A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations

Wang, Luming; Persson, Per-Olof

doi:10.1016/j.compfluid.2015.05.026

Title: A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations

Journal Article · Thu Jun 04 00:00:00 EDT 2015 · Computers and Fluids

DOI:https://doi.org/10.1016/j.compfluid.2015.05.026· OSTI ID:1524036

Wang, Luming ^[1]; Persson, Per-Olof ^[1]

Univ. of California, Berkeley, CA (United States)

We introduce a high-order accurate space-time discontinuous Galerkin method for solving two-dimensional compressible flow problems on fully unstructured space-time meshes. The discretization is based on a nodal formulation, with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. The meshes are produced by a mesh moving technique with element connectivity updates, and the corresponding space-time elements are produced directly based on these local operations. To obtain globally conforming tetrahedral meshes, we first derive the required conditions on a prism boundary mesh to allow for a valid local triangulation. Next, we present an efficient algorithm for finding a global mesh that satisfies these conditions. We additionally show how to add and remove mesh nodes, again using local constructs for the space-time mesh. Our approach is demonstrated on a number of test problems, showing the high-order accuracy for model problems, and the ability to solve flow problems on domains with complex large deformations.

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Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231; FA9550-10-1-0229

OSTI ID:: 1524036

Alternate ID(s):: OSTI ID: 1245264

Journal Information:: Computers and Fluids, Vol. 118, Issue C; ISSN 0045-7930

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 26 works

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References (52)

Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
A Cartesian grid embedded boundary method for hyperbolic conservation laws Colella, Phillip; Graves, Daniel T.; Keen, Benjamin J. Journal of Computational Physics, Vol. 211, Issue 1 https://doi.org/10.1016/j.jcp.2005.05.026	journal	January 2006
A parallel multilevel method for adaptively refined Cartesian grids with embedded boundaries Aftosmis, M.; Berger, M.; Adomavicius, G. 38th Aerospace Sciences Meeting and Exhibit https://doi.org/10.2514/6.2000-808	conference	February 2013
Robust and efficient Cartesian mesh generation for component-based geometry Aftosmis, M.; Berger, M.; Melton, J. 35th Aerospace Sciences Meeting and Exhibit https://doi.org/10.2514/6.1997-196	conference	February 2013
The immersed boundary method Peskin, Charles S. Acta Numerica, Vol. 11 https://doi.org/10.1017/S0962492902000077	journal	January 2002
Immersed boundary method for flow around an arbitrarily moving body Kim, Dokyun; Choi, Haecheon Journal of Computational Physics, Vol. 212, Issue 2 https://doi.org/10.1016/j.jcp.2005.07.010	journal	March 2006
A Fictitious Domain Approach to the Direct Numerical Simulation of Incompressible Viscous Flow past Moving Rigid Bodies: Application to Particulate Flow Glowinski, R.; Pan, T. W.; Hesla, T. I. Journal of Computational Physics, Vol. 169, Issue 2 https://doi.org/10.1006/jcph.2000.6542	journal	May 2001
Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows Wan, Decheng; Turek, Stefan Journal of Computational Physics, Vol. 222, Issue 1 https://doi.org/10.1016/j.jcp.2006.06.002	journal	March 2007
Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids Farhat, Charbel; Geuzaine, Philippe Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 39-41 https://doi.org/10.1016/j.cma.2003.09.027	journal	October 2004
A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains Lomtev, I.; Kirby, R. M.; Karniadakis, G. E. Journal of Computational Physics, Vol. 155, Issue 1 https://doi.org/10.1006/jcph.1999.6331	journal	October 1999
Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains Persson, P. -O.; Bonet, J.; Peraire, J. Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 17-20 https://doi.org/10.1016/j.cma.2009.01.012	journal	April 2009
Discontinuous Galerkin spectral element approximations on moving meshes Minoli, Cesar A. Acosta; Kopriva, David A. Journal of Computational Physics, Vol. 230, Issue 5 https://doi.org/10.1016/j.jcp.2010.11.038	journal	March 2011
On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes Mavriplis, Dimitri J.; Nastase, Cristian R. Journal of Computational Physics, Vol. 230, Issue 11 https://doi.org/10.1016/j.jcp.2011.01.022	journal	May 2011
A mesh adaptation framework for dealing with large deforming meshes: A MESH ADAPTATION FRAMEWORK Compère, Gaëtan; Remacle, Jean-François; Jansson, Johan International Journal for Numerical Methods in Engineering, Vol. 82, Issue 7 https://doi.org/10.1002/nme.2788	journal	November 2009
A New Changing-Topology ALE Scheme for Moving Mesh Unsteady Simulations. Olivier, Geraldine; Alauzet, Frederic 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition https://doi.org/10.2514/6.2011-474	conference	June 2012
Efficient Moving Mesh Technique Using Generalized Swapping Alauzet, Frédéric Proceedings of the 21st International Meshing Roundtable https://doi.org/10.1007/978-3-642-33573-0_2	book	January 2013
Simulation of flows with strong shocks with an adaptive conservative scheme Isola, D.; Guardone, A. Journal of Computational and Applied Mathematics, Vol. 236, Issue 18 https://doi.org/10.1016/j.cam.2012.04.004	journal	December 2012
Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping Guardone, A.; Isola, D.; Quaranta, G. Journal of Computational Physics, Vol. 230, Issue 20 https://doi.org/10.1016/j.jcp.2011.06.026	journal	August 2011
Space-time finite element methods for elastodynamics: Formulations and error estimates Hughes, Thomas J. R.; Hulbert, Gregory M. Computer Methods in Applied Mechanics and Engineering, Vol. 66, Issue 3 https://doi.org/10.1016/0045-7825(88)90006-0	journal	February 1988
Space-time finite element methods for second-order hyperbolic equations Hulbert, Gregory M.; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 84, Issue 3 https://doi.org/10.1016/0045-7825(90)90082-W	journal	December 1990
Space-time finite element computation of compressible flows involving moving boundaries and interfaces Aliabadi, S. K.; Tezduyar, T. E. Computer Methods in Applied Mechanics and Engineering, Vol. 107, Issue 1-2 https://doi.org/10.1016/0045-7825(93)90176-X	journal	August 1993
A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems Masud, Arif; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 146, Issue 1-2 https://doi.org/10.1016/S0045-7825(96)01222-4	journal	July 1997
Discontinuous Galerkin finite element methods for second order hyperbolic problems Johnson, Claes Computer Methods in Applied Mechanics and Engineering, Vol. 107, Issue 1-2 https://doi.org/10.1016/0045-7825(93)90170-3	journal	August 1993
Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains Sudirham, J. J.; van der Vegt, J. J. W.; van Damme, R. M. J. Applied Numerical Mathematics, Vol. 56, Issue 12 https://doi.org/10.1016/j.apnum.2005.11.003	journal	December 2006
Space–Time Discontinuous Galerkin Finite Element Method with Dynamic Grid Motion for Inviscid Compressible Flows van der Vegt, J. J. W.; van der Ven, H. Journal of Computational Physics, Vol. 182, Issue 2 https://doi.org/10.1006/jcph.2002.7185	journal	November 2002
Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows van der Ven, H.; van der Vegt, J. J. W. Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 41-42 https://doi.org/10.1016/S0045-7825(02)00403-6	journal	September 2002
Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations Klaij, C. M.; van der Vegt, J. J. W.; van der Ven, H. Journal of Computational Physics, Vol. 217, Issue 2 https://doi.org/10.1016/j.jcp.2006.01.018	journal	September 2006
A space–time discontinuous Galerkin method for the time-dependent Oseen equations van der Vegt, J. J. W.; Sudirham, J. J. Applied Numerical Mathematics, Vol. 58, Issue 12 https://doi.org/10.1016/j.apnum.2007.11.010	journal	December 2008
Space–time discontinuous Galerkin finite element method for two-fluid flows Sollie, W. E. H.; Bokhove, O.; van der Vegt, J. J. W. Journal of Computational Physics, Vol. 230, Issue 3 https://doi.org/10.1016/j.jcp.2010.10.019	journal	February 2011
Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations Klaij, C. M.; van der Vegt, J. J. W.; van der Ven, H. Journal of Computational Physics, Vol. 219, Issue 2 https://doi.org/10.1016/j.jcp.2006.04.003	journal	December 2006
h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations Klaij, C. M.; van Raalte, M. H.; van der Ven, H. Journal of Computational Physics, Vol. 227, Issue 2 https://doi.org/10.1016/j.jcp.2007.08.034	journal	December 2007
A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains Rhebergen, Sander; Cockburn, Bernardo Journal of Computational Physics, Vol. 231, Issue 11 https://doi.org/10.1016/j.jcp.2012.02.011	journal	June 2012
A space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations Rhebergen, Sander; Cockburn, Bernardo; van der Vegt, Jaap J. W. Journal of Computational Physics, Vol. 233 https://doi.org/10.1016/j.jcp.2012.08.052	journal	January 2013
Efficient Solutions of the Euler Equations in a Time-Adaptive Space-Time Framework Mani, Karthik; Mavriplis, Dimitri 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition https://doi.org/10.2514/6.2011-774	conference	June 2012
Conservative unsteady aerodynamic simulation of arbitrary boundary motion using structured and unstructured meshes in time: UNSTEADY AERODYNAMIC SIMULATION OF ARBITRARY BOUNDARY MOTION Rendall, Thomas C. S.; Allen, Christian B.; Power, Edward D. C. International Journal for Numerical Methods in Fluids, Vol. 70, Issue 12 https://doi.org/10.1002/fld.2756	journal	December 2011
Pitching Tents in Space-Time: mesh Generation for Discontinuous Galerkin Method ÜNgÖR, Alper; Sheffer, Alla International Journal of Foundations of Computer Science, Vol. 13, Issue 02 https://doi.org/10.1142/S0129054102001059	journal	April 2002
Layer based solutions for constrained space–time meshing Üngör, Alper; Sheffer, Alla; Haber, Robert B. Applied Numerical Mathematics, Vol. 46, Issue 3-4 https://doi.org/10.1016/S0168-9274(03)00037-0	journal	September 2003
Spacetime meshing with adaptive refinement and coarsening Abedi, Reza; Zhou, Yuan; Chung, Shuo-Heng Proceedings of the twentieth annual symposium on Computational geometry - SCG '04 https://doi.org/10.1145/997817.997863	conference	January 2004
Building spacetime meshes over arbitrary spatial domains Erickson, Jeff; Guoy, Damrong; Sullivan, John M. Engineering with Computers, Vol. 20, Issue 4 https://doi.org/10.1007/s00366-005-0303-0	journal	May 2005
Adaptive spacetime meshing for discontinuous Galerkin methods Thite, Shripad Computational Geometry, Vol. 42, Issue 1 https://doi.org/10.1016/j.comgeo.2008.07.003	journal	January 2009
Simplex space–time meshes in finite element simulations Behr, Marek International Journal for Numerical Methods in Fluids, Vol. 57, Issue 9 https://doi.org/10.1002/fld.1796	journal	July 2008
Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations Persson, P. -O.; Peraire, J. SIAM Journal on Scientific Computing, Vol. 30, Issue 6 https://doi.org/10.1137/070692108	journal	January 2008
Scalable Parallel Newton-Krylov Solvers for Discontinuous Galerkin Discretizations Persson, Per-Olof 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition https://doi.org/10.2514/6.2009-606	conference	June 2012
A Simple Mesh Generator in MATLAB Persson, Per-Olof; Strang, Gilbert SIAM Review, Vol. 46, Issue 2 https://doi.org/10.1137/S0036144503429121	journal	January 2004
Approximate Riemann solvers, parameter vectors, and difference schemes Roe, P. L. Journal of Computational Physics, Vol. 43, Issue 2 https://doi.org/10.1016/0021-9991(81)90128-5	journal	October 1981
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo SIAM Journal on Numerical Analysis, Vol. 39, Issue 5 https://doi.org/10.1137/S0036142901384162	journal	January 2002
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems Peraire, J.; Persson, P. -O. SIAM Journal on Scientific Computing, Vol. 30, Issue 4 https://doi.org/10.1137/070685518	journal	January 2008
Mesh deformation based on radial basis function interpolation de Boer, A.; van der Schoot, M. S.; Bijl, H. Computers & Structures, Vol. 85, Issue 11-14 https://doi.org/10.1016/j.compstruc.2007.01.013	journal	June 2007
Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements Stein, K.; Tezduyar, T.; Benney, R. Journal of Applied Mechanics, Vol. 70, Issue 1 https://doi.org/10.1115/1.1530635	journal	January 2003
A variationally consistent mesh adaptation method for triangular elements in explicit Lagrangian dynamics Lahiri, Sudeep K.; Bonet, Javier; Peraire, Jaume International Journal for Numerical Methods in Engineering, Vol. 82, Issue 9 https://doi.org/10.1002/nme.2784	journal	December 2009
Qualitative measures for initial meshes Field, David A. International Journal for Numerical Methods in Engineering, Vol. 47, Issue 4 https://doi.org/10.1002/(SICI)1097-0207(20000210)47:4<887::AID-NME804>3.0.CO;2-H	journal	February 2000
High-order Discontinuous Galerkin Simulations on Moving Domains using ALE Formulations and Local Remeshing and Projections Wang, Luming; Persson, Per-Olof 53rd AIAA Aerospace Sciences Meeting https://doi.org/10.2514/6.2015-0820	conference	January 2015

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