Abstract
This paper presents the construction of a conservative radiation hydrodynamics algorithm in two-dimensional (2D) spherical geometry. First, we discretize the radiation transport equation (RTE) in that geometry. The discretization preserves the conservation of photons by integrating the original RTE in 2D spherical coordinates over both angular and spatial control volumes. Some numerical results are provided to verify the discretization for both optically thin and thick circumstances. Second, we formulate the staggered Lagrangian hydrodynamics in that geometry. The formulation preserves the conservation of mass, momentum, and energy by integrating the original hydrodynamic equations in 2D spherical coordinates over their respective control volumes. The original edge-centered artificial viscosity in 2D cylindrical geometry is also extended to be capable of capturing shock waves in 2D spherical geometry. Several 2D benchmark cases are provided to verify the scheme. The subsequent construction of the conservative radiation hydrodynamics algorithm is accomplished by the combination of the staggered Lagrangian hydrodynamics scheme and the solution of the RTE in 2D spherical geometry. Several 2D problems are calculated to verify our radiation hydrodynamics algorithm at the end.
References
1.
Shestakov
,
A. I.
,
Prasad
,
M. K.
,
Milovich
,
J. L.
,
Gentile
,
N. A.
,
Painter
,
J. F.
, and
Furnish
,
G.
, 2000
, “
The Radiation-Hydrodynamic ICF3D Code
,” Comput. Methods Appl. Mech. Eng.
,
187
(1–2
), pp. 181
–200
.10.1016/S0045-7825(99)00117-62.
Castor
,
J. I.
, 2004
, Radiation Hydrodynamics
,
Cambridge University Press
,
New York
.3.
Ramis
,
R.
,
Schmalz
,
R. J.
, and
Meyer-ter-Vehn
,
J.
, 1988
, “
Multi-A Computer Code for One-Dimensional Multigroup Radiation Hydrodynamics
,” Comput. Phys. Commun.
,
49
(3
), pp. 475
–505
.10.1016/0010-4655(88)90008-24.
Hodge
,
N. E.
,
Ferencz
,
R. M.
, and
Solberg
,
J. M.
, 2014
, “
Implementation of a Thermomechanical Model for the Simulation of Selective Laser Melting
,” Comput. Mech.
,
54
(1
), pp. 33
–51
.10.1007/s00466-014-1024-25.
Soleimani
,
M.
,
Wriggers
,
P.
,
Rath
,
H.
, and
Stiesch
,
M.
, 2016
, “
Numerical Simulation and Experimental Validation of Biofilm in a Multi-Physics Framework Using an SPH Based Method
,” Comput. Mech.
,
58
(4
), pp. 619
–633
.10.1007/s00466-016-1308-96.
Hirt
,
C. W.
,
Amsden
,
A. A.
, and
Cook
,
J. L.
, 1974
, “
An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds
,” J. Comput. Phys.
,
14
(3
), pp. 227
–253
.10.1016/0021-9991(74)90051-57.
Maire
,
P. H.
, 2009
, “
A High-Order Cell-Centered Lagrangian Scheme for Compressible Fluid Flows in Two-Dimensional Cylindrical Geometry
,” J. Comput. Phys.
,
228
(18
), pp. 6882
–6915
.10.1016/j.jcp.2009.06.0188.
Margolin
,
L.
, and
Shashkov
,
M.
, 1999
, “
Using a Curvilinear Grid to Construct Symmetry-Preserving Discretizations for Lagrangian Gas Dynamics
,” J. Comput. Phys.
,
149
(2
), pp. 389
–417
.10.1006/jcph.1998.61619.
Caramana
,
E. J.
, and
Whalen
,
P. P.
, 1998
, “
Numerical Preservation of Symmetry Properties of Continuum Problems
,” J. Comput. Phys.
,
141
(2
), pp. 174
–198
.10.1006/jcph.1998.591210.
Kong
,
R.
,
Ambrose
,
M.
, and
Spanier
,
J.
, 2008
, “
Efficient, Automated Monte Carlo Methods for Radiation Transport
,” J. Comput. Phys.
,
227
(22
), pp. 9463
–9476
.10.1016/j.jcp.2008.06.03711.
Guo
,
Z.
,
Aber
,
J.
,
Garetz
,
B. A.
, and
Kumar
,
S.
, 2000
, “
Monte Carlo Simulation and Experiments of Pulsed Radiative Transfer
,” J. Quant. Spectrosc. Radiat. Transfer
,
73
, pp. 4411
–4417
.12.
Brooks
,
E. D.
,
Szőke
,
A.
, and
Peterson
,
J. D. L.
, 2006
, “
Piecewise Linear Discretization of Symbolic Implicit Monte Carlo Radiation Transport in the Difference Formulation
,” J. Comput. Phys.
,
220
(1
), pp. 471
–497
.10.1016/j.jcp.2006.07.01413.
Wawrenczuk
,
A.
,
Kuhnert
,
J.
, and
Siedow
,
N.
, 2007
, “
FPM Computations of Glass Cooling With Radiation
,” Comput. Methods Appl. Mech. Eng.
,
196
, pp. 4656
–4671
.10.1016/j.cma.2007.05.02514.
Dedner
,
A.
, and
Vollmoller
,
P.
, 2002
, “
An Adaptive Higher Order Method for Solving the Radiation Transport Equation on Unstructured Grids
,” J. Comput. Phys.
,
178
(2
), pp. 263
–289
.10.1006/jcph.2002.700115.
Carlson
,
B. G.
, and
Lathrop
,
K. D.
, 1968
, Transport Theory D the Method of Discrete Ordinates, Computing Methods in Reactor Physics
,
Gordon and Breach
,
New York
.16.
Fiveland
,
W. A.
, 1984
, “
Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures
,” ASME J. Heat Transfer
,
106
(4
), pp. 699
–706
.10.1115/1.324674117.
Kim
,
I. K.
, and
Kim
,
W. S.
, 2001
, “
A Hybrid Spatial Differencing Scheme for Discrete Ordinates Method in 2D Rectangular Enclosures
,” Int. J. Heat Mass Transfer
,
44
(3
), pp. 575
–586
.10.1016/S0017-9310(00)00114-918.
Akamatsu
,
M.
, and
Guo
,
Z. X.
, 2013
, “
Comparison of Transmitted Pulse Trains Predicted by Duhamel's Superposition Theorem and Direct Pulse Simulation in a 3-D Discrete Ordinates System
,” Numer. Heat Transfer Part B
,
63
(3
), pp. 189
–203
.10.1080/10407790.2013.75126819.
Ragusa
,
J. C.
,
Guermond
,
J. L.
, and
Kanschat
,
G.
, 2012
, “
A Robust SN-DG-Approximation for Radiation Transport in Optically Thick and Diffusive Regimes
,” J. Comput. Phys.
,
231
(4
), pp. 1947
–1962
.10.1016/j.jcp.2011.11.01720.
Ghosh
,
K.
, and
Menon
,
S. V. G.
, 2010
, “
Fully Implicit 1D Radiation Hydrodynamics: Validation and Verification
,” J. Comput. Phys.
,
229
(19
), pp. 7488
–7502
.10.1016/j.jcp.2010.06.03121.
Avila
,
M.
,
Codina
,
R.
, and
Principe
,
J.
, 2011
, “
Spatial Approximation of the Radiation Transport Equation Using a Subgrid-Scale Finite Element Method
,” Comput. Methods Appl. Mech. Eng.
,
200
(5–8
), pp. 425
–438
.10.1016/j.cma.2010.11.00322.
Raithby
,
G. D.
, and
Chui
,
E. H.
, 1990
, “
A Finite Volume Method for Predicting a Radiative Heat Transfer in Enclosures With Participating Media
,” ASME J. Heat Transfer
,
112
(2
), pp. 415
–423
.10.1115/1.291039423.
Chui
,
E. H.
,
Raithby
,
G. D.
, and
Hughes
,
P. M. J.
, 1992
, “
Prediction of Radiative Transfer in Cylindrical Enclosures With the Finite Volume Method
,” AIAA J. Thermophys. Heat Transfer
,
6
(4
), pp. 605
–611
.10.2514/3.1154024.
Chai
,
J. C.
, and
Moder
,
J. P.
, 2000
, “
Radiation Heat Transfer Calculation Using an Angular-Multiblock Procedure
,” Numer. Heat Transfer Part B
,
38
, pp. 1
–13
.25.
Kim
,
M. Y.
, 2008
, “
Assessment of the Axisymmetric Radiative Heat Transfer in a Cylindrical Enclosure With the Finite Volume Method
,” Int. J. Heat Mass Transfer
,
51
(21–22
), pp. 5144
–5153
.10.1016/j.ijheatmasstransfer.2008.03.01226.
Hunter
,
B.
, and
Guo
,
Z. X.
, 2011
, “
Comparison of the Discrete-Ordinates Method and the Finite Volume Method for Steady-State and Ultrafast Radiative Transfer Analysis in Cylindrical Coordinates
,” Numer. Heat Transfer Part B
,
59
(5
), pp. 339
–359
.10.1080/10407790.2011.57271927.
Giani
,
S.
, and
Seaid
,
M.
, 2016
, “
hp-Adaptive Discontinuous Galerkin Methods for Simplified PN Approximations of Frequency-Dependent Radiative Transfer
,” Comput. Methods Appl. Mech. Eng.
,
301
, pp. 52
–79
.10.1016/j.cma.2015.12.01328.
Olson
,
G. L.
, 2009
, “
Second-Order Time Evolution of PN Equations for Radiation Transport
,” J. Comput. Phys.
,
228
(8
), pp. 3072
–3083
.10.1016/j.jcp.2009.01.01229.
Edwards
,
J. D.
,
Morel
,
J. E.
, and
Knoll
,
D. A.
, 2011
, “
Nonlinear Variants of the TR/BDF2 Method for Thermal Radiative Diffusion
,” J. Comput. Phys.
,
230
(4
), pp. 1198
–1214
.10.1016/j.jcp.2010.10.03530.
Argyris
,
J.
,
Tenek
,
L.
, and
Oberg
,
F.
, 1995
, “
A Multilayer Composite Triangular Element for Steady-State Conduction/Convection/Radiation Heat Transfer in Complex Shells
,” Comput. Methods Appl. Mech. Eng.
,
120
(3–4
), pp. 271
–301
.10.1016/0045-7825(94)00775-I31.
Shen
,
W.
, and
Han
,
S.
, 2002
, “
Numerical Solution of Two-Dimensional Axisymmetric Hyperbolic Heat Conduction
,” Comput. Mech.
,
29
(2
), pp. 122
–128
.10.1007/s00466-002-0321-332.
Argyris
,
J.
, and
Szimmat
,
J.
, 1992
, “
An Analysis of Temperature Radiation Interchange Problems
,” Comput. Methods Appl. Mech. Eng.
,
94
(2
), pp. 155
–180
.10.1016/0045-7825(92)90145-A33.
Bialecki
,
R. A.
,
Burczynski
,
T.
,
Dlugosz
,
A.
,
Kus
,
W.
, and
Ostrowski
,
Z.
, 2005
, “
Evolutionary Shape Optimization of Thermoelastic Bodies Exchanging Heat by Convection and Radiation
,” Comput. Methods Appl. Mech. Eng.
,
194
, pp. 1839
–1859
.10.1016/j.cma.2004.07.00434.
Luo
,
X. P.
,
Wang
,
C. H.
,
Zhang
,
Y.
,
Yi
,
H. L.
, and
Tan
,
H. P.
, 2018
, “
Multiscale Solutions of Radiative Heat Transfer by the Discrete Unified Gas Kinetic Scheme
,” Phys. Rev. E
,
97
(6
), pp. 1
–14
.10.1103/PhysRevE.97.06330235.
Sun
,
W. J.
,
Jiang
,
S.
, and
Xu
,
K.
, 2015
, “
An Asymptotic Preserving Unified Gas Kinetic Scheme for Gray Radiative Transfer Equations
,” J. Comput. Phys.
,
285
, pp. 265
–279
.10.1016/j.jcp.2015.01.00836.
Sun
,
W. J.
,
Jiang
,
S.
,
Xu
,
K.
, and
Li
,
S.
, 2015
, “
An Asymptotic Preserving Unified Gas Kinetic Scheme for Frequency-Dependent Radiative Transfer Equations
,” J. Comput. Phys.
,
302
, pp. 222
–238
.10.1016/j.jcp.2015.09.00237.
Kim
,
M. Y.
,
Baek
,
S. W.
, and
Lee
,
C. Y.
, 2008
, “
Prediction of Radiative Heat Transfer Between Two Concentric Spherical Enclosures With the Finite Volume Method
,” Int. J. Heat Mass Transfer
,
51
(19–20
), pp. 4820
–4828
.10.1016/j.ijheatmasstransfer.2008.02.01638.
Vachal
,
P.
, and
Wendroff
,
B.
, 2016
, “
On Preservation of Symmetry in R-Z Staggered Lagrangian Schemes
,” J. Comput. Phys.
,
307
, pp. 496
–507
.10.1016/j.jcp.2015.11.06339.
Cheng
,
J.
, and
Shu
,
C. W.
, 2014
, “
Second Order Symmetry-Preserving Conservative Lagrangian Scheme for Compressible Euler Equations in Two-Dimensional Cylindrical Coordinates
,” J. Comput. Phys.
,
272
, pp. 245
–265
.10.1016/j.jcp.2014.04.03140.
Darbandi
,
M.
, and
Schneider
,
G. E.
, 2000
, “
Performance of an Analogy-Based All-Speed Procedure Without Any Explicit Damping
,” Comput. Mech.
,
26
(5
), pp. 459
–469
.10.1007/s00466000019441.
Scovazzi
,
G.
,
Shadid
,
J. N.
,
Love
,
E.
, and
Rider
,
W. J.
, 2010
, “
A Conservative Nodal Variational Multiscale Method for Lagrangian Shock Hydrodynamics
,” Comput. Methods Appl. Mech. Eng.
,
199
(49–52
), pp. 3059
–3100
.10.1016/j.cma.2010.03.02742.
Ramis
,
R.
,
Meyer-ter-Vehn
,
J.
, and
Ramírez
,
J.
, 2009
, “
MULTI2D-A Computer Code for Two-Dimensional Radiation Hydrodynamics
,” Comput. Phys. Commun.
,
180
(6
), pp. 977
–994
.10.1016/j.cpc.2008.12.03343.
Xia
,
G. H.
,
Zhao
,
Y.
,
Yeo
,
J. H.
, and
Lv
,
X.
, 2007
, “
A 3D Implicit Unstructured-Grid Finite Volume Method for Structural Dynamics
,” Comput. Mech.
,
40
(2
), pp. 299
–312
.10.1007/s00466-006-0100-744.
Zhao
,
Y.
,
Tai
,
J.
, and
Ahmed
,
F.
, 2002
, “
Simulation of Micro Flows With Moving Boundaries Using High-Order Upwind FV Method on Unstructured Grids
,” Comput. Mech.
,
28
(1
), pp. 66
–75
.10.1007/s00466-001-0271-145.
Markesteijn
,
A. P.
,
Karabasov
,
S. A.
,
Glotov
,
V. Y.
, and
Goloviznin
,
V. M.
, 2014
, “
A New Non-Linear Two-Time-Level Central Leapfrog Scheme in Staggered Conservation-Flux Variables for Fluctuating Hydrodynamics Equations With GPU Implementation
,” Comput. Methods Appl. Mech. Eng.
,
281
, pp. 29
–53
.10.1016/j.cma.2014.07.02746.
Caramana
,
E. J.
,
Burton
,
D. E.
,
Shashkov
,
M. J.
, and
Whalen
,
P. P.
, 1998
, “
The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy
,” J. Comput. Phys.
,
146
(1
), pp. 227
–262
.10.1006/jcph.1998.602947.
Caramana
,
E. J.
,
Shashkov
,
M. J.
, and
Whalen
,
P. P.
, 1998
, “
Formulations of Artificial Viscosity for Multi-Dimensional Shock Wave Computations
,” J. Comput. Phys.
,
144
(1
), pp. 70
–97
.10.1006/jcph.1998.598948.
Zeldovich
,
Y. B.
, and
Raizer
,
Y. P.
, 1966
, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena
, Vols. I and II,
Academic Press
,
New York
.49.
Pomraning
,
G. C.
, 1973
, The Equations of Radiation Hydrodynamics
,
Pergamon Press
,
New York
.50.
Mihalas
,
D.
, and
Mihalas
,
B. W.
, 1984
, Foundations of Radiation Hydrodynamics
,
Oxford University Press
,
New York
.51.
Asllanaj
,
F.
,
Feldheim
,
V.
, and
Lybaert
,
P.
, 2007
, “
Solution of Radiative Heat Transfer in 2-D Geometries by a Modified Finite Volume Method Based on a Cell Vertex Scheme Using Unstructured Triangular Meshes
,” Numer. Heat Transfer Part B
,
51
(2
), pp. 97
–119
.10.1080/1040779060076280552.
Kim
,
M. Y.
,
Baek
,
S. W.
, and
Park
,
J. H.
, 2001
, “
Unstructured Finite Volume Method for Radiative Heat Transfer in a Complex Two-Dimensional Geometry With Obstacles
,” Numer. Heat Transfer Part B
,
39
, pp. 617
–635
.10.1080/1040779015203485453.
Caramana
,
E. J.
, and
Shashkov
,
M. J.
, 1998
, “
Elimination of Artificial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pressures
,” J. Comput. Phys.
,
142
(2
), pp. 521
–561
.10.1006/jcph.1998.595254.
Loubere
,
R.
,
Shashkov
,
M. J.
, and
Wendroff
,
B.
, 2008
, “
Volume Consistency in a Staggered Grid Lagrangian Hydrodynamics Scheme
,” J. Comput. Phys.
,
227
, pp. 3731
–3737
.10.1016/j.jcp.2008.01.00655.
Lin
,
Z. W.
,
Jiang
,
S. E.
,
Wu
,
S. C.
, and
Kuang
,
L. Y.
, 2011
, “
A Local Rezoning and Remapping Method for Unstructured Mesh
,” Comput. Phys. Commun.
,
182
(6
), pp. 1361
–1376
.10.1016/j.cpc.2010.11.03456.
Lin
,
Z. W.
,
Jiang
,
S. E.
,
Zhang
,
L.
,
Kuang
,
L. Y.
, and
Li
,
H.
, 2018
, “
Maxis-A Rezoning and Remapping Code in Two Dimensional Cylindrical Geometry
,” Comput. Phys. Commun.
,
227
, pp. 148
–149
.10.1016/j.cpc.2018.02.00657.
Campbell
,
J. C.
, and
Shashkov
,
M. J.
, 2001
, “
A Tensor Artificial Viscosity Using a Mimetic Finite Difference Algorithm
,” J. Comput. Phys.
,
172
(2
), pp. 739
–765
.10.1006/jcph.2001.685658.
Vachal
,
P.
, and
Wendroff
,
B.
, 2014
, “
A Symmetry Preserving Dissipative Artificial Viscosity in an R-Z Staggered Lagrangian Discretization
,” J. Comput. Phys.
,
258
, pp. 118
–136
.10.1016/j.jcp.2013.10.03659.
Kuropatenko
,
V. F.
, 1967
, Difference Methods for Solutions of Problems of Mathematical Physics
,
American Mathematical Society
,
Providence, RI
, p. 116
.60.
Lindl
,
J. D.
,
Amendt
,
P.
,
Berger
,
R. L.
,
Glendinning
,
S. G.
,
Glenzer
,
S. H.
,
Haan
,
S. W.
,
Kauffman
,
R. L.
,
Landen
,
O. L.
, and
Suter
,
L. J.
, 2004
, “
The Physics Basis for Ignition Using Indirect-Drive Targets on the National Ignition Facility
,” Phys. Plasmas
,
11
(2
), pp. 339
–491
.10.1063/1.1578638Copyright © 2019 by ASME
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