Project Details
Projekt
High-Performance Cut Discontinuous Galerkin Methods for Flow Problems and Surface-Coupled Multiphysics Problems
Subject Area
Fluid Mechanics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 456365667
This application proposes a new solver for fluid flow and surface-coupled multiphysics problems, combining high-order discontinuous Galerkin (DG) spatial discretizations with cut finite element (CutFEM) technologies. High-order discontinuous Galerkin methods are more efficient than low-order methods in fluid flow for moderate to high accuracy requirements, especially in marginally resolved turbulent simulations, with much lower dispersion and dissipation errors. At the same time, upwind-like fluxes make DG also robust in difficult application problems. The CutFEM methodology enables to carry over these attractive features to a wider class of problems, namely those where the computational mesh does not align with the (potentially moving) boundary of the simulation domain. Besides a much simpler grid generation, this concept is ideal for coupled multiphysics configurations when the involved geometries are subjected to large deformations or even undergo complex topological changes during the simulation, such as in fluid-structure interaction or multiphase flow. Cut methods are fully accurate all the way to the boundary by design, which makes them far superior over immersed methods for problems where the interface dynamics play a crucial role. However, robust and optimally convergent schemes for fluid flow and multiphysics are today only available for low-order continuous Galerkin approximations. With this proposal, the mathematical theory will be extended to high-order DG methods, targeting robustness the context of multiphysics applications. In order to avoid ill-conditioning in the presence of small cuts, a ghost penalty stabilization will be developed and complemented by inverse estimates and stability proofs for high-order polynomial spaces. These developments will initially be tailored to the incompressible Navier-Stokes equations. The mathematical ingredients will then be complemented by high performance computing contributions, including new fast iterative solvers based on multigrid algorithms specifically adapted to the cut scenario. The proposed multigrid methods use p-multigrid concepts with appropriate smoothers that will first coarsen to low-order polynomial bases, for which robust algebraic multigrid are available. The novel algorithms will employ efficient matrix-free operator evaluation schemes based on sum factorization for high-order DG methods on the regular elements, which will be augmented by new strategies on cut elements. The proposed methods enable more detailed and efficient simulations in various engineering and biomedical applications where the resolution requirements are continuously rising.
DFG Programme
Research Grants
International Connection
Norway, United Kingdom, USA
Cooperation Partners
Professor Erik Burman; Dr. Andre Massing, Ph.D.; Professor Raymond S. Tuminaro, Ph.D.
