Mathematics

In the last decades optimal shape design problems have gained an increasing importance in many engineering fields and especially in structural mechanics and in thermo–fluid dynamics. The problems we consider, being related with optimal design and flow control, necessarily involve the study of an evolving system modelled by PDEs and the evaluation of functionals depending on the field variables, such as velocity, pressure, drag forces, temperature, energy, wall shear stress or vorticity. Especially in the field of shape optimization, where the recursive evaluation of the field solution is required for many possible configurations, the computational costs can easily become unacceptably high. Nevertheless, the evaluation of an "input/output" relationship of the system plays a central role: a set of input parameters identifies a particular configuration of the system and they may represent design or geometrical variables, while the outputs may be expressed as functionals of the field variables associated with a set of parametrized PDEs. The rapid and reliable evaluation of many input/output relationships typically requires great computational expense, and therefore strategies to reduce the computational time and effort are being developed. Among model order reduction strategies, reduced basis method represents a promising tool for the simulation of flow in parametrized geometries, for shape optimization or sensitivity analysis. An implementation of the reduced basis method is presented by considering different shape or domain parametrizations: from simple affine maps to non–affine ones, transforming an original parametrized domain to a reference one. Our analysis will focus on the general properties and performance of the reduced basis method by highlighting with several examples its special suitability and considering parametrized wavy or curvy geometries. The proposed approach includes also a geometric model reduction resulting from a suitable low–dimensional parametrization of the geometry based on free–form deformations technique. We focus on the possibility of handling very generic geometric parametrizations without requiring to create "ad hoc" affine representations necessary to solve the problem efficiently, but recovering this property by an empirical interpolation method in order to take advantage of an offline–online decomposition. We present in particular some examples of reduced basis method applied to steady incompressible viscous flows for shape optimization, parameter estimation and inverse problems in cardiovascular geometries. In collaboration with L. Iapichino, T. Lassila, A. Manzoni, and A. Quarteroni.