Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change
Abstract
A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method’s capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method’s accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- North Carolina State Univ., Raleigh, NC (United States)
- Univ. of California, Davis, CA (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1241930
- Alternate Identifier(s):
- OSTI ID: 1359281
- Report Number(s):
- LLNL-JRNL-669889
Journal ID: ISSN 0021-9991
- Grant/Contract Number:
- AC52-07NA27344; 13-SI-002
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 305; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Discontinuous Galerkin Method; High-Order Space-Time Discretization; Fully-Implicit Solvers; Newton-Krylov Algorithm; Fluid Dynamics; Phase-Change
Citation Formats
Nourgaliev, R., Luo, H., Weston, B., Anderson, A., Schofield, S., Dunn, T., and Delplanque, J. -P. Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change. United States: N. p., 2015.
Web. doi:10.1016/j.jcp.2015.11.004.
Nourgaliev, R., Luo, H., Weston, B., Anderson, A., Schofield, S., Dunn, T., & Delplanque, J. -P. Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change. United States. https://doi.org/10.1016/j.jcp.2015.11.004
Nourgaliev, R., Luo, H., Weston, B., Anderson, A., Schofield, S., Dunn, T., and Delplanque, J. -P. Wed .
"Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change". United States. https://doi.org/10.1016/j.jcp.2015.11.004. https://www.osti.gov/servlets/purl/1241930.
@article{osti_1241930,
title = {Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change},
author = {Nourgaliev, R. and Luo, H. and Weston, B. and Anderson, A. and Schofield, S. and Dunn, T. and Delplanque, J. -P.},
abstractNote = {A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method’s capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method’s accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.},
doi = {10.1016/j.jcp.2015.11.004},
journal = {Journal of Computational Physics},
number = ,
volume = 305,
place = {United States},
year = {Wed Nov 11 00:00:00 EST 2015},
month = {Wed Nov 11 00:00:00 EST 2015}
}
Web of Science
Works referenced in this record:
A second-order projection method for the incompressible navier-stokes equations
journal, December 1989
- Bell, John B.; Colella, Phillip; Glaz, Harland M.
- Journal of Computational Physics, Vol. 85, Issue 2
Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow
journal, June 2002
- Bijl, Hester; Carpenter, Mark H.; Vatsa, Veer N.
- Journal of Computational Physics, Vol. 179, Issue 1
Numerical solution of the Navier-Stokes equations
journal, January 1968
- Chorin, Alexandre Joel
- Mathematics of Computation, Vol. 22, Issue 104
Estimation of Sparse Jacobian Matrices and Graph Coloring Blems
journal, February 1983
- Coleman, Thomas F.; Moré, Jorge J.
- SIAM Journal on Numerical Analysis, Vol. 20, Issue 1
Modelling liquid-solid phase changes with melt convection
journal, August 1989
- Dantzig, Jonathan A.
- International Journal for Numerical Methods in Engineering, Vol. 28, Issue 8
Natural convection of air in a square cavity: A bench mark numerical solution
journal, May 1983
- De Vahl Davis, G.
- International Journal for Numerical Methods in Fluids, Vol. 3, Issue 3
Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations
journal, January 2010
- Dumbser, Michael
- Computers & Fluids, Vol. 39, Issue 1
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
journal, September 2008
- Dumbser, Michael; Balsara, Dinshaw S.; Toro, Eleuterio F.
- Journal of Computational Physics, Vol. 227, Issue 18
Efficient implementation of high order unstructured WENO schemes for cavitating flows
journal, November 2013
- Dumbser, Michael; Iben, Uwe; Munz, Claus-Dieter
- Computers & Fluids, Vol. 86
Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations
journal, October 2009
- Dumbser, Michael; Zanotti, Olindo
- Journal of Computational Physics, Vol. 228, Issue 18
High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method
journal, December 1982
- Ghia, U.; Ghia, K. N.; Shin, C. T.
- Journal of Computational Physics, Vol. 48, Issue 3
On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory
journal, October 1990
- Gresho, Philip M.
- International Journal for Numerical Methods in Fluids, Vol. 11, Issue 5
Numerical calculation of almost incompressible flow
journal, August 1968
- Harlow, Francis H.; Amsden, Anthony A.
- Journal of Computational Physics, Vol. 3, Issue 1
A numerical fluid dynamics calculation method for all flow speeds
journal, October 1971
- Harlow, Francis H.; Amsden, Anthony A.
- Journal of Computational Physics, Vol. 8, Issue 2
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
journal, September 2005
- Hernandez, Vicente; Roman, Jose E.; Vidal, Vicente
- ACM Transactions on Mathematical Software, Vol. 31, Issue 3
Differencing of the diffusion equation in Lagrangian hydrodynamic codes
journal, February 1981
- Kershaw, David S.
- Journal of Computational Physics, Vol. 39, Issue 2
Mesoscopic simulation model of selective laser melting of stainless steel powder
journal, November 2014
- Khairallah, Saad A.; Anderson, Andy
- Journal of Materials Processing Technology, Vol. 214, Issue 11
Jacobian-free Newton–Krylov methods: a survey of approaches and applications
journal, January 2004
- Knoll, D. A.; Keyes, D. E.
- Journal of Computational Physics, Vol. 193, Issue 2
A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids
journal, January 2006
- Luo, Hong; Baum, Joseph D.; Löhner, Rainald
- Journal of Computational Physics, Vol. 211, Issue 2
A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids
journal, October 2008
- Luo, Hong; Baum, Joseph D.; Löhner, Rainald
- Journal of Computational Physics, Vol. 227, Issue 20
A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids
journal, November 2012
- Luo, Hong; Luo, Luqing; Nourgaliev, Robert
- Communications in Computational Physics, Vol. 12, Issue 5
A Parallel, Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids
journal, February 2011
- Luo, Hong; Luo, Luqing; Ali, Amjad
- Communications in Computational Physics, Vol. 9, Issue 2
A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids
journal, September 2010
- Luo, Hong; Luo, Luqing; Nourgaliev, Robert
- Journal of Computational Physics, Vol. 229, Issue 19
A Parallel Reconstructed Discontinuous Galerkin Method for Compressible Flows on Arbitrary Grids
conference, June 2012
- Luo, Hong; Ali, Amjad; Nourgaliev, Robert
- 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids
conference, June 2012
- Luo, Hong; Luo, Luqing; Nourgaliev, Robert
- 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids
journal, June 2012
- Luo, Hong; Xia, Yidong; Li, Shujie
- Journal of Computational Physics, Vol. 231, Issue 16
A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids
journal, March 2013
- Luo, Hong; Xia, Yidong; Spiegel, Seth
- Journal of Computational Physics, Vol. 236
An Improved piso Algorithm for the Computation of Buoyancy-Driven Flows
journal, December 2001
- J. Oliveira, Raad I. Issa, Paulo
- Numerical Heat Transfer, Part B: Fundamentals, Vol. 40, Issue 6
On physics-based preconditioning of the Navier–Stokes equations
journal, December 2009
- Park, HyeongKae; Nourgaliev, Robert R.; Martineau, Richard C.
- Journal of Computational Physics, Vol. 228, Issue 24
Elemental: A New Framework for Distributed Memory Dense Matrix Computations
journal, February 2013
- Poulson, Jack; Marker, Bryan; van de Geijn, Robert A.
- ACM Transactions on Mathematical Software, Vol. 39, Issue 2
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986
- Saad, Youcef; Schultz, Martin H.
- SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems
journal, August 1987
- Voller, V. R.; Prakash, C.
- International Journal of Heat and Mass Transfer, Vol. 30, Issue 8
A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations: A THIRD-ORDER IMPLICIT DISCONTINUOUS GALERKIN METHOD BASED ON A HERMITE WENO RECONSTRUCTION FOR TIME-ACCURATE SOLUTION OF THE COMPRESSIBLE NAVIER-STOK
journal, June 2015
- Xia, Yidong; Liu, Xiaodong; Luo, Hong
- International Journal for Numerical Methods in Fluids, Vol. 79, Issue 8
An implicit Hermite WENO reconstruction-based discontinuous Galerkin method on tetrahedral grids
journal, June 2014
- Xia, Yidong; Luo, Hong; Nourgaliev, Robert
- Computers & Fluids, Vol. 96
A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids
journal, July 2014
- Xia, Yidong; Luo, Hong; Frisbey, Megan
- Computers & Fluids, Vol. 98
Works referencing / citing this record:
A reconstructed discontinuous Galerkin method for multi‐material hydrodynamics with sharp interfaces
journal, January 2020
- Pandare, Aditya K.; Waltz, Jacob; Bakosi, Jozsef
- International Journal for Numerical Methods in Fluids, Vol. 92, Issue 8
High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics
journal, November 2018
- Nourgaliev, R.; Greene, P.; Weston, B.
- Shock Waves, Vol. 29, Issue 5
A trust region approach for numerical modeling of non-isothermal phase change
journal, July 2019
- Amiri, Erfan A.; Craig, James R.; Hirmand, M. Reza
- Computational Geosciences, Vol. 23, Issue 5