参数化白车身多目标轻量化优化设计

OriginalArticleProc IMechE Part D:J Automobile Engineering2016,Vol.230(2)273–288ÓIMechE2015Reprints and permissions:/journalsPermissions.navDOI:10.1177/0954407015581937Design and application of lightweightmulti-objective collaborativeoptimization for a parametricbody-in-white structureChuan-Qing Wang,Deng-Feng Wang and Shuai ZhangAbstractThe static bending and torsional stiffnesses,the lower-order modal frequencies of a body-in-white structure and the full frontal-crash and side-impact passive safety performances are simulated with finite element models which are generated on the basis of the implicit parametric model.The implicit parametric model is established through SFE CONCEPT soft-ware.The simulation results are compared with tests to validate the simulation analysis results.It is proposed that the multi-objective optimization is divided into non-safety parts optimization,frontal-crash safety parts optimization and side-impact safety parts optimization,which is computationally more efficient than optimizing the non-safety parts,the frontal-crash safety parts and the side-impact safety parts simultaneously.In this paper,the lightweight multi-objective collaborative optimization design of the body-in-white structure is conducted for a passenger car by optimizing the thick-ness,the beam section shape and the size;while maintaining the performances of the static bending and torsional stiff-nesses,the lower-order modal frequency decreases to less than5%of the initial value,and the full-frontal-crash and side-impact passive safety performances remain almost the same.Structural modifications are applied by means of impli-cit parametric technology,providing changes in the geometry in a fully controllable manner.After comparison between the optimized body-in-white structure and the initial structure,the mass decreased in total by32.41kg(i.e.by as high as 7.63%).The decreases in the performances of the bending and torsional stiffnesses are less than2.54%;the bending and torsional frequencies increased a little,and the frontal-crash and side-impact passive safety performances underwent almost no change.KeywordsImplicit parametric body-in-white structure,lightweight design,multi-objective collaborative optimizationDate received:27October2014;accepted:20March2015IntroductionRecently,laws and regulations on the emissions,the fuel efficiency and the protection of the environment have become stricter.Design of a lightweight vehicle is one of the easiest methods to preserve energy and to reduce gas emissions.The mass of the BIW structure is about30%of the mass of the whole vehicle.1 Therefore,a reduction in the mass of the BIW structure plays an important role in decreasing the mass of the whole vehicle.Although multi-material mixes,assembling tech-niques and new manufacturing are promising areas, one of the main approaches to lightweight design is structure optimization.2Structure optimization often contains topology optimization,size optimization, shape optimization,single-objective optimization and multi-objective optimization.The aim of topology optimization is to improve the allocation efficiency of the material of the structure within the design domain.In recent decades,much State Key Laboratory of Automotive Simulation and Control,Jilin University,Changchun,Jilin,People’s Republic of China Corresponding author:Deng-Feng Wang,State Key Laboratory of Automotive Simulation and Control,Jilin University,Changchun,Jilin130022,People’s Republic of China.Email:419170738@work has been carried out by topology optimization.3–5 However,topology optimization is mainly focused on the concept design phase,for which the size of the structure is not known accurately.Thus,topology opti-mization,size optimization and shape optimization work together to determine the structure.6The size optimization and shape optimization methods have been extensively studied and utilized.7These methods consider parameters such as the plate thickness and the beam cross-section(e.g.the height and the width)as the design variables to conduct optimization design.8In the process of optimization,some geometrical shapes may be changed.In order to solve this problem,the morphing approach has been applied through software such as MeshWorks.The morphing approach takes the dimensions of the geometrical shape as variables in the process of optimization.It is used as a significant per-formance enhancement tool,which was originally used in the aerospace domain,9–11and then subsequently in the vehicle industry,which attracted much attention.In the vehicle industry,it has been used to improve the performances of the aerodynamics,the static stiffness and the crashworthiness.12,13This technique is rela-tively limited since the quality of the finite element(FE) mesh rapidly decreases as the amount of shape varia-tion increases,and the adjacent parts cannot change with the changing parts.Although remeshing technol-ogy is provided for the changed shape,the quality is poor in comparison with that of the parametric model. The implicit parametric model allows larger geometri-cal modifications,and the adjacent parts change with the shape variation.14The mesh of the implicit para-metric model is generated at each new generation,and so the quality is good.Therefore,the body-in-white (BIW)model used in this paper is an implicit para-metric model which is established by SFE CONCEPT software.Grujicic et al.15introduced topology optimization, size optimization and shape optimization.Then they obtained a new optimal structure with a polymer–metal hybrid material based on the stiffness and the strength, which was subject to bending.They also obtained the optimal structure by linearized eigenvalue buckling analysis and non-linear buckling analysis based on the buckling resistance,which was subject to axial com-pression loading.They found that the difference in the geometrical structure obtained from the linearized anal-ysis and the non-linear analysis is relatively small. Grujicic et al.16presented a multi-disciplinary optimiza-tion methodology,and they considered an inner door panel as an example for carrying out lightweight design based on this methodology.In the process of optimiza-tion,the number and orientation of the composite piles,the thickness of the local laminate and the shape of the panel were taken as variables,and the perfor-mances of noise vibration and harshness(NVH),the durability,the crashworthiness and the manufacturabil-ity were the constraints.Their research studies were divided into conceptual-design optimization and detailed-design multi-disciplinary optimization.In the conceptual-design optimization,they determined the make-up of the local composite laminate and the boundaries between different laminate patches using the free element sizing technique.In the detailed-design multi-disciplinary optimization,they took the car-door panel mass as the objective and made sure that the NVH,the crashworthiness and the durability met the requirements.Finally,they obtained the ply thickness and the inter-patch boundaries of the door panel.These research studies are very helpful in developing a new part structure of new material.So far,most stud-ies of BIW structure optimization have been mainly limited to the field of size optimization and utilize the performance of crashworthiness as the constraint for single-objective or multi-objective optimization.17–19 Studies which employ the thickness,the size and the beam section shape as variables and synthetically con-sider the performances of the static stiffnesses,the low-order natural modal frequencies and the frontal-crash and side-impact passive safety performances of the BIW structure are rare.This may obviously cause some performances of the BIW structure to decrease.While this paper is based on a parametric BIW model in order to conduct lightweight design.The aim is obtained through seeking the optimal combination of the thick-ness,the local section and the geometrical shapes.In the optimization,the BIW model is divided into non-safety parts,frontal-crash safety parts and side-impact safety parts.The optimization includes three phases based on these parts.This method can improve the computational efficiency and is described in more detail in the third section.After each optimization,the optimized model is compared with the initial model.Also,by maintaining the performances of the static bending and torsional stiffnesses,the lower-order modal frequencies decrease by less than5%of the initial values,and the full-fron-tal-crash and side-impact passive safety performances remain almost the same.However,in the process of multi-objective optimization,conflicting objectives arise mostly.Therefore,one performance is increased while another performance may decrease.The non-dominated sorting genetic algorithm(NSGA-II)when used for the multi-objective optimization problem (MOP)is a quite important method,which can deal with the above contradiction efficiently. Performance verification of the implicit parametric BIW modelIn the process of establishing the parametric BIW model,it is usually divided into three processes.The first process is to define the base points and the base-lines to indicate the position of the component ledges. The second process is to define the beam section and to assign the lines to it.Finally,the beams and compo-nents are created,and the corresponding components274Proc IMechE Part D:J Automobile Engineering230(2)are mapped;then the implicit parametric model is fin-ished.The structure can be easily modified by changing some base point coordinates,baselines and local sections.20The implicit parametric BIW model is shown in Figure 1.Before multi-objective collaborative optimiza-tion design,the FE model was generated on the basis of the implicit parametric model.The low-order modal frequencies,the static bending and torsional stiffnesses,the full-frontal-crash and side-impact passive safety performances were simulated and compared with tests to verify that the BIW model was correct.Verification of the low-order modal frequencies of the implicit parametric BIW modelThe low-order modal frequencies of the BIW model were analysed by NASTRAN,and the simulation results were compared with the tests to verify that theimplicit parametric BIW model was correct.In testing,the air pressure of the air spring was adjusted to make sure that the rigid vibration frequency of the BIW on the air spring brackets (shown in Figure 2)was less than 3Hz.The air spring was placed on the testbed,and the BIW structure was made horizontal using a wood block.The BIW object was tested in the frequency range for a burst random signal of 1–256Hz which was gen-erated with an electromagnetic vibrator and amplified with a power amplifier.It helped to analyse the global vibration frequencies and modes through LMS b structure testing software.19The vibrator force was given to two points of the left front longitudi-nal beam and right rear longitudinal beam,which are shown in Figure 3.The rear excitation force was verti-cally upwards,and the front excitation force had a gra-dient angle in the lateral direction and the longitudinal direction,which can fully indicate the three direction modes of the BIW structure.The vibration acceleration response of the BIW structure was collected through 180standard acceler-ometers which were pasted on the BIW.The geometry of the physical BIW structure was created according to the coordinates of 180accelerometers,which are shown in Figure 4.The BIW geometry in Figure 4was able to exhibit vibration modes in the tested BIW.These accel-erometers provided signals in three directions to LMS b.The ‘PolyMAX’of LMS b used the poly-reference least-squares complex frequency-domain method which could construct a stabilization diagram (shown in Figure 5)and identified the vibration modes through stable poles.21A comparison of the results on the principal modal frequencies for the tests and the simulations are shown in Table 1.As can be seen from Table 1,the relative errors of the simulations and the tests were less than 7.00%.Thus,it is acceptable to use this parametric BIW model to conduct modal analysis.The rear torsional mode and the first-order bending mode were overall modal,and the relative errors were very small.Therefore,they were used as constraints for lightweight optimizationdesign.Figure 3.The vibrator location of the BIWstructure.Figure 2.The free support of the BIWstructure.Figure 1.Implicit parametric BIW model.Wang et al.275Verification of the static bending and torsional stiffnesses of the implicit parametric BIW structureThe static bending and torsional stiffnesses of the BIW structure were analysed using NASTRAN,and the simulation results were compared with the tests to ver-ify that the implicit parametric BIW structure was cor-rect.In the simulations of the static bending stiffness,the front suspensions (points A and B)and the rear sus-pensions (points C and D)were all constrained.Static vertical forces of 2kN were exerted on the body floor around the B pillars,as shown in Figure 6.In the simu-lations of the static torsional stiffness,the BIW struc-ture was constrained except for the Z direction of the front suspensions (points A and B)and all the rearsuspensions (points C and D).The equal but opposite forces exerted on the front suspensions (points A and B)are shown in Figure 7.The force was 1703.5N,which is equivalent to a torque of 2kN m.The loca-tions of points A,B,C and D are shown in Figure 8.The bending stiffness and the torsional stiffnesses are expressed asBending stiffness =4000Z 1j j +Z 2j jð1ÞandTorsional stiffness =2000tan À1Z 3j j +Z 4j j L0ð2ÞT able parisons of the results for the major modal frequencies obtained from the tests and the simulations on the BIW structure.Value for the following modesRear torsional modeFrontal torsional mode First-order bending mode Major modal frequency,simulations (Hz)30.3336.1151.54Major modal frequency,tests (Hz)30.2638.8351.84Relative error0.23%7.00%0.58%Figure 5.The stabilizationdiagram.Figure 4.The BIW geometry in LMS Tb.Figure 8.Locations of the constraintpoints.Figure 6.Loading forces of the bendingmode.Figure 7.Loading forces of the torsional mode.276Proc IMechE Part D:J Automobile Engineering 230(2)respectively,where Z 1,Z 2,Z 3and Z 4are the displace-ment values of the loading location and L is the lateral distance between A and B.The test analyses were conducted under the same restrictions and loadings as the simulation analyses.When testing the static bending and torsional stiff-nesses,two adjustable height loading brackets with force sensors were used to constrain the front-suspension shock tower of the BIW structure,and two fixed brackets were used to constrain the rear-suspension spring holder,as shown in Figure 9.The loading brackets were linked with the front suspension through bolts.The fixed brackets were linked through bolts with the rear longitudinal beam,which was adja-cent to the rear suspension.When testing the static tor-sional stiffness,the left loading bracket was adjusted to an upper location,and the right loading bracket to a lower location,to make sure that the values shown by the two force sensors were 1703.5N.When testing the static bending stiffness,it was important to make sure that the BIW structure was horizontal,which kept the values shown by the two force sensors equal.A comparison of the stiffness results for the simula-tions and the tests are shown in Table 2.As can be seen from Table 2,the relative errors of the static bending stiffness and the torsional stiffness were 6.70%and 3.30%respectively.Therefore,the implicit parametric BIW model is acceptable for conducting bending and torsional analyses.Verification of the full-frontal-crash safetyperformance of the implicit parametric BIW modelIn this paper,both simulations and tests on a full fron-tal crash were carried out according to the China NewCar Assessment Program (C-NCAP).A full frontal crash is a vehicle crash into a full-width barrier with a velocity of 50km/h.In the simulations,the parametric BIW structure was connected with the chassis and the powertrain,as shown in Figure 10.The simulation analysis was carried out using LS-DYNA software.The vehicle needed a counterweight so that it could be considered without a body trim decoration,a dummy,etc.After the counterweight was added,com-parisons of the masses and the mass centre positions for the physical vehicle and the FE vehicle are shown in Table 3.As can be seen from Table 3,the relative errors of the masses and the mass centre positions were less than 2.01%,which accorded with the full-frontal-crash requirement.The upper left front-door hinge (ULH)intrusion,the lower left front-door hinge (LLH)intrusion,the upper right front-door hinge (URH)intrusion,the lower right front-door hinge (LRH)intrusion,the deformation mode and the acceleration curves of B pil-lars of both sides of the BIW structure are extracted in the simulations and compared with the tests.Figure 11shows a comparison of the deformation modes for the simulations and the tests.It can be seen that the defor-mation modes were identical.This comparison was only visual.It showed only that the frontal crash had no problems of fatalities.More detailed comparisons are shown in Figure 12and Table 4.As can be seen from Figure 12,the acceleration curves of both sides of the vehicle for the simulations and the tests fitted each other well.The peak value of the acceleration was 33g ,where g is the acceleration due to gravity.The crash time was 78ms.Comparisons of the front-door hinge intrusions for the simulations and the tests are shown in Table 4.AsFigure 9.Constraints of the BIW structure when testing.T able parisons of the results for the stiffnesses obtained from the tests and the simulations.Bending stiffness (N/mm)T orsional stiffness (N m/deg)Simulations 16071.8618308.32T ests15052.2418937.79Relative error6.70%3.30%Figure 10.The vehicle model,the chassis and the powertrain.Wang et al.277can be seen from Table 4,the maximum intrusion was the lower left door intrusion,and the relative errors were less than 9.97%.Therefore,the implicit para-metric BIW model and the connected parts are accepta-ble for conducting full-frontal-crash analysis.Verification of the side-impact safety performance of the implicit parametric BIW modelSide-impact analyses of both simulations and tests were made according to C-NCAP.Because the types and the numbers of dummies in a side impact are different from those in a full frontal crash,therefore,a counterweightwas again needed.After the counterweight was added,comparisons of the masses and the mass centre posi-tionsc for the physical vehicle and the FE model are shown in Table 5.As can be seen from Table 5,the rela-tive errors of the masses and the mass centre positions were less than 4.89%.Therefore,the implicit para-metric BIW model and the connected parts are accepta-ble for conducting side-impact analysis.In this paper,the moving deformable barrier is a commercial model (shown in Figure 13)from Engineering Technology Associates,Inc.Figure 14show a comparison of the deformation modes for the simulations and the tests.This was a visual comparison only to show that there was no prob-lems of fatalities.Therefore,comparisons of the intru-sion velocity curves and the acceleration curves were needed.The intrusion velocity curves,the acceleration curves of the head location,the beltline location,the chest location and the H point of the B pillar and the deformation modes were extracted from the simula-tions and compared with the tests.A comparison of the corresponding points of the B-pillar intrusion velocity curves for the simulations and the tests are shown in Figure 15.As can be seen in Figure 15,the maximum deviation was about 1.5m/s at the location of the H point.The minimum deviation was at the head location of theB pillar.The curves fitted each other well.A comparison of the corresponding points of the B-pillar acceleration curves for the simulations and the tests are shown in Figure 16.As can be seen in Figure 16,before 30ms all the acceleration curves fitted each other well,then the peaks and the troughs did not coin-cide and finally they all tended to zero.The implicit parametric BIW model and the connected parts are acceptable for conducting side-impact analysis.T able parisons of the masses and the mass centre positions of the physical car and the FE model.Value for the followingPhysical vehicleFE model Relative error Mass (kg)1603.801606.010.14%Mass centre coordinate X (mm)1040.761042.49 1.65%Mass centre coordinate Y (mm)7.00 6.86 2.01%Mass centre coordinate Z (mm)203.76203.860.05%FE:finite element.T able parisons of the front-door hinge intrusion values obtained from the tests and the simulations.Intrusion (mm)of the following hingesUHLLHL UHR LHR Simulations 2.86 3.16 1.30 1.77T ests2.683.51 1.19 1.96Relative error6.72%9.97%9.24%9.69%UHL;upper left front-door hinge;LHL:lower left front-door hinge;LHR:lower right front-door hinge;UHR:upper right front-doorhinge.Figure parison of the acceleration curves of the B pillars for the simulations and thetests.Figure parison of the deformation modes for the simulations and the tests.278Proc IMechE Part D:J Automobile Engineering 230(2)Lightweight optimization design for the implicit parametric BIW modelThis paper divided the components of the BIW model into non-safety parts,frontal-crash safety parts and side-impact safety parts.Thus,the optimization can be divided into three phases:non-safety parts optimiza-tion,frontal-crash safety parts optimization and side-impact safety parts optimization.This method can reduce the calculation time in comparison with the cal-culation time when these parts are considered together.For example,in this paper,the method takes only 21.17%of the time requires when the parts are consid-ered together.The number of non-safety part variables is 73,the number of frontal-crash part variables is 13and the number of side-impact part variables is 11.The following times for calculation are required:1.5h for the static bending and torsional stiffnesses;2h for the low-order modal frequencies;15h for the frontal-crash variables;20h for the side-impact variables.If the radial basis functions (RBFs)surrogate modelisFigure parison of the corresponding points of the B-pillar intrusion velocity curves for the simulations and the tests.T able parisons of the masses and the mass centre positions of the the physical car and the FE model.Value for the followingPhysical vehicleFE model Relative error Mass (kg)1551.801548.900.19%Mass centre coordinate X (mm)1150.551137.67 1.12%Mass centre coordinate Y (mm)–100.94–96.64 4.26%Mass centre coordinate Z (mm)234.28222.834.89%FE:finiteelement.Figure parison of the deformation modes for the simulations and thetests.Figure 13.The moving deformable barrier model.Wang et al.279adopted,at least 2n +1sample points are required.Therefore,this method needs a calculation time of only 1589.5h,in contrast with the calculation time of 7507.5h when the parts are considered together.For comprehensive consideration of the structural optimization design for crashworthiness,static stiff-nesses and natural vibration properties,the computa-tion time was extremely expensive.The surrogate model can reduce the computational cost.22The Latin hyper-cube samples (LHSs)were considered to have good space-filling properties and some degree of symmetry.23When the number of samples ranged from 20to 500,the optimal LHS algorithm was recommended.24The optimal LHS method can represent the higher order of non-linearity using relatively fewer sample points.To ensure the uniformity of the sampling points,a combi-natorial optimization algorithm was considered with an entropy criterion to minimize the bias of the mean square error.25Figure 17shows a comparison between the LHS method and the optimal LHS method to illus-trate the sampling strategy.Response surface methodology (RSM)and RBFs are two commonly used methods in constructing a sur-rogate model.Fang at al.26compared RSM and RBFs for multi-objective optimization of a BIW structure in the field of impact and found that RBFs performed bet-ter for optimization of highly non-linear objectives.The RBFs are a series of basic functions that are symmetric and centred at each sampling point and were originally developed for scattered multi-variate data interpola-tion.Let f (x )be the true value of the response function and f 0(x )the approximate value obtained from a classi-cal RBF with the general formf 0x ðÞ=X n i =1l i u x Àx i k k ðÞwhere n is the number of sampling points,x is the vec-tor of design variables,x i is the vector of design vari-ables at the i th sampling point,x Àx i k k is theEuclidean distance,u is a basis function and l i is an unknown weighting coefficient.Therefore,an RBF is in fact a linear combination of n basis functions with weighted coefficients.Some of the most commonly used basis functions include thin-plate spline,Gaussian,multi-quadric and inverse multi-quadric functions.In recent years,the NSGA-II has been widely used when dealing with multi-objective optimization.27Therefore,the optimal LHS was adopted to build RBF neural network surrogate models for each optimiza-tion.The MOP presented in this paper usedaFigure parison of the corresponding points of the B-pillar acceleration curves for the simulations and thetests.Figure 17.The sampling strategy:(a)the LHS method;(b)the optimal LHS method.280Proc IMechE Part D:J Automobile Engineering 230(2)non-normalized method,which was NSGA-II.NSGA-II employs both the elite-preserving strategy as well as an explicit diversity-preserving mechanism.The algo-rithm first randomly generates a population of prede-fined size N ,which undergoes conventional selection,crossover and mutation procedures to produce off-spring for the next generation with better solutions.From the second generation,the parent generation (size N )competes with its offspring generation (size N )to introduce elitism.Multi-objective optimizations of the three phases were carried out through the software ISIGHT.All the solutions were searched for using NSGA-II,which could be evaluated and compared graphically with Pareto frontier sets.28,29ISIGHT can also recommend one solution through a scale factor which was deter-mined by users.Non-safety part optimization for the implicitparametric BIW modelThe variable parts were selected through relative sensi-tivity analysis which was represented by the ratio of the direct sensitivity of the stiffness and the modal fre-quency to the direct sensitivity of the mass,as given byRS b ,RS t ,RS fb ,RS ft ÀÁ=S b ,S t ,S fb ,S ft ÀÁS m ð3Þwhere RS b ,RS t ,RS fb and RS ft are the relative sensitiv-ities of the static bending stiffness,the static torsional stiffness,the first-order bending modal frequency and the first-order torsional modal frequency respectively,S b ,S t ,S fb and S ft are the direct sensitivities of the static bending stiffness,the static torsional stiffness,the first-order bending modal frequency and the first-order tor-sional modal frequency respectively and S m is the direct sensitivity of mass.The non-safety part design variables were selected according to the following criteria.1.Select the intersection set of all the relative sensitiv-ities of these performances.2.Increase the thickness of the design variables to enhance the stiffness and the modal frequency per-formances if the relative sensitivities are high.3.Decrease the thickness of the design variables to reduce the BIW mass if the relative sensitivities are low.4.Do not select as a design variable if the mass of a single part is less than 0.5kg.Through relative sensitivity analysis,the design variables were divided into variables increasing the thickness and variables reducing the thickness.The design variables are shown in Figure 18.The larger relative sensitivity of 61design variables were chosen to reduce the thickness.The MOP for non-safety parts optimization can be expressed as follows.The objective isf x ðÞ=min f M ðÞ½ ,max f T ðÞ½ fg ð4Þand the design variables are given byÀ23t 0i 4t i Àt 0i 413t 0i ð5ÞÀ13t 0i 4t i Àt 0i 423t 0ið6Þsubject to0:95f 1B ðÞ,f 1BM ðÞ,f 1TM ðÞf g 4f B ðÞ,f BM ðÞ,f TM ðÞð7Þwhere f (M)and f (T)are the mass of the BIW and the mass of the static torsion stiffness respectively,t i is the thickness of the components,t 0i is the initial thickness of the components,f 1(B),f 1(BM)and f 1(TM)are the initial static bending stiffness,the initial first-order bending modal frequency and the initial first-order tor-sional modal frequency respectively and f (B),f (BM)and f (TM)are the static bending stiffness,the first-order bending modal frequency and the first-order tor-sional modal frequency of the sample points.The design space is shown in equation (5);the smaller rela-tive sensitivities of 12design variables were chosen to increase the thickness,and the design space is shown in equation (6).The non-safety parts were mainly the panels of the passenger compartment and had little influence on the crashworthiness.When optimizing,the objective functions were defined as the minimum mass and the maximum torsional stiffness in equation (4).The constraint conditions were defined as the static bending stiffness,and the first-order bending modal frequencies were not less than 95%of the initial values,as shown in equation (7).The generation process of the sample points by the design-of-experiments (DOE)method is shown in Figure 19.The frontal crash and the side impact were non-linear,and the maximum acceleration peak and the maximum intrusion were changing with time.Therefore,the performances were summarized into text,and the multi-objective optimization design pro-cess was based on the text,as shown in Figure 20.Although non-safety parts optimization did not con-sider crashworthiness,the multi-objective optimization design process in Figure 20was also adopted.Through iterations the front Pareto sets were obtained.Owing to frontal-crash safety parts optimization and the subse-quent side-impact safety parts optimization,the static stiffnesses were reduced more.Therefore,in this paper the middle location of the torsional stiffness intheFigure 18.Non-safe part design variables.Wang et al.281。