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Session Chair: Stefanie Reese, RWTH Aachen University Session Chair: Carolin Birk, Universität Duisburg-Essen
Location:Auditorium Wolfsburg
Presentations
Unfitted HDG-NEFEM method for Stokes flows
Stefano Piccardo1,3,4, Matteo Giacomini1,2, Antonio Huerta1,2
1Laboratori de Càlcul Numèric (LaCàN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, Barcelona, Spain; 2Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain; 3CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France; 4Inria, 2 rue Simone Iff, 75589 Paris, France
We consider the Stokes problem involving one or two immiscible, viscous, incompressible fluids. Each fluid splits the computational domain into multiple subdomains either void or occupied by a fluid governed by the Stokes equations. At the interface between two fluids, the fluid velocities are continuous, whereas the jump of the normal stress is proportional to the curvature of the interface according to Laplace's law. These equations define the so-called Stokes immersed problem (fluid-void) and Stokes interface problem (fluid-fluid), respectively.
An unfitted NURBS-enhanced finite element method (NEFEM) combined with a hybridizable discontinuous Galerkin (HDG) approach, named unfitted HDG-NEFEM is proposed.
The approach is denoted as unfitted because the employed mesh does not conform to the interface. As a result, the interface can cut arbitrarily through the mesh elements. The adoption of unfitted meshes offers a substantial advantage in simplifying the meshing procedure, especially for high-order approximations. The enforcement of inhomogeneous Dirichlet boundary conditions is then achieved by means of a consistent penalty technique inspired by Nitsche’s method.
Given by a CAD model, NURBS define boundaries and interfaces; thus, the exact geometry of the computational domain is considered. As a consequence, geometric errors resulting from polynomial approximations of boundaries and interfaces are eliminated. The exact treatment of geometry is thus integrated into the functional approximation via the NEFEM paradigm.
HDG methods rely on a mixed formulation of the incompressible Stokes problem employing mesh skeleton-based hybrid unknowns for the velocity and only element-based unknowns for the pressure and the gradient of velocity. The element-based unknowns are then erased via static condensation to reduce the number of globally coupled degrees of freedom. Note that the HDG mixed formulation is particularly suited to accurately approximate the stress since both pressure and gradient of velocity converge with the same order of the primal variable.
The Stokes immersed problem is considered first to examine the influence of badly-cut elements and badly-cut faces on the condition number of the discrete system and the resulting accuracy of the method. On one hand, to counter the effects of badly-cut elements, we propose an element extension technique coupled with local static condensation of the element nodes lying outside the physical domain. On the other hand, the negative effect of badly-cut faces is handled by employing modal basis functions for the face variable.
Then, the Stokes interface problem is solved for complex geometries and interfaces. In particular, we investigate the equilibrium problem between two fluids, that is zero normal velocity at the fluid interface, when a shear flow is prescribed in the far field to assess the suitability of the method to solve physically-relevant problems in microfluidics.
