Ruben Sevilla

Ruben Sevilla, Antonio Huerta

The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.

Ruben Sevilla, Sergio Zlotnik, Antonio Huerta

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.