Multi-Objective Optimisation of Expensive Objective Functions with Variable Fidelity Models

For the major part of real-life application, the formulation of an optimisation problem involves a lot of different objective functions, often coming from different disciplines or areas. In this context, the optimisation represents a meeting point for many specialists, each one focused his proper requirements, that is, criteria constraints and objective functions. Different disciplines could be involved, like Computational Fluid Dynamics (CFD), structural analysis etc.

Moreover, being different criteria involved, Multi-Objective (MO) techniques must be adopted, in order to control the enhancements of all the objective functions. By the way, designers are not interested in marginal improvements of the starting design, and only Global Optimisation (GO) techniques are able to guarantee a wide and exhaustive exploration of the design space. In conjunction to that, high-fidelity models must be applied during the optimisation process, in order to ensure the quality of the optimising design. This last feature is conflicting with the desiderata of the GO algorithms, that usually require a large amount of evaluations on the objective function in order to qualify the global optimum. Moreover, the design team needs a solution in a short time, and the total time needed by the application of reliable solvers in conjunction with GO algorithms may be unpractical if a single objective function evaluation takes hours or days, as for CFD computations.

In this context, the only way to make the process feasible is to perform a strong reduction on the number of calls to the high-fidelity models, adopting a cheaper one to be substituted to the high-fidelity solver for the most of the calls, without loosing the accuracy of the high-fidelity model. This goal can be obtained by different strategies, all referring to the concept of Variable Fidelity Model (VFM): solvers with different complexity (and cost) are applied together, in a framework in which the exchange of information between the models makes possible to correct the evaluations of the low-fidelity one, substituting efficiently the high-fidelity model.

Here an algorithm for the solution of optimum ship design problems is presented. The procedure, illustrated in the framework of multi-objective optimisation problems, make use of high-fidelity, CPU time expensive computational models, like a free surface capturing RANSE solver, coupled with analytical meta-models of the objective functions (low-fidelity).

The optimisation is composed by global and local phases. In the global stage of the search, few computationally expensive simulations (high-fidelity) are applied and surrogate models (metamodels) of the objective functions are produced (low-fidelity). After that, a large number of tentative design, placed uniformly on the Feasible Solution Set (F

SS

), are evaluated with the low-fidelity model. The most promising designs are clustered, then locally minimized and eventually verified with high-fidelity simulations. New exact values are used to enlarge the training points for the low-fidelity model and repeated cycles of the algorithm are performed. A Decision Maker strategy is adopted to select the most promising designs.