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# What we call Robust Design Optimization (IOSO RDO Application example)

Nowadays there are many technologies that people call Robust Design Optimization. Among them one can encounter for example such popular methods like six-sigma. But despite the absence of common terminology for RDO, one can agree that methods and technologies like six-sigma may be called as sensitivity analysis other than Robust Design Optimization. Real-life RDO technologies could be classified by the deepness and thoroughness of the development of their theory and their practical realization.

From our side we offer different concepts of fully elaborated Robust Design Optimization schemes with the introduction of the realization probability criteria (that means RDO always is a multiobjective optimization task) and with the direct solution of RDO problem in stochastic statement a) without b) with the usage of metamodels being built during the process of optimization.

Avoiding various theoretical details (which anyone can find in the full paper, referenced below) here let us offer you the description of some practical RDO task solution without the usage of metamodels. Let us consider the example of the multiobjective robust design optimization of the multistage axial flow compressor.

Purpose: To insure the maximum efficiency and maximum implementation probability of multistage axialflow compressor under preset level of production technology. Setting features: **140 independent design variables** (flowpath geometry); two objectives; three nonlinear constraints (mass flow, pressure ratio, surge margin). Optimization process features: Object under study – quasi-3D mathematical model. Implementation probability was calculated as the probability of assuring the preset constraints. IOSO optimization software was used as optimizer. Fig. 1 shows the main results of this problem. One can see that there is a compromise area (Pareto-front) between the compressor efficiency and the implementation probability. In general, designer can select any solution from the obtained set. Design No 10 has the best compressor efficiency but unsatisfactory realization probability, design No 1 has the best realization probability but poor efficiency. In this case the design No 4 was selected as the final design.

Fig. 1 Results of compressormulticriteria robust design optimization.

While solving this problem we used only 50 calls of mathematical model to approximately evaluate the probability criterion at each iteration. After 400 iterations optimization process was halted. Then we used 5000 additional calls of mathematical model to refine probability criteria for Pareto set found. Thus, total number of mathematical model calls was 25000. This is a not enormous value for the RDO stochastic problem taking into account 140 independent variables.

*Full paper link:** **http://www.iosotech.com/text/aiaa_4328.pdf** **(.pdf, 395Kb), 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 04 - 06 Sep. 2002, Atlanta, Georgia.*

In the next issue of our news we will show on real-life example how we use metamodels in addition to high-fidelity mathematical model to dramatically decrease the number of direct calls of high-fidelity model. Metamodels in our case are automatically built by Approx software (designed by Sigma Technology as a part of IOSO RM technology) and from our point of view could not be built in one shot; during the solution of RDO problem our metamodels are gradually verified, rebuilt and improved.

The simplified scheme of work for the multilevel optimization procedure can be represented as follows.

I) Building of low-fidelity (surrogate or meta) model on the basis of data set previously obtained.

II) Solving the multi-objective optimization problem based upon a metamodel.

III) For the obtained Pareto-set the objectives and constrained parameters are updated using the high-fidelity analysis tool.

IV) The refinement of the metamodel is performed.

V) Replacement of the metamodel and the return to step II).

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