Parametric solutions involving geometry integrated with computer-aided design
For engineers using CAD, this provides a general, problem-independent approach to compute geometrically parametrized solutions, but the impact is limited to specific domain applications.
This work presents a general framework for computing off-line parametric solutions for geometry described by NURBS, using proper generalized decomposition (PGD) to handle multiple geometric parameters and compute sensitivities efficiently. The method achieves optimal convergence rates for incompressible flows.
The main objective of this work is to describe a general and original approach for computing an off-line solution for a set of parameters describing the geometry of the domain. That is, a solution able to include information for different geometrical parameter values and also allowing to compute readily the sensitivities. Instead of problem dependent approaches, a general framework is presented for standard engineering environments where the geometry is defined by means of NURBS. The parameters controlling the geometry are now the control points characterising the NURBS curves or surfaces. The approach proposed here, valid for 2D and 3D scenarios, allows a seamless integration with CAD preprocessors. The proper generalised decomposition (PGD), which is applied here to compute explicit geometrically parametrised solutions, circumvents the curse of dimensionality. Moreover, optimal convergence rates are shown for PGD approximations of incompressible flows.