  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.
