Mat-1.3656 Seminar on
numerical analysis and computational science
Monday, Jan 25, 2010, room
U322 at 14.15, Eirola & Stenberg
Gianluigi Rozza,
Modelling and Scientific Computing,
Ecole Polytechnique Federale de Lausanne, Switzerland
Shape design in parametrized domains by reduced basis method
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 (including transfinite mappings),
transforming an original parametrized domain to a reference one.
The presentation 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. The focus is also 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 computational decomposition. We
present in particular some examples of reduced basis method applied to
external inviscid potential flow, internal viscous thermal flows in
channels and cavities, and steady incompressible Stokes flows for
shape optimization problems in cardiovascular geometries such as
bypass, stenosis and bifurcations.
Joint work in collaboration with L. Iapichino, T. Lassila and A. Manzoni.