Multiple-Gradient Descent Algorithm for isogeometric shape optimization

Giacomini, Matteo
Multiple-Gradient Descent Algorithm for isogeometric shape optimization
Monday 22nd July 2013
Formaggia, L.
Advisor II:
Désidéri, J.A.
Duvigneau, R.
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This thesis has been developed during a six-month internship within the OPALE research team at INRIA Méditerranée (Institut Nationale de Recherche en Informatique et Automatique - Sophia Antipolis, France) under the supervision of Jean-Antoine Désidéri and Régis Duvigneau. This work focuses on the research field of optimization. Moreover it deals with topics related to numerical methods for the approximation of PDEs and applications to computational mechanics. This work faces some theoretical issues in the cooperative phase of multiobjective optimization and applies the resulting methodology to a shape optimization problem in linear elasticity. In the first part of this thesis, a general paradigm for the treatment of multiobjective optimization is presented, introducing the concepts of Pareto-optimal solutions and Pareto fronts. Then we describe a methodology that extends classical Steepest-Descent Method to the case of concurrent optimization of several criteria by means of the so-called Multiple-Gradient Descent Algorithm. Moreover we present an application to a linear elasticity problem, formulated using Iso-Geometric Analysis. In particular, we propose to analyze a classical problem of shape optimization in structural engineering within a multiobjective optimization framework. Thus the following variants of Multiple-Gradient Descent Algorithm are tested: MGDA using gradients approximated by Finite Difference Method; MGDA assisted by statistical-based metamodels in order to predict the values of the objective functionals; MGDA enhanced by the information contained in the gradients analytically extracted from the NURBS -based formulation of the problem. Some numerical simulations for a test case are presented and the results are cross-validated using all the variants of this multiobjective optimization algorithm, a strategy based on shape derivatives and a genetic algorithm widely used in the literature. Eventually, an introduction to multiobjective competitive optimization is given and the general framework to formulate a proper Nash game is established. Keywords: Multiobjective optimization, Pareto-optimal solutions, gradient descent, IsoGeometric Analysis, shape optimization, shape gradient, kriging models