EU Regional School 2018 Part 2 with Prof. Antonio Huerta Ph.D.

Prof. Antonio Huerta, Ph.D.
Department of Applied Mathematics III, Universitat Politècnica de Catalunya, Spain

ABSTRACT:

Finite volume and finite element methods are well established computational frameworks to simulate complex engineering problems in solid and fluid mechanics. Efficient and reliable implementations of these techniques are available in several commercial and open-source softwares. Nonetheless, parametric studies and real-time simulations still represent a major challenge for today’s industry which demands the development of fast and accurate techniques to tackle such problems.

In the first part, an overview of recent advances on modern hybrid discretization approaches, namely the face-centered finite volume (FCFV) and the hybridizable discontinuous Galerkin (HDG) methods, are presented. The former is an efficient low-order approach that has been shown to be extremely robust to mesh distortion and stretching, which are usually responsible for the degradation of classical finite volume solutions [R. Sevilla, M. Giacomini, and A. Huerta. “A face-centred finite volume method for second-order elliptic problems” Int. J. Numer. Methods Eng. 115(8), pp. 986-1014 (2018). R. Sevilla, M. Giacomini, and A. Huerta. “A locking-free face-centred finite volume (FCFV) method for linear elasticity” arXiv:1806.07500 (2018)]. The latter is a high-order strategy originally proposed in [B. Cockburn, J. Gopalakrishnan, and R. Lazarov. “Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems" SIAM J. Numer. Anal. 47(2):1319–1365 (2009)]. Recently, an alternative high-order HDG formulation allowing the pointwise fulfillment of the conservation of angular momentum has been proposed. This aspect is crucial in the approximation of problems in computational solid and fluid mechanics in which quantities of engineering interest (e.g. compliance and aeronautical forces) have to be evaluated starting from the stress tensor. [R. Sevilla, M. Giacomini, A. Karkoulias, and A. Huerta. “A superconvergent hybridisable discontinuous Galerkin method for linear elasticity” Int. J. Numer. Methods Eng. 116(2), pp. 91-116 (2018). M. Giacomini, A. Karkoulias, R. Sevilla, and A. Huerta. “A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor” arXiv:1802.09394 (2018)].

In the second part, the proper generalized decomposition (PGD) is employed to devise efficient separated representations of the solution of parameterized engineering problems. The resulting PGD-based computational vademecums allow the fast evaluation of solutions involving user-supplied data, such as boundary conditions and geometrical configurations of the domain.

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