Hybridizable Discontinuous Galerkin: low and high-order approximations for computationally-demanding problems
Antonio Huerta, Universitat Politècnica de Catalunya, Barcelona, Spain
Matteo Giacomini, Universitat Politècnica de Catalunya, Barcelona, Spain
Ruben Sevilla, Swansea University, Wales, UK
Daily industrial practice requires fast and robust strategies to solve computationally-demanding problems. For instance, efficient strategies are needed to compute multiple queries of complex multi-physics and multi-disciplinary problems. Multiple queries are typical of parametric studies such as flow control, shape design or optimization, real-time monitoring of manufacturing processes and inverse analysis in medical imaging.
Discontinuous Galerkin (DG) methods have proven their applicability and efficiency in high-order approximations, today's computing architectures and adaptive strategies for non-uniform degree approximations. Nonetheless, the computational cost [1] has almost restricted their use to academic contexts. Recently, hybridization techniques [2] have opened new possibilities to devise efficient adaptive high-order DG strategies capable of treating large-scale engineering problems in a competitive way.
To further expedite such strategies in daily engineering computations a novel hybridizable discontinuous Galerkin (HDG) method for computational mechanics is proposed [3,4]. The proposed HDG strategy relies on strongly enforcing symmetry of the mixed variable. Post-processing leads to super-convergence of the primal variable (order k+2 for k≥1). Thus, an inexpensive local error indicator drives degree adaptivity [5]. This allows for accurate approximations via a high-order HDG strategy, see Figure 1. The incorporation of the NEFEM paradigm further improves the performance of the error indicator in the presence of curved boundaries without the costly communication with CAD systems to regenerate the mesh during the adaptive process.
Antonio Huerta, Universitat Politècnica de Catalunya, Barcelona, Spain
Matteo Giacomini, Universitat Politècnica de Catalunya, Barcelona, Spain
Ruben Sevilla, Swansea University, Wales, UK
Daily industrial practice requires fast and robust strategies to solve computationally-demanding problems. For instance, efficient strategies are needed to compute multiple queries of complex multi-physics and multi-disciplinary problems. Multiple queries are typical of parametric studies such as flow control, shape design or optimization, real-time monitoring of manufacturing processes and inverse analysis in medical imaging.
Discontinuous Galerkin (DG) methods have proven their applicability and efficiency in high-order approximations, today's computing architectures and adaptive strategies for non-uniform degree approximations. Nonetheless, the computational cost [1] has almost restricted their use to academic contexts. Recently, hybridization techniques [2] have opened new possibilities to devise efficient adaptive high-order DG strategies capable of treating large-scale engineering problems in a competitive way.
To further expedite such strategies in daily engineering computations a novel hybridizable discontinuous Galerkin (HDG) method for computational mechanics is proposed [3,4]. The proposed HDG strategy relies on strongly enforcing symmetry of the mixed variable. Post-processing leads to super-convergence of the primal variable (order k+2 for k≥1). Thus, an inexpensive local error indicator drives degree adaptivity [5]. This allows for accurate approximations via a high-order HDG strategy, see Figure 1. The incorporation of the NEFEM paradigm further improves the performance of the error indicator in the presence of curved boundaries without the costly communication with CAD systems to regenerate the mesh during the adaptive process.
Moreover, since optimal convergence of order k+1 (including the mixed variable) is obtained any k, degree of the polynomial discretization (including k=0), fast and robust computations of large-scale problems are performed with HDG of order zero. This is a face-centered finite volume method (FCFV) [6,7]. It is able to perform efficiently large-scale simulations on complex unstructured meshes. It is robust to cell distortion and stretching. It provides first-order accurate approximations for stresses/fluxes without the need of flux reconstruction procedures. It is locking-free in the incompressible limit and does not require any shock-capturing technique to compute non-oscillatory approximations of shock waves (see Figures 2 and 3). The potential of the FCFV discretization paradigm to treat computationally-demanding industrial problems will be described using examples including thermal and elastic phenomena, as well as incompressible and compressible flows.
REFERENCES
[1] A. Huerta, A. Angeloski, X. Roca, J. Peraire, "Efficiency of high-order elements for continuous and discontinuous Galerkin methods." Int. J. Numer. Methods Eng., 96(9):529–560, 2013.
[2] B. Cockburn, J. Gopalakrishnan, 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.
[3] R. Sevilla, M. Giacomini, A. Karkoulias, A. Huerta. “A superconvergent hybridisable discontinuous Galerkin method for linear elasticity.” Int. J. Numer. Methods Eng.,116(2):91–116, 2018.
[4] M. Giacomini, A. Karkoulias, R. Sevilla, A. Huerta. “A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor.” J. Sci. Comput., 77(3):1679–1702, 2018.
[5] R. Sevilla, A. Huerta. “HDG-NEFEM with degree adaptivity for Stokes flows.” J. Sci. Comput., 77(3):1953–1980, 2018.
[6] 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):986–1014, 2018.
[7] R. Sevilla, M. Giacomini and A. Huerta, A locking-free face-centred finite volume (FCFV) method for linear elastostatics, Comput. Struct., 212:43–57, 2019.
[1] A. Huerta, A. Angeloski, X. Roca, J. Peraire, "Efficiency of high-order elements for continuous and discontinuous Galerkin methods." Int. J. Numer. Methods Eng., 96(9):529–560, 2013.
[2] B. Cockburn, J. Gopalakrishnan, 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.
[3] R. Sevilla, M. Giacomini, A. Karkoulias, A. Huerta. “A superconvergent hybridisable discontinuous Galerkin method for linear elasticity.” Int. J. Numer. Methods Eng.,116(2):91–116, 2018.
[4] M. Giacomini, A. Karkoulias, R. Sevilla, A. Huerta. “A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor.” J. Sci. Comput., 77(3):1679–1702, 2018.
[5] R. Sevilla, A. Huerta. “HDG-NEFEM with degree adaptivity for Stokes flows.” J. Sci. Comput., 77(3):1953–1980, 2018.
[6] 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):986–1014, 2018.
[7] R. Sevilla, M. Giacomini and A. Huerta, A locking-free face-centred finite volume (FCFV) method for linear elastostatics, Comput. Struct., 212:43–57, 2019.
