skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Abstract

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.

Authors:
 [1];  [1]
  1. Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1524036
Alternate Identifier(s):
OSTI ID: 1245264
Grant/Contract Number:  
AC02-05CH11231; FA9550-10-1-0229
Resource Type:
Accepted Manuscript
Journal Name:
Computers and Fluids
Additional Journal Information:
Journal Volume: 118; Journal Issue: C; Journal ID: ISSN 0045-7930
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; discontinuous Galerkin; space-time; high-order accuracy; deformable domains; Navier-Stokes

Citation Formats

Wang, Luming, and Persson, Per-Olof. A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations. United States: N. p., 2015. Web. https://doi.org/10.1016/j.compfluid.2015.05.026.
Wang, Luming, & Persson, Per-Olof. A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations. United States. https://doi.org/10.1016/j.compfluid.2015.05.026
Wang, Luming, and Persson, Per-Olof. Thu . "A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations". United States. https://doi.org/10.1016/j.compfluid.2015.05.026. https://www.osti.gov/servlets/purl/1524036.
@article{osti_1524036,
title = {A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations},
author = {Wang, Luming and Persson, Per-Olof},
abstractNote = {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.},
doi = {10.1016/j.compfluid.2015.05.026},
journal = {Computers and Fluids},
number = C,
volume = 118,
place = {United States},
year = {2015},
month = {6}
}

Journal Article:

Citation Metrics:
Cited by: 3 works
Citation information provided by
Web of Science

Figures / Tables:

Figure 1 Figure 1: Space-Time Mesh Generation. The left figure illustrates two mesh layers at time $t$ and $t$ + $∆t$, and the right figure shows a corresponding 3D space-time mesh between the two layers. The blue faces show a cross-section of the tetrahedral mesh.

Save / Share:

Works referenced in this record:

A Cartesian grid embedded boundary method for hyperbolic conservation laws
journal, January 2006

  • Colella, Phillip; Graves, Daniel T.; Keen, Benjamin J.
  • Journal of Computational Physics, Vol. 211, Issue 1
  • DOI: 10.1016/j.jcp.2005.05.026

A parallel multilevel method for adaptively refined Cartesian grids with embedded boundaries
conference, February 2013

  • Aftosmis, M.; Berger, M.; Adomavicius, G.
  • 38th Aerospace Sciences Meeting and Exhibit
  • DOI: 10.2514/6.2000-808

Robust and efficient Cartesian mesh generation for component-based geometry
conference, February 2013

  • Aftosmis, M.; Berger, M.; Melton, J.
  • 35th Aerospace Sciences Meeting and Exhibit
  • DOI: 10.2514/6.1997-196

The immersed boundary method
journal, January 2002


Immersed boundary method for flow around an arbitrarily moving body
journal, March 2006


A Fictitious Domain Approach to the Direct Numerical Simulation of Incompressible Viscous Flow past Moving Rigid Bodies: Application to Particulate Flow
journal, May 2001

  • Glowinski, R.; Pan, T. W.; Hesla, T. I.
  • Journal of Computational Physics, Vol. 169, Issue 2
  • DOI: 10.1006/jcph.2000.6542

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows
journal, March 2007


Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids
journal, October 2004

  • Farhat, Charbel; Geuzaine, Philippe
  • Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 39-41
  • DOI: 10.1016/j.cma.2003.09.027

A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains
journal, October 1999

  • Lomtev, I.; Kirby, R. M.; Karniadakis, G. E.
  • Journal of Computational Physics, Vol. 155, Issue 1
  • DOI: 10.1006/jcph.1999.6331

Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains
journal, April 2009

  • Persson, P. -O.; Bonet, J.; Peraire, J.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 17-20
  • DOI: 10.1016/j.cma.2009.01.012

Discontinuous Galerkin spectral element approximations on moving meshes
journal, March 2011

  • Minoli, Cesar A. Acosta; Kopriva, David A.
  • Journal of Computational Physics, Vol. 230, Issue 5
  • DOI: 10.1016/j.jcp.2010.11.038

On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes
journal, May 2011

  • Mavriplis, Dimitri J.; Nastase, Cristian R.
  • Journal of Computational Physics, Vol. 230, Issue 11
  • DOI: 10.1016/j.jcp.2011.01.022

A mesh adaptation framework for dealing with large deforming meshes: A MESH ADAPTATION FRAMEWORK
journal, November 2009

  • Compère, Gaëtan; Remacle, Jean-François; Jansson, Johan
  • International Journal for Numerical Methods in Engineering, Vol. 82, Issue 7
  • DOI: 10.1002/nme.2788

A New Changing-Topology ALE Scheme for Moving Mesh Unsteady Simulations.
conference, June 2012

  • Olivier, Geraldine; Alauzet, Frederic
  • 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
  • DOI: 10.2514/6.2011-474

Simulation of flows with strong shocks with an adaptive conservative scheme
journal, December 2012

  • Isola, D.; Guardone, A.
  • Journal of Computational and Applied Mathematics, Vol. 236, Issue 18
  • DOI: 10.1016/j.cam.2012.04.004

Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping
journal, August 2011


Space-time finite element methods for elastodynamics: Formulations and error estimates
journal, February 1988

  • Hughes, Thomas J. R.; Hulbert, Gregory M.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 66, Issue 3
  • DOI: 10.1016/0045-7825(88)90006-0

Space-time finite element methods for second-order hyperbolic equations
journal, December 1990

  • Hulbert, Gregory M.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 84, Issue 3
  • DOI: 10.1016/0045-7825(90)90082-W

Space-time finite element computation of compressible flows involving moving boundaries and interfaces
journal, August 1993

  • Aliabadi, S. K.; Tezduyar, T. E.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 107, Issue 1-2
  • DOI: 10.1016/0045-7825(93)90176-X

A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems
journal, July 1997

  • Masud, Arif; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 146, Issue 1-2
  • DOI: 10.1016/S0045-7825(96)01222-4

Discontinuous Galerkin finite element methods for second order hyperbolic problems
journal, August 1993


Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains
journal, December 2006

  • Sudirham, J. J.; van der Vegt, J. J. W.; van Damme, R. M. J.
  • Applied Numerical Mathematics, Vol. 56, Issue 12
  • DOI: 10.1016/j.apnum.2005.11.003

Space–Time Discontinuous Galerkin Finite Element Method with Dynamic Grid Motion for Inviscid Compressible Flows
journal, November 2002

  • van der Vegt, J. J. W.; van der Ven, H.
  • Journal of Computational Physics, Vol. 182, Issue 2
  • DOI: 10.1006/jcph.2002.7185

Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows
journal, September 2002

  • van der Ven, H.; van der Vegt, J. J. W.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 41-42
  • DOI: 10.1016/S0045-7825(02)00403-6

Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations
journal, September 2006

  • Klaij, C. M.; van der Vegt, J. J. W.; van der Ven, H.
  • Journal of Computational Physics, Vol. 217, Issue 2
  • DOI: 10.1016/j.jcp.2006.01.018

A space–time discontinuous Galerkin method for the time-dependent Oseen equations
journal, December 2008


Space–time discontinuous Galerkin finite element method for two-fluid flows
journal, February 2011

  • Sollie, W. E. H.; Bokhove, O.; van der Vegt, J. J. W.
  • Journal of Computational Physics, Vol. 230, Issue 3
  • DOI: 10.1016/j.jcp.2010.10.019

Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations
journal, December 2006

  • Klaij, C. M.; van der Vegt, J. J. W.; van der Ven, H.
  • Journal of Computational Physics, Vol. 219, Issue 2
  • DOI: 10.1016/j.jcp.2006.04.003

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations
journal, December 2007

  • Klaij, C. M.; van Raalte, M. H.; van der Ven, H.
  • Journal of Computational Physics, Vol. 227, Issue 2
  • DOI: 10.1016/j.jcp.2007.08.034

A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains
journal, June 2012


A space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations
journal, January 2013

  • Rhebergen, Sander; Cockburn, Bernardo; van der Vegt, Jaap J. W.
  • Journal of Computational Physics, Vol. 233
  • DOI: 10.1016/j.jcp.2012.08.052

Efficient Solutions of the Euler Equations in a Time-Adaptive Space-Time Framework
conference, June 2012

  • Mani, Karthik; Mavriplis, Dimitri
  • 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
  • DOI: 10.2514/6.2011-774

Conservative unsteady aerodynamic simulation of arbitrary boundary motion using structured and unstructured meshes in time: UNSTEADY AERODYNAMIC SIMULATION OF ARBITRARY BOUNDARY MOTION
journal, December 2011

  • Rendall, Thomas C. S.; Allen, Christian B.; Power, Edward D. C.
  • International Journal for Numerical Methods in Fluids, Vol. 70, Issue 12
  • DOI: 10.1002/fld.2756

Pitching Tents in Space-Time: mesh Generation for Discontinuous Galerkin Method
journal, April 2002

  • ÜNgÖR, Alper; Sheffer, Alla
  • International Journal of Foundations of Computer Science, Vol. 13, Issue 02
  • DOI: 10.1142/S0129054102001059

Layer based solutions for constrained space–time meshing
journal, September 2003


Spacetime meshing with adaptive refinement and coarsening
conference, January 2004

  • Abedi, Reza; Zhou, Yuan; Chung, Shuo-Heng
  • Proceedings of the twentieth annual symposium on Computational geometry - SCG '04
  • DOI: 10.1145/997817.997863

Building spacetime meshes over arbitrary spatial domains
journal, May 2005

  • Erickson, Jeff; Guoy, Damrong; Sullivan, John M.
  • Engineering with Computers, Vol. 20, Issue 4
  • DOI: 10.1007/s00366-005-0303-0

Adaptive spacetime meshing for discontinuous Galerkin methods
journal, January 2009


Simplex space–time meshes in finite element simulations
journal, July 2008

  • Behr, Marek
  • International Journal for Numerical Methods in Fluids, Vol. 57, Issue 9
  • DOI: 10.1002/fld.1796

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations
journal, January 2008

  • Persson, P. -O.; Peraire, J.
  • SIAM Journal on Scientific Computing, Vol. 30, Issue 6
  • DOI: 10.1137/070692108

Scalable Parallel Newton-Krylov Solvers for Discontinuous Galerkin Discretizations
conference, June 2012

  • Persson, Per-Olof
  • 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition
  • DOI: 10.2514/6.2009-606

A Simple Mesh Generator in MATLAB
journal, January 2004


Approximate Riemann solvers, parameter vectors, and difference schemes
journal, October 1981


Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
journal, January 2002

  • Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo
  • SIAM Journal on Numerical Analysis, Vol. 39, Issue 5
  • DOI: 10.1137/S0036142901384162

The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems
journal, January 2008

  • Peraire, J.; Persson, P. -O.
  • SIAM Journal on Scientific Computing, Vol. 30, Issue 4
  • DOI: 10.1137/070685518

Mesh deformation based on radial basis function interpolation
journal, June 2007


Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements
journal, January 2003

  • Stein, K.; Tezduyar, T.; Benney, R.
  • Journal of Applied Mechanics, Vol. 70, Issue 1
  • DOI: 10.1115/1.1530635

A variationally consistent mesh adaptation method for triangular elements in explicit Lagrangian dynamics
journal, December 2009

  • Lahiri, Sudeep K.; Bonet, Javier; Peraire, Jaume
  • International Journal for Numerical Methods in Engineering, Vol. 82, Issue 9
  • DOI: 10.1002/nme.2784

Qualitative measures for initial meshes
journal, February 2000


High-order Discontinuous Galerkin Simulations on Moving Domains using ALE Formulations and Local Remeshing and Projections
conference, January 2015

  • Wang, Luming; Persson, Per-Olof
  • 53rd AIAA Aerospace Sciences Meeting
  • DOI: 10.2514/6.2015-0820

    Works referencing / citing this record:

    A locally conservative and energy-stable finite-element method for the Navier-Stokes problem on time-dependent domains: HDG for Navier-Stokes on time-dependent domains
    journal, January 2019

    • Horváth, Tamás L.; Rhebergen, Sander
    • International Journal for Numerical Methods in Fluids, Vol. 89, Issue 12
    • DOI: 10.1002/fld.4707

    Weight‐adaptive isogeometric analysis for solving elastodynamic problems based on space‐time discretization approach
    journal, May 2019

    • Izadpanah, E.; Shojaee, S.; Hamzehei‐Javaran, S.
    • International Journal for Numerical Methods in Engineering, Vol. 119, Issue 10
    • DOI: 10.1002/nme.6082

    Simplex space‐time meshes in compressible flow simulations
    journal, June 2019

    • Danwitz, Max; Karyofylli, Violeta; Hosters, Norbert
    • International Journal for Numerical Methods in Fluids, Vol. 91, Issue 1
    • DOI: 10.1002/fld.4743