A Meshless Method for Aeroelastic Applications in ASTE-P

Toolset

Patrick Hu1, Ramji Kamakoti

2, Liping Xue

3, Zhen Wang

4, Qingding Li

4

Advanced Dynamics Inc., Lexington, KY 40511, USA

and

Peter Attar5, Prakash Vedula

6

University of Oklahoma, Norman, OK

Advanced Dynamics has developed a comprehensive and integrated software toolset,

ASTE-P, for mutidisciplinary, multiphysics, multiscale and multifidelity analysis and design

optimization (4MAO) of aerospace vehicles, and the module of pure particle method (PPM) is

one of the key modules in ASTE-P for fluid-structure interaction applications. PPM is similar

to the Smooth Particle Hydrodynamic (SPH) method in a sense that the solutions are solved

on the particles. But unlike SPH, instead of a kernel function, PPM uses least square

technique to interpolate properties to the particles. PPM can handle complex fluid flows and

it is also very convenient to treat the fluid-solid boundaries, thus making it a very powerful

method for solving fluid-structure interaction and aeroelastic problems. In addition, a novel

Reduced-Order Model (ROM) based on the Harmonic Balance (HB) methodology is

developed in the context of the meshless, particle-based methods. The use of such techniques

can greatly reduce the computational time for problems that can be considered time-periodic,

which is frequently observed in aeroelastic problems such as flutter, limit cycle oscillations

(LCO), etc. Such methods improve the robustness of the particle solver as well as making it

computationally efficient.

I. Introduction

The pure particle method is a class of meshless methods that was developed at ADI in the context of the ASTE-P

integrated toolset. It is similar to the Smooth Particle Hydrodynamic (SPH) [1-2] or the Material Point (MPM)

[3,4,5] method in a sense that the solutions are solved on the particles. However, unlike SPH, instead of a kernel

function, PPM uses least square technique to interpolate properties to the particles. Also, in MPM, the momentum

equations are solved on the background mesh which needs interpolating functions to transfer information between

particles and mesh, which is eliminated in PPM, thereby leading a robust method for fluid-structure interaction

problems.

In PPM, material body is discretized into a collection of particles. As the dynamic analysis proceeds, the solution is

tracked on the particles by updating all required properties such as position, velocity, density, pressure, stress state,

etc. PPM solves the entire system using a unified set of governing equations with the exception of different

constitutive laws being used for different materials (i.e. fluid and structure materials). At each time step, fitting

functions are solved using least square method, and then used to calculate the change of properties on each particle.

The particles can freely move and different particles can interact through the least square. PPM uses interface

particles to communicate information between fluid and solid. Since interface particles are used to provide boundary

conditions for both fluid and solid, it is convenient to do the coupling.

1 President, Principal Scientist, Senior Member AIAA 2 Senior Research Scientist, corresponding author: ramjik@gmail.com 3 Principal Scientist 4 Research Scientist 5,6 Assistant Professor

AIAA Modeling and Simulation Technologies Conference2 - 5 August 2010, Toronto, Ontario Canada

AIAA 2010-7604

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

This novel method has no numerical diffusion and avoids the time-consuming grid generation for complex

geometries and mathematically solving the fluid and structure compatibility as well as time-synchronization

problems that are typically encountered by standard CFD/CSD coupling approaches for fluid-structure interaction,

leading to the best accuracy and computational efficiency achievable. More importantly, the particle-based method

allows for computing the geometrically nonlinear structure dynamics (i.e. large structure/control surface

deformations and/or motions) in the natural and efficient way because particles can freely move. It is an innovative

approach that offers significant more advantages than conventional methods.

A novel time-reduced order model technique is also added to the PPM solver. The ROM is based on the high-

dimension harmonic balance (HDHB) method that has been developed by Hall and colleagues at Duke University

[6]. This technique is tuned towards problems with periodicity in time, which is frequently encountered in the

aeroelasticity community, such as flutter, buffet and limit cycle oscillation (LCO). These methods have been

used in large-scale CFD-based aeroelastic simulations to overcome the computationally intensive issue

associated with time-marching schemes. In the HDHB method, which is a variant of the classical harmonic-

balance (HB) method, the problem variables are cast in the form of a Fourier series in time. This step is

equivalent to the HB method of solution. However, in order to be easily implemented into large-scale

computational codes which are based upon time-domain variables, i.e. codes which have been developed for

time-marching techniques, the discrete Fourier transform matrices are used to cast the problem back into the

time-domain. Although this technique has been successfully implemented in the context of grid-based solvers,

this is the first attempt at implementing this methodology in the context of particle based solvers.

II. Theory

In this section, we briefly describe the underlying theory and the equations associated with the pure particle method

as well as theory related to implementing the HDHB method into PPM.

A. Governing Equations for PPM

The governing equations are standard conservation equations for mass, momentum and energy

0D

Dt

ρρ+ ∇ ⋅ =v (1)

ρ ρ= ∇ ⋅ +a σ b (2)

De

k TDt

ρ = +∇ ∇σε& � (3)

In these equations, ρ is the mass density, a is the acceleration, v is the velocity, σ is a symmetric stress tensor, b

is the specific body force, e is the internal energy, ε& is the strain rate tensor, k is the thermal conductivity, and T

is the temperature.

In addition to these equations, particle constitutive response must be updated with the deformation. A

constitutive equation for the stress is required. For simplicity, constitutive equations are in terms of small

deformation theory. Thus the strain rate tensor can be defined as

( ){ }1

2

Tε = ∇ + ∇v v& (4)

Fluids and solids are distinguished by the constitutive equation relating stress to strain or strain rate.

B. Fluid Constitutive Equations

Take Newtonian fluid for example, the stress tensor for a fluid particle is given as

( )22 tr

3pσ µε µ ε= − −I I& & (5)

where I is the second order unit tensor, µ is shear viscosity, p is the pressure which is determined from an equation

of state.

To implement the fluid model, Eq. (5) is applied on the fluid particles. The strain rate is calculated by Eq. (4). The

equation of state for polytropic gas is

( )1p eγ ρ= − (6)

where p is the pressure, e is specific internal energy, and γ is the ratio of specific heats.

C. Solid Constitutive Equations

The mechanical response of solids can be quite different, from elastic to plastic to viscoelastic or viscoplastic. Any

of these responses can be implemented in our PPM code. For example, for the hypoelastic material in our code, the

Cauchy stress is decomposed into a volumetric and a deviatoric part as shown in Eq. (7).

p s= − +σ I (7)

where σ is the Cauchy stress, p is the hydrostatic pressure, I is the second-order identity tensor, and s is the

deviatoric part of the stress.

The hydrostatic pressure is computed using the hypoelastic constitutive model as shown in the following equation

( ) ( ) ( ) ( )2 3 trp t t p t K G t+ ∆ = + − ∆D (8)

where K is the bulk modulus, G is the shear modulus, D is the rate of deformation tensor, and t∆ is the time

increment.

The deviatoric stress can be divided into elastic and plastic contributions. For the hypoelastic material, there is no

plastic part. The elastic part is computed uses a forward Euler time discretization. The procedure used is as

( ) ( ) 2dev( )s t t s t G t+ ∆ = + ∆D (9)

where dev(D) is the deviatoric part of the rate of deformation tensor.

D. Least Square Fitting and Interface Boundary Conditions

In order to update the properties of particles, spatial derivatives of some variables are needed. They are calculated

from the fitting functions which are the results of least square fitting.

In addition, the interface boundary condition is also treated by the least square method. As a result, the fitting

functions obtained from least square method will satisfy the interface boundary condition automatically.

Before the discussion of our least square method, a new type of particles should be defined. We call it interface

particle because it is placed on the interface. In our fluid-structure coupling solver, an interface particle is neither

fluid particle nor solid particle. It moves with solid particle and transforms interactions between the fluid and solid.

They are critical in the implementation of the interface boundary condition. Furthermore, the solid interface can be

precisely tracked by the interface particles.

The least square method requires cloud points that are defined before the start of the computation. Since we have the

background grid to fill particles, using this background grid to define cloud points is convenient. For each particle,

the cloud points are determined using 3x3 stencil as shown in Figure 1. Only the particles with the same material

type and the interface particles that fall in the stencil are included.

Figure 1. Cloud points for fluid particles

Another problem for least square fitting is the chosen of fitting function. Generally speaking, higher order means

higher accuracy, while lower order means higher stability and efficiency. Since the accuracy can be improved by

using finer grid, the orders of the fitting functions are chosen as lower as possible to improve the stability of the

algorithm.

Then we can perform the least square curve fitting in the cloud. If the cloud points are all fluid particles, or all solid

particles, it’s very easy to get the fitting functions. However, if there are interface particles in the cloud points, the

interface boundary conditions should be treated simultaneously. Considering the difference between Dirichlet

condition and Neumann condition, we should treat them separately.

1. Dirichlet condition

The Dirichlet boundary condition specifies the value a solution needs to take on the boundary of the domain. Since it

gives the values on the boundary, it’s natural to assign these values on the interface particles. Then we can treat

these interface particles as normal fluid or solid particles. The fitting functions can be easily solved by least square

method.

The least square normal equations can be written as:

0

2

1

2

2

2

3

i i i i

i i i i i i i i

i i i i i i i i

i i i i i i i i

am x y z f

ax x x y x z x f

ay x y y y z y f

az x z y z z z f

=

∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑

(10)

where m is the total number of normal(fluid/solid) particles and interface particles, [1, ]i m∈ , ( ), ,i i i

x y z is the

position of each particle, i

f is the property value at the particle position.

2. Neumann condition

The Neumann boundary condition specifies the values that the derivative of a solution is to take on the boundary of

the domain. In order to implement this condition, some other equations should be constructed.

Particle to be

calculated

Interface

particles

Solid

particles

Cloud

points

Fluid

particles

a. Cloud points for fluid near boundary b. Cloud points for fluid

Take a simple situation for example. The variables are assumed to vary according to the function:

0 1 2 3

f a a x a y a z= + + + (11)

And a Neumann boundary condition should be implemented on the interface:

0f∂

=∂n

(12)

It might also be noted that we do not have the exact interface; we only have the interface particles instead. Each

particle has its own position and normal. Assuming we have an interface particle with the position ( ), ,x y z and the

normal ( ), ,x y zn n n . Performing the Neumann boundary condition (12) on this particle results:

1 2 3

0x y z

a n a n a n+ + = (13)

If there are m normal (fluid/solid) particles and n surface particles in the cloud, the overdetermined linear system of

equations will be:

0 1 2 3

1 2 3

[1, ]

0 [1, ]

i i i i

xj yj zj

a a x a y a z f i m

a n a n a n j n

+ + + = ∈ + + = ∈

(14)

And the least square normal equations can be written as:

0

2 2

1

2 2

2

2 2

3

i i i i

i i xj i i xj yj i i xj zj i i

i i i xj yj i yj i i yj zj i i

i i i xj zj i i yj zj i zj i i

m x y z a f

x x n x y n n x z n n a x f

y x y n n y n y z n n a y f

z x z n n y z n n z n a z f

+ + + = + + + + + +

∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

(15)

After the least square normal equations are constructed, Cholesky decomposition is used to solve the equations. As

we all know, the least square normal equations are usually very ill-conditioned. Some treatment should be

performed when constructing the equations, such as translation, scaling. If these equations are still ill-conditioned,

Cholesky decomposition might be a bad choice. Some other method, such as SVD(Singular Value Decomposition)

and regularization techniques, should be considered.

E. Formulation of HDHB-PPM without Particle Regularization

In this subsection we will demonstrate the formulation of the high dimensional harmonic balance (HDHB) method

[7,8] for the pure particle method. The governing equations of PPM can be written in a form similar to solving CFD

equations, such as

0t

∂+ =

∂Q R (16)

where,

p

p

p

u

e

ρ

=

Qr

, R = Rint

+ Rext

(17)

( )

int

01

01

ext

k T

ρ

ρ

ρ

∇ ⋅ = − ∇ ⋅ = − − +∇ ∇

v

R σ R b

σε& �

(18)

3. Fourier Series Expansion

While solving HDHB system, we consider the problem to be strictly periodic in time with a fundamental frequency

of ω . Based on this assumption the terms in Eq. (17) can be expanded in a Fourier series as follows

0

1 1

ˆ ˆ ˆcos( ) sin( )H H

c s

N Nm m

m m

m t m tω ω= =

= + +∑ ∑Q Q Q Q (19)

0

1 1

ˆ ˆ ˆcos( ) sin( )H H

c s

N Nm m

m m

m t m tω ω= =

= + +∑ ∑R R R R (20)

In reality, infinite harmonics are present but for all practical numerical simulations it is usually enough to limit the

number of harmonics to a finite number. Here the number of harmonics is limited to ‘NH’.

4. Fourier Coefficients

Substituting, Eqs. (19) and (20) in Eq. (16) yields,

0

1 1 1 1

ˆ ˆ ˆ ˆ ˆsin( ) cos( ) cos( ) sin( ) 0H H H H

c s c s

N N N Nm m m m

m m m m

m m t m m t m t m tω ω ω ω ω ω= = = =

− + + + + =∑ ∑ ∑ ∑Q Q R R R (21)

By equating terms of like harmonics in Eq. (21), following equations are obtained for 1m = to H

N

0 ˆ ˆˆ ˆ ˆ0; 0; 0c s s cm m m mm mω ω= − + = + =R Q R Q R (22)

These equations may be arranged in a matrix form as follows

{ } { } { }ˆ ˆ[ ] 0Aω + =Q R (23)

where { }Q̂ and { }R̂ are vectors containing the Fourier harmonics for the corresponding flow conservation variables

as shown below

{ }

0

1

1

( * )

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

H

H

T V N

c

N

c

s

N

s N N N×

=

Q

Q

Q Q

Q

Q

M

M

{ }

0

1

1

( * )

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

H

H

T V N

c

N

c

s

N

s N N N×

=

R

R

R R

R

R

M

M

(24)

0 0

0 0 1 0 0

0 2

0

[ ] 0 0 0 0

1 0 0 0 0

0 2

0

0 0 0 0 0T T

H

H N N

A N

N×

=

− − −

L L L L

M L L L

M O M M

M O M M O

M L L L

M L L L

M M O M

M M O M O M

L L L

(25)

Here NV is the number of variables in {Q}, NN is the total degrees of freedom in the system and 2 1T H

N N= + is the

total number of “harmonic terms”.

The time period of flow can be sub-divided into N equal intervals or time locations. The number of time locations

required is dependent on the harmonics retained in the solution.

5. Time Domain Variables

Let { }Q% and { }R% be the vectors containing the values for the flow conservation variables and residual variables at

those N time locations.

{ }0

1

2 NH

t

t

t

=

Q

QQ

Q

%

M { }

0

1

2 NH

t

t

t

=

R

RR

R

%

M (26)

The inverse Fourier transformation matrix 1[ ]E − converts the terms in the frequency domain to time domain as

{ } { }1 ˆ[ ]E −=Q Q% ; { } { }1 ˆ[ ]E −=R R% (27)

and the matrix [ ]E performs the reverse operation

{ } { }ˆ [ ]E=Q Q% ; { } { }ˆ [ ]E=R R% . (28)

After further manipulation, one gets the standardized form of the HDHB equation given by

[ ]{ } { } {0}D + =Q R% % (29)

where 1[ ] [ ] [ ][ ]D E A E−=

0 0 0 0

1 1 1 11

2 2 2 2

1 cos( ) cos( ) sin( ) sin( )

1 cos( ) cos( ) sin( ) sin( )[ ]

1 cos( ) cos( ) sin( ) sin( )H H H H

T T

H H

H H

N H N N H NN N

t N t t N t

t N t t N tE

t N t t N t

ω ω ω ωω ω ω ω

ω ω ω ω

−

×

=

K L

K L

M M M M M M M

L K

(30)

0 1 2

0 1 2

0 1 2

0 1 2

1 1 1

2cos( ) 2cos( ) 2cos( )

12cos( ) 2cos( ) 2cos( )[ ]

(2 1)2sin( ) 2sin( ) 2sin( )

2sin( ) 2sin( ) 2sin( )

H

H

H

HT T

N

H H H N

H

N

H H H NN N

t t t

N t N t N tEN

t t t

N t N t N t

ω ω ω

ω ω ω

ω ω ω

ω ω ω×

=

+

L

L

M M M M

L

M

M M L M

L

(31)

In order to solve this, we use a pseudo time stepping procedure. A pseudo time term is added to the equation so that

the conventional CFD time marching technique may be used to achieve the “steady state” solution

{ }[ ]{ } { } {0}D

τ∂

+ + =∂Q

Q R%

% % (32)

III. Results

A. Subsonic Flow Past a 3D Model Aircraft Using PPM

In our first test case, we demonstrate the robustness of the PPM solver in solver problems with complex geometries.

Here, we perform a subsonic flow calculation past an Aluminum Fighter Aircraft (AFA) model. It is a generic

aircraft with a wing aspect ratio of 4.8, taper ratio 30%, leading edge sweep angle 37 degrees and a weight of 30000

lbs. The total wing span was about 9 units with a length of 18 units. The geometry of the aircraft is shown in Figure

2. The domain size used for this computation was 24x16x8 with 3 levels of grid refinement. The coarsest

background mesh has 120x80x40 points, which leads to a total of 9.5 million particles initially. We performed 2

calculations, one corresponding to an angle of attack of 0 degrees and one with 5 degrees. The Mach number for

both cases was 0.8. The pressure contours on the particles, both on the surface of the aircraft as well as the

surrounding flow pertaining to the zero degree angle of attack case are shown in Figure 3. As can be observed from

the plot, high pressure regions are seen in front of the nose of the aircraft as well as the wings of the aircraft, which

is in good agreement with theory. For the 5 degree angle of attack case, the pressure distribution on the surface of

the aircraft along with the velocity streamlines in the y-z plane is shown in Figure 4. The v-w velocity streamlines

depict the wing tip vortex, which is in good agreement with theory. The overall velocity contours on the particles

past the entire aircraft in the x-z plane is shown in Figure 5, which is also in good agreement with theory. This case

demonstrates promise in the robustness of the particle methods to deal with complex geometries.

Figure 2: Schematic of Aluminum Fighter Aircraft (AFA).

Figure 3: Pressure distribution of the aircraft surface and surrounding fluid

Figure 4: Streamlines past AFA at 5 degrees angle of attack.

Figure 5: Streamwise velocity contours past AFA at AOA = 5 degrees.

B. Pitch and Plunge Airfoil Using PPM

In this example we demonstrate the robustness of the PPM solver for moving geometry problem. Particularly we

will demonstrate it using a 2D pitch and plunge airfoil at Mach 0.7 and a Reynolds number of 3.1x105

. The pitch-

plunge motion is given as follows

0

0

0 0

0 0

( ) cos(2 )

( ) sin(2 )

cos sin

sin cos

t f

h t h ft

x x y

y x y h

α α π ϕ

π

α α

α α

= +

=

= −

= + +

where a denotes the pitch motion and h denoted the plunge motion. Values of 0 0

0 04 , 0.1, 100, 90h fα ϕ= − = = = is

used for this simulation. Plot of the lift and drag coefficient is shown in Figure 6, wherein one can observe that the

periodicity in the aerodynamic coefficients in reproduced as per theory. The u-velocity contours are two different

time instants are shown in Figure 7, which are also in agreement with theory.

Figure 6. Lift and drag coefficient past a pitch-plunge airfoil at Mach 0.7

(a) t = 0.0033 sec

(b) t= 0.0066 sec

Figure 7. U-velocity contours for pitch and plunge airfoil at 2 different time instants.

C. Flow Past Subsonic Plunging Airfoil Using HDHB-PPM

In this example, we will validate the HDHB-PPM code via a forced-velocity on the boundary of an RAE2822 airfoil

at Mach 0.3 and a Reynolds number of 1.3x105

. Characteristic boundary conditions are applied at all four

boundaries of the computational domain. The plunge velocity imposed on the surface of the airfoil is given by

0

0

2 sin(2 )

u

v fh ftπ π=

=

where h0 is the amplitude of velocity oscillation and f is the frequency in hertz. For this case, we chose a value of h0

= 0.1 and f = 100hz. Both time-marching solution as well as HDHB solution with 1 and 2 harmonics (totaling 3 and

5 harmonic time levels, respectively) was sought for this problem to compare the time dependent results with the

discrete frequency-dependent HDHB solution. A plot of the lift coefficient history comparison is shown in Figure 8

for this problem and it can be seen from the figure that the agreement between the time-marching solution and the

HDHB solution is very good. A contour plot of the pressure for the 1 harmonic case at the 2 different time instants is

shown in Figure 9 and the upward and downward strokes of the airfoil are well depicted based on the high and low

pressure regions seen above and below the airfoil, respectively. This further demonstrates the validity and robustness

of the HDHB-PPM solver in seeking an accurate solution by eliminating the need to march in time, which can be a

very time consuming simulation. This validates the use of the HDHB method in the context of a particle method,

which has never been attempted before as per our knowledge.

Figure 8. Comparison of lift coefficient for plunging airfoil using HDHB-PPM using 1 and 2 harmonics.

Figure 9. Pressure contours at two time instants for plunging airfoil case.

D. Transonic Flow past a Vibrating ONERA M6 Wing

In this example, we demonstrate the use of the harmonic balance method for 3D geometries. The case considered

here is a forced vibrating 3D ONERA M6 wing at Mach 0.789. The plunge and twist motion is given by

0

0

0 0

0 0

( ) cos(2 )

( ) sin(2 )

cos sin

sin cos

t ft

h t h ft

x x y

y x y h

α α π

π

α α

α α

=

=

= −

= + +

where 00 02 , 0.02, 100h fα = − = = . The solution is run using 1 harmonic (corresponding to a total of 3 harmonic

time levels) and the solution is compared to that of time marching solution. The plot of lift and drag coefficient

comparison between the explicit time marching solution and HB solution is shown in Figure 10 where it can be seen

that the solutions are in good agreement. The flow field as well as surface pressure coefficient on the wing is also

compared for a particular time instant, as depicted in Figure 11. Again, the comparison is excellent between the two

methods. From these findings, it is evident that the multiple shooting HB method can predict the solution with the

same accuracy as that of time marching methods but with a fast turn around time as we do not solve the entire

unsteady solution and also, the HB solver can be solved in parallel for each of the harmonic time levels, thus making

it a very attractive method for problems in aeroelasticity.

Figure 10. Lift and Drag coefficients for vibrating M6 wing geometry

Figure 11. Comparison of time marching and HB method for vibrating M6 wing

E. Fluid-Structure Interaction using Harmonic Balance

As a last example, we demonstrate the fluid-structure interaction capability using the HB-particle solver. In this

case, both fluid and solid particles are involved in the computation, as opposed to only fluid in the previous cases.

The flow field and geometrical movement are the same as that used for the airfoil case mentioned in Sec B of

results. However in this case the solid particles inside the airfoil also move with the surface. This computation was

also run using 1 harmonic (corresponding to 3 total time levels). Plot of lift coefficient comparison with the explicit

time marching method is shown in Figure 12 and the agreement is found to be good between the two methods. The

u-velocity contours at the 3 different time levels obtained using the HB methodology are shown in Figure 13. While

it appears that the solid particles all have zero velocity, this is due to the choice of contour levels which were chosen

to highlight the flow field solution. Shown in Figure 14 is the u- and v-velocity contours for the solid particles alone

and it can be seen that the pitch and plunge motions are clearly depicted.

Figure 12. Lift coefficient comparison for 2D FSI RAE2822 airfoil

Figure 13. u-velocity contours for 2D FSI examples: (a) t= 0.00167sec, (b) t= 0.0033sec, (c) t= 0.005sec

Figure 14. u- and v-velocity contours on the solid particles

IV. Summary and Conclusions

The pure particle method was successfully incorporated into the ASTE-P toolset for fluid-structure interaction and

aeroelastic problems for a wide range of flow parameters. It is a unique and powerful method for solving complex

fluid-structure interaction and aeroelastic problems, due to the ability in treating nonlinear large structure

deformation, and straight-forward coupling procedure. The high dimension harmonic balance (HDHB) was

formulated and implemented in the context of PPM as well.

Results obtained using the PPM method was found to be encouraging, particularly for complex geometries and

moving boundary problems. The comparison of the results for HDHB methods and time accurate methods indicates

that inclusion of the HDHB method as a substitute for time-accurate methods will greatly speedup the overall

simulation and analysis process by still preserving the spatial accuracy. Therefore, the HDHB method can be used in

both concept design and final design verification because it can provide fast turn-around time and high-fidelity

information.

V. Acknowledgement

This work was supported by National Aeronautics and Space Administration SBIR Phase I contract NNX07CA39P

and Phase II contract No. NNX08CA39C under Mr. Martin Brenner as COTR and Kentucky State SBIR Matching

Fund Phase I Contract No. KSTC-184-512-07-018 and Phase II Contract No. KSTC-184-512-07-032 under Mr. Ken

Ronald as program manager. This work was also partially supported by National Aeronautics and Space

Administration SBIR Phase I contract NNX10CE75P under Mr. Walter Silva as COTR.

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Aerospace Sciences Meeting and Exposition. Orlando, FL. AIAA Paper 2010-1464

4 Sulsky, D., S. Zhou, and H.L. Schreier (1995) “Application of a Particle-In-Cell Method to Solid Mechanics”. Comp. Phys.

Comm., 87: p. 236-252.

5 Hu, G.P., et al. (2009) “Unified Solver for Modeling and Simulation of Nonlinear Aeroelasticity and Fluid-Structure

Interactions”, 2009 Atmospheric Flight Dynamics Conference, Invited Session on Modeling and Simulation, Chicago, IL,

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Balance Technique,” 9th International Symposium on Unsteady Aerodynamics, Aeroacoustics, and Aeroelasticity of

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8 Thomas, J.P., Dowell, E.H., Hall, K.C., “Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit

Cycle Oscillations,” AIAA Journal, Vol. 40, No. 4, 2002, pp. 638-646.