1
PARALLEL COMPUTATIONS OF 3D UNSTEADY COMPRESSIBLE
EULER EQUATIONS WITH STRUCTURAL COUPLING
Master’s Candidate
Zhenyin Li
Advisor: Dr. H. U. Akay
Department of Mechanical Engineering
Computational Fluid Dynamics Laboratory
Indiana University Purdue University Indianapolis
July 19, 2002
3D Unsteady Compressible Euler Equations with Structural Coupling
Outline

Introduction to Fluid-Structure Coupling

Fluid-Structure Coupling Procedure

Computational Fluid Dynamics Solver – USER3D

Computational Structural Dynamics Solver– SAP4

Test Cases

Conclusions and Recommendations

Acknowledgements
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
2/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Introduction to Aeroelasticiy



“Aeroelasticity is the phenomenon which exhibits appreciable
reciprocal interactions (static or dynamic) between
aerodynamic forces and the deformations induced in the
structure of a flying vehicle, its control mechanisms, or its
propulsion system.” Bisplinghoff (1975)
Two major concerns in aeroelasticity are stability and
response problem.
Experiments and computer simulations are two basic ways to
reveal the characteristic of various phenomena in
aeroelasticity study.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
3/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Studies done in this research




Develop a procedure based coupling of on independent CFD
(Computational Fluid Dynamics and CSD (Computational
Structural Dynamics) solvers to resolve static and dynamic
aeroelasticity problems.
The developed procedure was demonstrated by AGARD
wing 445.6.
A dual zone mesh movement method developed for large
mesh movements when solving unsteady aerodynamic
problems.
Parallel computation performance was studied.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
4/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM

A basic procedure to obtain an aeroelastic solution includes
following steps:
1. Get pressure on CFD mesh nodes from flow calculation
2. Pass the load information to CSD domain
3. Calculate nodal displacements with CSD code
4. Feedback the structure deformation to CFD domain
5. Deform the CFD mesh
6. Repeat steps 1 through 5
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
5/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

Mesh-based Parallel Code Coupling Interface (MPCCI), is
used to exchange information between CFD and CSD codes
and administer both in-code and out of code communications
Process I
Process II
CFD
CSD
fluid solver
structure solver
Application
Interface
Application
Interface
MPCCI
MPCCI Configuration
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
6/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

The current version of MPCCI works well with Message
Passing Interface (MPI)-based parallel as well as serial
computing programs.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
7/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

A global communication ID (GID) is assigned to each of the
processes involved in the coupled computation, and a local
communication ID (LID) is assigned to the processes of the
current code.
MPCCI Control Process
GID=0 LID=N/A
CODE II Process 1
GID=i+1 LID=0
CODE I: Process 1
GID=1 LID=0
CODE I: Process 2
GID=2 LID=1
MPCCI
CODE II Process 2
GID=i+2 LID=1
CODE II Process j
GID=i+j+1 LID=j-1
CODE I: Process i
GID=i LID=i-1
MPCCI communications ID settings
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
8/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

Any CSD/CFD code must define its coupling region at the initial
stage. The coupling regions do not need to be identical in either
size of the region or the density of the elements.
Fluid Model
Solid Model
MPCCI
Non-matching meshes
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
9/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

Information Exchange : Pressure and displacements
need to be exchanged during the coupling process.
Qt  (1  u )(1  v)Q1
Qt  Q1u  Q2 v  Q3 w
 u (1  v)Q2  uvQ3  (1  u )vQ
Q3
Q4
Q2
Q3
Q1
u
Q1
w
Q2
v
w
u
v
Qt
Qt
Triangular element interpolations
u
v
Quadrilateral element interpolations
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
10/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

Exchanging Quantities
Virtual CSD
Surface Mesh
Mid-surface
Structural Mesh
Fu
Mc
CFD surface Mesh
Match Virtual CSD
Surface Mesh
Real Surface
Central Surface
Fc
Fb Central surface transformations
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
11/57
3D Unsteady Compressible Euler Equations with Structural Coupling
AEROELASTIC COUPLING ALGORITHM (Cont.)

Time Integrations of Coupled System
 Here, the same ∆t is used for fluid and structure
Fluid
Solid
Pn-1
Δt
Step n-1
Un
Step n
Pn
Δt
Un+1
Step n+1
Time integration
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
12/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Construct CFD Mesh
Steady State Solution for
rigid body
Construct CSD virtual
surface mesh
Calculate new CFD flow field
Put pressure on
virtual surface
Extract fluid surface mesh
MPCCI
Calculate node pressure
on surface mesh
Put the displacements on
surface mesh
Calculate dynamic forces on
CSD virtual surface mesh
Transform the dynamic forces to
structure mesh and solve
equilibrium equation
MPCCI
Deform the CFD mesh
Map the displacements to
CSD virtual surface mesh
Finish
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
13/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D

Background of USER3D
• A parallel finite-volume based unstructured Euler solver;
• Serial version of User3D was developed by Oktay (1994) ;
• Parallel version of User3D was developed at CFD Laboratory
at IUPUI (2000);
• This solver was validated in previous studies.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
14/57


{Q}dV3D
  FUnsteady
 nˆdS  0

t 

Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D (Cont.)

Governing Equations for USER3D
The Arbitrary Lagrangian-Eulerian formulation of the threedimensional time-dependent inviscid fluid-flow equations is
expressed in the following form:
 Where
Q is the vector of conserved flow variables
 F is the normal component of the convective flux vector
 N is the unit normal vector to the boundary
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
15/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D (Cont.)

The time integration employed in the flow solver is the cellcentered finite volume formulation. The volume-averaged
values are adopted to represent the flow variables.
n
n


V

V
{Q}n
   F (Q)  n dS  {Q}n
t
t


An implicit time integration scheme is used to solve flow field
at each time step.
n

V
[ A]n {Q}n  {R}n  {Q}n
t
n
n
V
{

R
}
[ A]n 
[I ] 
t
{Q}n


n
{R}    F (Q ) n  ndS

Zhenyin Li, Master’s Thesis Defense, July 19, 2002
16/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D (Cont.)

Mesh-Movement Algorithm
The mechanism of this method is that any two neighboring nodes in the
mesh are connected by a spring and the spring stiffness is inversely
proportional to the distance of the two nodes.
Stiffness K
km  [( x j  xi )2  ( y j  yi )2  ( z j  zi )2 ]1/ 2
Displacement
xi
n 1

 k x ,
k
m
y i
n 1
m
Zhenyin Li, Master’s Thesis Defense, July 19, 2002

 k y ,
k
m
m
z i
n 1

 k z
k
m
m
17/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D (Cont.)

Limitation of the current scheme
• The spring technology needs a large amount of CPU time and memory;
• The small size cells near the inner boundary can not afford large amplitude motion;

A simple dual-zone smoothing approach is proposed to improve
the performance of the current spring system
II
Region I: The inner zone is moving
rigidly with the body ;
I
Region II: The outer zone is
deformed by general mesh
deformation method .
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
18/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Fluid Dynamics Solver - USER3D (Cont.)

Geometric Conservation Law :
The geometry conservation equation is required to solve
simultaneously with other conservation equations.

t
 
 dV   Ws  ndS


where Ws denotes the local velocity on the boundary surface S
The cell volume can be calculated by
Vi n1  Vi n  t  (xtnm1 Ax  ytnm1 Ay  ztnm1 Az )
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
20/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Structural Dynamics Solver – SAP4

The finite element discrete aeroelasticity element
equation for a structural system can be written as:
M (e) {q}(e)  C (e) q(e)  K (e) q(e)  R(e)
[M], [C] and [K] are system mass, damping and stiffness matrix

For static analysis, equation can be rewritten as:
K q  R

For dynamic analysis, equation can be rewritten as:
M {q}  Cq K q  R(t )
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
21/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Structural Dynamics Solver – SAP4 (Cont.)

Mode superposition method
1. Get the generalized eigenvalue solution
[ K ]{}   2 [ M ]{}
2. Use first n modes to simulate structural response
{q(t )}  [{1},{2 },......,{n }]{ X (t )}  [ A]{ X (t )}
3. Get the generalized displacement solution
X  2  X   2 X  F *
i
i i
i
i
i
i
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
22/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Computational Structural Dynamics Solver – SAP4 (Cont.)

A Newmark-family of time integration scheme is used
to obtain the solution at the (n+1) time step:
2
(1  2 )
1
*
*
[
M
]

[
C
]

(
 2    )[ K ]* ]{ X }n 
2
t
t
2
1
(1  2 )
1
*
*
*
n 1
[
[
M
]

[
C
]

(




)[
K
]
]{
X
}
 {F } *
t 2
t
2
[ M ]*{ X }n 1  [
α
β
Galerkin method
3/2
4/5
Always
The backward
difference method
3/2
1
Always
The constant
acceleration method
1/2
1/4
The linear
acceleration method
1/2
1/6
Stable
Condition
Initial Condition: { X }t 0  { X 0 }
{
dX
}t 0  { X 0 }
dt
For Flutter Analysis
Always
t 
Either
2
3
{ X 0 }  0 or { X 0 }  0
{ X 0 }  0
{ X 0 }  0
i
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
23/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES

Aeroelastic Research Wing
(AGARD Wing 445.6)
1.208
ft
5.2 ft
45O
1.833
ft
AGARD wing 445.6 panel dimensions
The CFD grid consists of 147,547 cells and 26,228 nodes. The
CFD wing surface has 2020 elements and 1077 nodes
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
24/57
3D Unsteady Compressible Euler Equations with Structural Coupling
 In
the present application:
 n processors are used for CFD solution
 One processor for CSD solution
 One processor for communication management with
MPCCI
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
25/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Modal Analysis of Wing 445.6
Table 5.2 Modal frequencies of AGARD wing 445.6
SAP4
400 eles.
SAP4
200 eles.
SAP4
100 eles.
ANSYS
100 eles
Experim
ent
f1
9.60
9.60
9.60
10.85
9.59
f2
39.77
39.81
39.86
44.57
38.16
f3
50.88
50.20
48.30
56.88
48.35
f4
95.37
95.40
95.01
109.10
91.55
Comparison of AGARD wing 445.6
modal frequencies
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
26/57
3D Unsteady Compressible Euler Equations with Structural Coupling
SAP4 Modal Shape
TEST CASES (Cont.)
MODE 1
MODE 3
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
MODE 2
MODE 4
27/57
3D Unsteady Compressible Euler Equations with Structural Coupling
ANSYS Modal Shape
TEST CASES (Cont.)
Mode 1
Mode 3
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
Mode 2
Mode 4
28/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Steady Solution of the Rigid Body
Steady State Transonic Flow at M∞ = 0.96 and M∞ = 1.141
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
29/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
30/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
31/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Rigid Body Result
Static Aeroelastic
Analysis at Mach = 0.8
1. The coupling iteration starts from
the steady-state solution of the rigid
body.
2. In practice, a load factor is used to
control the force loaded on the
structural system.
3. An alternate approach also
performed here is using dynamic
analysis to simulate steady case.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
32/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
The tip deflection at the trailing
edge was computed to be 0.40 inch
which is very close to 0.39 inch
from MDICE
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
33/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Deformed
Mesh
Undeformed
Mesh
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
34/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
35/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Dynamic Aeroelastic Analysis
Mach = 0.8, AOA =1.0 degree
In this section, the previous
steady-state solution is used as a
sudden load at time zero. The
wing motion is entirely
determined by the structural
response. The time increment is
1.0e -4
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
36/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
37/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Deformed
Mesh
Undeformed
Mesh
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
38/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Flutter Analysis
Dynamic instability
where-by the system
extracts energy from the
free stream flow
producing a divergent
response. The computed
flutter characteristics are
presented in terms of
velocity index Vf which is
defined as
Stable
Neutral
V f  U  / b 
Unstable
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
39/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
 Mach=0.957, Vf = 0.349 , U∞=14400 inch/s
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
40/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Mach=0.957, Vf = 0.250 , U∞=10200 inch/s
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
41/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Mach=0.957, Vf = 0.262 , U∞=10800 inch/s
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
42/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Comparison of Results
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
43/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Initial Velocity Effect
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
44/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Parallel Aerodynamic
Studies
A standard research configuration
for missile geometries, is studied
under forced pitching conditions.
The computational mesh used
consists of 144,216 nodes and
706,105 cells, 24 Blocks
The steady case was performed
with M∞ = 1.58, angle of attack
(AOA) = 0.0.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
45/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
46/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

This case is the basic finner performing a sinusoidally
pitching motion about its center of gravity. The angle of
attack varies as:
 (t )   m   p sin(  t )
For this test case, the reduced frequency k = 2.53165, freestream
Mach number M∞ = 1.58, the mean angle of pitching αm = 0.0 degree
and the amplitude of pitching is 10 degrees. The results were obtained
using 2000 steps per cycle of the motion. The time increment of 2e-4
was used
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
47/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
48/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
49/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)

Parallel Efficiency Study
The parallel efficiency study
performed here is based on
Indiana University’s IBM SP
clusters and Compaq Linux
clusters.
The speedup is defined as
S p  T1 / T p
Efficiency E is defined as
UNIX
LINUX
Model
IBM RISC System
/ SP6000
POWER3+
Thin Node
Compaq ProLiant
1850R rackmounted
compute
nodes
CPU
(Each Node)
4 CPU, 375
MHz clock cycle
Memory
2GB
256 MB
Cache
8MB
512 KB
Network
10/100Mb
"Fast" Ethernet
(100 TX)
E  100  S p / p
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
Dual Intel 400
MHz Pentium
II processors
50/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
144,216 nodes and 706,105 cells
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
51/57
3D Unsteady Compressible Euler Equations with Structural Coupling
TEST CASES (Cont.)
144,216 nodes and 706,105 cells
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
52/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Conclusions

A loosely coupled procedure is developed by using parallel Euler
equations solver USER3D and finite element structural solver
SAP4. The advantage of current method is to provide a flexible
and easy implementation for coupling CFD and CSD codes
without a large amount of works in existing codes.
 In steady aeroelastic problems, due to the limitation of mesh
deformation scheme, a load factor was used to increase the load
gradually. The results are quite consistent with other researcher’s
work. Using dynamic aeroelastic solutions with damping the
results of static problem is also validated.
 Dynamic aeroelastic problems were solved using the coupled
CFD-CSD procedure. Significant aeroelastic effects were
observed in this study.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
53/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Conclusions (Cont.)



Flutter analysis was implemented by choosing initial perturbation
of the structural system and examining whether the initial
perturbation will decay, grow or maintain neutral conditions to
determine the flutter conditions. The results compared well with
previous works and experimental results.
A dual-zone dynamic mesh system was successfully employed to
solve unsteady aerodynamic problems. High computational
efficiency was obtained.
Both steady-state solution scheme and unsteady solution showed
good speedup and efficiency for multi-block cases.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
54/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Future Works



The present dynamic grid scheme can prevent two nodes colliding with
each other. And the dual-zone scheme can only deal with known
motion. This scheme works well with small motion or large simple
motion such as sinusoidal motion. Problems will occur when solving
aeroelastic problems with large motion.
Time increment in the present scheme is same on both CFD and CSD
solvers. But, CSD solver usually requires larger time increments than
the CFD solver. In the future work, the effect of time sub-cycle should be
studied. Another problem in current scheme is only that only the CFD
code is a parallel code. In the future study, a parallel CSD code may be
required to improve the computational efficiency, especially for large
structures such as a complete aircrafts or missiles.
The information exchange between CFD and CSD solvers is based on
bi-linear interpolations. Although its accuracy is enough for the current
problem, a more complex interpolation scheme maybe required for
future applications.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
55/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Future Works (Cont.)


One remaining problem in this procedure is that MPCCI requires that
each sub process must define its own coupling region, but some CFD
blocks which are partitioned by GD do not include such coupling regions.
As the result, the current procedure may be limited to a few blocks which
depend on how GD divides a grid.
Although reasonable results are obtained for flutter analysis, there are
still some differences between the present results and experiments.
One possible way to improve the accuracy is to refine the mesh to get
more accurate fluid solutions. Another way to improve the accuracy is
by improving the present bilinear interpolation scheme to get more
accurate quantities exchanging.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
56/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Acknowledgement




First, I would like to thank my advisor and committee
chairman, Dr. Hasan U. Akay. His invaluable guidance
helped me in realizing this research throughout the course of
my studies.
I also would like to extend my thanks to Dr. Hasan U. Akay
and Dr. Erdal Oktay for giving me the opportunity to work on
this research project; to Dr. Akin Ecer for providing me the
opportunity to use the facilities of the CFD Laboratory and
serving in my thesis committee; and to Dr. Andrew T. Hsu for
serving in my thesis committee.
Valuable assistance from Mr. Resat U. Payli contributed a lot
to the computational work in this research to which I am
grateful.
Finally, I would thank to my lovely wife, Jing, without her,
none of this would have been possible.
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
57/57
3D Unsteady Compressible Euler Equations with Structural Coupling
Question?
Zhenyin Li, Master’s Thesis Defense, July 19, 2002
58/57