PARALLEL COMPUTATIONS OF 3D UNSTEADY COMPRESSIBLE EULER EQUATIONS WITH STRUCTURAL COUPLING

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PARALLEL COMPUTATIONS OF 3D UNSTEADY COMPRESSIBLEPARALLEL COMPUTATIONS OF 3D UNSTEADY COMPRESSIBLEEULER EQUATIONS WITH STRUCTURAL COUPLINGEULER EQUATIONS WITH STRUCTURAL COUPLING

Master’s Candidate

Zhenyin LiAdvisor: Dr. H. U. Akay

Department of Mechanical Engineering Computational Fluid Dynamics Laboratory

Indiana University Purdue University IndianapolisJuly 19, 2002

Zhenyin Li, Master’s Thesis Defense, July 19, 2002 2/57

3D Unsteady Compressible Euler Equations with Structural Coupling

OutlineOutline

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



Introduction to AeroelasticiyIntroduction 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.



Studies done in this researchStudies 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.



AEROELASTIC COUPLING ALGORITHMAEROELASTIC 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



AEROELASTIC COUPLING ALGORITHM (Cont.)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

CFDfluid solver

Application Interface

Process II

CSDstructure solver

Application Interface

MPCCI

MPCCI Configuration




The current version of MPCCI works well with Message Passing Interface (MPI)-based parallel as well as serial computing programs.




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 ProcessGID=0 LID=N/A

CODE I: Process 1GID=1 LID=0

CODE I: Process 2GID=2 LID=1

CODE I: Process iGID=i LID=i-1

CODE II Process 1GID=i+1 LID=0

CODE II Process 2GID=i+2 LID=1

CODE II Process jGID=i+j+1 LID=j-1

MPCCI

MPCCI communications ID settings




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




Information Exchange : Pressure and displacements need to be exchanged during the coupling process.

wQvQuQQt 321

Q3

Q2Q1

Qt

u

u

w

v

w

v

Triangular element interpolations

uv

Q2

Qt

Q3

Q1

Q4

Quadrilateral element interpolations

vQuuvQQvu

QvuQt

)1()1(

)1)(1(

32

1




Exchanging Quantities Virtual CSD Surface Mesh

Mid-surface Structural Mesh

Real Surface

Central Surface

Central surface transformations

Fu

Fb

Fc

Mc

CFD surface Mesh Match Virtual CSD

Surface Mesh




Time Integrations of Coupled System Here, the same ∆t is used for fluid and structure

Fluid Solid

Pn-1

Un

Pn

Un+1

Step n-1

Step n

Step n+1

Δt

Δt

Time integration



Steady State Solution for rigid body

Calculate new CFD flow field

Calculate node pressure on surface mesh

Construct CSD virtual surface mesh

Put pressure on virtual surface

Calculate dynamic forces on CSD virtual surface mesh

Transform the dynamic forces to structure mesh and solve

equilibrium equation

Map the displacements to CSD virtual surface mesh

Put the displacements on surface mesh

Deform the CFD mesh

Extract fluid surface mesh

Finish

MPCCI

MPCCI

Construct CFD Mesh



Computational Fluid Dynamics Solver - USER3DComputational 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.



Computational Fluid Dynamics Solver - USER3D (Cont.)Computational Fluid Dynamics Solver - USER3D (Cont.) Governing Equations for USER3D

The Arbitrary Lagrangian-Eulerian formulation of the three-dimensional time-

dependent inviscid fluid-flow equations is expressed in the following form:

0ˆ}{ dSnFdVQt

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



Computational Fluid Dynamics Solver - USER3D (Cont.)Computational Fluid Dynamics Solver - USER3D (Cont.) The time integration employed in the flow solver is the cell-

centered finite volume formulation. The volume-averaged values are adopted to represent the flow variables.

t

VQdSnQF

t

VQ

nn

nn }{)(}{

An implicit time integration scheme is used to solve flow field at each time step.

dSnQFR

Q

RI

t

VA

t

VQRQA

nn

n

nnn

nnnnn

)(}{

}{

}{][][

}{}{}{][



Computational Fluid Dynamics Solver - USER3D (Cont.)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.

2/1222 ])()()[( ijijijm zzyyxxk

m

mni

m

mni

m

mni k

zkz

k

yky

k

xkx 111 ,,

Stiffness K

Displacement



Computational Fluid Dynamics Solver - USER3D (Cont.)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

I

Region I: The inner zone is moving

rigidly with the body ;Region II: The outer zone is deformed by general mesh deformation method .



Computational Fluid Dynamics Solver - USER3D (Cont.)Computational Fluid Dynamics Solver - USER3D (Cont.)

The cell volume can be calculated by

dSnWdVt s

where Ws denotes the local velocity on the boundary surface S

Geometric Conservation Law :

The geometry conservation equation is required to solve simultaneously with other conservation equations.

)( 1111z

nty

ntx

nt

ni

ni AzAyAxtVV

mmm



Computational Structural Dynamics Solver – SAP4Computational Structural Dynamics Solver – SAP4

The finite element discrete aeroelasticity element equation for a structural system can be written as:

)()()()()()()( }{ eeeeeee RqKqCqM

[M], [C] and [K] are system mass, damping and stiffness matrix

For static analysis, equation can be rewritten as:

RqK

For dynamic analysis, equation can be rewritten as:

)(}{ tRqKqCqM



Computational Structural Dynamics Solver – SAP4 (Cont.)Computational Structural Dynamics Solver – SAP4 (Cont.)

Mode superposition method

}]{[}]{[ 2 MK

1. Get the generalized eigenvalue solution

3. Get the generalized displacement solution

*2 2iiiiiii FXXX

2. Use first n modes to simulate structural response

)}(]{[)}(}]{{},......,{},[{)}({ 21 tXAtXtq n



Computational Structural Dynamics Solver – SAP4 (Cont.)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(][

)21(][

1[

}]{])[22

1(][

)21(][

2[}{][

1***2

***2

1*

FXKCt

Mt

XKCt

Mt

XM

n

nn

α β Stable Condition

Galerkin method 3/2 4/5 Always

The backward difference method

3/2 1 Always

The constant acceleration method

1/2 1/4 Always

The linear acceleration method

1/2 1/6i

t 32

Initial Condition:

For Flutter Analysis

Either or

}{}{

}{}{

00

00

Xdt

dX

XX

t

t

0}{

0}{

0

0

X

X 0}{

0}{

0

0

X

X



TEST CASESTEST CASES

Aeroelastic Research Wing (AGARD Wing 445.6)

1.833 ft

1.208 ft

45O

AGARD wing 445.6 panel dimensions

5.2 ft

The CFD grid consists of 147,547 cells and 26,228 nodes. The CFD wing surface has 2020 elements and 1077 nodes



In the present application: n processors are used for CFD solutionOne processor for CSD solutionOne processor for communication management with

MPCCI



TEST CASES (Cont.)TEST CASES (Cont.)

Modal Analysis of Wing 445.6

Table 5.2 Modal frequencies of AGARD wing 445.6

SAP4 400 eles.

SAP4200 eles.

SAP4100 eles.

ANSYS100 eles

Experiment

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




SA

P4

Mod

al S

hape

MODE 1 MODE 2

MODE 3 MODE 4



TEST CASES (Cont.)TEST CASES (Cont.)A

NS

YS

Mod

al S

hape

Mode 1 Mode 2

Mode 3 Mode 4




Steady Solution of the Rigid Body

Steady State Transonic Flow at M∞ = 0.96 and M∞ = 1.141










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.

Rigid Body Result




The tip deflection at the trailing edge was computed to be 0.40 inch

which is very close to 0.39 inch from MDICE




Undeformed Mesh

Deformed Mesh







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







Deformed Mesh

Undeformed Mesh




Flutter Analysis

Dynamic instability where-by the system extracts energy from thefree stream flow producing a divergent response. The computed flutter characteristics are presented in terms of velocity index Vf which is defined as

Stable

Neutral

Unstable

bUV f /



TEST CASES (Cont.)TEST CASES (Cont.) Mach=0.957, Vf = 0.349 , U∞=14400 inch/s




Mach=0.957, Vf = 0.250 , U∞=10200 inch/s




Mach=0.957, Vf = 0.262 , U∞=10800 inch/s




Comparison of Results




Initial Velocity Effect




Parallel Aerodynamic StudiesA 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.







This case is the basic finner performing a sinusoidally pitching motion about its center of gravity. The angle of attack varies as:

)sin()( tt pm 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










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

UNIX LINUX

Model IBM RISC System / SP6000

POWER3+ Thin Node

Compaq ProLiant 1850R rack-

mounted compute

nodes

CPU(Each Node)

4 CPU, 375MHz clock cycle

Dual Intel 400 MHz Pentium II processors

Memory 2GB 256 MB

Cache 8MB 512 KB

Network 10/100Mb "Fast" Ethernet (100 TX)

pp TTS /1

Efficiency E is defined as

pSE p /100




144,216 nodes and 706,105 cells



TEST CASES (Cont.)TEST CASES (Cont.)144,216 nodes and 706,105 cells



ConclusionsConclusions

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.



Conclusions (Cont.)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.



Future WorksFuture 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.



Future Works (Cont.)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.



AcknowledgementAcknowledgement

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.



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PARALLEL COMPUTATIONS OF 3D UNSTEADY COMPRESSIBLE EULER EQUATIONS WITH STRUCTURAL COUPLING