Nonlinear Programming Approaches in the Multidisciplinary Design Optimization of a Sailing Yacht Keel Fin

Reliable Computational Fluid Dynamics (CFD) solvers, as long as design databases, are a modern way to reduce the number of experimental tests on models. However, due to the increasing interest on the design optimization, mature CFD analysis should also be used in a larger context, where the simultaneous effectiveness of different solvers is sought. The problem complexity has so far prevented from assessing a satisfactory reformulation of the overall design problem, into a unique mathematical programming formulation. In fact, traditional approaches to shape design have often focused on the satisfaction of feasibility constraints of the problem in hand, rather than tackling optimal solutions. As a result, when several disciplines are involved in the design problem, different heuristics have been used, which address individual disciplinary optimization. The growing complexity of modern engineering systems has spurred designers to provide more efficient heuristics. Unfortunately, the latter are often based on designers personal skills on the specific problem treated, instead of relying on exact and self-adaptive techniques. These reasons motivate our interest for the systematic numerical approach to MDO (Multidisciplinary Design Optimization). In our case the multidisciplinarity refers to the design of a ship, which encompasses interacting physical phenomena as hydrodynamics, structural mechanics, and control (see also [4]). Recently a larger number of real industrial applications have included complex optimization approaches, where efficient solutions were claimed. Aircraft and spacecraft engines design are among the latter applications, which intrinsically yield challenging MD formulations (see e.g. [1]). Observe that in most of the cases, MDO methodologies substantially imply a process of parallelization and coupling of different independent optimization schemes (disciplines). Moreover, a distinguishing feature of the MDO formulations is that the interaction among the standard optimization approaches, each related to a discipline, is non-trivial. On this guideline, observe that most of the typical issues considered for nonlinear programming formulations (e.g. feasibility, optimality conditions, sensitivity analysis, duality theory, etc.), require a suitable adaptation when considered in an MDO framework. This work briefly reviews the main results of MDO literature, and details some more recent MDO formulations based on multilevel programming. In the latter schemes, the overall MDO formulation is decomposed into a master level problem and a set of optimization subproblems. The master (i.e. the system level) problem depends on the optimal solutions of subproblems; conversely each subproblem includes a set of unknowns provided by the master level. Then, we study and solve the MDO formulation of a sailing yacht keel fin design problem, where the derivatives of the objective functions are unavailable. The latter problem is a hydroelastic design optimization problem for a race yacht, where the fin is used to sustain the bulb, adopted to increase the stability of the sailing yachts during a competition (e.g. the America's Cup). Unlike a pure fluid dynamic approach, in a multidisciplinary framework, the shape of the keel fin is influenced by both the weight of the bulb and the hydrodynamic forces arising from the different sailing positions. Therefore, the final performances of the yacht are undoubtedly affected by the dynamic structural behavior of the fin, which bends according with both the hydrodynamic forces and the sailing positions. We study and compare a suite of different MDO formulations underlying the latter problem. The interaction between our two disciplines is approached with the application of derivative-free optimization methods. We highlight that for several MDO problems the real functionals to be minimized are described by expensive simulations and the derivatives are unavailable. This strongly motivates the interest for effective derivative-free techniques. We apply a modified Particle Swarm Optimization (PSO, see [3]) method, which belongs to the family of evolutionary algorithms. PSO [8] owes its popularity to the reasonable balance it often yields, between its overall computational cost and the quality of the final solution it provides.