multilevel distributed structure optimization€¦ · multilevel distributed structure optimization...

24th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

MULTILEVEL DISTRIBUTED STRUCTURE OPTIMIZATION

J.O. Entzinger*, R. Spallino** , W. Ruijter** University of Twente, Mechanical Engineering Department, ** AIRBUS Deutschland GmbH

Keywords: Genetic Algorithms, Neural Networks, Optimisation, Composite Structures, AerospaceStructures

Abstract

An iterative optimisation routine for aircraftstructures using Genetic Algorithms (GAs) andNeural Networks (NNs) is presented. In thissetup the NNs form a response surface, approx-imating the key mechanical properties of sub-structures. NNs are updated every iteration. TheGA uses these NNs in the optimisation to quicklydetermine the feasibility of different variants. Allfound optimal substructures are checked using aFinite Element (FE) calculation. When the FEoutputs differ too much from the NN approxima-tions the solution is added to the NN training set,thus improving the NN’s performance.

Main advantages of the proposed strategy arefirstly the possibility to take into account manytopologically distinct designs and secondly theflexibility to quickly evaluate the influence of up-dated loads or different design restrictions (e.g.materials, access holes) on the optimum. Thebenefit of the feedback of inaccurately estimatedsubstructure properties (according to the FE veri-fication) is the improvement of accuracy and con-vergence. Also this principle drastically reducesthe number of datasets (i.e. FE calculations)needed to train the NNs initially.

Two levels are implemented: a global levelcontaining the structure as a whole, and a locallevel to describe the substructures (e.g. compos-ite panels the structure is made of) more accu-rately. On the global level a coarse mesh can beused, for it is only needed to derive the loadingof the panels. On the local level more detail isneeded, for linear static buckling and local strainsmust be analysed accurately.

On the local level all substructure optimisa-tions (including the time-consuming FE checks)are mutually independent and can therefore beparallelised. This speeds up the process signif-icantly when using multiple computers to dis-tribute the workload. In our setup 7-27 machineswere used, enabling overnight optimisation of alarge structure.

Full scale optimisations on a major substruc-ture of a representative aircraft design understatic loadcases have been performed success-fully.

1 Introduction

The goal of this research project is to develop aprogram which enables overnight optimisation ofan aircraft stucture in order to quickly estimatethe possible weight savings due to new designconsiderations. A combination of Genetic Al-gorithms (GAs) and Neural Networks (NNs) hasbeen employed to iteratively optimise the aircraftstructure by optimising the (more or less similar)panels it consists of. In this setup the NNs serveas a response surface, mapping the key mechani-cal properties of the panels. They are used in theevaluation function of the GA to determine thefeasibility of different solutions very fast.

The GAs are chosen for their robustness andflexibility. Because they do not need any gradi-ent information, GAs allow a mixture of continu-ous and discrete parameters so different topolo-gies and even different design principles (plateor trusswork) can be handled. The drawback ofthis algorithm is the high number of evaluationsneeded, which urges for a response surface such

1

J.O. ENTZINGER*, R. SPALLINO** , W. RUIJTER*

as a NN to approximate the outputs of Finite El-ement (FE) analyses in a fast and accurate way.

When NNs with fixed architecture and train-ing sets had been trained with a more or lessarbitrary set of FE solutions, the errors of theNN estimates appeared quite large. Therefore,a setup with feedback of optima found by theGA has been implemented, so accurate FE val-ues are added to the NN’s training set when theNN’s accuracy is insufficient. This means thatless initial datasets can be used and the NN isonly trained to a higher accuracy in the placeswhere the GA expects an optimum. The iterativecharacter of the routine increases the benefit of a(locally) accurate response surface, for the pan-els to be evaluated in successive iterations differin loading only.

The proposed setup is more broadly usable tohandle problems which have the following prop-erties:

• The structure can be divided into smaller,approximately independent components.

• Parametric models are available for boththe structure and its components.

• Load paths (or component loads) are tosome extent dependent of component de-signs.

• Optimisation constraints are applied at thecomponent level.

• Multiple topologies may be allowed for acomponent.

The paper is organised as follows. Sections 2and 3 provide some backgrond information onNeural Networks and Genetic Algorithms respec-tively. In section 4 the structural mechanics prob-lem will be explained. The proposed strategy ispointed out in section 5. Section 6 deals with thefull scale tests while test results are presented insection 7. Finally the main conclusions are drawnin section 8.

2 Neural Networks

A (feedforward) neural network can be seen asa curve-fitting technique based on an analogywith the (human) brain. A set of known inputsand outputs (the training set) is used to set thecoefficients in the NN. This process, calledtraining, is a form of non-linear least squaresfitting. A reason to choose NNs as a responsesurface is the fact that they can map virtuallyany function without any a priori knowledgeof the relationship between inputs and outputs.When sigmoid functions are used as transferfunctions, the NN’s ability to generalize (i.e.accurately map function samples outside thetraining dataset) from a small amount of inputsis much larger than conventional techniqueslike fitting with polynomials or fourier-series.Another difference is that sigmoid functions arecapable of handling discontinuities in the output,but it is suspected that the quality of the fit isbetter with ‘sleek’ functions.

Most function mapping artificial NNs are ofthe Feed-Forward (FF) type and have neuronsgrouped in layers. Each neuron has a transferfunction (e.g. a sigmoid function) operating onthe neuron’s input. In FF-NNs only neuronsin successive layers are interconnected. Thisresults in the typical shape shown in figure 1with one input layer (receiving an input vectorx typically representing design variables), oneoutput layer (producing output vector y which isan approximation of the mechanical response ofa component, the exact value of which are targetsvector t) and an arbitrary number of hiddenlayers. As the lines in the figure point out, theresponse of a neuron transfer function dependson weighted inputs (by weights matrices w) anda bias value (by bias matrix b) which acts as anoffset to the neuron’s input. The NNs employedhave so-called back-propagation training algo-rithms, this means that the contribution of eachconnection weight is determined by passing er-rors in a backward manner through the network.The errors are then minimised by employing aniterative scheme such as Levenberg-Marquardt.

2

Multilevel Distributed Structure Optimization

PSfrag replacements

Hidden Layer

Signal processing direction

x y t

Error BP direction

wH wM

bH bM

Fig. 1 Typical shape of an artificial neural net-work

The topic of neurocomputing (i.e. the com-puter simulation of neural networks) has beenextensively analysed in literature, the reader isreferred to [1] and [2] for more specific informa-tion on the background and operation of NeuralNetworks (NN) in engineering applications.

3 Genetic Algorithms

Genetic algorithms are guided random searchtechniques that work analogous to biological evo-lution. Much of the GA terminology is derivedfrom the biological counterpart, e.g. there are in-dividuals, populations, genes and fitnesses. Anindividual represents an element in the solutionspace. The individual’s genes determine its loca-tion (i.e. specify all parameter values) and its fit-ness indicates the performance of the individualwith respect to the objective and the constraints.Individuals are grouped in populations.

Assuming that a recombination of geneticmaterial from well performing individuals cre-ates favourable offspring, a GA uses selectionand variation to reach a solution to an optimi-sation problem. Selection is applied by assign-ing a higher probability to the genetic materialof favourable individuals of getting passed on tothe next generation. Variation is accomplished bymeans of crossover and mutation, i.e. recombina-tion of genetic material from selected individuals,respectively random changes in genes.

The algorithm starts by creating an initialpopulation consisting of individuals with ran-

domly assigned genes. The fitness of eachindividual is determined using an appropriateevaluation function, which is a value depend-ing on the objective (weight in this project)and penalties (which are given depending onconstraint violation). Selection, crossover andmutation are applied consecutively to generatea new population (the next generation). Thisprocess is repeated until convergence is reached,that is, until a certain solution dominates thepopulation.

A GA can handle more general classes offunctions than most traditional mathematicalprogramming search techniques. Whereasthe latter use characteristics of the problem(e.g. gradients and continuity), GAs do notrequire such assumptions. They can handle alsonon-differentiable and discontinuous functions.Because of the use of mutation and the fact thatGAs operate on a population instead of a singlestarting point, GAs are less likely to get stuck ina local optimum. However, increased random-ness comes with decreased convergence speed,therefore GAs typically require more functionevaluations than gradient based optimisers. Thisis the reason for using a response surface such asNNs instead of direct FE calculations.

A genetic algorithm has been proven to bean effective way of solving a wide range of non-differentiable problems. The reader is referred to[3] for an application in composite structures, and[4] for a civil engineering application. The GAoperators used for this paper are derived from aMatlab implemented GA by Houck et. al. [5].

4 Structural Mechanics Problem

Two mechanics problems have to be analysed inthe optimisation process:

1. A global level structure analysis with rela-tively coarse meshes, used to obtain load-ings for components

2. A local level component analysis with finermeshes, used to obtain the buckling multi-plier and the maximum local strain level

3


All analyses are done automatically using para-metric models, for wich linear static FE calcula-tions are performed.

The assembly of an aircraft’s vertical tailplane (VTP) structure consists of skin, rib andspar panels, as shown in figure 2, is analysed.

PSfrag replacements

Rib

Skin

Spar

Fig. 2 VTP structure assembly of rib, skin, andspar panels

A parameterised panel is shown in figure 3.The panels are made of composite material andanalysed using layered shell elements. Typi-cal design parameters are length, width, mate-rial properties, topology, stiffener/hole positions,stiffener heights and panel thickness. Topologiesare distinct in the number of transverse stiffeners,the number of longitudinal stiffeners, the numberof holes and their ordering on the panel. Lengthand width become height and width respectivelyin the global structure. All panels are consideredsymmetric.

For spar panels the optimisation parametersare: topology, thickness, positions of the holesand/or stiffeners and stiffener height. For rib pan-els the thickness and number of transverse stiff-eners and their positions are optimised. For skinpanels only thickness is optimised.

PSfrag replacements Transverse stiffenerBending

Width

Shear

x1 Lengthx2

x3

Longitudinalstiffener

xyz

Fig. 3 Parameterised panel

5 Optimisation Strategy

Every optimisation run starts with initialisingthe geometry and the NNs. NNs can eitherbe reloaded from a previous optimisation (thusmaintaining previously acquired knowledge) orbe initialised by training data from new FE cal-culations.

As stated before, the optimisation routine canbe split in a local (component optimisation) leveland a global (panel loading derivation) level.

5.1 Local level

Every panel in the structure is optimised for mini-mal structural weight using the genetic algorithmas displayed in figure 4. The NNs are used in theevaluation function for estimating the bucklingload and maximum local strain for each panelproposed by the GA. When the buckling load islower than the applied load (derived on the globallevel), or when the local strain exceeds its max-imum allowed value, the solution is penalised.This way the panel’s geometry is less likely topersist in the population, so the algorithm tendsto panels that do not violate the constraints. Eachtopology (see figure 5) is represented by one NN,so the evaluation function always has a completeset of NNs at its disposal.

On termination of the GA loop the best panelfound is checked with a FE analysis, to makesure the NN’s estimate is accurate. If this isnot the case, the NN is retrained with the infor-mation just obtained from the FE check. TheNN aims to achieve a certain error level (a userdefined threshold), if this value is not reachedwithin a given number of training cycles the num-

4


ber of neurons in the NN hidden layer is in-creased. After all panels have been checked andthe NNs have been updated with the feedback ofthe FE analyses, the optimisation is started againwith improved NNs, or returns to the global levelstructure analysis if all found optimal panels ap-peared accurately estimated by the NNs.

Genetic

Algorithm

(§3)

Neural

Networks

2)(§Generate

Initial population

Selection

Cross-over

Mutation

Terminate?

Optimum

matches

FE analysis?

Yes YesNo

No

TrainingInitial FE-data

Generation

Evaluation

functionBasic

Neural Networks

Fig. 4 Component level layout

5.2 Additional rules

To avoid repetition of equal FE analyses in subse-quent feedback iterations, the following rules areapplied:

• The topology of the optimal panel solutionis excluded in the next feedback iteration,forcing the optimiser to search in differentregions of the design space. This improvesthe accuracy of the set of NNs.

• Once a NN for a certain topology is re-trained, all panels for which this topol-

ogy was excluded re-add it to their searchspace.

• Once a feasible solution (judging from FE)has been found, the next solution must belighter. This way, the optimiser can findoptima that are approximated too conser-vative by the NN. Example: Normally alighter panel which does not satisfy theconstraints (according to the NN approxi-mation of its properties) gets a penalty, so itis found unfavourable. However, when allsolutions which are not lighter than the op-timum also get a penalty, the lighter panelmight still be found as optimum. Whenthis optimum is checked by a FE analy-sis, it might appear feasible (i.e. not vio-lating any constraints) and badly approxi-mated by the NN.

5.3 Global level

No optimisation takes place on the global struc-tural level. On this level all components (panels)are assembled and the new loadings of each com-ponent are derived from the global model for anumber of different loadcases. A representativeload is calculated for each panel. This load willbe a constraint in the local level optimisation.

After all loads have been obtained from thestructure analysis, the components are optimisedagain using the iterative procedure of optimising,checking and retraining the NNs as described in§5.1. This process of load derivation and optimi-sation goes on as long as the loads obtained fromthe global structure change significantly.

5.4 Distributed computing implementation

Since all panel optimisations are independent,the local level process (§5.1) can be parallelisedto a large extent using multiple computers, thusspeeding up the overall optimisation process.This means that the GA panel optimisation, butmore importantly, the time-consuming FE checksof optima and NN (re)training cycles are dis-tributed among several computers in a LAN.

5


6 Optimisations

Three optimisation runs are performed to showsome characteristic features and behaviour. Re-sults are obtained using a Java implementation ofthe proposed distributed computing optimisationalgorithm written at the University of Twente.

6.1 Optimisation setups

1. A (large) series of components only (frontand rear spar panels) using the additionalrules in §5.2. PATRAN/NASTRAN anal-yses are performed on 7 HP-UX machines@400MHz.

2. A VTP-like structure run (as introducedin §5.3) using simple skin optimisation(thickness only) for 18 skin, 9 rib and 18spar panels, which are fully optimised (pa-rameters are stated in §4). ANSYS analy-ses are performed on 20 MS windows ma-chines @2.6GHz.

3. As 2. but for 36 skin, 18 rib and 36 sparpanels, using 27 MS windows [email protected].

6.2 Optimisation constraints

Linear buckling and local strain constraints haveto be satisfied for all panels. Extra constraints forthe respective setups are:

1. In all spar panels except the 6 lower ones(front and rear) access holes are required.

2. Topologies 9b and 19-21 as shown in figure5 are not taken into account.

3. In all spar panels except the 4 upper ones(front and rear) access holes are requiredand no longitudinal stiffeners are allowed.Topologies 9b and 19-21 as shown in figure5 are not taken into account.

It must be mentioned that for the structureoptimisations (setup 2 and 3) non-realistic globalloadcases and constraints are used, they are for

testing and illustrative purposes only. The com-ponents only run is an early actual optimisationrun.

Loading:

• Spar panels: combined shear and bending

• Rib panels: shear only

• Skin panels: combined compres-sion/tension and shear

• Structure: global torsion and bending load-cases

Topologies used for spar panels are depicted infigure 5. Panels are rotated 90 degrees comparedto the global structure. For ribs topologies withtransverse stiffeners only (no holes and no lon-gitudinal stiffeners) are taken into account. Ribtopologies differ in no. of stiffeners only.

PSfrag replacements

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

9b 19 20 21

1

Fig. 5 Spar panel configurations that can be chosenin optimisation

New topologies can easily be added. Alsotopologies using trusswork instead of compositepanels could be implemented, however this is be-yond the scope of the current research.

7 Results

Some of the optimisation results are highlighted.

6


7.1 Convergence

A components only optimisation (setup 1) withapplication of the extra rules mentioned in §5.2typically converges in 50-80 feedback iterations(depending on the problem complexity) and takesabout 18 hours to complete on 7 HP-UX worksta-tions.

The structure optimisations (setups 2 and 3)needed 2-3 structure iterations to converge, usinga fixed number of 35 component (NN feedback)iterations. With an average FE solving time of ca.1 minute per panel the optimisations completedin 8-9 hours.

Runtimes are indications. During the processsome machines were not available all the time.Also the speed is strongly dependent on the meshsizes used. Nevertheless, it is clear that overnightoptimisation of a complete structure is possible,however only when enough computers are avail-able.

7.2 Optimised design

A picture of the optimised spars is shown in fig-ure 6. Keep in mind that all configurations arerotated 90 degrees compared to the ones depictedin figure 5. It is clear that for larger panels, morestiffeners are chosen to prevent buckling. A re-markable fact is the appearance of holes in panelswhere they were not required, probably due to theweight the hole itself saves. Surrounding a holeby stiffeners seems by far the best option whena hole is required. Therefore addition of newtopologies with one or more holes, more than 2transversal stiffeners and 2 or more longituninalstiffeners might be attractive for upper panels inthe rear spar, according to this result.

In figure 7 the optimised structure designsare depicted. As can be seen more stiffenersare chosen in the lower panels, for they typicallyhave a higher loading. The influence of stiffenerplacement can be clearly seen in the rear spar ofthe first structure, which has longitudinal stiffen-ers placed divergently towards the lower end (to-wards higher bending).

The outcomes of multiple optimisation runs,

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y Z

X

Y Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y Z

(a) Front spar

X

Y Z

X

Y Z

X

Y

Z

X

Y

Z

X

Y Z

X

Y Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

(b) Rear spar

Fig. 6 Optimised designs (with realistic load-cases)

7


(a) Setup 2 (b) Setup 3

Fig. 7 Optimised designs (loadcases are not re-alistic)

sometimes with slightly changed paramaters, arevery similar. Though not proven scientifically,the proposed method seems quite robust. Thisis not a surprise, considering the robustness andthoroughness of the GA and the learning and gen-eralisation capabilities of the NNs.

8 Conclusions

From several experiments with the proposed pro-gram, some conclusions can be drawn:

• The proposed program can be a powerfuldesign tool as it allows overnight evalua-tion of design decisions such as:

– Use of different materials

– Different hole placement constraintsthroughout the structure

– Use of different variables (i.e. fixedor variable stiffener heights)

– Allowance of different topologies.

• The effect of new or different loadcases onthe optimal design can be quickly evalu-ated, for knowledge is stored in re-usableNNs.

• The feedback of the FE outputs of prelimi-nary optima to the NN is essential to reachan accurate optimum

• The use of additional rules (see §5.2)greatly improves the efficiency and effec-tiveness of the optimisation.

• The robustness of the routine is not proven,though the fact that optima can be repro-duced whithin narrow bounds with differ-ent initial NN training sets and other GArandom seeds indicates convergence to aglobal optimum.

• Integrated structure optimisation is rela-tively efficient, for later iterations benefitof (locally) well trained NNs from previousiterations (due to the fact that only loadingchanges).

• Neural Network training time can becomedominant for NNs with large datasets (typ-ically favourable topologies with manyvariables).

• Distributing time consuming operationssuch as FE calculations and NN trainingamong several computers is essential for apractical concept tool.

9 Acknowledgement

The contributions of Jeroen Hol and Jelmer Windare highly appreciated.

References

[1] W.M. Jenkins. Neural network-based approxi-mations for structural analysis. Developmentsin Neural Networks and Evolutionary Computingfor Civil and Structural Engineering, pages 25–35, 1995.

8


[2] L. Ziemianski Z. Waszczyszyn. Neural networksin mechanics of structures and materials - new re-sults and prospects of applications. Proceedingsof European Conference on Computational Me-chanics, München, 1999.

[3] S. Rizzo R. Spallino. Optimal design of lami-nated composite plates. Computer Aided Opti-mum Design of Structures IV, Boston, 1999.

[4] G. Giambanco R. Spallino, S. Rizzo. Evolu-tion strategies for the shakedown optimal designof r.c. framed structures. European Congress onComputational Methods in Applied Science andEngineering, Barcelona, 2000.

[5] M.G. Kay C.R. Houck, J.A. Joines. A GeneticAlgorithm for Function Optimization: A MatlabImplementation. GAOTv5 Manual (Genetic Al-gorithm Optimization Toolbox), 1998.

9

multilevel distributed structure optimization€¦ · multilevel distributed structure optimization...

Documents