Research Article Advanced Aerostructural Optimization ...downloads.hindawi.com/journals/mpe/2015/753042.pdf · cedure to get the jig shape from cruise shape is called jig shape correction

Research ArticleAdvanced Aerostructural Optimization Techniques forAircraft Design

Yingtao Zuo1 Pingjian Chen2 Lin Fu1 Zhenghong Gao1 and Gang Chen3

1National Key Laboratory of Aerodynamic Design and Research Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China2AVIC China Helicopter Research and Development Institute Jingdezhen Jiangxi 333001 China3State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Gang Chen aachengangmailxjtueducn

Received 10 May 2015 Revised 13 September 2015 Accepted 29 September 2015

Academic Editor Francesco Pesavento

Copyright copy 2015 Yingtao Zuo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Traditional coupled aerostructural design optimization (ASDO) of aircraft based on high-fidelity models is computationallyexpensive and inefficient To improve the efficiency the key is to predict aerostructural performance of the aircraft efficiently Thecruise shape of the aircraft is parameterized and optimized in this paper and a methodology named reverse iteration of structuralmodel (RISM) is adopted to get the aerostructural performance of cruise shape efficiently A new mathematical explanation ofRISM is presented in this paper The efficiency of RISM can be improved by four times compared with traditional static aeroelasticanalysis General purpose computing on graphical processing units (GPGPU) is adopted to accelerate the RISM further and GPU-accelerated RISM is constructed The efficiency of GPU-accelerated RISM can be raised by about 239 times compared with that ofthe loosely coupled aeroelastic analysis Test shows that the fidelity of GPU-accelerated RISM is high enough for optimizationOptimization framework based on Kriging model is constructed The efficiency of the proposed optimization system can beimproved greatly with the aid of GPU-accelerated RISM An unmanned aerial vehicle (UAV) is optimized using this frameworkand the range is improved by 467 after optimization which shows effectiveness and efficiency of this framework

1 Introduction

Coupled ASDO has reached a high academic level and nowit is widely utilized in enterprises and universities Manyresearchers construct complex multifidelity optimizationframework to satisfy the various requirements of differentphases of aircraft design [1 2] Various coupled ASDOframeworks are developed in the past decades [3ndash5] Typicalapplications include the detailed wing optimization wingletdesign [6] supersonic aircraft design [7 8] and so forth

Low-fidelity models are widely used in coupled ASDO[9ndash11] Optimization based on low-fidelity models can reflectthe complex interaction of aerodynamics and structure Itis widely used in conceptual design to reach the optimalsolution space rapidly Optimization based on high-fidelitymodels is necessary in detail design phase to enhance theoptimization quality further High-fidelity models includethe compressible EulerNS equations and structural finite

element methodThe huge expense of optimization based onhigh-fidelity models encumbers its engineering applicationTherefore one of the main tasks is to increase the optimiza-tion efficiency Numerous efforts have been made and theseefforts include the following

(1) Tightly coupled solving strategy is adopted to solvethe static aeroelastic problem which can reduce thecomputational expense [12]

(2) Many optimization frameworks [13 14] based on var-ious kinds of surrogatemodel are developed to reducethe computational expense

(3) Gradient-based optimization [15ndash18] is utilized toimprove the optimization efficiency

(4) GPGPU is adopted to speed up the optimizationThe most popular aerostructural prediction method is

loosely coupled aeroelastic analysis for its simplicity It solves

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 753042 12 pageshttpdxdoiorg1011552015753042

2 Mathematical Problems in Engineering

the aerodynamic problem until it converges and then thestructural problem and then the aerodynamic problem untilit converges again This procedure continues until the defor-mation of wing converges It needs five iterations of repeatedfull-cycle aerodynamicstructural analysis at least which isvery time-consuming and inefficient On the contrary thetightly coupled analysis method is very efficient [17] Theflow-solid system consists of structure and flow subproblemsand they are successively solved at each time step Withtime marching the flow and deformation converge andthe aerostructural performance is achieved Tightly coupledstrategy is relatively complex for implementation and is rarelyfound in commercial software packages So it is seldom usedin engineering

Gradient-based optimization is very efficient The key isto get the gradient of objective with efficiency Usually thisis accomplished by adjoint method in which the expenseis independent on the number of design variables Adjointmethod is widely used in aerodynamic design optimizationHowever the derivation of adjoint equations in aerostruc-tural design optimization is very complex and the conver-gence of the adjoint equations needs to be improved greatly[18] before it is widely used

Driven by the insatiable market demand of calculationGPGPU has experienced a fast development in recent yearsMany applications have been done in computational fluiddynamics (CFD) [19ndash24] and twenty times of accelerationhave been achieved at least compared with conventionalcomputation based on CPU [25] Generally speaking thecomputational expense of ASDO is much larger than thatof the conventional aerodynamic optimization Thus opti-mization based on GPU provides a very bright prospectfor ASDO and ASDO based on GPU architecture is aninnovative research area Some work has been done Typicalwork includes Pattersonrsquos conceptual design studies of electricaircraft [24] with a higher order vortex lattice solver run onGPU to predict the aerodynamic performance No researcheshave been done in coupled CFDCSD design optimization upto now

This paper aims to construct an efficient multidisciplinedesign optimization framework based on CFDCSD cou-pling Twomeasures are taken to ensure the efficiency Firstlywe propose to directly optimize the cruise shape insteadof the jig shape A novel methodology named RISM isadopted to evaluate the stress and strain of the aircraft [26]and a new mathematical explanation of RISM is presentedThe advantage of RISM is that it avoids the repeated andtime-consuming aerodynamic and structural iteration andthe efficiency can be improved greatly Secondly a GPU-accelerated CFD solver is developed which can greatlyraise the efficiency Based on this novel methodology andGPU-acceleratedCFD solver a high-efficiency aerostructuraloptimization framework is constructed

The following part of this paper is divided into 4 sectionsSection 2 provides an introduction of the tools and methodsused in the paper and the novel aerostructural performanceprediction methodology is described in detail Section 3provides optimization framework used in this work Section 4validates the proposed methodologies and an aerostructural

optimization design case is done Finally Section 5 gives theconclusions

2 Methodology

21 Flow Solver In this context a multiblock viscous flowsolver named LMNS3D [25] for steady and unsteady tur-bulent flows based on GPU parallel methodology underthe finite volume frame is employed The nondimensionalNavier-Stokes equations are solved on body-fitted structuredmesh All the variables are stored and operated in the cellcenter

Several approximate Riemann solvers for example ROEHarten-Lax-van Leer with contact discontinuities (HLLC)and Simple Low-dissipation AUSM (SLAU) are availablefor inviscid flux calculation [27] SLAU is mainly exploiteddue to its parameter-free property and all-speed simula-tion capability In order to obtain high order third-orderMonotone Upwind Schemes for Scalar Conservation Laws(MUSCL) and fifth-order Weighted Essentially Nonoscilla-tory (WENO) scheme are utilized to reconstruct primitivevariables on the cell interface Second-order and fourth-ordercentral scheme can be chosen to evaluate the viscous flux ina conservative way Considering the turbulent model one-equation Spalart-Allmaras (SA) and two-equation Menterrsquos119896-119908 SST models are adopted for steady RANS simulationsThese eddy turbulent models are discretized and updatedin a loosely coupled way from the mean governing equa-tion Furthermore scale-resolving methods for exampleDetached Eddy Simulation (DES) and Scale-Adaptive Simu-lation (SAS) are constructed for high-fidelity turbulent flowsimulations

Data-Parallel LowerndashUpper Relaxation (DP-LUR) [28]method is chosen as the implicit time marching method asit eliminates the data dependency in the LU-SGS methodby replacing forward and backward sweeps as point-wiseiterations This key property enables highly efficient GPUcards based parallel simulations Second-order temporalaccuracy is enforced by imposing dual time-stepping strat-egy The heterogeneous multiple CPU + GPU coprocessingsystem is established by implementing NVIDIArsquos CUDAas well as Message Passing Interface (MPI) programmingmodels The data transfer time is hidden by a carefullydesigned concurrent coping and execution algorithm Onthe other hand multithreaded OpenMp parallel techniqueis also developed for shared-memory simulations Throughextensive numerical experiments the robustness accuracyand efficiency of current in-house solver are well validatedbased on consumer-market oriented GPU cards and pro-fessional GPU cards Double precision arithmetic is alwayskept through the entire residual computations with the helpof the latest GPU hardware and careful design of CFDcodesThe aerodynamic performance calculated by theGPU-accelerated solver is the same as the result calculated by thecorresponding CPU solver

22 The Proposed Aerostructural Performance PredictionMethodology Jig shape is the aircraft shape that does not

Mathematical Problems in Engineering 3

Jig shape

CSD

CFD

CFD

Deformationconverged

forceTransfer of

Transfer of displacement

Cruise shape and aerostructural

performance

CFD grid of initial cruise shape

No

Yes

Figure 1 Procedure of loosely coupled aeroelastic analysis

undergo aerodynamic load gravity and the propulsive forceIt deforms into cruise shape at cruise condition by appliedforce Traditional coupled aerostructural design optimizationdeals with the jig shape of the aircraftThe jig shape is param-eterized and static aeroelastic analysis is adopted to get thecorresponding aerostructural performance which is used toevaluate the objective function in multidisciplinary designoptimization We also get the corresponding cruise shapeafter static aeroelastic analysis Usually the loosely coupledCFDCSD simulation is adopted in static aeroelastic analysis[29] The procedure of loosely coupled aeroelastic analysis ispresented in Figure 1

Cruise shape can be corrected into jig shape The pro-cedure to get the jig shape from cruise shape is called jigshape correction and it is widely used in aircraft design It isnotable that the jig shape correction procedure provides notonly the jig shape but also the distribution of the stress andstrain In light of it we put forward to optimize the cruiseshape of aircraft directly For a given cruise shape the aero-dynamic performance of the aircraft at cruise condition canbe achieved directly by aerodynamic analysis The repeatedaerodynamicstructural iterations to the jig shape as done intraditional coupled aerostructural design are avoided Nowwhat we need to do is to get the stress and strain of the aircraftefficiently

We can apply the jig shape correction to get the structuralperformance of the aircraft Alyrsquos methodology to get initial

jig shape is considered firstly [30] The procedure of Alyrsquosmethod is listed as follows

(1) Get the aerodynamic load of cruise shape(2) Applying the above aerodynamic load and gravity

in the reverse direction to the cruise shape conductstructural analysis to get the displacement of theaircraft

(3) Add the displacement to the cruise shape to get thedeformed aircraft

If finite element method is adopted the structural nodescoordinate Xjig of jig shape can be achieved by solving thefollowing

X119889 = [119870]minus1 F (1)

Xjig = X119888 + X119889 (2)

where [119870] represents the stiffness matrix of the cruise shapeX119889 means the unknown vector of the structural nodesdisplacement and F means the forces including the aero-dynamic force and force of gravity acting on the structuralnodes X119888 is the structural nodes coordinates of cruise Thestructural performance of the aircraft can also be achieved bysolving (1)

If we apply the aerodynamic load of cruise shape andforce of gravity to the jig shape the deflected jig shape hassome difference with the cruise shape as pointed by Aly [30]The distribution of stress and strain also has some differencewith the former structural performance Alternatively ifstatic aeroelastic analysis is done to the above jig shape thedeflected jig shape does not coincide with the cruise shapetoo This is mainly caused by the difference of the stiffnessmatrices of these two configurations Inaccurate jig shapeleads to inaccurate structural performance of the aircraft Toget the correct structural performance of the cruise shapeimprovements must be done

The general improved jig shape correction or the con-ventional jig shape correction (CJSC) is precise It gives thecorrect jig shape as well as the accurate structural perform-ance The main improvement of CJSC compared with Alyrsquosmethod is that static aeroelastic analysis is performed tothe jig shape and the jig shape is adjusted according to thedifference between the deflected jig shape and cruise shapeUsually CFD and CSD method is adopted in CJSC Theprocedure can be summarized in detail as follows [31]

(1) Call the CFD solver to feature the aerodynamic char-acteristics of the cruise shape and we get the aerody-namic load of the aircraft at cruise condition

(2) Get the initial jig shape Xjig by Alyrsquos methodology(3) Aeroelastic analysis as mentioned above is conducted

to get the deflected jig shape Xdis(4) Compare the configurations of coordinates Xdis and

X119888 Their difference is achieved by ΔX = X119888 minus Xdis(5) If the 2-norm of ΔX is small enough finish Other-

wise the jig shape is updated by X119895 = X119895 + 120596ΔXwhere 120596 is a factor between 0 and 1 Go to (3)


Cruise shape CFD CSD

Initial jig shape

Aeroelastic analysis

Comparison of the deflected jig shape and

the cruise shape

Converge

Stress and strain distribution of the

aircraft

Aerodynamic performance of the cruise shape

Reverse application

of load

Alyrsquosmethod

Geometry difference

process

Yes

No

Figure 2 Conventional jig shape correction method

The procedure can be represented by Figure 2The aeroelastic analysis is needed only once to get the aer-

ostructural performance of an aircraft in conventional aero-structural optimization However the abovemethod involvesseveral times of static aeroelastic analysis to get the aerostruc-tural performance of an aircraft Obviously it is inefficientand not suitable for use in aerostructural optimizations

It is noticed in the above procedure that as the deflectedjig shape converges to the cruise shape the aerodynamic loadacting on the jig shape converges to the aerodynamic load ofthe cruise shape In the last iteration the aerodynamic loadacting on the jig shape equals the aerodynamic load of thecruise shape This prompts a way of using the loads of thecruise shape to get the deflected jig shape directly withoutiteratively calling the CFD solver Here a novel aerostructuralperformance analysis methodology named RISM is adoptedas follows [26]

(1) Call the GPU-accelerated CFD solver to obtain theaerodynamic characteristics of the cruise shape andwe get the aerodynamic load of the aircraft

(2) Get the initial jig shape Xjig by Alyrsquos methodology(3) Apply the aerodynamic forces of the cruise shape and

the force of gravity in the right direction to the jigshape to get the deflected jig shape X1015840dis

(4) Compare the displacement of every structural nodeof X119888 and X1015840dis Calculate the coordinate differencevector of the corresponding structural nodes andwe get ΔX where ΔX = X119888 minus X1015840dis Add 120596ΔX tothe structural nodes of the jig shape and we get theupdated jig shape

(5) Go to step (3) unless the 2-norm of ΔX is smallenough

About 15 iterations are needed in this procedure through-out which the aerodynamic load is invariableThis procedurecan be presented by Figure 3

As can be seen in Figure 3 the CFD solver is calledonly once to get the aerodynamic performance of the cruiseshape and CSD solver is called iteratively to get the structuralperformance Generally structural models composed of shelland beam elements are used to represent a stiffened aircraftwing in aerostructural optimization The expense to conductone time of structural analysis is negligible compared withthat of aerodynamic analysis based on CFD The CFDsolver is called at least five times in general loosely coupledstatic aeroelastic analysis Therefore the efficiency to get theaerostructural performance by RISM can be improved by atleast four times comparedwith the loosely coupled aeroelasticanalysis

The coordinate difference vector of finite element meshof the jig shape and cruise shape is denoted by U Now thatthe jig shape will deform into the cruise shape under theaerodynamic load of cruise shape and force of gravity thestrain of the jig shape should be U The coordinate vectorof the finite element mesh of the jig shape is X119888 minus U Thedeformation of the jig shape under the aerodynamic load ofcruise shape and force of gravity can be modeled as follows

K (X119888U)U minus F = 0 (3)

where K(X119888U) is the structural stiffness matrix of the jigshape It depends on the finite element mesh of the cruiseshapeX119888 the displacementU between the jig shape and cruiseshape and the property of material F is the resultant force ofthe aerodynamic load of the cruise shape and force of gravity

Equations (3) are nonlinear algebraic equations Theoverrelaxation iteration method is adopted to solve them


Cruise Shape CFD CSD

Initial jig shape

CSD


the cruise shape

Converge


aircraft


Reverse application

of load

Geometry difference

process

Yes

No

Positive application

of load

Alyrsquosmethod

Figure 3 Aerostructural analysis of aircraft

Firstly an appropriate initial value U0 is provided by Alyrsquosmethod Then it can be solved by the following iterations

U119896+1 = (1 minus 120596)U119896 + 120596K (X119888U119896)minus1 F (119896 = 0 1 ) (4)

where 120596 is the relaxation factor between 0 and 1There lies an interesting contrast between the loosely

coupled static aeroelastic analysis and RISM The finiteelement mesh is fixed in loosely coupled static aeroelasticanalysis and the CFD grid as well as aerodynamic loads ofFEM updates during iterations But in RISM the CFD gridand aerodynamic loads are fixed and the finite element meshupdates during iterations

To accelerate the above procedures the former GPU-accelerated CFD solver can be adopted and the GPU-accelerated RISM is formed

23 Geometry Parameterization The free form deformation(FFD) method is used to parameterize the cruise shape inthis paper The FFD was proposed by Sederberg and Parry[32] in 1986 and then used in graphics processing andnow it is widely used by CAD and cartoon Later it wasintroduced into aircraft design [33 34] FFD constructs a1198773rarr 11987710158403 mapping function X = 119891(1198831015840) from physical space

to parameter space where1198831015840 is logic coordinate of parameterspace and X is coordinate of physical space The deformationof aircraft in physical space is controlled by the movementof the control points of the parameters 119904 119905 and 119906 by thefollowing formula

X (119904 119905 119906) + ΔX (119904 119905 119906)

=

119897

sum

119894=1

119898

sum

119895minus1

119899

sum

119896=1

[119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906)]

sdot [P119894119895119896 + ΔP119894119895119896]

(5)

where P119894119895119896 and ΔP119894119895119896 are the matrixes representing theoriginal coordinates and the displacements of the node point(119894 119895 119896) of the control box respectivelyThe number of controlpoints of the hexahedral control box is 119897 times119898times119899 (119904 119905 119906) is thelocal curvilinear coordinates mapped into the control boxand it is also called lattice coordinates 119861119894minus1

119897minus1(119904) is the (119894 minus 1)th

Bernstein polynomial of degree 119897 minus 1 defined as follows

119861119894minus1

119897minus1(119904) =

(119897 minus 1)

(119894 minus 1) (119897 minus 119894)119904119894minus1(1 minus 119904)

119897minus119894 (6)

In the matrix form equation can be written as follows

ΔX = 119861 (119904 119905 119906) sdot ΔP

(

1205751199091

1205751199092

1205751199093

)

= (119861111 sdot sdot sdot 119861119897119898119899)

[[[[

[

Δ1198751111 Δ1198751112 Δ1198751113

Δ1198751198971198981198991 Δ1198751198971198981198992 Δ1198751198971198981198993

]]]]

]

(7)

where

119861119894119895119896 = 119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906) (8)

The procedures of FFD are listed as follows

(1) Construct control lattice around the objective to beparameterized

(2) Calculate the logic coordinate(3) Move the control point of lattice(4) Calculate the deformation of the aircraft with (5)


3 Multidisciplinary DesignOptimization Framework

Direct optimization based on GPU-accelerated RISM is stilltime-consuming despite its high efficiency Thus surrogatemodel is adopted in this paper which is computationallycheap The widely used optimization framework based onthe surrogate model and the genetic algorithm is adopted Inthis framework most high-fidelity simulations are replacedby recursively updating surrogate models Latin hypercubeis selected as the sampling method since it is the mostpopular version of stratified sampling ones The design spacefor each factor is uniformly divided by this technique Theoptimization is organized as follows

(1) Initial samples are generated by Latin hypercube sam-pling method The responses of these sample pointsare evaluated by high-fidelity models such as GPU-accelerated RISM

(2) Construct Kriging surrogatemodel based on the sam-ple points and corresponding responses

(3) Search the model to get the optimum by genetic algo-rithm Validate the optimum by high-fidelity models

(4) If the variation of objective function is small enoughstop otherwise the new results are added to the sam-ple dataset and go to step (2)

The procedure can be represented by Figure 4

4 Testing and Analysis

41 Validation of the CFD Solver The numerical accuracy ofin-house solver LMNS3D is validated very carefully in [25]and the computational efficiency is also tested on consumer-market oriented GPU cards Now it is validated for efficiencyin a new workstation provided by Beijing Rongtian HuihaiTechnology Company Limited The workstation is a multi-GPU cluster It contains three NVIDIA Tesla K10 GPUsand two six-core Intel Xeon Processor E5-2620 CPUs Thememory is 40GB For benchmarking the NS solver wasfirstly run on CPU and then it was run on GPU

The same DLR F6-WBNP configuration and grid used in[25] is tested at first The Mach number is 075 the angle ofattack is 10 degree and the Reynolds number is 40 times 106The grid is shown in Figure 5 and the grid size is 82 millionWENO 5th reconstruction scheme coupled with SLAU fluxscheme for inviscid terms and 4th-order central differencingscheme for viscous terms are adopted Menterrsquos 119896-119908 SSTmodel is adopted The DP-LUR method is used for timeadvancement

Table 1 shows the elapsed time and speedup factor for 500time steppings As can be seen the program is very effectiveThe CFD results are in line with that of [25] The speedupfactor is different because the test is performed on a differentplatformThe platform in [25] was a personal computer withNVIDIAGeForce 560Ti graphic card It is a workstation withNVIDIA Tesla K10 GPU in this paper

A high altitude long endurance UAV is tested too Theflight speed is 06 Mach The Reynolds number is 15 times 106

Aerodynamic and structural parameterization

Definition of objective function and constraints

Sampling design space and evaluating aerostructuralperformance of samples

Construct Kriging model

Searching with genetic algorithm

Validate the optimum

Converge

Finish

Add the validated results to sample space

Yes

No

Figure 4 Optimization based on Kriging model

Figure 5 Surface grid of DLR F6-WBNP

The size of the CFD grid is 31 millionThe classical SAmodelis adopted to describe turbulenceWENO 5th reconstructionscheme coupled with SLAU flux scheme for inviscid termsand 4th-order central differencing scheme for viscous termsare adopted Figure 6 displays part of the CFD grid of theaircraft Table 2 shows the elapsed time and speedup factorfor 500 time steppings As can be seen from this table 278timesspeedup is achieved by single GPU platform compared tosingle core (CPU)

42 Validation of RISM Methodology The prediction ofaerostructural performance of the aircraft is the basis of


Table 1 Elapsed wall-time and speedup factor for 500 time step-pings

Single core (CPU) Single K10 Triple K10Time (minutes) 50115 1839 857Speedup factor 10 2724 5848



Table 3 Structural performance of the UAV predicted by differentmethodologies

Prediction method Maximumdisplacement m

Mass of thewing Kg

Maximumstress MPa

CJSC 0286 815623 9324Alyrsquos methodology 0291 816183 8656GPU-accelerated RISM 0291 816074 9564SAA 02871 816074 9510

the multidisciplinary design optimization Therefore theeffectiveness of RISM should be carefully validated

The function of GPU cards is to speed up the CFD cal-culation It does not affect the final aerodynamic force andthe results given by GPU-accelerated RISM are the same asthat of RISM alone So we do not compare the results givenby GPU-accelerated RISM with that of RISM alone in thefollowing of this paper

The above UAV is used to validate the precision andefficiency of the proposed methodologyThe semispan of theUAV is 80m and the aspect ratio of this UAV is 176 Theflight altitude of this UAV is 20Km

In [26] the surface grid of CFD grid did not coincide withthat of the FEM Mesh Therefore fluid-structure interaction(FSI) was adopted and error was introduced This makes itdifficult to judge the accuracy of RISMTo avoid this problemthe CFD surface grid of the UAV is directly extracted toconstruct the surface grid of FEM in this paper as shown inFigure 7 The main load-carrying components of the wingbox are considered including skins ribs wing spars andstringers The front and rear spar are defined at 15 and65 along the chord respectively The spars and ribs areassumed to be ldquoT-beamsrdquo The aluminium alloy adopted hasthe elasticity modulus of 70GPa and Poissonrsquos ratio of 033

The aerostructural performance prediction of this UAVis done by the GPU-accelerated RISM and the jig shape isachieved at the same time The result is presented in Table 3Two measures are taken to check the result Firstly the CJSCis accomplished and we get the corresponding jig shape andthe deflected jig shape (Config1) Secondly static aeroelasticanalysis (SAA) of the jig shape achieved by GPU-acceleratedRISM is conducted and the deflected jig shape (Config2) is

Figure 6 CFD grid of UAV

Figure 7 Wing structural model

Table 4 Wing tip twist angle of jig shape predicted by differentmethods

Predictionmethod CJSC Alyrsquos

methodologyGPU-accelerated

RISMTwist (degree) minus03865 minus04050 minus03813

Table 5 Aerodynamic characteristics of different configurations

Configuration 119862119871 119862119863 119862119898

Cruise shape 0550 0028735 minus00804Config1 0550 0028740 minus008035Config2 05495 0028727 minus008033

achieved Table 3 presents all the structural performance andwing mass of these right-hand side jig shapes achieved by allthese methods Table 4 shows the twist angle of the wing tipsection Figure 8 shows the twist angle of jig shape predictedby different methods along the spanwise Table 5 presents theaerodynamic performance of different configurations andthe pressure contour and sectional pressure distribution arecompared in Figure 9


01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y

CJSCAlyrsquos methodRISM

Figure 8 Wing twist angle predicted by different method

Cruise shapeConfig2

Figure 9 Pressure contour and sectional pressure distribution ofthe cruise shape and the deflected jig shape

As can be seen from Table 3 the maximum displace-ment difference between the jig shapes achieved by GPU-accelerated RISM and CJSC is 0005m The maximum stresspredicted by RISM is 25 larger than that of the CJSCThe maximum stress predicted by Alyrsquos method is 72smaller than that of the CJSC Table 4 shows that the twistanglesrsquo difference of jig shapes predicted by CJSC and RISMis 00052∘ which is negligible while the difference of jig shapepredicted by CJSC andAlyrsquos method reaches 00185∘ It can be

5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07

Figure 10 Convergence history of RISM with different 120596

seen from Table 5 that the aerodynamic performance of theaircraft predicted by different methods is almost the same

With the aid of GPU architecture the efficiency can beimproved by about 110 times (with single K10 GPU) or 239times (with triple K10 GPU) compared with loosely coupledstatic aeroelastic analysis running on a single CPU coreThis result is impressive The objective function used inaerostructural optimization consists of the mass of the wingand the drag of the cruise shape The maximum stress anddisplacement are usually used as constraints As can be seenfrom the above the mass and drag of the aircraft predicted byRISM are very precise indicating that the objective functiongiven by RISM will be very precise The error of maximumstress is relatively large but it is not as critical for optimizationas the objective function All these results show that theproposed methodology can be adopted in aerostructuraloptimization

Figure 10 shows convergence history of RISMwith differ-ent 120596 where the longitudinal axis represents the maximumdistance between the corresponding structural nodes ofdeflected jig shape and the cruise shape Figure 11 shows thecruise shape and jig shape achieved by RISM

43 UAV Optimization FFD approach is used to parameter-ize the above UAV All the parameters such as the weightof the UAV and reference area refer to that of the right-hand side of the UAV in the following for simplicity Thecontrol framework is shown in Figure 12The fuselage is fixedduring optimization The wing geometry variables includechord of root chord of tip semispan and twist of the wingTwist is defined at four sections across the wing A graphicalrepresentation of the geometric design variables is shown inFigure 12 The number of aerodynamic design variables is7 In this work the topology of the wing structure remainsunchanged which means that the number of spars and ribsand their planform-view locations are all fixed The FEM isdivided into three segments along the spanwise direction asshown in Figure 7 The thickness of skins area of spar capof the front spar and rear spar of each segment are selected


Cruise shape

Jig shape

Figure 11 Comparison of the jig shape and the cruise shape

2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4

Figure 12 FFD control framework

as design variables Therefore the number of structural vari-ables is 9 and the total number of design variables is 16

The range is chosen to be the objective function whichis given by Breguet equation This equation considers thetradeoff between drag and structural weight very well It canbe written as follows

119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)

where 119877 is the range 119881 is the cruise velocity 119862 is the specificfuel consumption of the powerplant 119871119863 is the lift-to-dragratio and 1198821 and 1198822 are the initial and final weights ofthe UAV during cruise respectively The initial weight ofthe aircraft consists of the structural weight fuel weightand so on It is fixed to be 123578N in this optimizationThe final weight is simply the structural weight of wingplus a given constant weight We try to lighten the weight

Table 6 Operating conditions

State Mach Altitude (Km) Lift coefficient Load factorCruise 060 20 055 10Maneuver 060 10 0326 25

of the wing to increase the fuel (1198821-1198822) The actual lift-to-drag ratio varies over the different cruise stages due toaerostructural effects produced by changes in aircraft weightand the inertial fuel load distribution In order to simplifythe analysis we maintain a constant lift-to-drag ratio and 119862The cruise velocity and the flight altitude are fixed so we onlyneed to maximize (119871119863) ln(11988211198822)

Constraints must be enforced in optimization Here thelift in cruise condition is fixed to be 111328N The referencearea to calculate the lift coefficient and the drag coefficientis fixed to be 727m2 for convenience which is the projectedwing area of the initial right-hand side wing Thus the liftcoefficient is fixed to be 055 in optimization which can beachieved by adjusting the angle of attack periodically duringaerodynamic calculation The maximum von Mises stress ofthe UAV must be below an allowable value at maneuver andcruise condition The load at cruise condition is multipliedby a load factor equal to 25 to consider the peak loadsencountered during various flight maneuvers or caused byturbulent air The operating condition is listed in Table 6

Now we can summarize the optimization problem asfollows

Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)

where 120575max1 represents the maximum von Mises stress ofthe wing at cruise condition 120575max2 represents the maximumvon Mises stress of the wing at maneuver condition 119878wingis the projected area of the right-hand side wing The UAVis optimized by the above optimization framework and thenumber of initial samples is 280 Three K10 GPU cards areadopted in this optimizationThe aerostructural performanceat cruise condition is predicted by GPU-accelerated RISMand the aerostructural performance at maneuver conditionis predicted by GPU-accelerated loosely coupled static aeroe-lastic analysis

Totally 295 times of GPU-accelerated RISM and staticaeroelastic analysis are called respectively during this opti-mization A comparison of the geometric design variablesthe projected wing area and aerostructural performance ofthe UAV before and after optimization is listed in Table 7 Ascan be seen the drag is reduced by 21 counts and the massof the wing is reduced by 435 Kg The range of the UAV isincreased by 467


Table 7 Optimization results

Initial OptimizedTwist 1 (degree) 0 minus0653Twist 2 (degree) 0 minus07634Twist 3 (degree) 0 minus157Twist 4 (degree) 0 minus159Chord root (m) 1210 1290Chord tip (m) 0556 0533Semispan (m) 80 7602119878wing (m

2) 6576 643119862119863 00287 002849120575max 1 (MPa) 9324 7642120575max 2 (MPa) 2250 1830Mass of wing (Kg) 816 7725119865 4235 4433

Table 8 Optimization results validation

CJSC GPU-acceleratedRISM SAA

119862119871 0550 0550 04982119862119863 002849 002849 002846119862119898 minus00642 minus00642 minus00641Maximum stress (MPa) 7625 7776 7743Maximum displacement (m) 02178 02198 02162

To check the optimization results similar validation isconducted as in Section 42 Table 8 lists the results of RISMCJSC and static aeroelastic analysis of the jig shape achievedby RISM As can be seen the difference is negligible Thecorrectness of RISM is validated again

The pressure contour of the wing before and after opti-mization is given in Figure 13 Figures 14 and 15 show thecomparisons of the von Mises stress and planform beforeand after optimization separately The convergence history isshown in Figure 16 A comparison of the deflections of theinitial and optimized design of the aircraft at cruise conditionand a 25 gmaneuver condition is given in Figure 17 All theseresults validate the proposed optimization framework

5 Conclusions

An efficient multidisciplinary design optimization frame-work is proposed in this paper Message Passing Interface(MPI) and CUDAwere used to accelerate the flow simulationin this framework GPU-accelerated RISM was proposedto predict the aerodynamic and structural performance ofthe aircraft The efficiency to predict the aerostructuralperformance of the aircraft at cruise condition is raised 4times by RISM alone compared with conventional looselycoupled static aeroelastic analysis It can be further improvedby about 110 times (with single K10 GPU) or 239 times (withtripleK10GPU)with the aid ofGPUandMPI Tests show thatRISM has almost the same fidelity as that of the conventional

Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t

Figure 13 Pressure contour comparison before and after optimiza-tion

932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum

Figure 14 Von Mises stress before and after optimization

Y

X

5

4

2 4 6 8

BaselineOptimum

Figure 15 Planform comparison before and after optimization

445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number

Figure 16 Convergence history of the optimization


Initial jig shape

Optimized jig shape

25 g of initial project25 g of optimized project

Figure 17 Front view showing comparison of the jig shape andconfiguration at maneuver condition before and after optimization

method Optimization of an UAV proved the effectiveness ofthe optimization framework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (11272005) the Open Projectof State Key Laboratory for Strength and Vibration ofMechanical Structures of Xirsquoan Jiaotong University (SV2014-KF-10) and the Advanced Programs of Jiangxi PostdoctoralScience Foundation This research used resources of the Bei-jing Rongtian Huihai Technology Company Limited Thesesupports are gratefully acknowledgedThe authors would liketo thankDrs YingnanGuo QingHan ZhengpingWang andJiechu Jiang of Northwestern Polytechnical University and BoZhang of Xirsquoan Jiaotong University for their assistance withthis paper

References

[1] P Piperni A DeBlois and R Henderson ldquoDevelopment ofa multilevel multidisciplinary-optimization capability for anindustrial environmentrdquoAIAA Journal vol 51 no 10 pp 2335ndash2352 2013

[2] L Cavagna S Ricci and L Travaglini ldquoNeoCASS an integratedtool for structural sizing aeroelastic analysis and MDO atconceptual design levelrdquo Progress in Aerospace Sciences vol 47no 8 pp 621ndash635 2011

[3] L Cavagna S Ricci and L Riccobene ldquoStructural sizingaeroelastic analysis and optimization in aircraft conceptualdesignrdquo Journal of Aircraft vol 48 no 6 pp 1840ndash1855 2011

[4] S S Ghoman R K Kapania P C Chen D Sarhaddi and D HLee ldquoMultifidelity multistrategy and multi-disciplinary designoptimization environmentrdquo AIAA 2013-4672 2013

[5] R P Liem C A Mader E Lee and J R P A Martin ldquoAero-structural design optimization of a 100-passenger regional jetwith surrogate-based mission analysisrdquo in Proceedings of the13th AIAA Aviation Technology Integration and OperationsConference AIAA 2013-4372 Los Angeles Calif USA August2013

[6] S A Ning and I Kroo ldquoMultidisciplinary considerations in thedesign of wings and wing tip devicesrdquo Journal of Aircraft vol47 no 2 pp 534ndash543 2010

[7] K Alston S Doyle T Winter H Kim and S Ragon ldquoHighfidelitymultidisciplinary optimization (HFMDO)rdquo AIAA 2010-9319 2010

[8] D L Allison C CMorris and J A Schetz ldquoAmultidisciplinarydesign optimization framework for design studies of an efficientsupersonic air vehiclerdquo in Proceedings of the 14th AIAAISSMOMultidisciplinary Analysis and Optimization Conference AIAAPaper 2012-5492 Indianapolis Ind USA 2012

[9] J M Schweiger M Busing and J Feger ldquoA novel approachto improve conceptual air vehicle design by multidisciplinaryanalysis and optimization models and methodsrdquo in Proceedingsof the 12th AIAA Aviation Technology Integration and Oper-ations (ATIO) Conference and 14th AIAAISSMO Multidisci-plinary Analysis andOptimization Conference AIAA 2012-5450Indianapolis Ind USA September 2012

[10] K A James G J Kennedy and J R R A Martins ldquoConcurrentaerostructural topology optimization of a wing boxrdquoComputersamp Structures vol 134 pp 1ndash17 2014

[11] S Rajagopal and R Ganguli ldquoMultidisciplinary design opti-mization of a UAV wing using Kriging based multi-objectivegenetic algorithmrdquo AIAA no 2009-2219 2009

[12] Y Kim Y-H Jeon and D-H Lee ldquoMulti-objective and mul-tidisciplinary design optimization of supersonic fighter wingrdquoJournal of Aircraft vol 43 no 3 pp 817ndash824 2006

[13] R M Paiva A R D Carvalho C Crawford and A SulemanldquoComparison of surrogate models in a multidisciplinary opti-mization framework for wing designrdquoAIAA Journal vol 48 no5 pp 995ndash1006 2010

[14] S G Lehner L B Lurati G C Bower et al ldquoAdvancedmultidisciplinary optimization techniques for efficient subsonicaircraft designrdquo in Proceedings of the 48th AIAA Aerospace Sci-ences Meeting Including the New Horizons Forum and AerospaceExposition AIAA 2010-1321 Orlando Fla USA January 2010

[15] A Fazzolari N R Gauger and J Brezillon ldquoEfficient aero-dynamic shape optimization in MDO contextrdquo Journal ofComputational and Applied Mathematics vol 203 no 2 pp548ndash560 2007

[16] I Ghazlane G Carrier A Dumont and J A Desideri ldquoAero-structural adjoint method for flexible wing optimizationrdquo inProceedings of the 53rdAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA Paper2012-1924 Honolulu Hawaii USA April 2012

[17] M Barcelos and K Maute ldquoAeroelastic design optimizationfor laminar and turbulent flowsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 19-20 pp 1813ndash18322008

[18] Y Zuo G Chen Y Li and Z Gao ldquoEfficient aeroelasticdesign optimization based on the discrete adjoint methodrdquoTransactions of the Japan Society for Aeronautical and SpaceSciences vol 57 no 6 pp 343ndash351 2014

[19] C Obrecht F Kuznik B Tourancheau and J-J Roux ldquoA newapproach to the lattice Boltzmannmethod for graphics process-ing unitsrdquo Computers and Mathematics with Applications vol61 no 12 pp 3628ndash3638 2011

[20] J Appleyard and D Drikakis ldquoHigher-order CFD and interfacetracking methods on highly-parallel MPI and GPU systemsrdquoComputers amp Fluids vol 46 no 1 pp 101ndash105 2011

[21] V Esfahanian B Baghapour M Torabzadeh and H ChizarildquoAn efficient GPU implementation of cyclic reduction solver forhigh-order compressible viscous flow simulationsrdquo Computersamp Fluids vol 92 pp 160ndash171 2014


[22] R Lohner F Camelli and J D Baum ldquoLarge-scale blastcalculations on GPU clustersrdquo in Proceedings of the 50th AIAAAerospace Sciences Meeting including the New Horizons Forumand Aerospace Exposition AIAA 2012-0565 Nashville TennUSA January 2012

[23] D A Jacobsen and I Senocak ldquoA full-depth amalgamatedparallel 3D geometric multigrid solver for GPU clustersrdquoin Proceedings of the 49th AIAA Aerospace Sciences MeetingIncluding the New Horizons Forum and Aerospace ExpositionAIAA 2011-946 Orlando Fla USA January 2011

[24] M D Patterson M J Daskilewicz and B J German ldquoCon-ceptual design of electric aircraft with distributed propellersmultidisciplinary analysis needs and aerodynamic modelingdevelopmentrdquo in Proceedings of the 52nd AIAA AerospaceSciences Meeting AIAA 2014-0534 National Harbor Md USAJanuary 2014

[25] L Fu Z Gao K Xu and F Xu ldquoA multi-block viscous flowsolver based on GPU parallel methodologyrdquo Computers ampFluids vol 95 pp 19ndash39 2014

[26] Y T Zuo ZHGao G Chen X PWang andYM Li ldquoEfficientaero-structural design optimization coupling based on reverseiteration of structural modelrdquo Science China TechnologicalSciences vol 58 no 2 pp 307ndash315 2015

[27] E Shima and K Kitamura ldquoOn new simple low-dissipationscheme of AUSM family for all speedsrdquo AIAA 2009-136 2009

[28] M J Wright G V Candler and M Prampolini ldquoData-parallellower-upper relaxationmethod for the navier-stokes equationsrdquoAIAA Journal vol 34 no 7 pp 1371ndash1377 1996

[29] M Barcelos H Bavestrello and K Maute ldquoA Schur-Newton-Krylov solver for steady-state aeroelastic analysis and designsensitivity analysisrdquo Computer Methods in Applied Mechanicsand Engineering vol 195 no 17-18 pp 2050ndash2069 2006

[30] A Sherif M Ogot R Pelz and M Siclari ldquoJig-shape staticaeroelastic wing design problem a decoupled approachrdquo Jour-nal of Aircraft vol 39 no 6 pp 1061ndash1066 2002

[31] W Huang Z Lu T Guo F Xue and M Zhang ldquoNumericalmethod of static aeroelastic correction and jig-shape design forlarge airlinersrdquo Science China Technological Sciences vol 55 no9 pp 2447ndash2452 2012

[32] TW Sederberg and S R Parry ldquoFreeform deformation of solidgeometric modelsrdquo Computer Graphics vol 22 no 4 pp 151ndash160 1986

[33] C B Allen andT C S Rendall ldquoCFD-based shape optimizationof hovering rotors using global and local parametersrdquo inProceedings of the 28th AIAA Applied Aerodynamics ConferenceAIAA 2010-4236 Chicago Ill USA June-July 2010

[34] W K Anderson S L Karman and C Burdyshaw ldquoGeometryparameterization method for multidisciplinary applicationsrdquoAIAA Journal vol 47 no 6 pp 1568ndash1578 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the aerodynamic problem until it converges and then thestructural problem and then the aerodynamic problem untilit converges again This procedure continues until the defor-mation of wing converges It needs five iterations of repeatedfull-cycle aerodynamicstructural analysis at least which isvery time-consuming and inefficient On the contrary thetightly coupled analysis method is very efficient [17] Theflow-solid system consists of structure and flow subproblemsand they are successively solved at each time step Withtime marching the flow and deformation converge andthe aerostructural performance is achieved Tightly coupledstrategy is relatively complex for implementation and is rarelyfound in commercial software packages So it is seldom usedin engineering

Gradient-based optimization is very efficient The key isto get the gradient of objective with efficiency Usually thisis accomplished by adjoint method in which the expenseis independent on the number of design variables Adjointmethod is widely used in aerodynamic design optimizationHowever the derivation of adjoint equations in aerostruc-tural design optimization is very complex and the conver-gence of the adjoint equations needs to be improved greatly[18] before it is widely used

Driven by the insatiable market demand of calculationGPGPU has experienced a fast development in recent yearsMany applications have been done in computational fluiddynamics (CFD) [19ndash24] and twenty times of accelerationhave been achieved at least compared with conventionalcomputation based on CPU [25] Generally speaking thecomputational expense of ASDO is much larger than thatof the conventional aerodynamic optimization Thus opti-mization based on GPU provides a very bright prospectfor ASDO and ASDO based on GPU architecture is aninnovative research area Some work has been done Typicalwork includes Pattersonrsquos conceptual design studies of electricaircraft [24] with a higher order vortex lattice solver run onGPU to predict the aerodynamic performance No researcheshave been done in coupled CFDCSD design optimization upto now

This paper aims to construct an efficient multidisciplinedesign optimization framework based on CFDCSD cou-pling Twomeasures are taken to ensure the efficiency Firstlywe propose to directly optimize the cruise shape insteadof the jig shape A novel methodology named RISM isadopted to evaluate the stress and strain of the aircraft [26]and a new mathematical explanation of RISM is presentedThe advantage of RISM is that it avoids the repeated andtime-consuming aerodynamic and structural iteration andthe efficiency can be improved greatly Secondly a GPU-accelerated CFD solver is developed which can greatlyraise the efficiency Based on this novel methodology andGPU-acceleratedCFD solver a high-efficiency aerostructuraloptimization framework is constructed

The following part of this paper is divided into 4 sectionsSection 2 provides an introduction of the tools and methodsused in the paper and the novel aerostructural performanceprediction methodology is described in detail Section 3provides optimization framework used in this work Section 4validates the proposed methodologies and an aerostructural

optimization design case is done Finally Section 5 gives theconclusions

2 Methodology

21 Flow Solver In this context a multiblock viscous flowsolver named LMNS3D [25] for steady and unsteady tur-bulent flows based on GPU parallel methodology underthe finite volume frame is employed The nondimensionalNavier-Stokes equations are solved on body-fitted structuredmesh All the variables are stored and operated in the cellcenter

Several approximate Riemann solvers for example ROEHarten-Lax-van Leer with contact discontinuities (HLLC)and Simple Low-dissipation AUSM (SLAU) are availablefor inviscid flux calculation [27] SLAU is mainly exploiteddue to its parameter-free property and all-speed simula-tion capability In order to obtain high order third-orderMonotone Upwind Schemes for Scalar Conservation Laws(MUSCL) and fifth-order Weighted Essentially Nonoscilla-tory (WENO) scheme are utilized to reconstruct primitivevariables on the cell interface Second-order and fourth-ordercentral scheme can be chosen to evaluate the viscous flux ina conservative way Considering the turbulent model one-equation Spalart-Allmaras (SA) and two-equation Menterrsquos119896-119908 SST models are adopted for steady RANS simulationsThese eddy turbulent models are discretized and updatedin a loosely coupled way from the mean governing equa-tion Furthermore scale-resolving methods for exampleDetached Eddy Simulation (DES) and Scale-Adaptive Simu-lation (SAS) are constructed for high-fidelity turbulent flowsimulations

Data-Parallel LowerndashUpper Relaxation (DP-LUR) [28]method is chosen as the implicit time marching method asit eliminates the data dependency in the LU-SGS methodby replacing forward and backward sweeps as point-wiseiterations This key property enables highly efficient GPUcards based parallel simulations Second-order temporalaccuracy is enforced by imposing dual time-stepping strat-egy The heterogeneous multiple CPU + GPU coprocessingsystem is established by implementing NVIDIArsquos CUDAas well as Message Passing Interface (MPI) programmingmodels The data transfer time is hidden by a carefullydesigned concurrent coping and execution algorithm Onthe other hand multithreaded OpenMp parallel techniqueis also developed for shared-memory simulations Throughextensive numerical experiments the robustness accuracyand efficiency of current in-house solver are well validatedbased on consumer-market oriented GPU cards and pro-fessional GPU cards Double precision arithmetic is alwayskept through the entire residual computations with the helpof the latest GPU hardware and careful design of CFDcodesThe aerodynamic performance calculated by theGPU-accelerated solver is the same as the result calculated by thecorresponding CPU solver

22 The Proposed Aerostructural Performance PredictionMethodology Jig shape is the aircraft shape that does not


Jig shape

CSD

CFD

CFD


forceTransfer of



performance


No

Yes










X119889 = [119870]minus1 F (1)

Xjig = X119888 + X119889 (2)











Initial jig shape



the cruise shape

Converge


aircraft


Reverse application

of load

Alyrsquosmethod

Geometry difference

process

Yes

No













K (X119888U)U minus F = 0 (3)





Initial jig shape

CSD


the cruise shape

Converge


aircraft


Reverse application

of load

Geometry difference

process

Yes

No


of load

Alyrsquosmethod









X (119904 119905 119906) + ΔX (119904 119905 119906)

=

119897

sum

119894=1

119898

sum

119895minus1

119899

sum

119896=1

[119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906)]

sdot [P119894119895119896 + ΔP119894119895119896]

(5)




119861119894minus1

119897minus1(119904) =

(119897 minus 1)


119897minus119894 (6)


ΔX = 119861 (119904 119905 119906) sdot ΔP

(

1205751199091

1205751199092

1205751199093

)

= (119861111 sdot sdot sdot 119861119897119898119899)

[[[[

[

Δ1198751111 Δ1198751112 Δ1198751113

Δ1198751198971198981198991 Δ1198751198971198981198992 Δ1198751198971198981198993

]]]]

]

(7)

where

119861119894119895119896 = 119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906) (8)























Converge

Finish


Yes

No












Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Jig shape

CSD

CFD

CFD


forceTransfer of



performance


No

Yes










X119889 = [119870]minus1 F (1)

Xjig = X119888 + X119889 (2)











Initial jig shape



the cruise shape

Converge


aircraft


Reverse application

of load

Alyrsquosmethod

Geometry difference

process

Yes

No













K (X119888U)U minus F = 0 (3)





Initial jig shape

CSD


the cruise shape

Converge


aircraft


Reverse application

of load

Geometry difference

process

Yes

No


of load

Alyrsquosmethod









X (119904 119905 119906) + ΔX (119904 119905 119906)

=

119897

sum

119894=1

119898

sum

119895minus1

119899

sum

119896=1

[119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906)]

sdot [P119894119895119896 + ΔP119894119895119896]

(5)




119861119894minus1

119897minus1(119904) =

(119897 minus 1)


119897minus119894 (6)


ΔX = 119861 (119904 119905 119906) sdot ΔP

(

1205751199091

1205751199092

1205751199093

)

= (119861111 sdot sdot sdot 119861119897119898119899)

[[[[

[

Δ1198751111 Δ1198751112 Δ1198751113

Δ1198751198971198981198991 Δ1198751198971198981198992 Δ1198751198971198981198993

]]]]

]

(7)

where

119861119894119895119896 = 119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906) (8)























Converge

Finish


Yes

No












Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Initial jig shape



the cruise shape

Converge


aircraft


Reverse application

of load

Alyrsquosmethod

Geometry difference

process

Yes

No













K (X119888U)U minus F = 0 (3)





Initial jig shape

CSD


the cruise shape

Converge


aircraft


Reverse application

of load

Geometry difference

process

Yes

No


of load

Alyrsquosmethod









X (119904 119905 119906) + ΔX (119904 119905 119906)

=

119897

sum

119894=1

119898

sum

119895minus1

119899

sum

119896=1

[119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906)]

sdot [P119894119895119896 + ΔP119894119895119896]

(5)




119861119894minus1

119897minus1(119904) =

(119897 minus 1)


119897minus119894 (6)


ΔX = 119861 (119904 119905 119906) sdot ΔP

(

1205751199091

1205751199092

1205751199093

)

= (119861111 sdot sdot sdot 119861119897119898119899)

[[[[

[

Δ1198751111 Δ1198751112 Δ1198751113

Δ1198751198971198981198991 Δ1198751198971198981198992 Δ1198751198971198981198993

]]]]

]

(7)

where

119861119894119895119896 = 119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906) (8)























Converge

Finish


Yes

No












Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Initial jig shape

CSD


the cruise shape

Converge


aircraft


Reverse application

of load

Geometry difference

process

Yes

No


of load

Alyrsquosmethod









X (119904 119905 119906) + ΔX (119904 119905 119906)

=

119897

sum

119894=1

119898

sum

119895minus1

119899

sum

119896=1

[119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906)]

sdot [P119894119895119896 + ΔP119894119895119896]

(5)




119861119894minus1

119897minus1(119904) =

(119897 minus 1)


119897minus119894 (6)


ΔX = 119861 (119904 119905 119906) sdot ΔP

(

1205751199091

1205751199092

1205751199093

)

= (119861111 sdot sdot sdot 119861119897119898119899)

[[[[

[

Δ1198751111 Δ1198751112 Δ1198751113

Δ1198751198971198981198991 Δ1198751198971198981198992 Δ1198751198971198981198993

]]]]

]

(7)

where

119861119894119895119896 = 119861119894minus1

119897minus1(119904) 119861119895minus1

119898minus1(119905) 119861119896minus1

119899minus1(119906) (8)























Converge

Finish


Yes

No












Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Converge

Finish


Yes

No












Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Mass of thewing Kg

Maximumstress MPa














Configuration 119862119871 119862119863 119862119898




01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01

0

minus01

minus02

minus03

minus04

Twist

1 2 3 4 5 6 7 8

Y



Cruise shapeConfig2



5 10 15

Iteration

0

0005

001

0015

Max

imum

diff

eren

ce

120596 = 03

120596 = 04

120596 = 05

120596 = 06

120596 = 07







Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Cruise shape

Jig shape


2 chords

4 twist angles

Twist 1

SemispanTwist 2

Twist 3Twist 4




119877 =119881

119862

119871

119863ln(1198821

1198822

) (9)







Maximize 119865 =119871

119863ln(1198821

1198822

)

Subject to 119862119871 = 055

63m2 lt 119878wing lt 67m2

120575max 1 lt 95 MPa

120575max 2 lt 225MPa

(10)












5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















5 Conclusions


Baseline Optimum108

06

04

02

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus22

minus2

Pres

sure

coeffi

cien

t


932 + 007

871 + 007

809 + 007

747 + 007

686 + 007

624 + 007

562 + 007

501 + 007

439 + 007

377 + 007

316 + 007

254 + 007

192 + 007

131 + 007

692 + 006

752 + 005

764 + 007

714 + 007

663 + 007

613 + 007

563 + 007

513 + 007

462 + 007

412 + 007

362 + 007

312 + 007

262 + 007

211 + 007

161 + 007

111 + 007

606 + 006

104 + 006

Baseline Optimum


Y

X

5

4

2 4 6 8

BaselineOptimum


445

44

435Obj

ectiv

e fun

ctio

n

0 5 10 15

Iteration number



Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Initial jig shape

Optimized jig shape






Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Advanced Aerostructural Optimization ...downloads.hindawi.com/journals/mpe/2015/753042.pdf · cedure to get the jig shape from cruise shape is called jig shape correction