scholarly journals Solving Galbrun's Equation with a Discontinuous galerkin Finite Element Method

2019 ◽  
Vol 105 (6) ◽  
pp. 1149-1163 ◽  
Author(s):  
Marcus Maeder ◽  
Andrew Peplow ◽  
Maximilian Meindl ◽  
Steffen Marburg
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Early Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linearized Euler Equations ◽  
Displacement Based ◽  
Spurious Modes

Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving Galbrun's equation in a mixed and pure displacement based formulation. The results achieved with the proposed approach are compared with results from previous attempts to solve Galbrun's equation. The numerical determination of acoustic modes and the identification of vortical modes is discussed. Furthermore, case studies for a lined-duct and an annulus supporting a rotating shear-flow are investigated.

scholarly journals A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2316
Author(s):  
Laura Río-Martín ◽  
Saray Busto ◽  
Michael Dumbser
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Finite Element ◽  
Mach Number ◽  
Finite Volume ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Massively Parallel ◽  
Navier Stokes ◽  
Navier Stokes Equations

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

Exponential convergence of mixed hp-DGFEM for the incompressible Navier–Stokes equations in ℝ2

2020 ◽  
Author(s):  
Dominik Schötzau ◽  
Carlo Marcati ◽  
Christoph Schwab
Keyword(s):  
Discontinuous Galerkin ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Rates Of Convergence ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Hp Spaces ◽  
Galerkin Approximations

Abstract In a polygon $\varOmega \subset \mathbb{R}^2$ we consider mixed $hp$-discontinuous Galerkin approximations of the stationary, incompressible Navier–Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and $hp$ spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in $\varOmega$, we prove exponential rates of convergence of the mixed $hp$-discontinuous Galerkin finite element method, with respect to the number of degrees of freedom, for small data which is piecewise analytic.

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