calculation in this method,extra function evaluations are introduced and more CPU time is cost consequently.On the other hand,when considering multiple structure cases,some structural mass will be lost due to the changes between different structure cases.For example,for the solar array panels,the total mass in the deployed-panel case is slightly smaller than that under the compacted-panel case,which is due to the mass loss of the initiating explosive devices.When the total structural mass is treated as an objective in the optimization design,it will be different in multiple structure cases even with the same design variables.However,even though the optimization system has the preliminary capability by handling multiple structural cases,it could only deal with problems with mass being unchanged in different structural cases.Thus,the efficiency as well as the capability of this optimization method needs to be enhanced.

With the use of this TMA method,Chen and Huang8conducted the optimization design to the main frame of a satellite platform,while Peng et al.9developed an efficient truss structure optimization framework and optimized the similar main frame structure.The main difference of these two satellite structures lies in the different payloads,which results in different requirements in the designs.If the optimization results obtained in one payload case were used for another case with a different payload,this optimization design could be infeasible to satisfy all constraints.Therefore,an efficient optimization strategy should also be proposed by considering multiple payload cases.

Thus,the main objective of the present study is to present an optimization scheme for solving the structural optimization problem considering multiple structure cases and payload cases.Since the previous optimization scheme developed in Ref.8exhibits good performance in engineering optimization designs and it has the preliminary capability to address multiple structure case problems,this method is employed and the n improved in this work.By using analytical method for sensitivity calculation in this study,the computational efficiency and result accuracy are increased to some extent.Considering the common characteristic for multiple structure and payload case problems,which is that the design domain keeps the same no matter what the structure cases or payload cases are,these two kinds of optimization problems are formulated into the unified expressions.Based on the unified problem formulations,an efficient optimization system is the n established by involving both types of optimization problems.Numerical examples are the n conducted to demonstrate its feasibility and efficiency.

The outline of this paper is organized as follows.Section 2 formulates the problem expressions with a unified form by involving both multiple structure cases and multiple payload cases.Section 3 introduces the optimization strategy followed by the establishment of optimization system presented in Section 4.Numerical examples,including an optimization of solar array panels with two structure cases and a design for a satellite platform with three payload cases,are shown in Section 5 and a brief conclusion is drawn in Section 6.

2.Problem formulation

Structure mass is often treated as an objective in structural optimization problems and it is also the objective in the present work.For structures with multiple structure cases or states,the total mass will be different in different structure cases,which is due to the mass loss of some devices used for the changing the structure case.Similarly,for a structure platform with varied payloads,the total structural mass also varies along with different payload masses.For these two types of problems,what the y have in common is that the design domain is unchanged.For multiple structure cases,the devices used for changing structure case always do not need to be optimized and the y are not involved in the design domain;while for multiple payload cases,only the platform needs optimization design for reducing its design period and costs as well as being suitable for each payload case.By sharing the same design domain,it means the design variables are the same in different structure cases or payload cases.Actually,for a structure which is to be designed,it involves two parts,i.e.design domain and non-design domain.In a given structure case or payload case,the mass for the non-design domain is fixed as a constant.For this reason,the structure mass of the design domain instead of the total structure mass can be calculated as the objective in the optimization design.Thus,by gathering all structural response constraints,the optimization problem by considering multiple structure cases or multiple payload cases is formulated as

where X is the design variable vector and n is the number of design variables;the objective function f(X)is the structure mass of the design domain,gjk(X)is the jth structural response constraint in the kth structure case or payload case,such as stiffness and stress constraints,mkis the number of constraints under the kth structure case or payload case,K is the number of considered structure cases or payload cases,andandare lower and upper bounds on the ith design variable xi.

Based on the previous work8where only sizing variables were considered,the present work also only takes sizing variables into account,and the design variables X are linked cross-sectional dimensions of beams and thicknesses of structure shells,etc.The method used in the previous work8has been extended for continuum structure topology optimization,13truss topology optimization,14stacking sequence optimization,15–18as well as actuator placement optimization.19Thus,even though only sizing variables are considered in this paper,it can be expected to be improved for handling the aforementioned other types of problems.

3.Optimization scheme

3.1.Optimization method

The optimization problem expressed in Eq.(1)is always complex,implicit and nonlinear to the design variable X,and its solution by directly using general mathematical programming methods is nearly impossible for its large computational costs.Approximation concept approaches or surrogate-based methods were usually introduced to solve structural optimization problems,20,21and quite a few of the approximations were constructed with the use of the gradients for the objectives and constraints.Huang and Xia10presented a TMA method,which proved to be a powerful structural optimization method and the n was used to develop an efficient optimization system.8On the basis of inheriting the original performance of the TMA method,this method is extended in this work to address both multiple structure case and multiple payload case optimization problems as formulated in Eq.(1).This extended capability further illustrates the excellent performance of this method in dealing with various optimization problems.

3.1.1.The first-level approximate problem

To solve the implicit problem in Eq.(1),it is firstly transformed into a series of nonlinear approximate problems.By using constraint elimination techniques,after the pth structural analysis and sensitivity analysis,the first-level approximate problems is formulated as

where

and H is the number of the points to be counted and its upper bound Hmaxcan be given.When the number of known points is more than Hmax,only the last Hmaxpoints are taken into consideration in Eq.(3).The exponent rtis an adaptive parameter to control the nonlinearity of the approximate function.When there is only one known point,i.e.H=1,the value of rtis-1,and the approximate function in Eq.(3)becomes a Taylor series in the reciprocal design variable space.When Hgt;1,rt(t=1,2,...,H-1)is the solution of the following equation:

where t=1,2,...,H-1 and XHis the last known point.For rH,rH=rH-1.

With structure mass as the objective f(X),it can be expressed in an explicit function and it is still adopted in the first-level approximate problem,i.e.

It should be noted that f(X)can also be similarly approximated aswhen f(X)is implicit.

3.1.2.The second-level approximate problem

Considering the number of active constraints is usually fewer than that of the design variables,it is reasonable to use the dual method to efficiently solve the first-level approximate problem.However,as the constructed first-level approximate problem is still complex and nonlinear,it is difficult to establish the functional relations between the design variables and the dual variables,which is required in dual methods.Thus,a second-level approximate problem is built to approximate the first-level approximations and it can be easily solved by using the dual method.By expanding the objective function and the constraint functions in the first-level approximate problem Eq.(2)into linear Taylor series in the variable space X and its reciprocal variable space,respectively,at the qth step,the second-level approximate problem is stated as

where

The dual problem in Eq.(13)can be solved by using the variable metric method(BFS algorithm).After finding the optimal solution λ*,the solution to the qth second-level approximate problem in Eq.(12)can be obtained by the relations in Eq.(14)and this solution can be treated as an expansion point of the (q+1)th step.This process is repeated until the solution to the pth first-level approximate problem in Eq.(2)is obtained.After that,the structural analysis and sensitivity analysis are conducted again and the (p+1)th first-level approximate problem in Eq.(2)is reconstructed.Proceeding as above,the primal optimization problem in Eq.(1)is solved after iterations.

3.2.Improved sensitivity analysis for structural mass

Sensitivity analysis plays an important role in structural optimization and design.22,23With the use of the commercial finite element analysis software MSC.Patran/Nastran,an optimization system was developed in Refs.8,12based on the TMA method.When constructing the first-level approximate problem,the structural response constraint values and their sensitivities were derived from Nastran analysis which was in default of using a semi-analytical method,and the sensitivities for the constraint functions in the second-level approximate problem were obtained from the derivatives of the first-level approximate functions expressed in Eq.(3).As for the objective of the structure mass,its exact value was calculated by using relevant module from Patran.The derivatives∂f(X(q))/∂xiused in the second-level approximate problem in Eq.(12)were obtained with the use of a backward difference method,as expressed below.

where Δxi(q)is a small value and it takes 0.01xi(q)in Refs.8,12.As a kind of finite difference scheme,this method is generally faced with the dilemma of using a small Δxi(q)to minimize the truncation error versus avoiding a small Δxi(q)because of the subtractive cancellation error.A method using complex variables proves to be an effective way in avoiding the subtractive cancellation errors.24,25Additionally,as the number of parameter adjustment is 2n to obtain the sensitivities in the finite difference method above,extra function evaluations are introduced and more CPU time is cost.Analytical method for sensitivities can be another common approach to improve the computational efficiency as well as to increase numerical accuracy and it is adopted in this work.As sizing variables(including the shell thickness and beam dimensions,etc.)are either linear or quadratic to the structural mass,these derivatives can be explicitly obtained with specific expressions so as to obtain these sensitivities,and these sensitivity values will replace those obtained with the backward difference method in Eq.(16).

4.Development of optimization system for multiple structure cases and multiple payload cases

Based on the commercial finite element software MSC.Patran/Nastran,an optimization system is developed for multiple structure cases and multiple payload cases,and this optimization system mainly consists of three parts:pre-processing,numerical calculation(structural analysis and optimization)and post-processing.An intuitive and easy way to use GUI(Graphical User Interface)was developed for pre-amp;postprocessing,which would be very easy to operate for users who have been familiar with Patran/Nastran.Users can establish FE models in the original Patran,and the n define the corresponding optimization models and input related parameters in the developed interfaces.The main program of the numerical calculation part was developed with PCL(Patran Command Language).The optimizer was coded in FORTRAN language and called by the main program as an executablefile.Nastran was also called to conduct structural analysis as well as sensitivity analysis by the main program according to therequirements of optimization algorithm.

According to the principle of the optimization strategy,the system structure and its solving procedure were designed as follows:

Step 1.For each structure case or payload case,the FE model of the whole structure and its optimization model are established with Patran pre-processing functions.Meanwhile,using these pre-processing functions,information files that contain the information of the finite element model as well as the optimization model are created for each case which will be used for structural analysis.Then,the optimization task is submitted to the calculation core and the next steps will be executed automatically.

Step 2.From the first to the last structure case or payload case,submit related information file to Nastran to execute structural analysis and sensitivity analysis sequentially for each FE model under each case.

Step 3.After Nastran calculation is finished,read the values of design variables,the objective function(structure mass for the design domain),the constraint functions and their sensitivities from the result files,and organize these data into files which can be called and modified by self-defined optimizer.

Step 4.Call self-defined optimizer to search optimum with the TMA method.

Step 5.Evaluate whether the second-level approximate problem is converged or not.If it is converged,go to Step 6;otherwise,from the first to the last structure case or payload case,sequentially modify the related structural parameters that are contained in the information files according to the current design variables and calculate the objective values and its sensitivities,and the n go to Step 4.

Step 6.Evaluate whether the first-level approximate problem is converged or not.If it is converged,stop calculation;otherwise,go to Step 2.

The implementation of the whole system is schematically shown in Fig.1.

5.Numerical examples

Numerical examples are introduced in this section to demonstrate the feasibility and efficiency of the improved method in dealing with multiple structure case and multiple payload case problems.The first example is to optimize a structure with three solar array panels considering two structure cases,i.e.deployed case and compacted case,and the second example is to optimize a satellite platform with three payload cases.To illustrate the efficiency of the improved TMA method when using analytical sensitivity for the objective function(designated as ITMA method hereafter),comparisons are firstly conducted with both TMA and ITMA methods in two examples when considering single structure case or payload case.Meanwhile,optimization results obtained from simultaneously considering multiple structure cases or payload cases are compared with the results from single case to show the necessity of simultaneous optimizations of multiple structure cases or payload cases.All of the calculations are implemented in a computer with CPU 3.30 GHz/RAM 8.00 G.

5.1.Solar array panels with two structure cases

5.1.1.Case 1(optimization design of the deployed solar array panels)

The FE model of the structure consisting of three solar array panels is shown in Fig.2 and the main body of each panel is made of honeycomb sandwich plates.The length of each panel is 2.581 m and its width is 1.755 m.Three panels are connected with beams in a length of 0.058 m.The panels are strengthened with beam frames surrounded on all sides,and the leftmost panel is connected with a cradle to be fixed at the base.The cradle is modeled with beams in box cross-section as shown in Fig.3,where tube and bar cross-sections are used in the second example.Non-structural masses are uniformly distributed on each panel and their related beam frames,and the triangles shown in Fig.2 represent lumped mass points that are attached to the solar panels.Each panel is divided into four regions as shown in Fig.4 and each region is allowed to have different designs,i.e.different honeycomb core thicknesses.Meanwhile,as an entirety,three panels are required to have the same designs.

In this deployed case,the objective is to minimize the structure mass and the constraint is that the first-order natural frequency should be more than 0.25 Hz.The design variables include the core thicknesses for the four regions,as well as the dimensions of box cross-section in the cradle,as listed in Table 1.

With the use of TMA and ITMA methods,the respective results are given in Table 2.It should be pointed out that when TMA method is used,the total structure mass rather than the mass for the design domain is treated as the objective,while the structure mass of the design domain is considered if the ITMA is applied.The structure mass in Table 2 refers to the mass of the design domain.It can be seen that the optimization results in this deployed case(Case 1)are almost the same for both methods.However,as there is no need to call Patran mass calculation tools for sensitivity calculations of the mass objective function in the ITMA method,CPU time is reduced from 0.22 min to 0.12 min consequently.Additionally,it is shown that the optimization process is stopped after only three iterations,which seems to be a premature optimization.However,based on our numerical tests by changing the criterion from small to large values,even though the optimization may proceed further by consuming more iterations,it is found that the objective value varies within ±0.0002 kg.This variation range is so small that this optimization can be taken as the optimized solution without premature.

Fig.1 Flowchart of developed system.

Fig.3 Beam cross-section type used in examples.

Fig.4 Each panel divided into four regions.

Table 1 Initial designs as well as lower and upper bounds of variables.

Table 2 Optimization results for structure of three solar array panels.

5.1.2.Case 2(optimization design of the compacted solar array panels)

Fig.5 shows the FE model for the structure of three solar array panels in the compacted state.The structure configuration parameters are the same with those in Case 1.As some initiating explosive devices will be used for deploying the solar array panels,the structure total mass in the compacted case is slightly larger than the total mass in the deployed case.This kind of mass loss is reflected with different masses of the lumped mass point,as shown in Fig.2 with triangles,in different structures.However,the masses of the design domain will not be changed if the same values of design variables are assigned for different structures.

Fig.5 Structure of three solar array panels in a compacted state.

In this compacted case,the objective is also to minimize the structure mass and the constraint is that the first-order natural frequency should be more than 25.00 Hz.The design variables are the same with those in Case 1 as listed in Table 1.The optimization results with both TMA and ITMA methods are also listed in Table 2.It can also be seen that the CPU time is largely saved when the ITMA method is used.Besides,compared with the results obtained with TMA method,the number of iterations is also reduced and the optimized objective value is a bit smaller while the constraint value is the same.

5.1.3.Case 3(optimization design of the solar array panels by simultaneously considering deployed and compacted cases)

In this case,the optimization design is conducted by simultaneously considering two structure cases as described above.The objective and the design variables are the same with Case 1 and Case 2.The constraints are that the first-order natural frequency in the deployed case should be more than 0.25 Hz,and meanwhile,it should be more than 25.00 Hz in the compacted case.The optimization results in this case are also summarized in Table 2.As the structural analysis and sensitivity analysis are conducted twice at each iteration point in this case,the CPU time is consequently much larger than the time cost in both Case 1 and Case 2.With the ITMA method,the iteration histories for the objectives of the three cases are plotted in Fig.6,and iteration histories for normalized constraints are shown in Fig.7.As for the normalized constraint,its definition in this work is expressed as

Fig.6 Objective iteration histories of three cases with ITMA method.

Fig.7 Constraint iteration histories of three cases with ITMA method.

To further illustrate the necessity of simultaneous optimizations of multiple structure cases,the results obtained from one structure case are tested in another case.For example,when the results obtained from Case 1(where only the deployed state is considered)with the ITMA method is applied in the compacted case,the structure mass of the design domain is still 14.75 kg,but the first-order frequency in the compacted case is 17.75 Hz,which does not satisfy the constraint value of 25.00 Hz.If the results obtained from Case 2(where only the compacted state is considered)with the ITMA method is used in the deployed case,the structure mass of the design domain is not changed as 19.23 kg,while the first-order frequency in the deployed case is only 0.19 Hz.Back to the optimization results of Case 3 listed in Table 2,it can be seen that both of the considered constraints are satisfied,even with a small increase in the mass value compared with the result obtained from Case 2.

5.2.Satellite platform with three payload cases

Fig.8 FE model of entire satellite.

Fig.9 FE model and parameters of the main frame.

A satellite structure is often composed of two parts:the main structure platform and the payload cabin,as shown in Fig.8.To reduce the design period and to improve its versatility for different payload cabins,this satellite platform is to be designed by simultaneously considering three payload cases with different masses.The design domain for the platform involves the main frame,and its location is highlighted in Fig.8.The design variables mainly include the cross-sections of the main frame in Fig.9,where the element marked with 11 represents rod elements and others are beam elements.The design variables and their initial values as well as related lower and upper bounds are shown in Table 3.With this initial design,the total mass in these three payload cases are 7000,7800 and 6000 kg,respectively,and these differences are caused by different masses of the payload cabins.Correspondingly,the centers of mass for the whole structure system are different in the three cases and the y are summarized in Table 4.For this problem,the constraints require that the first-order frequency should not be less than 11.00 Hz(Case1),10.50 Hz(Case 2)and 12.00 Hz(Case 3)for the respective three payload cases,and the objective is to minimize the structure mass of the design domain in the main frame.

Separate optimizations are firstly conducted where only one constraint is considered in its relevant payload case.Both TMA and ITMA methods are used in these separate optimizations and the optimization results are listed in Table 5.It should also be noted that when the TMA method is used,the total satellite mass is taken into account,and only the massin the design domain is considered when the ITMA method is applied.As the structure mass of the design domain takes less than 1%of the total satellite mass,the calculation accuracy is lost to some extent when the TMA method is used.This leads to fewer iteration numbers as shown in Table 5.However,the CPU time in ITMA methods is always less than that in TMA methods,and the optimized objective values are quite fewer.These are also due to the fact that the original backward difference method for the mass sensitivity calculation is replaced by the use of the analytical calculation.Thus,the efficiency of the ITMA method can be clearly observed.The simultaneous optimization of the three considered payload cases are carried out with the ITMA method(Case 4),and the optimization result is also given in Table 5.It can be seen that the CPU time is the longest among all the cases conducted.This is because the structural analysis and sensitivity analysis are conducted three times at each iteration point,and 63(21X3)times of structural analysis and sensitivity analysis are executed consequently.With the ITMA method,the iteration histories for the objectives of the four cases are shown in Fig.10,and iteration histories for normalized constraints are shown in Fig.11.

Table 3 Initial design as well as lower and upper bounds on cross-sectional dimensions.

Table 4 Summary for center of mass in three payload cases.

Table 5 Optimization results for satellite platform with three payload cases.

For illustrating the necessity of simultaneous optimizations of multiple payload cases,the results obtained from one payload case are tested in another case.For example,when the results obtained from Case 3 with the ITMA method are applied in the case with the total mass of 7000 kg,the firstorder frequency in this payload case is 10.72 Hz,which does not satisfy the constraint value of 11.00 Hz.Moreover,if these results are adopted in the case with the total mass of 7800 kg,the first-order frequency in this case becomes 9.87 Hz,which means the considered constraint is also not satisfied.Similar outcomes will be produced if the obtained results from Case 1 are applied in Case 2 or the results from Case 2 are used for Case 1.When all payload cases are optimized simultaneously,the considered constraints in each payload case are satisfied in the meantime.To achieve this goal,the structure mass in Case 4 is larger than the other three cases as a result.After this comparison,the effectiveness of the multiple payload case optimizations is demonstrated.

Fig.10 Iteration histories of four cases with ITMA method.

Fig.11 Constraint iteration histories of four cases with ITMA method.

Besides sizing optimization,TMA method has been extended for continuum structure topology optimization,13truss topology optimization14as well as composite stacking sequence optimization.15–18From small to large scale problems,the TMA method has exhibited its high efficiency with a few dozens of structural analyses in solving numerical examples as well as engineering applications which involve hundreds of and thousands of design variables.13Considering that the proposed optimization strategy in this work is developed on the basis of TMA method,it can be expected to be applicable for handling other types of problems,like topology optimization and composite structure problems,when the y consider multiple structure cases or multiple payload cases,and accordingly,it could also be effective and efficient in more complicated engineering problems with thousands of design variables,even though the examples presented above are not complicated enough.Additionally,the non-probabilistic reliability-based structural optimization is also a typical twolevel optimization problem26;or more exactly speaking,it is a challenging problem with nested optimization where the calculation for updating uncertain variables(second-level optimization)is nested into the solution of design variables(first-level optimization).According to the extensions of the TMA method for truss topology optimization14and composite stacking sequence optimization,15–18both discrete and continuous variables are involved in the first-level approximate problem and the solving process for continuous variables is nested into the solution of discrete variables.If similar techniques are used to involve both design variables and uncertain variables in the first-level approximate problems,it is also possible for this method to be extended for solving non-probabilistic reliability-based structural optimizations,where the solving processes for design variables and uncertain variables are likely to be achieved in optimizations at different levels.

6.Conclusions

Based on the TMA method,a structural optimization scheme is developed by considering multiple structure cases and multiple payload cases.The optimization method is firstly improved by replacing the backward difference method for the mass sensitivity calculation with the use of the analytical calculation,which proves to take less CPU time and produce more reasonable optimization results.Considering the shared characteristic for multiple structure case and multiple payload case problems,that is,the design domain keeps the same in different cases,a unified problem formulation is the n established.On the basis of the problem formulation and the improved optimization method,an optimization system is developed with the commercial finite element software MSC.Patran/Nastran.Results of the numerical examples demonstrate its feasibility and efficiency,which indicates that it can handle practical engi-neering problems.Moreover,by making comparisons of the single and multiple case results,the necessity of the simultaneous optimizations for multiple structure cases and multiple payload cases is illustrated.

Acknowledgements

This work was supported by the Innovation Foundation of Beihang University for Ph.D.Graduates.

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An Haichaois a Ph.D.candidate at School of Astronautics,Beihang University (BUAA).Currently he is also a graduate research trainee at Department of Mechanical Engineering,McGill University.He received his B.S.degree from Beihang University in 2011.His area of research includes composite optimization,topology optimization for discrete structures and multidisciplinary design optimization(MDO).

Chen Shenyanis an associate professor at BUAA.She received her Ph.D.degree from BUAA in 2005.Her main research interests include structural optimization and its engineering application.

Huang Haiis a professor at BUAA.He received the Ph.D.degree from BUAA in 1990.His current research interests include concept design of spacecraft,structural and multidisciplinary optimization,adaptive/smart structuresamp;mechanisms and control,and the related applications in aerospace engineering.

19 September 2015;revised 5 February 2016;accepted 5 April 2016

Available online 27 August 2016

Multipoint approximation;

Multiple payload cases;

Multiple structure cases;

Sensitivity analysis;

Structural optimization

©2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 10 82338404.

E-mail addresses:anhaichao@buaa.edu.cn(H.An),chenshenyan@buaa.edu.cn(S.Chen),hhuang@buaa.edu.cn(H.Huang).

Peer review under responsibility of Editorial Committee of CJA.