Toward a Simple, Accurate Lagrangian Hydrocode

203 Pages·2015·12.33 MB·English

Toward a Simple, Accurate Lagrangian Hydrocode
by
Tyler B. Lung
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Aerospace Engineering)
in The University of Michigan
2015
Doctoral Committee:
Professor Philip L. Roe, Chair
Assistant Professor Karthik Duraisamy
Associate Professor Krzysztof J. Fidkowski
Professor William R. Martin
(cid:13)c Tyler B. Lung 2015
All Rights Reserved
“For my thoughts are not your thoughts, neither are your ways my ways,” declares
the Lord. “As the heavens are higher than the earth, so are my ways higher than
your ways and my thoughts than your thoughts.” - Isaiah 55:8-9
ii
ACKNOWLEDGEMENTS
I have encountered many wonderful people during my graduate studies and they
have made large contributions to my personal and professional development. To
begin, I want to thank Professor Roe for agreeing to take me on as a student and for
serving as a superb teacher, a patient mentor, a bright colleague, and a good friend
over the past years. I have learned many things, been offered many opportunities,
and traveled to many places that would not have been possible without him. I will
always be grateful for his investment in me. I am also indebted to the other members
of my dissertation committee, Karthik Duraisamy, Chris Fidkowski, and Bill Martin
for all of the useful feedback and constructive criticism they have provided.
Many individuals at Los Alamos National Laboratory deserve my thanks. Don
Burton originally recruited Professor Roe to work on Lagrangian hydrodynamics,
who in turn recruited me, and he has provided much insight and support since.
Scott Runnels and Nathaniel Morgan have each spent large amounts of their time
and talent working with me through the Computational Physics Student Summer
Workshop. Their mentoring has contributed immensely to this project and their
hospitality contributed to two excellent summers at Los Alamos. Rob Lowrie also
provided helpful critiques and encouragement, which were very welcome.
Andy Barlow from the Atomic Weapons Establishment in the United Kingdom
has provided significant feedback on this work. His expertise and commentary was
invaluable. He was also instrumental in securing travel funding for me to visit the
Centre for Scientific Computing at the University of Cambridge and attend a con-
iii
ference in Barcelona, Spain. Additionally, my graduate studies would not have been
possible without the financial support I received at various times from Los Alamos
National Laboratory, a Dobbins Fellowship, and the Department of Defense High
Performance Computing Modernization Program via a National Defense Science and
Engineering Graduate Fellowship.
My fellow students here at the University of Michigan have made graduate school
much more enjoyable than it would have otherwise been. Kyle Ding, Brad Maeng,
Rohan Morajkar, and Kevin Neitzel made the process of preparing for Preliminary
Examinations bearable and I have learned much from each of them. I also want to
thank Johann Dahm, Steve Kast, Loc Khieu, and Yimin Lou for many interesting
discussions.
Nothing that I have achieved to this point in my life would have been possible
without my parents, Darrin and Jill Lung. I am thankful beyond words for their
teaching, love, and encouragement. The support I have received from the rest of
my family has also been a blessing: Thank you Greg, Denise, Zack, Jessica, Caitlin,
Kurt, Alyssa, Katelyn, Noelle, and Bethany. Finally, I want to thank my beautiful
wife Miranda. I imagine that it is not easy being married to a Ph.D. student and I
am grateful for her patience and love over the past four years. This thesis would not
have been completed without her support.
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . xix
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
CHAPTER
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Foundations of Lagrangian Hydrocodes . . . . . . . . . . . . 2
1.2 An Overview of Staggered-grid Hydrodynamics Methods . . . 5
1.3 A Persistent Failing: Spurious Mesh Movement . . . . . . . . 7
1.4 Recent Progress: Cell-centered Hydrodynamics Methods . . . 9
1.5 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 A New Proposal . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Research Strategy . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Broader Contributions . . . . . . . . . . . . . . . . . . . . . . 19
II. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 The Euler Equations in the Lagrangian Frame . . . . . . . . 23
2.2 Information Propagation . . . . . . . . . . . . . . . . . . . . 25
2.3 Vorticity Transport . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The Acoustic Equations . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Relationship with the Lagrangian Euler Equations . 29
v
III. Second-order Methods for Acoustics . . . . . . . . . . . . . . . 33
3.1 Notation and Test Problems . . . . . . . . . . . . . . . . . . 35
3.2 Vorticity Control . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Parameterization of the Lax-Wendroff Family . . . . 39
3.3 Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
IV. Third-order Methods for Acoustics . . . . . . . . . . . . . . . . 51
4.1 Third-order Accuracy . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Third-order Vorticity Preserving Methods for Acoustics . . . 55
4.2.1 Dispersion Analysis . . . . . . . . . . . . . . . . . . 56
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
V. Temporal Flux Limiting . . . . . . . . . . . . . . . . . . . . . . . 68
5.1 Limiting Review . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 A Vorticity Preserving Flux-corrected Transport Scheme . . . 71
5.3 Flux Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
VI. First-order Methods for Lagrangian Hydrodynamics: Part I 91
6.1 From Acoustics to a Simple Lagrangian Method (SLaM) . . . 92
6.1.1 Initial Flux Interpolation . . . . . . . . . . . . . . . 93
6.1.2 Characteristic Cell Size . . . . . . . . . . . . . . . . 94
6.1.3 Q-parameter Selection . . . . . . . . . . . . . . . . . 95
6.1.4 Differentiation Operators . . . . . . . . . . . . . . . 95
6.1.5 Mesh Movement and Flux Integration . . . . . . . . 96
6.1.6 Time Step Selection . . . . . . . . . . . . . . . . . . 97
6.1.7 SLaM-A Update Procedure . . . . . . . . . . . . . . 99
6.2 SLaM-A Numerical Results . . . . . . . . . . . . . . . . . . . 99
6.2.1 Convergence Analysis . . . . . . . . . . . . . . . . . 100
6.2.2 Sedov . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.3 Noh . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.4 Saltzman . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
VII. First-order Methods for Lagrangian Hydrodynamics: Part II 113
7.1 Face Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.1.1 Q-parameter and Time Step Selection . . . . . . . . 114
vi
7.1.2 Flux Integration . . . . . . . . . . . . . . . . . . . . 115
7.2 Flux Formula Robustness . . . . . . . . . . . . . . . . . . . . 116
7.2.1 Control Volume for Driver Estimation . . . . . . . . 117
7.2.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . 118
7.3 SLaM-B Update Procedure . . . . . . . . . . . . . . . . . . . 118
7.4 SLaM-B Numerical Results . . . . . . . . . . . . . . . . . . . 119
7.4.1 Convergence . . . . . . . . . . . . . . . . . . . . . . 119
7.4.2 Sedov . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.4.3 The Riemann Solver Pitfall . . . . . . . . . . . . . . 120
7.4.4 Noh . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.4.5 Triple Point . . . . . . . . . . . . . . . . . . . . . . 124
7.4.6 Saltzman . . . . . . . . . . . . . . . . . . . . . . . . 128
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
VIII. A Second-order Method for Lagrangian Hydrodynamics . . . 133
8.1 Second-order Accuracy . . . . . . . . . . . . . . . . . . . . . 134
8.2 General Limiting Approach . . . . . . . . . . . . . . . . . . . 134
8.3 SLaM-TFL Update Procedure . . . . . . . . . . . . . . . . . 138
8.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.1 Sedov . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.2 Noh . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4.3 Limited Convergence . . . . . . . . . . . . . . . . . 142
8.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
IX. Concluding Remarks and Future Work . . . . . . . . . . . . . . 147
9.1 Suggestions for Future Work . . . . . . . . . . . . . . . . . . 149
9.1.1 Lax-Wendroff Methods . . . . . . . . . . . . . . . . 149
9.1.2 Limiting . . . . . . . . . . . . . . . . . . . . . . . . 149
9.1.3 SLaM Method . . . . . . . . . . . . . . . . . . . . . 150
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
vii
LIST OF FIGURES
Figure
1.1 StencilsrepresentativeofthoseusedforSGHmethods(left)andCCH
methods (right) are shown. . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Riemann problem from gas dynamics is an initial value prob-
lem in which the unsteady interaction of two uniform, discontinuous
statesisdetermined. Riemannsolversareusedtocomputefacefluxes
in Godunov-type methods. . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Mesh imprinting, which is accompanied by spurious vorticity, de-
stroys a Lagrangian computation that should possess perfect radial
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Shown here is a conceptual representation of Maire’s CCH method.
The nodal solver computes the half-face pressures (p ) and nodal
hf
velocity (V ) in a single step. . . . . . . . . . . . . . . . . . . . . 11
node
1.5 An illustration to reinforce the connection between the RR scheme
and its Lagrangian analog. The primary difference in the case of
a Lagrangian algorithm is the need to move the mesh. Note that
unique nodal velocities can be defined from the nodal fluxes. . . . . 15
1.6 A graphical representation of Roe’s proposed CCH method. . . . . 17
3.1 The initial pressure distributions for the discontinuous test problem
(a) and smooth test problem (b) are shown. . . . . . . . . . . . . . 36
3.2 Compact vorticity contours predicted by LW (a) and RR (b) for the
discontinuous test problem are plotted at t = 3. The computations
were run with ν = 0.6. The LW method generates spurious vorticity,
but the Rotated Richtmyer (RR) method maintains zero vorticity to
double precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
viii
3.3 LW and RR comparison: Pressure (a) and velocity magnitude (b)
profiles for the discontinuous test problem are plotted at t = 3. The
computations were run with ν = 0.6. . . . . . . . . . . . . . . . . . 39
3.4 LW and RR comparison: Pressure (a) and velocity magnitude (b)
profiles for the Gaussian test problem are plotted at t = 3. The
computations were run with ν = 0.6. . . . . . . . . . . . . . . . . . 40
3.5 An example parameterization for the approximation of the second
derivative of a generic state variable U with respect to x is shown. . 41
3.6 RR and VPLW1 comparison: Pressure (a) and velocity magnitude
(b) profiles for the discontinuous test problem are plotted at t = 3.
The computations were run with ν = 0.6. . . . . . . . . . . . . . . 46
3.7 RR and VPLW2 comparison: Pressure (a) and velocity magnitude
(b) profiles for the discontinuous test problem are plotted at t = 3.
The computations were run with ν = 0.6. . . . . . . . . . . . . . . 47
3.8 The phase and damping relationships for the VPLW2 method are
plotted for the propagation directions ψ = 0, ψ = π/4, and ψ = π/8
when ν = 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 The phase and damping relationships for the VPLW2 method are
plotted for the propagation directions ψ = 0, ψ = π/4, and ψ = π/8
when ν = 0.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.10 Results from the VPLW2 method for the discontinuous test problem
show that the spurious features in the solution become much more
severe when the CFL number is lowered to ν = 0.15. . . . . . . . . . 50
4.1 ItissomewhatdisappointingthatnoneoftheVPFCTO3methodsare
optimally stable. It is evident here that the θ = π/8 wave traveling
with direction ψ = π/4 will be unstable by ν ≈ 0.8 regardless of the
choice for q . The functions q = ν, q = ν2, and q = 1 are plotted
C C C C
for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The amplification (top) and phase (bottom) relationships for the
propagation directions ψ = 0, ψ = π/8, and ψ = π/4 are plotted for
the second-order VPLW2 method and the VPFCTO3-FUP method
when ν = 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
ix