Integrated and Multi-fidelity Software Package for Aero-

Servo-Thermo-Elasticity and Propulsion (ASTE-P) of Aero-

space Vehicles from Subsonic to Hypersonic Flight

Patrick Hu1, Liping Xue

2, Kan Ni

2, Hongwu Zhao

3, Handan Liu

4

Advanced Dynamics Inc. Lexington, KY

Marty Brenner5

NASA/Dryden Flight Research Center, Edwards, CA

Advanced Dynamics has developed an Integrated Variable-Fidelity Tool Set - ASTE-P

for Modeling and Simulation of Aero-Servo-Thermo-Elasticity and Propulsion (ASTE-P) of

Aerospace Vehicles Ranging from Subsonic to Hypersonic Flights. The ASTE-P software

tool set is developed in the state-of-the-art and commercial standard and enables accurate

integration and tight/loose coupling of the fluid, structural and control field simulation with

variable fidelity options available. The ASTE-P software tool can be applicable to modeling

and simulation of aerodynamics, structural dynamics, flight control and propulsion dynam-

ics as well as more important interactions of these dynamics. All flight regimes from subson-

ic to hypersonic are covered. The interface of structural/control surface motion and vibra-

tion modes with fluid flows is modeled using either unified particle-based methods

(MPM/PPM) or FVM/FEM based tight/loose coupled fluid/structure solving algorithms. The Euler/RANS/LES/DES solvers enable the accurate prediction of nonlinear coupled fluid-

structure problems in aeroelasticity and the embedded fluid and structural dynamics solvers

make the software self-contained and not require the integration of two separate third-party

fluid and structure solvers for aeroelastic modeling and simulation. Three levels of simula-

tion environments are included in ASTE-P tool set: (1) the bottom level of high-fidelity and

full-order simulation environment, (2) middle level of fast analysis and evaluation environ-

ment which is based upon reduced order models (ROM) and provides fast turn around time,

and (3) top level of rapid design and optimization environment. In several our previous

AIAA papers we have reported the integrated framework of ASTE-P and some validation

cases, and in this paper we will update the status and enhancement of the ASTE-P recently,

and present some new cases we have run to demonstrate that the functionalities of ASTE-P are compatible or even more efficient than the similar software that is currently available.

I. Introduction

HE development of modern transportation vehicles (i.e. high-speed train and aircraft) is driving the need to develop advanced multi-disciplinary analysis methods and software design tools. Methods and software

capability for predicting aerodynamics, structural and control dynamics have been extensively developed during last

decades. For example, for computational fluid dynamics (CFD) only there are tens types of software available in

both commercial standard (i.e. Ansys-Fluent, Ansys-CFX, Start-CD, etc.) and in-house codes (Cart3D, Overflow,

CFL3D, Fun3D, etc.). The leading software for computational structural dynamics (CSD) includes Nastran, Ansys,

LS-Dyna and others. However, understanding and quantifying the coupling and interaction among these dynamics

still require development of advanced and accurate numerical methods and coupling algorithms guided by the

underlying physics. Therefore, the present paper for developing the full predictive capability of integrated Aero-

Servo-Thermo-Elasticity and Propulsion analysis and design for transportation vehicles will have a great potential in

supporting such vehicle evaluations and designs methodology.

_______________________ 1 Chief Scientist, 2 Principal Scientist, 3,4Senior Scientist, 5Aerospace Engineer

T

AIAA Atmospheric Flight Mechanics Conference08 - 11 August 2011, Portland, Oregon

AIAA 2011-6367

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

It is a well known fact that a linear aerodynamic method, such as vortex-lattice and doublet-lattice based panel

method, is not suitable for compressible flows with shocks, and that a nonlinear aerodynamic method, such as com-

putational fluid dynamics (CFD) using Euler equations and Euler equations with boundary layer correction, is not

capable of predicting separated flows. For simplicity and computational efficiency, some of the current software

tools still use these methods or these methods with some nonlinear correction terms. Commercial codes such as

MSC/Aeroelasticity, ZAERO/ASE developed by ZONA Technology, Inc., and the STARS code developed by NASA are favored in current aeroelastic (AE)/aeroservoelastic (ASE) modeling and simulation because of the inte-

gration with other analysis tools (e.g. NASTRAN for structures, MATLAB for controls). However, the fidelity of

computing the flight environment around a vehicle will strongly influence the forces (pressure and skin friction),

energy flux (conduction and radiation heat), and mass flux (ablation, if any) on the vehicle. These effects are inte-

grated over the complete vehicle configuration to determine the total aerodynamic forces (lift, drag, pitching mo-

ment, control surface effectiveness) and the TPS sizing requirements for the vehicle. One of the objectives for de-

velopment of the ASTE-P software tool set is to enable a high-fidelity yet computationally-efficient capability for

simulating highly-nonlinear aerodynamics and structural dynamics and their interactions.

The significant innovation of the present work is the ASTE-P software tool set1-2 which enables accurate integra-

tion and coupling of the fluid, structure and control field simulation in aeroelastic system with variable fidelity op-

tions available at different analysis and design stages, resulting in significant time saving for aerospace vehicle eval-

uations and designs. Moreover, the ASTE-P software tool set will be applicable to all flight regimes ranging from subsonic to hypersonic. The interface of structural/control surface motion and vibration modes with aerodynamic

flows is modeled using the powerful state-of-the-art CFD/CSD coupling algorithm and particle-based methods (Ma-

terial Point Method-MPM and Pure Particle Method-PPM) 1-5 in a static and dynamic fluid-structure interaction en-

vironment. By using the particle methods, the information between different field solvers passes through momentum

and energy exchange terms in conservation laws. Consequently, the overall accuracy will maintain the same order as

field solvers, being not compromised by information passing at interface that is typically encountered by standard

finite element based methods. The PPM/MPM is essentially a “mesh-free” and highly parallelizable method which

avoids dealing with the time-varying mesh distortions and boundary variations due to structural deformations and/or

control surface motions, thus being significantly more robust and computationally efficient than other methods with

moving-boundary and mesh-regeneration.

Full-coupled and full-order computational models based upon the Euler and Navier-Stokes fluid models and nonlinear structural models are computational extensive and generally take weeks or months to obtain the solution

for real engineering problems, and recent advances in rapid-solution techniques for such models – using the harmon-

ic balance (HB) or time periodic method7-14, and/or the expansion of the flow field and/or structure in terms of prop-

er orthogonal decomposition (POD) modes15-19- has made the use of such high fidelity models potentially feasible

for design and pretest evaluation studies of aerospace vehicles. See Refs.[9] and [20] for an overview of these me-

thods. The computational efficiency can be further enhanced by massive parallelization, in-situ residual monitoring

and computational steering. The ASTE-P tool set presented in this paper will encompass a detailed evaluation capa-

bility that includes the full solution benefits of the powerful state-of-the-art CFD/CSD coupling algorithm and the

particle-based MPM/PPM method with full flight control system (FCS) definition, propulsion module, fast evalua-

tion environment and fast design and optimization environment.

Advanced Dynamics is dedicated to the development of Multi-disciplinary, Multi-physics, Multi-scale and Mul-

ti-fidelity (4M) Analysis and Optimization (4MAO) modeling and simulation software tools21-22, including aeroelas-ticity (AE), aeroservoelasticity (ASE), aeroservothermoelasticity and propulsion (ASTE-P), as well as fluid dynam-

ics, structural mechanics, system dynamics and control, multiphase flows and combustions, etc. The software tools

developed by Advanced Dynamics are variable-fidelity, thus the low-to-mid fidelity has fast turn-around time for

concept design, and the high-fidelity can be used for final design verification. The integrated tools can be routinely

applied to realistic engineering design and physics-based real or near-real time modeling and simulation for complex

aerospace systems that are commonly researched and manufactured by NASA, U.S. Air Force, Navy and Army, as

well as major defense contractors such as Boeing, Lockheed Martin, Pratt Whitney, Rolls-Royce, General Electric,

General Dynamics and Textron, etc.

II. Integrated Analysis and Design Environments of ASTE-P Tool Set

The current ASTE-P software toolset allows for the tight/loose coupling of fluid, structural and control states in

an aeroservothermoelastic-propulsion simulation. A variety of fidelity levels are available ranging from DNS/LES

simulation to reduced order modeling (POD/Volterra series). The ASTE-P software tool can be applied to the mod-

eling and simulation of aerodynamics, structural dynamics, flight control and propulsion dynamics and the coupling

of all these disciplines is also possible. All flight regimes from subsonic to hypersonic are covered. Three levels of

simulation environments with variable-fidelity are included in ASTE-P tool set: (1) the bottom level of high-fidelity

and full-order modeling and simulation environment for computational fluid dynamics, computational structural

dynamics, flight control, computational propulsion and aeroacoustics, as well as their coupled simulations, (2) mid-

dle level of fast analysis and evaluation environment which is based upon on reduced order models (i.e. Volterra

series and POD-ROM) and provides fast turn-around time, and (3) top level of rapid design and optimization envi-ronment. In addition, various flight control systems (FCS) designs and design optimization modules are encom-

passed in the fully developed ASTE-P software toolset. The main window of the ASTE-P 2.0 is shown in Figure 2.

Figure 1. The Integrated Software Configuration of

ASTE-P 2.0.

Figure 2. The Main Window of the ASTE-P 2.0 Tool

Set.

III. Particle-based MPM/PPM [23-25] Capability in ASTE-P Toolset

As shown in Figure 3, the bodies of fluid and structure are discretized into a collection of material points / par-

ticles. In the material point method (MPM), all material points, both in the fluid and structure, carry stress. It is the

stress which results in internal forces at each background grid node in the vicinity of the fluid-structure interface.

Instead of solving the equations of each individual material point, the conservation of the coupled fluid-structure

system is solved at the grid nodes by taking into account the internal forces from all nearby materials. However, the source data for the solution is taken from a number of material points which surround that node. Different materials

within the same computational element at a given time step will see the same increment in strain, or strain rate, but

can have different stress since each material point follows its own constitutive law. Therefore, all material points are

represented by unconnected Lagrangian points that are treated in the same way with exception of constitutive rela-

tions that relate stress to strain or strain rate for different materials. The effect of fluid-structure interaction is deter-

mined on grid nodes when the coupled equation is solved at each node. The coupling of the fluid and structure is

indirect in the sense that the pressure from fluid and the displacement from the structure are not directly mapped to

the neighboring grid nodes of different field solvers that traditional FEA-based methods must utilize. Instead, the

forces from both fluid and structural material points are calculated together at grid nodes where the divergence of

the stress from all material points is summed. Note that the coupling procedure is substantially different than the

traditional FEA-based methods that must utilize force and displacement mapping or interpolation. With the proce-

dure presented here, there is no explicit interface-tracking procedure that traditional FEA-based methods must utilize to divide the whole computational domain into two separate domains upon which the fluid and structure solvers dis-

cretize their own conservation laws. With the MPM method, the procedure for dealing with the fluid and structure is

the same except that different types of materials follow different constitutive laws. Thus, the same formulation and

procedure are followed at the nodes around the interface as well as the internal nodes that are away from the inter-

face. The overall numerical accuracy, therefore, maintains at the same order of each field solver, and is not degraded

by the coupling procedure.

These novel methods contain no explicit numerical diffusion and avoid the time-consuming grid re-generation

and deforming and avoid the fluid and structure compatibility as well as time-synchronization difficulties that are

typically encountered in standard CFD/CSD coupling approaches for fluid-structure interaction. In addition, the par-

ticle-based methods include geometrically nonlinear behavior in structural dynamics (i.e. large structure/control sur-

face deformations and/or motions) in a natural and efficient way as particles can freely move. It is an innovative

approach that potentially offers significant advantages over conventional methods.

fσ

sσ

ext

if

External Force

Structural Surface

Structure Material Point Carries Stress sσ

Fluid Material Point Carries Stress fσ

Grid Node

Figure 3. Schematic of Fluid-Structure Interaction Using MPM.

The example presented below is the PPM simulation result for AGARD 445.6 wing case. In this simulation, the

standard 3D AGARD 445.6 wing is placed in a freestream at Mach 0.7 and subjected to a prescribed translation and

twisting degree of freedom. The root of the wing is fixed and the remainder of the wing is subjected to a first bend-

ing type translation motion and a linearly varying twist along the span of the wing. The wing shapes at the maximum

and minimum deflection points is shown in Figure 4. The surface pressure contours at these deflection points are

depicted in Figure 5. This case was selected to show that the particle method can simulate the complex unsteady

flow caused by large deformation of a 3D wing.

Figure 4. AGARD 445.6 Deflection Depicting Maximum and Minimum Deflection Shapes.

In order to demonstrate the particle method to handle complex geometry and large motions, a calculation was

performed for the Advanced Fighter Airplane(AFA) using the MPM method as well. The domain size used for this

computation was 24x16x8 with 3 levels of grid refinement. The coarsest background mesh has 100x80x40 points,

which amounts to 9.5 million particles initially. We performed two calculations, one corresponding to an angle of

attack of 0 degree and one with 5 degrees. The Mach number for both cases is 0.8. Plots pertaining to zero degree

angle of attack are shown in Figure 6(a) and those pertaining to 5 degrees AOA are shown in Figure 6(b). We can

clearly see the counter-rotating vortices over the wing tip region for the 5 degrees AOA case, which is consistent

with the theory.

Figure 5. Surface Pressure Contours at the Maximum (Left) and Minimum (Right) Deflection Points.

In addition, we performed a forced plunging movement computation using the AFA model. The aircraft was sub-

jected to a forced plunge movement, given by

0( ) sin(2 )h t h ftπ=

where 0 0.02 100h and f= = . The lift coefficient versus time for the AFA model is shown in Figure 7. As can be seen, the result is periodic as one expects for the enforced plunge problem. In addition, the pressure contours on the surface material points along with the u-velocity contours and streamlines at two different planes for three different time instants (highest, lowest and mean plunging time instants) is shown in Figure 8. The differences in the surface pressure contours caused by the enforced movement of the aircraft are clearly observed. Since the main idea of this example was to demonstrate the particle method for handling complex geometries with large movement, more simulation results are not given here. More details of MPM and PPM in ASTE-P can be found in Refs. [23-25].

Figure 6 (a). Pressure Distribution on the Aircraft Surface and Surrounding Fluid

Figure 6(b). Streamlines past AFA at 5 Degrees Angle of Attack

Figure 7. Lift coefficient for plunging AFA model.

Figure 8. u-velocity contours and streamlines at an x-z and y-z plane as well as surface pressure contours on

the AFA model at 3 different time instants.

IV. Grid-based Aeroelastic Modeling and Simulation Capability of ASTE-P

In addition to particle-based methods, ASTE-P also provides grid-based, coupled CFD/CSD capability for nonli-

near aeroelasticity. The options for Computational Fluid Dynamics (CFD) include solutions of the Reynolds-

averaged Navier-Stokes (RANS) equations with various turbulence models (algebraic models, Spalart-allmaras one

equation model, several k-epsilon and k-omega models), Large-Eddy Simulation (LES, see below) and Detached-

Eddy Simulation (DES). Both ideal as well as chemically and thermally equilibrium and non-equilibrium gas mod-

els are included. The CFD solver includes both structured and unstructured grid-based CFD solvers.

The finite element-based computational structural

dynamics solver in ASTE-P includes:

• Static and transient (linear and geometrically

nonlinear) structural analysis.

• FE-based aeroelasticity analysis.

• FE-based thermal analysis.

• Standard (finite-difference based) and non-

standard (HDHB) time- discretization schemes.

Currently the library of finite elements includes:

• 2D and 3D bars.

• 2D and 3D beam elements.

• Plane-stress elements (3-node and 4-node).

• Plate and shell elements (4-node and 8-node).

• 3D continuum elements (8-node, 20-node and

27-node).

3D continuum elements (8-node, 20-node and 27-node).In addition, ASTE-P has developed fast and robust nu-

merical algorithms for deforming the computational grid in aeroelastic analysis. This grid deformation algorithm

uses a method of background grid interpolation / projection26. The background grid is connected via a network of linear translational and nonlinear torsional springs. The background grid is independent from the CFD grid which

covers the same computational domain and typically consists of many fewer points. This background grid needs

only be deformed, rather than regenerated, at each time step thereby improving the computational efficiency. This

method of grid deformation can be used for all types of CFD grid, no matter of structured, unstructured or hybrid

CFD grids.

The powerful grid-based coupled CFD/CSD capability of ASTE-P for modeling and simulation of nonlinear

aeroelasticity has been demonstrated by the test case of aeroleastic AFA aircraft model presented here. The analysis

is performed, using the grid-based aeroelastic capability in ASTE-P, of an AFA aircraft model with main wing for

aspect ratio of 4.8, taper ratio 30%, leading edge sweep 37 degrees and a weight of approximately 30000 lbs. Three

simulations are performed which include 1) the symmetric modes only 2) the anti-symmetric modes only and 3)

both symmetric and anti-symmetric modes. The flow conditions are Mach 0.95, a dynamic pressure of 153,125 Pa, and the angle of attack is zero. Figure 9 presents the modal displacements for all 8 symmetric and anti-symmetric

modes and Figure 10 presents a snapshot of an instant air flow field and aircraft structure in flutter/LCO. From Fig-

ure 9 it is apparent that this dynamic pressure is above that of the flutter dynamic pressure which eventually results

in a limit cycle response for the aircraft which is caused by moving shock waves. The red color streamlines and sur-

face pressure contours in Figure 10 clearly show these shock waves along with strong vortical structures which inte-

ract with these shocks.

Figure 9. Modal Displacement for AFA simulation. Figure 10. Streamlines and pressure contours.

V. LES Capability of ASTE-P for Complex Flow and Acoustic Problems

In the ASTE-P implementation of large eddy simulation (LES), the Favre-filtered, unsteady, three-dimensional,

compressible form of Navier-Stokes equations is numerically discretized in curvilinear coordinate system. Fifth-

order WENO schemes are used to discretize inviscid fluxes. The sixth-order compact schemes by Lele27 are used for

viscous fluxes at the interior grid points (away from the boundaries) and third-order one-sided schemes are used at

the boundaries. An eighth-order implicit filter28 is used to remove saw-tooth-like numerical oscillations which are

due to the non-dissipative nature of the compact difference schemes. ES-WENO schemes29 are incorporated into

ADI’s LES solver for enhancing the numerical accuracy and stability of shock-boundary interaction.

In addition to the widely used standard and dynamic versions of the Smagorinsky model, the LES code of

ASTE-P also features the new Vreman’s SGS model30 for turbulence-shock interactions31. A multi-block approach based on a 5-point overlap is implemented to extend the solver to complex geometries. Another advantage of the

multi-block approach is that the spatially implicit compact schemes are computed locally in each block which im-

proves the efficiency of the solver. The 4- and 5-stage Runge-Kutta method is used for temporal integration. The

code is implemented to run in parallel using Message-Passing-Interface (MPI) as the inter-process communication

library, and full three-dimensional domain decomposition is used to allow for scaling to large problems.

Figure 11. Representative Results for LES Simulation of Single Jet Impingement.

A high-speed jet impingement case31 was run to demonstrate the strong capability of the LES solver in ASTE-P

to resolve the complex underlying shock-turbulence interactions and aeroacoustics in Ref.[32]. The computational

set-up for these simulations is based on the experiments conducted by Krothapalli et al.33. Experiments were con-

ducted for a range of distance between the nozzle exit and impinging wall at a Reynolds number of 7x105. The ex-

periments with a fully expanded jet at a Mach number of 1.5 impinging on a flat plate at a distance of 8 diameters

away from the nozzle exit were considered for these simulations. The nozzle is held in place using a flat plate re-

ferred here as “lift plate”. Grid resolution of 645x254x128 was specified with grid stretching in the near-wall re-

gions. The computational domain was decomposed into 32 blocks and the simulations were run on 32 processors of

a Linux Cluster.

Important compressibility effects including the formation of shock and emission of acoustic waves are clearly

shown using instantaneous contours of dilatation in a x-r plane as presented in Figure 11(a). The turbulent jet im-pinging on the plate and reflecting thereafter is shown in Figure 11 (b) and (c). Dilatation contours in the x-r plane

are plotted as three-dimensional surface contours in Figure 11 (d) which clearly shows large values in the shock re-

gion. Turbulent effects and acoustic wave propagation can also be observed in Figure 11 (d). Three-dimensional

surface contour plots of pressure loading on the impinging plate are presented in Figure 11 (e). High pressure levels

in stagnation region are observed which is similar to the observations made in experimental studies33.

Although the case illustrated here is for an impinging jet, there are many features which are present in this case

and which are also relevant in the aeroelastic modeling of transonic flows past a wing structures such as the shock-

turbulence interactions and the complex unsteady flow turbulence that may generate strong aeroacoustics. Therefore,

we believe the capability of the LES solver in ASTE-P combined with the robust grid deformation algorithms incor-

porated in ASTE-P should be sufficient for resolving the unsteady fluid dynamics in the aeroelastic problems.

VI. Reduced Order Modeling Techniques in ASTE-P

Full-order coupled CFD/CSD approaches for aeroelastic modeling and simulation for complex geometry are

most often very computationally time-intensive. For parametric study and design, reduced order models (ROM) that

capture the dominant features of the full system while significantly reduce the number of model states are highly

desired. In general, ROM analysis in ASTE-P involves several steps including: (1) generation of training data (snap-shots or time-histories of loading excited by prescribed inputs) by conducting full-order simulations, (2) generation

of the ROM model by utilizing methods such as eigenmode based methods and system identification methods, and

(3) deployment of the ROM model for the full system analysis. Variety of ROM models were extensively studied by

Advanced Dynamics research team members, including linearization about a nonlinear steady-state condition, linear

model fitting (such as the ARMA model), representation of the aeroelastic system in terms of its eigenmodes, and

linearized representation of a ROM for nonlinear aeroelastic/aeroservoelastic systems. Both Volterra series and

POD-ROM models have been incorporated into ASTE-P toolset. Although many studies have investigated aeroelas-

tic/aeroservoelastic phenomena using POD-ROM and Voletrra series representation, few have addressed the near-

real, real-time and continuous aeroelastic/aeroservoelastic dynamics simulation, even for simple configuration and

trajectory. Using the ROMs implemented in the ASTE-P toolset, Advanced Dynamics has successfully conducted

pioneering research in this field. For more details on this work please see Ref. [34-36].

A. Multiple-Shooting Scheme Applied to the Material Point Method As with most particle methods, “regularization” of particles is needed in regions of high strain (rate) to allow for a stable spatial discretization. The process of regularization involves replacing the strained grid with a new Lagrangian regular grid and simultaneously transporting (accurately) the new global flow quantities from the old grid to the new grid. In order to be able to perform this regularization while still having an efficient time discretization for problems which involve a limited number of relevant time scales, a new procedure was developed which is called the Multiple-Shooting method23. In the Multiple-Shooting method given the general set of ordinary differential equations, similar to harmonic-balance methods based on Fourier series expansions (i.e. HDHB), the problem states ( )tQ are expanded (interpolated) using a set of basis functions ( , )jh tω and vectors jQɶ :

1

() , )(N

j j

j

t h t=

=∑Q Q ω (1)

The basis functions are functions of time and the vector of fundamental frequencies ωwhich could possibly be unknown. A simple example of this would be the Fourier interpolation. The number of problem unknowns depends on whether the problem is autonomous, where t does not appear explicitly in the set of differential equations or non-autonomous, where it does. For autonomous systems the number of problem unknowns would correspond to Ndof NM P R= + + where M is the dimension of each jQɶ , P is the dimension of ω and R is a number (possibly equal to zero) which corresponds to the number of parameters in the problem which determine the nature of the solution. For non-autonomous problems, the vector ω is known and determines the nature of solution and there are no free parameters

hence the number of unknowns reduces to Ndof NM= . So in order to compute the Ndof unknowns in the problem a set of Ndof equations must be set up. To do this first the time domain of interest T is broken up into N time intervals. The value of T corresponds to the minimum period of the frequencies contained in ω :

1 2

2

( , ,.., )P

minT

π

ω ω ω= (2)

For autonomous problems this domain will change during the solution process while for non-autonomous sys-tems it will be fixed. There are N +1 discrete points in time which correspond to these N intervals. To start the so-lution process we prescribe both the jQ and vector of frequencies ω . Once again for non-autonomous systems ω is known apriori where for autonomous systems this would be an initial “guess”. With these initial values prescribed through the use of Eq. we are now able to prescribe ( )tQ at each of the discrete times jt i.e. we have the vector

1 2 1( ), ( ),.., ( )T

Nt t t + Q Q Q . The next step in the solution process is to use these values to initiate a time-marching solution within each of the time intervals. In other words we use 1( )tQ as the initial condition for a time-marching solution starting at 1t , 2( )tQ as the initial condition for a time-marching solution starting at 2t , etc. A key component of the method is that each one of these time-marching solution can be done independently and hence can be done in parallel. Once each of the time-marching solutions has reached the next time level the march-ing solution is stopped. We will call the solution which initiated at time-level i and ended at time-level 1i +

1( )

it +Q⌢

. We now acquire NM equations by requiring that the difference between the solution at time level 1i + ,which is found through the interpolation in Eq. (10), be equivalent to that found using a time marching solution from it to

1it + , 1

( )i

t +Q⌢

1

2 2

1) ( ) 0(

(

(

) ( ) 0

) ( ) 0

N

N N

t

t

t

t

t

t

+− =

− =

− =

Q

Q

Q Q

Q

Q⌢

⌢

⋮⌢

(3)

where the first equation in (3) is a constraint which is due to an assumption of solution periodicity. Obviously in general Eq. (1) will not be satisfied initially. Hence putting all of the terms on the right-hand side into a vector Rɶ we will have

0≠Rɶ (4)

If the problem to be solved is non-autonomous Eq. (3) closes the system of equations. However if the problem is autonomous an additional P R+ equations must be generated. To accomplish this, P R+ values of the solution are prescribed such that we have

=0 1,2..., , 1,2,...,( , ) j

i ij i q j r q r Pt Rβ = = + =− +Q (5)

where the superscript j on Q corresponds to the jth state. Adding these equations to Eq. (3) gives a closed system for autonomous problems. In order to drive the vector Rɶ , which is a function of the unknowns jQ and (possibly) ω , to zero an iterative scheme can be used. In the validation problems to be described shortly, the solution was found by adding a pseudo-time term / τ∂ ∂Qɶ

0τ

∂+ =

∂

QR

ɶɶ (6)

and then marching solution in “pseudo-time” until the L2-norm of Rɶ is smaller than a specified tolerance. If the Jacobian matrix /∂ ∂R Qɶɶ can be found, a full Newton scheme can be used otherwise various “Jacobian-free” Newton-Krylov (JFNK) schemes could be used. A particular form of the JFNK scheme has been implemented into a multi-shooting code developed the Advanced Dynamics to test the method on canonical problems (Duffing’s, van de Pol, etc.) and it appears to be an efficient method of computing the solution.

In the present study, a number of problems were solved using the method just described. Each of these problems

was non-autonomous and the interpolation used in Eq. (1) corresponded exactly to a Fourier series interpolation. A

few examples using this method for both 2D and 3D problems will be shown here.

Example 1: The first example is a problem where the surface of the RAE2822 airfoil is given a prescribed pitching/plunging motion in a free-stream flow which corresponds to M=0.7 and Re=3.1x105. The kinematics for the plunge ( ( )y t ) and pitch ( ( )tα ) motions are given by

0 0

0 0

sin(2 sin(2) ( ) )

4 degrees 0.1 m 10 0.0 Hz

ft t ft

h

y h

f

π α α π

α

=

= =

=

= − (7)

A two-level AMR grid-refinement was used for the spatial discretization with the initial number of material

points of 6 x 105, which corresponds to 1.5 x 105 background grid points. Three different simulations were run with

number of Fourier modes kept as 1, 2 and 3 (which corresponds to N=3, 5 and 7 in Eq. (1)). The time-step used for

the time-marching solution was dt=3 x 10-5. Good agreement in the lift coefficient between the time-marching solu-

tion and multiple-shoot solution are found for all the simulations, as can be seen in Figure 12(a). The u-velocity con-

tour comparison at two particular time instants using multiple shoot HB-MPM and explicit time marching is shown

in Figure 12(b).

(a) (b)

Figure 12. (a) Comparison of the Lift Coefficient Results for the Prescribed Pitch/Plunge Motion of a

RAE22822; (b) U-velocity Contours of the Multiple-shooting Method Solution at 3 Time-levels.

Example 2: In the next example, we demonstrate the use of the multiple shooting HB-MPM method for 3D geometries. The case considered here is that of an enforced vibrating 3D ONERA M6 wing at Mach 0.789. The plunge and twist motion for this case 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 HB-MPM solver 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 of MPM and HB-MPM solution is shown in Figure 13. As can be inferred from the figure, the solutions are in good agreement with each other. The

flow field and surface pressure coefficient on the wing were also compared for a particular time instant, as depicted in Figure 14. Again, the comparison is excellent between the time marching of MPM and HB-MPM methods. From these findings, it is evident that the multiple shooting HB-MPM 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-MPM can be solved in parallel for each of the harmonic time levels, thus making it a very attractive method for problems in aeroelasticity.

Figure 13. Lift and Drag Coefficients for Enforced Vibrating M6 Wing Geometry.

Figure 14. Comparison of Time Marching MPM and HB-MPM Methods for Enforced Vibrating M6 Wing.

Example 3: As a last example, we demonstrate the full fluid-structure interaction capability using the HB-MPM solver. In this case, both fluid and solid particles are involved in the computation, as opposed to only fluid in the previous cases. The flowfield and geometrical movement are the same as that used for the airfoil case mentioned above with the added fact 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 15 and the agreement is found to be good between the two methods. The u-velocity contours at the 2 different time levels obtained using the HB-MPM methodology are shown in Figure 16. Shown in Figure 17 is the u- and v-velocity contours in just the solid particles alone and it can be seen that the pitch and plunge motions are clearly depicted.

Figure 15. Lift coefficient comparison for 2D FSI RAE2822 airfoil.

(a) t=0.0033s

Figure 16. u-velocity contours for 2D FSI examples using HB-MPM method:

Figure17. u- and v-velocity Contours on the Solid Particles.

VII. V. Conclusions

In this paper we summarized the recent development of ASTE-P tool set and presented some test cases to show

that the ASTE-P tool set is very powerful for modeling, simulation, evaluation and design of various transportation

vehicles and complex mechanical systems. The release of ASTE-P enables (1) the accurate and efficient prediction

of nonlinear FSI/Aeroelastic problems using an integrated CFD and CSD solvers and (2) robust particle methods for

analysis and design for complex configurations. The middle level ROM models of fast analysis and evaluation envi-

ronment in ASTE-P can provide a reasonable good computational fidelity and fast turn-around time compared to the

bottom level high-fidelity and full order simulation environment.

Acknowledgement

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

NNX07CA39P, Phase II contract No. NNX08CA39C, and Phase III Contract No. NND10AM05P, Mr. Marty Bren-

ner was the COTR. This work was also partially supported by NASA SBIR Phase I contract No. NNX10CE75P, and

Walter Silva is the COTR, and Peter Attar and Prakash Vedula at University of Oklahoma are the University Part-ners. Many thanks should be given to Earl Dowell at Duke University for his great help during the development of

ASTE-P in these years. Also, many thanks should be given to Brian Froist, Brent Whiting and Dale Pitt for their

strong support on Advanced Dynamics Inc. in recent years. Any opinions, findings, conclusions, or recommendation

expressed here are those of the authors and do not necessarily reflect the views of NASA and our University and

Industry partners.

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