Certified real-time shape optimization using isogeometric analysis,PGD model reduction,and a posteriori error estimation

In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer-aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off-line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two- and three-dimensional numerical experiments.