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Cost-Based Methodologies forDesign Optimization

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technique designed to circumvent thisproblem. The objective is to construct anapproximate model (the meta-model) thatis computationally cheaper to evaluate andwhich approximates the output (objectivefunction) from the input parameters (designvariables) with reasonable accuracy. Newdata is periodically added to refine themeta-model gradually giving betterapproximations and a near optimal designat the end of the search. In this specific case, the objective is to mini-mize manufacturing cost subject to VonMises stress being less than 200 MPa. Twodimensions of the component (Arc Radius(r) and Thickness (t)) are selected as thedesign variables to be modified in thesearch. The meta-model is constructed bygenerating an initial set of candidatedesigns using the LPτ sampling technique.A radial basis function (RBF) is used toapproximate the actual relationshipbetween r, t, Cost and Von Mises stress usingthe data generated from the candidatedesigns. Asimulated annealing algorithm isthen employed to search the meta-modelover 5,000 design points before everyupdate. In total, the meta-model is updatedat fifty points before the optimal design ispredicted. Figure 5 shows the search spaceat which the full problem code was used toevaluate different candidate designs. Thefeasible designs are shown in blue circlesand the designs that violate the imposedconstraints are denoted by red asterisks.

Figure 5. The design concepts evaluatedby the optimizer.

Asolid model representation of the geome-try achieved after optimization is shown inFig. 6. Figure 7 shows the variations of costand Von Mises stress with respect to thedesign variables using the final meta-model generated after fifty updates.

Figure 6. Minimum cost geometry

Figure 7. Response Surfaces of Cost andVon-Mises Stress against the DesignVariables

Multiobjective Optimizationof Stress and CostThe present problem can also be formulat-ed with two objectives by simultaneouslytrying to minimize both stress and cost.This leads to the construction of a Paretofront and the idea of Pareto Optimization.APareto front is formed from a set of designsolutions to a single design problem whereeach member of the set is an optimal solu-tion for an aggregate goal. This aggregategoal can be formulated by assigningweights to each objective and taking theweighted sum. Results from this analysisled to the construction of a Pareto curve asshown in Fig.8.

Figure 8. The Pareto curve plottedthrough five points of evaluation.

This Pareto curve has five points, all ofthem optimal combinations of the parame-ters r and t for different values of weightingbetween stress and cost. A designer cannow easily move along this surface tochoose the best trade-off that fits into thespecific requirements of his product andcompany. In practice, many more combina-tions would have to be evaluated to form adense Pareto curve which may make thisstrategy computationally expensive. The entire process is automated as compu-tational time may run into hours. In future,we plan to develop a manufacturabilitymodel to reflect the relative ease of manu-facture of a design as a metric and use it inoptimizing designs in a multiobjectiveframework. This method will also beapplied in the design of more sophisticatedparts than the present component and atdifferent stages of the design process.

AcknowledgementsThis work is part of the Design AnalysisTool for Unit Cost Modeling (DATUM)research project headed by Rolls-Royce plc,University of Southampton and theUniversity of West of England.

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Abhijit. R. Rao, A. J. Keane,J. P. Scanlan,Computational Engineering &Design Centre,University of Southampton,SO17 1BJ, U.K

Feature based CostingCost estimation in this project is based oncalculating the cost of a ‘manufacturing fea-ture’. Amanufacturing feature is defined asa change in the state of a component. Thisstate change is often a change in geometrycaused by a machining process. The finalproduct geometry is achieved after a set ofmanufacturing features are applied to theraw material. The cost of a manufacturingfeature is the cost of resources expended inmaking the transition from state n-1 to n asshown in Fig. 3. The total cost is a summa-tion of the constituent manufacturing fea-ture costs. Since this method of costingrelies on component geometry, it providesthe incremental cost incurred by embed-ding a geometric feature within a design.

Figure 3. State Transition andManufacturing Feature Costs

The costing method explained above hasbeen encapsulated within DecisionPro™, adecision support software tool instrumen-tal in building detailed cost models forcomplex products. DecisionPro provides aclear and logical format in the form of ahierarchical tree structure for capturing thevarious cost computations used in themodel. This offers easy readability to devel-opers and simplified audit procedures forend users (designers), unlike spreadsheetswhere the logic is often difficult to follow asthe calculations assume greater complexity.It is also possible to include cost libraries offrequently used entities or objects. The costmodels can also be uploaded to a serverand queried remotely allowing better inte-gration in an existing MDO environment.Figure 4 shows a snapshot from the costmodel.

Figure 4. Detail from cost model

Two different optimization strategies havebeen tested so far on this system: (1) meta-model based optimization, and(2) multiobjective optimization and con-struction of a Pareto front for stress and cost.

Meta-model basedoptimizationThe presence of multiple software and com-putationally intensive tools in this integrat-ed system prohibit search using the fullproblem code over a very large designspace. Meta-model based optimization is a

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Life cycle cost is one of the key issues inaerospace manufacturing as business mod-els change from selling products to provid-ing a service, for example; the concept of“Power by the hour” and “Total Care” con-tracts. Reliable and accurate cost predic-tions have to be made as early as possiblewithin the design cycle and traded withother product attributes, as it becomes pro-gressively more difficult and expensive tomake modifications. This project aims todevelop methods for integrating cost mod-els within optimization processes to searchfor trade-offs between weight, stress andcost of an emerging design.The research till date has focused on devel-oping a system to perform manufacturingcost based optimization as shown in Fig.1.The four different elements essential to theprocess are: (1) a parameterized solidmodel of the component (2) a finite elementanalysis (FEA) tool (3) a cost model reflect-ing changes in cost as geometry is modifiedand (4) a robust optimizer.

Figure 1. An overview of the proposedcost optimization methodology

The component used to demonstrate thisconcept is a three dimensional geometry ofa Rear Mount link used in one of the Rolls-Royce civil aircraft engines. The link geom-etry is modeled parametrically in CATIAV5™. The input values to the parameter-ized solid model are controlled by the opti-mizer. Figure 2 shows a range of geometriesdeveloped by varying the inputs. Each ofthe candidate geometries is analyzed inANSYS 6.1™ to extract the maximum VonMises stress in the part for a predefined setof loading conditions. The inputs to the fea-ture based cost model are the weight, vol-ume, and surface area of the solid model.The outputs (stress and cost) are thenpassed back to the optimizer. The optimizerthen uses a specified algorithm to calculatethe input parameters for the subsequentiteration by comparing the outputs againstthe objective and constraint functions. Thisprocess is continued iteratively until theoptimum design solution is found.

Figure 2. Different geometriesdeveloped parametrically