PARALLEL COMPUTATIONS OF SOLIDFLUID INTERACTIONS USING ALE AND RELATIVE COORDINATE FORMULATIONS

1
PARALLEL COMPUTATIONS OF SOLID-FLUID INTERACTIONS
USING ALE AND RELATIVE COORDINATE FORMULATIONS

• Masters Candidate
• Xiaoyin He
• Department of Mechanical Engineering
• Computational Fluid Dynamics Laboratory
• Indiana University Purdue University Indianapolis
• April 21, 2004

2
Outline

• Background
• Governing equations of fluid solver and solution
  methodologies used in ALE and RC formulations
• Governing equations of structural solver
• Coupling procedures of ALE and RC formulations
• Test cases for aerodynamic and aeroelastic
  problems
• Introduction to I-Light and test case
• Conclusion
• Acknowledgements

3
Background

• Three remaining areas of
• interaction
• Elasticity Dynamics
• (Structural Dynamics)
• Aerodynamics Elasticity
• (static aeroelasticity)
• Elasticity Aerodynamics
• Dynamics
• (Dynamic Aeroelasticity)

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Why Study Aeroelastic Problems

• Aeroelastic problems would not exist if airplanes
  were perfectly rigid
• No aircraft is made by completely rigid materials
  
• Aircraft will deflect for a small amount to
  counteract the effect of the dynamic forces, and
  the deflection also changes the fluid field
  around the aircraft
• Aeroelastic phenomena have played a major role
  throughout the history of aircraft design

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Why Study Aeroelastic Problems (cont.)

• O/400 bomber experienced violent tail
  oscillations as the result of the lack of a
  torsion rod connection between the port and
  starboard elevators in 1916

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Aeroelastic Analysis

• 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.

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Studies Done in Current Research

• A relative coordinate formulation is used to
  solve the aerodynamic problems
• Generalize the coupling procedure for solid-fluid
  interaction (SFI) problem
• A relative coordinate formulation is also
  achieved to solve aeroelastic problems
• Parallel computation performance was studied for
  aeroelastic problems
• I-Light was used as communication medium in
  metacomputing and the performance of the I-Light
  was studied

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Governing Equations of Fluid Dynamics Analysis
for ALE formultaion

• The Arbitrary Lagrangian-Eulerian (ALE)
  formulation of the three-dimensional
  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 on the boundary

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Governing Equations of Fluid Dynamics Analysis
for ALE formlation (Cont.)
All of the variables used in the ALE formulation
are calculated under the absolute coordinate
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Methodologies used in ALE Formulation

• 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

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

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Methodologies used in ALE Formulation (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
Displacement
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Relationship between Relative and Absolute
coordinates

• Relationship of velocities between relative and
  absolute coordinates

• Local derivatives for scalar and vector

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Governing Equations of Fluid Dynamics Analysis
for RC Formulation

• The Relative Coordinate (RC) formulation of the
  three-dimensional 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 on the boundary
• S is the source term

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Governing Equations of Fluid Dynamics Analysis
for RC formulation (Cont.)
All of the variables used in the RC formulation
are calculated under the relative coordinate
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Methodologies used in RC Formulation

• The source term need to be calculate in the
  relative coordinate formulation

• Each component is defined as

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Methodologies used in RC Formulation (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

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

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Governing Equations of Structural Dynamics

• The field of structural dynamics addresses the
  dynamic deformation behavior of continuous
  structural configurations
• The finite element equations for dynamic response
  of a structural element can be expressed as

M is mass matrix C is damping matrix K
is stiffness matrix
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Methodologies used in Structural Dynamics
Analysis

• Mode superposition method

• 1. Get the generalized eigenvalue solution

2. Use first n modes to simulate structural
response
3. Get the generalized displacement solution
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Aeroelastic coupling algorithm

• Mesh-based Parallel Code Coupling Interface
• ( MpCCI ) is used to exchange information
  between CFD and CSD codes and administer both in
  and out code communications

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Aeroelastic coupling algorithm (Cont.)

• Three meshes involved in the current coupling
  procedure for both ALE and RC formulations
• Fluid mesh
• Virtual structural mesh
• Mid-surface structural mesh

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Aeroelastic coupling algorithm (Cont.)

• Pressures and displacements are transferred
  between fluid mesh and structural mesh at the
  solid-fluid interface.
• Bilinear interpolation is used in the current
  studies

• Quadrilateral elements at the structural surface

• Triangle elements at the fluid boundary

Triangle element interpolation
Quadrilateral element interpolation
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Aeroelastic coupling procedures with ALE
Formulation

• A basic procedure to obtain an aeroelastic
  solution includes following steps
• Get pressure on CFD mesh nodes from flow
  calculation
• Pass the load information to CSD domain
• Calculate nodal displacements with CSD code
• Feedback the structure deformation to CFD domain
• Deform the CFD mesh
• Repeat steps 1 through 5

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Aeroelastic coupling procedures with RC
Formulation

• A basic procedure to obtain an aeroelastic
  solution in RC formulation includes following
  steps
• Start with initial conditions which were obtained
  from steady state calculations
• Get nodal pressures for CFD mesh from flow
  calculation by using relative coordinate
  formulation
• Transfer the load information to nodes on the CSD
  mesh
• Calculate the nodal displacements with CSD code
• Feed back the structure normal change on the
  interface to interface nodal in CFD domain
• Repeat steps 1 through 5

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Background of fluid and structural solvers

• Fluid solver - USER3DP
• A parallel finite-volume based on unstructured
  Euler solver
• Serial version of USER3D was developed by Dr.
  Oktay (1994)
• Parallel version of USER3D was developed by Uzun
  at CFD Lab, IUPUI (2000)
• Coupled version of USER3D was validated by
  Zhenyin Li (2002)

• Structural solver - Modal
• A finite element solver based on the time
  integration of modal dynamic equation
• The original version of CSD was developed by
  Zhenyin Li Xiaoyin He (2002)

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Test cases

• Dynamic analysis NACA0012 Airfoil

Coarse mesh consists 11,448 cells and 2,895
nodes. Fine mesh consists 45903 cells and
11,395 nodes
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Test cases (Cont.)

• Case 1 Steady state calculation

• Coarse mesh reaches the steady state at 100 time
  steps, and fine mesh reaches the steady state at
  300 time steps

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Test cases (Cont.)

• Unsteady transonic flow in ALE and RC formulations

• ALE formulation uses the absolute coordinate
• RC formulation uses the relative coordinate fixed
  on the airfoil at the quarter of the chord length

Pitching oscillation
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Test cases (Cont.)

• Variation of lift coefficient comparison for ALE
  and RC formulations

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Test cases (Cont.)

• Comparison of pressure coefficient distribution

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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6

• Quarter-chord sweep angle of 45 degree
• Panel aspect ratio 1.65
• Taper ratio of 0.66
• NACA0065A004 airfoil section

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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6

• Material properties and frequency of modes used
  in the flutter analysis

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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6

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
Unstable
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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6 in ALE
  formulation

• Flutter speed index comparison for AGARD wing
  445.6 with ALE formulation

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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6 in RC
  formulation

• The boundary condition on the interface is
  enforced through the normal momentum equation

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Test cases (Cont.)

• Flutter analysis for AGARD Wing 445.6

• Flutter speed index comparison for AGARD wing
  445.6 with RC formulation

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Test cases (Cont.)

• Subiteration Scheme

• In order to use large time increment to get the
  accurate results and eliminate the phase lag in
  current single coupled scheme, subiteration
  scheme can be used.

Without subiteration
With subiteration
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Test cases (Cont.)

• Subiteration Scheme

• Different time increments are tested
• Relative errors are compared based on the
  smallest time increment used in the current study

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Parallelization for Metacomputing

• I-Light is a high speed optical fiber network
  connecting Indiana University Bloomington, IUPUI,
  and Purdue University, West Lafayette. I-Light
  also connects all three campuses to the national
  Internet infrastructure.
•  
• Purdue and Indiana University are limited in
  network capacity.  The current data access speed
  between Purdue and IU is 30 million bits per
  second.  I-Light increases access speed initially
  to 1 billion bits per second and is expandable to
  100s of billions of bits per second.

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Parallelization for Metacomputing (Cont.)
IU Bloomington IBM SP2 Processors (Unix)
Internet 2

• CFDL
• IBM RS/6000 Processors (Unix)

Processor
Processor
Processor
Processor
Processor n-1
Processor n
MODAL
n Blocks
MpCCI

• The Metacomputing is focused on efficient ways to
  utilize the resources of many computers that are
  connected by a network to complete the required
  work by the required time.

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Test Case for metacomputing

• Total elapsed time and CPU time are tested with
  200 time steps for three processor distribution
  schemes

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Test Case for metacomputing (Cont.)

• Total communication time and waiting time are
  tested by using (elapsed time CPU time)

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Test Case for I-Light (Cont.)

• Elapsed time speedup comparison for aeroealtic
  metacomputing solution

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Advantages and restrictions of current RC
formulation studies

• Advantages
• Economical and accurate approach
• Valid for aerodynamic and aeroelastic problems

• Restrictions
• Current RC formulation only valid on single frame
  of reference
• For two or more bodies moving relative to each
  other, multiple moving frames need to be used and
  the quantities on the boundary need to be
  coupled. But it is hard to realize.
• ALE formulation is easy to extend for multiple
  bodies moving problems

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Test cases

• Elapsed time comparison of RC and ALE formulation
  for aerodynamic and aeroelastic problems

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Conclusions

• A solution in relative coordinate formulation is
  developed by adding source term to the Euler
  equations in the absolute coordinate. In fluid
  dynamic cases, calculation results are good in
  agreement with the measured experimental values
  and the results obtained from ALE formulation.
• The elapsed computer time of running with
  relative coordinate formulation is much less than
  the elapsed time running with ALE formulation in
  the absolute coordinate due to the skipping the
  dynamic mesh movement algorithm, which is a time
  consuming part in the ALE formulation.
• Dynamic aeroelastic problems were solved by using
  a coupled procedure in both ALE formulation and
  relative coordinate formulation. The results
  obtained from RC formulation generally agree with
  the experimental results and existing numerical
  solutions done by other researchers.

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Conclusions (Cont.)

• Large time increments with inner code
  sub-iteration can be used in the coupled
  procedure. Five times larger time increment can
  be used to get the no phase lag results. For the
  same time increment, more accurate results can be
  obtained by using sub-iterations than without
  sub-iterations.
• Parallelization of metacomputing based on the
  program USER3dP-SFI was achieved by using
  Mesh-based parallel Code Coupling Interface
  (MpCCI) library. Program successfully ran on the
  clusters located in different campus. Good speed
  up and efficiency are shown for mutiblock cases.

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Acknowledgement

• I would like to express my deepest gratitude and
  thanks to my advisor Dr. Hasan U. Akay for his
  guidance, motivation, patience and support
  throughout the course of this research.
• I warmly thank Dr. Akin Ecer and Dr. Andrew Hsu
  for being on my advisory committee and for their
  careful reviews and helpful suggestions of the
  thesis.
• I would like to express my special thanks to
  Zhenyin Li for his generous and invaluable help
  when I started to work on the CFD method.
• Valuable assistance from Mr. Resat U. Payli of
  CFD lab and Mary Papakhian of UITS to give me
  valuable assistance for my computational work in
  this reseach. I also would like to thank all my
  friends and colleagues in the CFD lab and in
  Department of Mechanical Engineering.
• I am deeply indebted to my wife, Lei Chen, and my
  parents, for their love, encouragement and
  understanding.

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