[American Institute of Aeronautics and Astronautics 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 47th AIAA

1 CFD Research Corporation

A Reduced Order Method for Grid Deformation in Aeroelasticity Analysis

H. Q. Yang1 CFD Research Corp., Huntsville, AL 35805

A highly accurate and efficient solution algorithm for Navier-Stokes equations by unstructured grid for aeroelastic analysis of complex aircraft configurations is important in further design of modern aircraft and space vehicles. To handle the Lagrangian portion of the fluid equation moving mesh, we proposed a reduced order method (ROM) to model the grid motion. By this method, the fluid mesh is modeled as having structural properties: Young's modulus density and Poisson's ratio, but it acts passively to the structural deformation (wing or space vehicle deformation). The modal shape of both structure and CFD grid can be determined from FEM solver. By utilizing the modal superposition property of structure dynamics, the reduced model for the structure and CFD grid can be derived and the efficiency of the grid deformation can be significantly increased by preparing first a specified grid modal shape corresponding to the structural modal shape. The present ROM for grid deformation was applied to solve several aeroelastic problems. Comparisons with benchmark data, analytical solution or experimental data have shown good agreement.

Nomenclature CFD = Computational Fluid Dynamics CSD = Computational Structural Dynamics

I. Introduction Aeroelasticity Phenomena and the analysis The science of aeroelasticity is concerned with interaction between the deformation of an elastic structure in an

airstream and the resulting aerodynamic reactions. Aeroelastic phenomena occur on a daily basis in the nature, e.g. the swaying of trees, grass, etc. in the wind. They can result in dangerous static and dynamic deformations and instabilities and, thus, have very important practical consequences in many areas of technology. Especially in the design of modern aircraft and space vehicles-characterized by the demand for extremely light-weight structures - the solution of many arising aeroelastic problems is a basic requirement for operationally reliable and structurally optimal constructions.

Numerical aeroelastic models are built composing those of Computational Fluid Dynamics (CFD) and the computational structural dynamics (CSD) domain. Since the fluid and structural model differ in their foundation and discretization, an interface model has to be introduced that represents the connectivity and physical connection between the two single domain models.

A fluid-structure interaction model, which treats the coupled aeroelastic system, obtains an adequate numerical distribution of aerodynamic loads at the structural nodes of the finite element model - using the aerodynamic pressure given in finite volumes, volume elements, or panel of the discretized flow field or surface — as well as an adequate definition of the aerodynamic shape — using the displacements and rotations given at the nodes of the FEM model. For the solution of the coupled problem, existing well-established numerical solvers have been used in each domain. The interaction between the fluid and structural models have been limited to the exchange of surface loads and surface deformation information, using partitioned or staggered procedures and updating the related boundary conditions. This approach allows an easier extension to multi disciplinary problems.

1 Chief Scientist, Research, 215 Wynn Drive, Huntsville, AL 35805

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-890

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


II. Theoretical Background In this chapter, we will discuss the fundamental formulation for fluid dynamics, structural dynamics, interfacial

condition, and solution procedures.

A. Fundamentals of Governing Equations Unified Approach Continuum mechanics has been conventionally subdivided into solid mechanics and fluid mechanics. As a result,

problems involving an interaction between fluid flow and resulting deformations and stresses in the solid structure in contact with the flow (aircraft body and wing, wind loaded structure, heat exchange) are typically treated separately, in a decoupled manner, and often using completely different solution techniques (finite element method, finite volume method, or finite differencing method). There is however, a wide range of problems that requires a simultaneous solution of fluid flow and solid body deformation, and requires a uniform approach to the coupled problems. This is what will be addressed in this study.

Unified Governing Equation Regardless of the solution method (FEM, FVM, or FDM), the solid body and fluid mechanics actually share the

same governing equations, and differ only in constitute relations. The governing equation for both a fluid and a solid are the momentum equations:

ij,iji fv +σ=ρ& (2.1)

where ρ is the density, vi is the velocity, σij is the stress tensor, fi is the internal body force, a superscript dot designates a total derivative, a comma a partial derivative with respect to the following variable. Repeated indices denote summation over the appropriate range.

To close the system in Equation (2.1) the information about the response of particular material to applied force is necessary. Here, we will take a compressible gas and elastic solid as example of fluid and solid materials. For the fluid:

Equation of state:

( )T,pρ=ρ (2.2)

where p is the pressure, T is the temperature. For incompressible fluid

ρ = constant (2.3)

and for ideal compressible gas:

RTp

=ρ (2.4)

The constitutive relation between stress and rate of deformation for fluids is given by Stoke’s law of:

( ) ijk,kiji,jj,iij pv32vv δ−δμ−+μ=σ (2.5)

where μ is the dynamic viscosity. For the elastic solid, the constitutive relationship is given by Hooke's law of:

( ) k,kiji,jj,iij uuu λδ++η=σ (2.6)

where η and λ are the Lame constants, ui is the displacement vector and


ii uv &= (2.7)

We may notice, in the fluid, the stress is expressed in terms of velocity, whereas in the solid it is expressed in terms of displacement.

Consistent Interface Boundary Condition As for interface boundary condition, it is required that displacement, velocity, and stress are continuous, i.e.

( ) ( )fisi uu = (2.8)

( ) ( )sisi vv = (2.9)

( )( ) ( )( )( )( ) ( )( )

fijsij

fnijsnij

ττσ=σ

σ=σ (2.10)

with subscript s and f representing solid and fluid domains. n and τ are the normal and tangential directions of the interface.

B. Reference Frame and Mesh Systems Before an equation is discretized, it is important to select appropriate reference frame. In the classic solid

mechanics, the dynamics equation is formulated in a Lagrangian frame, where:

dt

dvv ii ρ=ρ& (2.11)

Here one moves/follows with the structure. In the classic fluid dynamics, the conservation equation is formulated in an Eulerian frame, where:

jj,ii

i vvt

vv +∂

∂ρ=ρ& (2.12)

It is the second nonlinear term that has given rise to many difficulties in fluid dynamics. However, in fluid dynamics approach, the Eulerian frame is necessary. To appreciate the difference between the two reference frames, we denote the space (Eulerian) coordinate by xi, the material (Lagrangian) coordinate by Xi, and mesh coordinate by χi, then if our mesh is given by

ii x=χ (2.13)

we have a Eulerian mesh, while

when ii X=χ (2.14)

it gives a Lagrangian mesh. Eulerian-Lagrangian (Fluid) - Lagrangian (Solid) Approach Some critical requirements for reference frames are: 1. preserve fluid-structure interface, and prevent cutting and crossing. This will ensure that correct interface

condition is supplied; 2. less distortion of computational mesh; and


3. efficient moving mesh algorithm. The Eulerian-Lagrangian (fluid) - Lagrangian (solid) approach has the above properties. By this formation, a grid

velocity is introduced. and the momentum equation will become:

( ) ij,ijj,iGjji fvvvt

v+σ=−+

∂∂

ρ (2.15)

Here we set grid mesh velocity as:

vGj = vj in solid (2.16)

( ) ( )fsjfsGj vv = at solid-fluid interface (2.17)

and

mesh distortion minimum (vGj) in fluid. (2.18)

One of our objectives is to develop a technology to generate a mesh velocity field so that mesh distortion is minimum.

C. Moving Mesh Algorithm In order to be able to perform aeroelasticity computations using Lagrangian-Eulerian formulations, a body-

conforming mesh (conforming to the fluid-structure interface) has to be generated either globally at each time step, or the existing grid can be allowed to deform. The former option is expensive, especially in three dimensions. Most of the current approaches have been using incremental procedures to reconfigure an existing grid when the boundaries move. Tension spring analogy (Batina, 1991; Palmerio, 1994; Singh et al., 1995), or other physical analogies, such as potential flow (Kennon et al., 1992), have typically been used to move the mesh point. In the former case, the distribution of the spring stiffness is critical, since all these techniques try to maintain connectivity of the grid. Other possibilities for grid movement are methods in use in the moving finite element method (Miller & Miller, 1981), where evolution equations are derived for grid point motion from the governing equations.

Spring Analogy The widely used dynamics mesh algorithm is due to Batina (1991). Basically, it assumes that each mesh point is

connected to each other by spring, such that the original mesh corresponding to the undeformed structure is moved to conform to the instantaneous shape of the structure at the fluid-structure interface, the rest of the points are moved through reaction to the interface motion. The following linear system equation is solved by a Jacobi method for the displacement xi in the field:

( )∑ =Δ−Δj

jiij 0xxK (2.19)

The spring stiffness Kij is typically taken to be l-Pij where lij is the length of the edge nodes i and j.

Venkatakrishnan and Mavriplis (1995) found the value of P=2 to work well. When boundary displacements are presented, the grid becomes entangled and thus invalid. The above equation attempts to untangle the grid. However, there are many limits of the spring analogy (Venkatakrishnan and Mavriplis, 1995):

1) when large boundary displacements are prescribed, the method does not guarantee that the grid lines will not cross;

2) when relative motion is present, it results in excessive skewing of grid lines and eventually the lines do cross; 3) near the wall where maintaining grid spacing is desirable, the spring analogy is adequate for maintaining the

normal spacing, since the spring is stiff in this direction. But streamwise spacing can be controlled to a much less degree because the spring stiffness are much smaller, resulting in skewness of the grids; and

4) Displacements in x, y, and z are independent, such resulting grid crossing.


D. Present Solid-Brick Analogy for Unstructured Grid Remeshing Here we propose to develop, evaluate and demonstrate an innovative remeshing methodology for aeroelasticity

computations using unstructured grids. Our meshing method utilizes the following features of fluid-structure interaction:

1) CFD codes are formulated in Lagrangian-Eulerian frame, and CSD are in Lagrangian frame. It is possible to group the Lagrangian part together and solve simultaneously, and hence bring in a unified approach; and

2) Solid material can sustain shear stress, and hence can reduce distortion of the grid, otherwise encountered in the spring analogy.

As such, we will treat the fluid mesh as part of the CSD region, and solve by Lagrangian formulations. As a result, we have the equilibrium equation for fluid mesh as:

j,gijgit

vσ=

∂

∂ρ (2.20)

where ( ) k,gkiji,gjj,gigij uuu λδ++η=σ (2.21)

The boundary conditions to be satisfied are the continuity of displacement and velocity of fluid structure interface:

vgf-s = vsf-s (2.22)

ugf-s = usf-s (2.23)

Since equations (2.19) and (2.20) are casted in the same form as structure itself, it can be solved by the same FEM code as structure. There are several ways of solving the system:

1) Simultaneous solution: By assigning different physical properties, the fluid mesh and structure itself can be solved simultaneously. The requirement is that the inclusion of fluid mesh will not change the original structural property. This can be satisfied by assigning:

Young’s modules: Eg = Small Density: ρg = 0.0 The boundary conditions of (2.22) and (2.23) are automatically satisfied. 2) Segregated solution: Since fluid mesh will not impose any constrain to the structure itself, we can solve fluid

mesh dynamic equations after the structure equation is solved, with prescribed boundary conditions of (2.22-2.23).

3) Modal analysis approach: In many aeroelasticity computations, the structural solver is performed using modal analysis. The advantage of using modal analysis is that usually only a relatively small number of the natural frequencies will be excited to a significant degree. Thus accurate solutions with many fewer degree of freedom can be obtained. By modal analysis, one assumes that the system response can be written as weighted sum of the mode shape:

( ) ( ) ( ) ( ) nn3322l1 dtqdtqdtqdtqu ++++= K (2.24)

where di is the modal shape, and qj(t) is a time-dependent weighting factor to be applied to the modal shape. With each modal shape, di, we have the displacement at fluid structure interface as:

( ) ( )issfisgf du −− = (2.25)

With this boundary condition, if a deformation is solved for each of the first n modes, it is clear that the deformation in the domain will simply be the summation of all the modes:


( )( )∑=η

=1igiig dtqu (2.26)

were dgi is the structure "modal shape" of fluid mesh deformation due to the ith modal deformation at the fluid-

structure interface. Merit of Solid-Brick Analogy The present solid brick analogy for moving unstructured grid mesh has the following advantage: 1) Reduced grid distortion: Since the grid movement is solved as a displacement field from a structural solver,

it can reduce the grid distortion in that the Navier equation which governs structure displacement can sustain shear stress and hence can sustain the large shear distortion of the grid. Figure 1shows the point.

2) Preventing grid crossing: Unlike spring analogy, where x, y and z components of mesh movement, are independent, solid-brick analogy has a physically realistic deformation mechanism built in, as a result,

( )jjjiii y,y,x,z,yfx ΔΔΔΔΔ=Δ (2.27)

through the stiffness matrix. This could prevent grid crossing during the mesh motion.

3) Consistent FEM approach: By using the same FEM methodology for Lagrangian part of fluid structure interaction, it brings consistency of the solution in space and time domains, and the satisfaction of continuity at fluid structure interface.

4) Flexibility in Grid System and Discretization: The solid-brick analogy can be applied to many different types of grid systems: triangular, quadralateral, 3D tetrahedral, 3D hexahedral, 3D prism, … all the cells FEM can handle. It is also suitable for any type of CFD discretization method: Finite Element Method, Finite Volume Method, or Finite Difference Method.

E. Present Technical Approach During this project, a CFD code, CFD-ACE+, and a CSD code, FEMSTRESS will be coupled to study

aeroelasticity problems. Their salient features will be described in the following sections. CFD-ACE+

Spring Analogy for Mesh Movement; Brick Analogy for Mesh Movement; Equilibrium is required at each node; Equilibrium is set up for the element; The deformed element can be distorted The distortion can be sustained by the internal shear stress Figure 1. Equilibrium Diagram for Spring Analogy and for Brick Analogy for Mesh Movement


CFD-ACE+ solves Navier-Stokes equations in a Lagrangian-Eulerian frame. The continuity and momentum equation can be generally written as:

( )( ) 0dsvvddtd

sg =∫ ∫ ⋅−ρ+∀ρ

∀ (2.28)

( ) ∫ ∫ ∀+⋅=∫ ∫ ⋅−ρφ+∀ρφ∀

φ∀ ss

g dSdsqdsvvddtd

(2.29)

where φ are the Cartesian velocity components, v is the absolute fluid velocity, q the diffusive flux and Sφ the volume sources. ∀ is the computational cell volume and S are bonding cell surfaces, vg is the grid velocity. As the grid is moving with time, a space conservation law (SCL) must be satisfied,

∫ ∫=∀∀

dsvddtd

g (2.30)

FEMSTRESS FEMSTRESS is a structural dynamics code developed at CFDRC. Its main capabilities include:

• triangular, quad, tetrahedral, prisms, or brick element; • linear or high-order isoparametric element; • small or large deformation; • elastic or plastic stress; • isotropic or anisotropic materials; • thin to thick shell/plate elements; • modal analysis and eigenvalue solution; and • steady and dynamic analysis.

FEMSTRESS uses the finite element method and solves the equation of motion

[ ]{ } [ ]{ } [ ]{ } { }FqKqCqM =++ &&& (2.31)

where {q} is the displacement vector, [M] is mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, and {F} is the force vector due to the fluid dynamic load and shear stresses.

FEMSTRESS supports several boundary conditions specified from a Graphic User Interface (GUI): • specified load: pressure or concentration loads; • specified body force; • specified displacements: zero (fixed) or prescribed motion; • symmetry condition; and • contact condition: contacting surfaces and target surfaces for elastic-rigid contact, and, elastic-elastic

contact.

III. BENCHMARK TEST AND VALIDATION STUDY This chapter will benchmark the results of FEMSTRESS against STARS code on several structural dynamics problems. 2D and 3D aeroelasticity problems are then presented and comparisons are made wherever possible with experimental data or others results.

A. Geometrically Nonlinear Analysis of a Clamped Square Plate Figure 2 shows a square plate with all edges fixed under uniform load. For symmetric purposes, only one quarter

of the plate is considered. The mesh size is 5 x 5 with second order shell elements. For comparison purposes, Table 3-1 listed the results of present FEMSTRESS, analytical and STARS results.


Important data parameters: Thickness, h = 1.0 Young’s modulus, E = 2.0 x 1011

Poisson’s ratio, ν = 0.3 Length, a = b = 50 Uniform pressure, q = 6.0 x 104

Table 1. Center Deflection for a Clamped Square Plate with a Uniform Load

hWmax

FEMSTRESS STARS

qa4/Dh Theory Compute Difference percent Compute Difference

percent

109.3 (p) 1.20 1.1976 0.2 1.21 0.8

218.6 (2p) 1.66 1.6718 0.71 1.70 2.4

327.9 (3p) 2.00 1.9870 0.65 2.03 1.5

One can see that FEMSTRESS matches well with analytical results at

all three loads, and it gives slightly better accuracy than STARS. This may be due to the fact that STARS uses an incremental approach with localized linearization method to account for large deformation, whereas FEMSTRESS solves the nonlinear problem directly with Newton-Raphson method. When the incrementals of loads used in STARS are increased, so will the accuracy. Figure 3 shows a distribution of the deflection at 3p conditions.

B. Aeroelasticity of Flapping Wing Most flying animals use flapping wings to provide lift and propulsion

force. From the mechanics point of view, such action is an interactive process of fluid (air) with a flexible or rigid moving surface (wing). The process itself can be modeled with the present proposed fluid-structure interaction model.

Here a structured grid is used as it is more convenient for this problem. The computational grid and flapping wing model are shown in Figure 4. The computational grid is generated from the CFD-GEOM module. The wing has a rigid bar perpendicular to the rotation axis, and its motion is prescribed as sinusoidal at frequency of 10 Hz. The wing itself is elastic and responds to boundary movement (the bars rotation) and to the fluid dynamic force acting on it. Figure 5 shows the fluid field and wing deformation during the flapping process. For comparison purposes, the experiment's results (courtesy of Prof. Chi-Ming Ho, UCLA) are shown in Figure 6. As we can see, the predicted wing deformation features are the same as that from the experiment.

Figure 2. Clamped Plate with Uniform Load

Figure 3. Distribution of Deflector at qa4/Dh = 327.9 Calculated by FEMSTRESS


3.3 Aeroelasticity Computation of 2D Panel Flutter in Mach 3.0 Panel flutter is a self-excited oscillation of the external surface skin of a flight vehicle which results from the

dynamic instability of the aerodynamic inertia and elastic forces of the systems. Extensive flutter calculations for a rectangular panel employing approximate aerodynamic theory and comparison with experimental findings has been reported by Dixon (1966). Gupta (1996) used these data as validation cases. The 3D panel with aspect ratio of 2.0 will be reported in the next section, while here we will concentrate on the demonstration of solid-brick analogy for a 2D panel.

3.3.1 Model Definition Here we consider the flutter of an elastic beam in supersonic flow of Μ = 3.0 at zero angle of attack. The model is

shown in Figure 3-6. The beam is clamped at the ends and has the following properties.

Figure 5. Computational Results of Insect Flying

(a) computational grid (b) boundary conditions

Figure 4. Computational Model of Insect Flying

Figure 6. Computational Results of Microbat Flying in Comparison with Experiment


Length L = 1 m Thickness t = 0.005 m Young’s Modulus E = 0.6894 x 1011 N/m2 Density ρ = 2750 kg/m3 Poisson Ratio ν = 0.30 The supersonic flow is in the direction parallel to the beam. The computational grid for the beam and the fluid

domain is also shown in Figure 7. Here the fluid and structure share the same nodes, and have the same interface. Once the beam deforms, it will change the fluid grid around it. Our brick analogy is to model the structure (beam) and the fluid region as a solid-elastic body so that the new fluid grid will deform with the beam.

Reduced Order method

To accomplish unstructured grid deformation, we set the Young's modulus of the fluid mesh as: E = 10-2 N/m2 and Density = 0.0 kg/m2 A parametric study of the fluid mesh Young's Modulus was performed and it was found to have little effect on the

natural frequency of the beam. Once the modal analysis is made on the above domain, (beam and fluid mesh) the grid deformation corresponding to the first, second, third, fourth, fifth and sixth bending modes can be obtained. Figure 78 shows all the beam bending modes and the fluid deforming meshes.

(a) flutter model for a 2D beam with two ends clamped at Mach number 3.0

(b) triangular grid for structure and fluid (unstructured) with one to one interface match

Super Sonic Flow at M=3

Figure 7. Brick Analogy for 2D Beam Fluttering


Aeroelasticity Analysis The above configuration was analyzed to determine the critical flutter point. First the steady solution at a certain

dynamic pressure is obtained from the CFD solver, CFD-ACE+. The CFD solution used second-order upwind in space with 10% first-order upwind blending for stability. Higher-order scheme in space provides a better shock resolution as shown in Figure 9, where density contours are displayed. Then a disturbance in the form of constant pressure acting on the upper surface of the beam is applied for the first few time steps. The deflection of the beam at the mid point in the subsequent calculation was monitored to determine if the initial disturbance will grow or will decay and the corresponding rate. During the computation, 20 iterations between fluid and structure were used at each time step, which involved 20 time updates of the fluid mesh. With the current algebraic summation of the first 6 modes of the fluid mesh, the fluid mesh update is rather quick compared to other method (such as spring analogy). For the structure we assume there is a 2% material damping. The damping matrix:

[ ] [ ]KMC β+α= (3.1) is calculated by assigning 2% damping on the first two structure vibration modes.

Figure 8 Modal Shapes for Structural Deformation and Surrounding Fluid Grid Motion


To obtain the critical fluttering point, several computations were made with different free-stream dynamic pressures, in our case by change free-stream static pressure while maintaining the same Mach number of 3.0. Defining flutter parameter λ1/3 as:

DLq2 3

β=λ (3.2)

where:

1M 2 −=β ∞ , ( )2

3

112EtD

ν−= , q = free stream dynamic pressure (3.3)

We can plot the typical displacement at mid point with time at different flutter parameters in Figure 10. As we

can see, increasing flutter parameters results in an increase in fluid density and then an increase in the virtual mass for the structure system. The coupling effect can be appreciated for the decrease in the responding frequency, meaning a virtual mass contribution for the fluid has to be added in the structural response. One also notes that beyond critical flutter point, the freestream adds energy into the system, enough to overcome the structure damping, and hence cause the displacement to grow with time and eventually cause structure failure.

Figure 9. Density Distribution at Steady State of Mach Number 3.0


Figure 11 shows the damping plot as function of flutter parameters. To see the grid deformation at various

instances, Figure 12 show the density fields during a cycle of the fluttering along with the grid. One can see that during the panel deformation refraction waves form at several locations near the structure.

Figure 10. 2D Panel Displacement at Various Values of the Flutter Parameter

Figure 11. Damping Plot for 2D Panel


Figure 13. Model for 3D Panel

C. Aeroelasticity Analysis of 3D Panel The configuration considered here involved a flat plate 2 units long

and 1 unit wide. The fluttering data have been well documented (Dixon, 1966; Gupta, 1996). The model is shown in Figure 13. Since the geometry is rather regular, we will use hexahedral elements. The properties are:

Poisson’s Ratio ν = 0.3 Young’s Modulus E = 0.6894 E11 N/m2 Panel Density ρ = 275.00 kg/m3 Panel Thickness t = 0.005 Mach Number M∞ = 3.0

Free Vibration Analysis Free Vibration Analysis is first carried out for the first ten modes of

the plate. Their deformation are given in Figure 14. To validate our natural frequency, Table 2 compares the value against STARS code.

Figure 12. Density Distribution and Grid Deformation During Non-linear Aeroelasticity

Computation of a 2D Panel


Table 2. Natural Frequency of 2 x 1 Panel Frequencies Hz

Mode FEMSTRESS (present code) STARS (J. of Aircraft) 1 14.8548 14.7965 2 23.7507 23.5866 3 38.6067 38.3150 4 50.5702 50.3578 5 59.4357 58.9320 6 59.4617 59.0684

Figure 14. First Ten Modes of 3D Panel


As seen, both codes give almost the same results.

Solid-Brick Analogy As the whole computational grid is quite large, only small portions of the fluid grid around the plate is considered

for grid remeshing. Here the outer boundary is specified as zero displacement. The corresponding grid deformation for each of the first six modes are shown in Figure 15. This can significantly reduce time required to prepare the modal data.

Fluid Dynamic Data First the steady state fluid solution is obtained. Since it is supersonic flow, a 3D bow shock is observable as seen

in Figure 16.

Figure 15. Modal Shape for Grid Deformation


Aeroelasticity Analysis The initial disturbance is introduced in form of a pulsed pressure force on one side of the panel. Once the pressure

is released, the vibration of the mid point of the panel is monitored. The flutter speed is located by carrying out a series of computations with different dynamic pressure values. The time step size is selected to have 60 steps in the first dominate mode. Using the same flutter parameter λ1/3 b/a

where:

Daq2 3

β=λ , 1M 2 −=β ∞ , ( )2

3

112EtD

ν−= (3.4)

The typical displacement response at three flutter parameters is shown in Figure 17. At sub-critical state, the

distribution will decay with time, and it will grow with time at supercritical flutter parameter. The growth rate can be calculated from the plot in Figure 17 and they are plotted in Figure 18.

Positive value of damping means delays of displacement and velocity with time, and negative value of damping

indicates divergence of the solution. Table 3-3 shows the critical flutter parameter.

Figure 16. Three Dimensional Perspective of Shock Front Around the 3D Panel

Figure 17. Displacement Plots for 2 x 1 Panel at Various Values of the Flutter Parameter


Table 3. Critical Flutter Parameter for 2x1 Panel Dixon STARS Present 5.30 5.03 5.007

Figure 19 shows the density distribution at the mid-place of the 3D panel. The present calculation shows a good

agreement in flutter parameter with others' results which are validated with experiments.

Figure 18. Damping Plot for 3D 2x1 Panel

Figure 19. Density Response to the Fluttering of 2 x 1 Panel at M = 3.


D. Nonlinear Aeroelasticity Analysis of Cantilever Wing This case was taken directly from STARS user manual, as it has all the calculation parameters and flutter

parameters. The configuration is shown in Figure 20, with a NACA 0012 wing clamped on one end, and free at the other end, subject to flow at 0.5 Mach number and zero angle of attack. The important data parameters are as follows:

Wing span = 2.0 (m) Wing chord length = 1.0 (m) Mach number = 0.5

Angle of Attack = 0° Speed of Sound of Infinity = 340.129 m/sec Structural Data: Young's modulus = 6.8967 × 1010 N/m2 Poisson's ratio = 0.3 Density = 2764.925 kg/m3 Material damping = 2% The tetrahedral element cells are generated using CFD-GEOM with advancing front method and they are shown

in Figure 21.

Figure 20. Cantilever NACA 0012 Wing Model


Structural Model Analysis In the STARS' manual (Gupta, 1997), the NACA 0012 wing was modeled as plate with constant thickness of

0.0127, while we are modeling the wing as a solid structure. For validation purposes, we first set up a separate plate/shell model for the NACA 0012 wing to simulate the same condition as STARS. The mode shape and the frequencies are shown in Figure 22. The corresponding mode shape and frequencies for the 3D tetrahedral element are also given in Figure 22. We can conclude that:

a. FEMSTRESS predicts the excellent frequencies agreement with STARS for the plate/shell model for all first six modes;

b. The 3D hexahedral element predicts very close agreement with the plate model in bending mode (first, second, and third) but has a higher frequency in torsion mode (first, second, and third). This is due to the fact that variable thickness of the wing along the chord directly has an enhancing effect on the torsion stiffness; and

c. 3D model predicts 2 more horizontal bending nodes which do not exist in plate/shell mode. Apparently, in the plate/shell mode only two rotation angles (vertically bending and torsion) are considered.

Figure 21. Tetrahedral Cells Around NACA 0012 Wing


(a)


In our following computations, we will use the 3D solid tetrahedral model with one to one surface and grid match

for the aeroelasticity computation.

Reduced Order method For each of the first six mode shapes of the 3D model, we can obtain the corresponding fluid mesh deformation.

Figure 23 illustrates all the mesh movements for each mode. The summation of all these modes will give the instantaneous grid deformation at each grid point.

(b)

Figure 22. The First Six Mode Shape and Frequencies for a NACA 0012 Wing using 3D Solid Model and using Plate/Shell Model


Steady-State Flow Field First the steady-state solution is obtained at ground level with M=0.5 and angle of attack α=00. The pressure

distribution on the wing surface is given in Figure 24. The general feature is similar to that by STARS.

Figure 23. Moving Grid Mode Shape


Stability Analysis We assume that material has 2% damping which in turn leads to

KMC β+α= (3.5)

and α, and β value can be determined from

ξω=βω+α i2

i 2 (3.6)

For the first two ωi values, we can solve for α and β. The wing is distributed by an impulse pressure at the upper surface, and the following responses are monitored at

the wing tip leading edge point. The damping values are then calculated from the displacement with time curve, with typical values shown in Figure 25. The computations were repeated for several values of density ratios and then results are plotted in Figure 26.

Figure 24. Steady-State Pressure Distribution over a Cantilever NACA 0012 Wing at M=0.5, α=00


The critical point here is 0.85, which is slightly different from that predicted by STARS, which is 0.92. This may

be due to: 1. different torsion natural frequency; or 2. different material damping value. It appears that STARS used a damping value of ≥ 3%.

E. AGARD 445.6 Wing Model AGARD wing 445.6 model has been used by many to validate computational aeroelasticity. The plan view of the

wing is shown in Figure 27 and is called AGARD Standard Aeroelasticity Configuration. It was tested in the Transonic Dynamic Tunnel at NASA Langley Research Center (Yates, 1987; Yates, et al., 1963). It is a semispan wall-mounted model having a quarter-chord sweep angle of 45° (see Figure 28), a panel aspect ratio of 1.65 and taper ratio of 0.66. The wing had a NACA65A004 airfoil section and was constructed of laminated mahogany. To reduce the stiffness, some wings had holes drilled through them and filled with foam. A series of wings was flutter tested both in air and in heavy gas in the TDT at NASA Langley Research Center. Since the majority of published

Figure 25. Displacement Response of an Internal Distribution for NACA 0012 Wing at M=0.5, α=0°

Figure 26. Aeroelasticity Stability Plot for a Cantilever NACA 0012 Wing at M=0.5, α=0°


calculations for this model are for the "weakened Model #3", tested in air, as the test covered the largest transonic speed range and showed a significant transonic dip effect, we will use this model and data in our current study.

Structural Vibration Analysis The measured modal frequencies and computed modal shape are reported by Yates (1987) and Yates, et al.

(1963). These modal data are used by several investigators directly for structural analysis (Bennett et al., 1989; Edward, 1996; Rausch et al., 1993). Since we model structure and fluid simultaneously, we will first calibrate the vibration modes of the wing. We used the same assumption as Yates (1987): the wing is solid, and is made of a homogeneous orthotropic composition. The density is calculated based on experimental panel mass and calculated volume from our FEM, as 418.65 Kg/m3. We used the same elastic moduli and Poisson ratio, which represent the anisotropic character of the laminated-mahogany wings used by Yates (1987).

G = 0.059745 x 106 psi, E = 0.47072 x 106 psi; ν = 0.310 The tetrahedral cells were used to compute the first five modes and they matched with benchmark results very

well. Figure 29 shows each modal shape and the corresponding values from STARS (Gupta, 1996) and Yates (1987) which have been widely used in computational aeroelasticity. The slight difference in torsion frequency may be due to the difference in FEM model. We used 3D tetrahedral elements whereas Yates used plate model. Figure 30 shows the top view of the modal deflections, they match well with those shown by Yates.

Figure 27. Plan view of AGARD Wing 445.6 Standard Aeroelasticity Configuration

Figure 28. Wing Panel Dimensions


Reduced Order method for Grid Deformation Figure 31 shows the surface grid and the background grid used to generate 3D tetrahedral cells around and inside

the AGARD wing. The unstructured mesh is generated by an advancing front method built into CFD-GEOM software package. The computational mesh extends 10 root chord lengths above/below and upstream/downstream of the wing surface to rectangular outer boundaries. At the plane of symmetry the cells are stretched away from the wing. The complete mesh contains 92,700 tetrahedrals. The unstructured fluid mesh was deformable and will move with the wing itself. Since only the first four modes are used to represent structure deformation, we subsequently only used the corresponding mode shapes to represent the grid motion. The computations were made using modal

Figure 29. Calculated Modal Deflection and their Frequencies for 2.5 Foot Weakened Model #3

Figure 30. Contours of Calculated Modal Deflections for 2.5 Foot Weakened Model #3


analysis combining structure and fluid meshes. Figure 32 shows the fluid grid mesh for the first four modes of the wing deformation. The superposition of the first four modes will give the fluid grid at any instance in time.

Steady State Flow Field The tabular data for critical flutter parameter was listed from the experiments of Yates et al., (1963). We used the

same Mach number and the same geometric size. Streamwise Chord Length at Wing Root = 1.833 ft Streamwise Chord Length at Wing Tip = 1.208 ft Mach Number Tested M = 0.499; M = 0.678; M = 0.901; M = 0.960; M = 1.072; M = 1.14; Angle of Attack = 0° First the steady state CFD solutions were obtained for each Mach number. One of the sample pressure

distributions is shown in Figure 33 at M = 1.14. At this supersonic flow condition, there is a shock front on the upper surface and the lower surface of the wing. The shocks were well captured in the present computation. The pressure distribution is similar to those obtained by Rausch and Batnia (1993).

Figure 31. Computational (Tetrahedral) Grid Around the AGARD Wing

Figure 32. Fluid Meshes for the First Four Modes of Wing Deflection


Aeroelasticity Analysis The calculation of each flutter point was started by obtaining a steady-rigid solution at a specified Mach number

condition. As the wing is symmetric and the angle of attack is zero, there is no need for the static-aeroelastic solution. The dynamic aeroelastic calculation was started by perturbing the wing using uniform pressure on the upper surface. To bracket the flutter point, time-marching calculations were performed for several values of dynamic pressure. The aeroelastic responses resulting from the dynamic aeroelastic calculation was analyzed to determine the damping and frequency components. Typically, we used 100 time steps per dominant cycle, within each time step there are 10-20 iterations between fluid-structure coupling to ensure convergence. General number of cycles per flutter point is about 5. It takes approximately 20 hours of CPU time on a Dell PC 700E machine to complete each flutter point. The typical response of displacement at several flutter parameters is shown in Figure 34. These responses were analyzed to determine the growth or decay rate of the disturbance, and they are plotted in Figure 35 for all of the Mach numbers studied. By interpolation we can determine the critical flutter speed index as a function of Mach number. This is plotted in Figure 36. The experimental data and the result from STARS are also shown in Figure 36. It can be seen that the present result were able to predict reasonable flutter boundary and typical transonic dips.

Figure 33. Steady State Pressure Distribution at M = 1.14 on AGARD Wing

Figure 34. Aeroelastic Response of AGARD 445.6 Wing at Mach Number 1.072


Figure 35. Damping Properties at Various Mach Numbers for AGARD 445.6 Wing

Figure 36. Comparison of Flutter Result for AGARD 445.6 Wing


From the previous section, we have demonstrated the feasibility of reduced order method for grid deformation in dealing with different types of grid systems, including: 2D triangular grid, 3D tetrahedral, 3D hexahedral, and 3D prism grids. It also should be emphasized that the present reduced order method can be applied to various discretizations for CFD, including, FEM, FVM, or FDM.

VI. Conclusion A unified FEM algorithm for Eulerian-Lagrangian (fluid) - Lagrangian (structure) formulation is developed to

handle aeroelasticity problems using unstructured Navier-Stokes equation solvers. In the present approach, fluid mesh is assigned with solid properties and behave passively as a solid-brick. By solving structural dynamics and fluid mesh dynamics, grid distortion can be significantly reduced. In addition, using a modal superposition approach greatly reduces the CPU time required for regridding. Validation cases were made for insect flying with flexible flapping wing in incompressible flow; 2D panel fluttering under supersonic crossflow; 3D panel instability under Mach number 3.0; nonlinear aeroelastic analysis of cantilever NACA0012 wing, and aeroelastic analysis of AGARD 445.6 wing.

Comparison with analytical, benchmark, other published data and experimental data were made when possible. The conclusion from the present study can be summarized as follows.

1. A unified FEM is developed using solid-brick analogy to solve the Lagrangian part of fluid dynamics (mesh motion) and structural dynamics in a consistent manner to satisfy interface continuity of displacement, velocity and stress.

2. Solid-brick analogy can sustain shear deformation and hence can prevent negative volume in comparison with widely used spring analogy.

3. Computational efficiency of solid-brick analogy can be significantly increased by using modal superposition characteristics of structural dynamics.

4. Validation of FEMSTRESS against STARS for nonlinear large deformation problems, 3D free vibration problem and plate/shell problem all showed excellent agreement.

5. Solid-brick analogy showed great flexibility in moving complex unstructured grids including: 2D triangular grid, 3D tetrahedral, 3D hexahedral, 3D prism cells, and it is applicable to various discretization methods of CFD, FEM, FVM, or FDM.

6. For AGARD 445.6 wing, essentially transonic tip was captured, and reasonably good agreement with experiments was obtained.

Acknowledgments This study was supported by a NASA SBIR project. The author would like to thank Mr. Lenard Voelker of NASA Dryden Flight Center for active participation, guidance and encouragement during this project.

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Rausch, R.D., Batina, J.T., Yang, H.T. Y., 1993, "Three-Dimensional Time-Marching Aeroelastic Analysis Using an Unstructured Grid Euler Method," AIAA Journal, vol. 31, pp. 1626-1633.

Venkatavkrishnan, V. and Mavriplis, D.J., 1995, "Implicit Method for the Computation of Unsteady Flows on Unstructured Grids," AIAA-95-1705-CP.

Yates, E.C., Jr., 1963, "AGARD Standard Aeroelastic Configuration for Dynamic Response. Candidate Configuration I. - Wing 445.6", NASA TM100492.

Yates, E.C., Jr., Land, N.S., and Foughner, J.T., 1987, "Measured and Calculated Subsonic and Transonic Flutter Characteristics of a 45 Degree Sweptback Wing Planform in Air and in Freon-12 in the Langley Transonic Dynamics Tunnel," NASA TN D-1616.

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[American Institute of Aeronautics and Astronautics 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 47th AIAA