Download Material Point Method Applied to Fluid

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2010, Orlando, Florida
AIAA 2010-1464
Material Point Method Applied to Fluid-Structure
Interaction (FSI)/Aeroelasticity Problems
Patrick G. Hu 1, Liping Xue 2, Shaolin Mao 3, Ramji Kamakoti3, Hongwu Zhao3,
Nagendra Dittakavi 4, Zhen Wang 5, Qingding Li5, and Kan Ni 6
Advanced Dynamics Inc., Lexington, KY 40511, USA
and
Marty Brenner 7
Dryden Flight Research Center, Edwards, CA 93523
The material point method (MPM) combined with adaptive mesh refinement (AMR)
technique is applied to investigate complicated fluid-structure interactions (FSI) such as
aircraft wing flutter and other aeroelastic problems. The advantage of MPM over traditional
fluid-structure interaction (FSI) methods is its computational efficiency, accuracy and
robustness. MPM avoids explicit discretization of convection terms in momentum equations
and the fluid-structure interaction is realized by coupling the fluid and solid stress at the
interface. The states of material points are updated through the solution on background
Cartesian grid node which is fixed during the calculation, thus no mesh distortion problem
needs to be dealing with in the case of large deformation. The dynamic AMR can effectively
decrease the total number of background mesh nodes and material points for complicated
FSI problems. Preliminary results indicate that MPM with dynamic AMR can be a powerful
tool to simulate FSI/aeroelastic problems for aerial vehicles. A 3D MPM code (a part of
ASTE-P toolset) has been developed at Advanced Dynamics Inc. (ADI) with the ability to
model complex fluid-structure interaction (FSI) problems.
I. Introduction
T
HE Material point method (MPM) originally evolved from the particle-in-cell (PIC) method and now has being
widely applied in modeling of solid mechanics problems such as body impact/contact, localization of big
gradient values, crack propagation, penetration, perforation, fragmentation, and interactions of different material
phases.1 The pioneering work of MPM has been credited to Sulsky and co-workers.2-5 The essential idea is to take
advantage of both the Eulerian and Lagrangian methods. Recently, MPM has gathered interest in solving fluidstructure interaction (FSI) and multi-phase flow problems. However, MPM has encountered some difficulties to
solve large-scale fluid-solid interaction problems, primarily due to higher requirement of memory compared to
traditional computational fluid dynamics (CFD) methods. This can be circumvented by employing adaptive mesh
refinement (AMR) and parallelization strategies which is currently being implemented in our work.
This paper reports the latest development of MPM to model fluid-structure interaction with applications to
aerodynamics and aeroelasticity. The development of a multi-physics numerical code, a part of ASTE-P toolset6-9, is
underway at Advanced Dynamics, Inc. (ADI) based on material point method. Initial testing shows the advantage of
modeling complex deformation involved with FSI. The dynamic adaptive mesh refinement (AMR) technique and
MPI parallelization technique have been successfully employed in the MPM code in order to improve the
1
President, Principal Scientist, Senior Member AIAA
Principal Scientist, corresponding author, [email protected], Member AIAA
3
Senior Research Scientist, Member AIAA
4
Research Scientist, Member AIAA
5
Research Scientist
6
Principal Scientist, Member AIAA
7
Aerospace Engineer, Senior Member AIAA
2
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American Institute of Aeronautics and Astronautics
Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
computational accuracy and efficiency for large deformation, boundary treatment and capturing shocks10. In addition,
most widely used turbulence models are integrated to ASTE-P toolset for general purpose applications.6-9. Recently,
more effort has been involved with FSI process to improve the accuracy of interface between fluid and solid10. To
the best of the authors’ knowledge, there are no open publications to report MPM applications for complex
FSI/aeroelasticity problems.
The objective of this study is to investigate the fluid-structure interaction problem for aerospace applications
using MPM. Special focus is given on fluid-structural coupling process and boundary treatment. The numerical
results are compared with available experiment data in order to benchmark the MPM.
II. Mathematical Models
The momentum equation for MPM is solved on a Cartesian background grid (Eulerian frame). The fluid and
solid domains are discretized into a collection of material points. For time evolution of the solution, the information
from the material points are projected onto the background Cartesian grid nodes using shape functions. The gridbased solution is then used to update material points using the appropriate shape functions. The material points are
free to move and interact with each other, thereby eliminating any mesh distortions/skewness encountered by
traditional grid-based CFD/CSD coupled methods.
A. Governing Equations
The mass and momentum conservation of fluid and solid are given by the same equations:
dρ

+ ρ ∇ • v =0
dt
(1)


ρ a = ∇ •σ + ρ b
(2)

 


where ρ (x, t ) is the density of material point, a (x , t ) is the acceleration, v (x , t ) is the velocity, σ (x, t ) is a stress
 
tensor, and b (x , t ) is the specific body force. The mass, velocity, and external force of material points are
interpolated to the background grid nodes by
mg = ∑ S g p m p
(3)
p
1

vg =
mg
f gext =
∑S
gp

mp vp
(4)
p
∑S
ext
gp f p
(5)
p
where the subscript (g) represents quantities on the background Cartesian grid node and subscript (p) indicates
quantity on the material point. The summation shows the scope of impact region as indicated by grid shape function
S g p . The gradient of velocity at a material point is computed by the interpolation formula,


∇ vp =
∑ Gg p v g
(6)
p
where G g p is the gradient of shape function of node (g) evaluated at the position of material point (p). Equation (6)
is used to compute the Cauchy stress at the material point. The internal force at the background grid node is
determined by

f gint = ∑ Gg pσ pV p
 ext  int

mg=
ag f g + f g
(7)
p
(8)
The velocity and stress or strain rate is updated based on time evolution scheme, for example, the forward Euler time
integration, as given below:
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


v g (t + ∆t ) = v g (t ) + a g ∆t



v p (t + ∆t ) = v p (t ) + ∑ S g p a g ∆t
(10)



x p ( t +=
∆t ) x p ( t ) + ∑ S g p vg ∆t
(11)
(9)
g
g
=
∆ε p
{
T
∆t


Gg p vg ( t + ∆t ) + ( Gg p vg ( t + ∆t ) )
∑
2 g
(
ρp =
( t + ∆t ) ρ p ( t ) / 1 + ∆t ⋅ tr ( ∆ε p )
}
)
(12)
(13)
B. Fluid and Solid Constitutive Equations
Fluid and solid are distinguished by the constitutive equation that gives the relation between stress and strain rate.
The stress tensor for compressible flow is given by
σ = 2 µ ε − µ tr (ε )I − pI
where ε is the strain rate tensor defined as
ε =
2
3
{

T
1
∇ v + (∇ v )
2
}
(14)
(15)
The mechanical response of solids is quite different from elastic to plastic to viscoelastic etc. The Cauchy stress is
decomposed into a volumetric and a deviatoric part for hypoelastc material:
σ =
− pI + s
(16)
Equations (4)-(16) combined with suitable initial and boundary conditions form the complete system of equations to
simulate FSI/aeroelasticity problems.
III. Fluid Structure Coupling Process and Boundary Treatment
As discussed in our previous work10 there are two coupling procedures to solve fluid structure coupling problems
using material point method – implicit and explicit procedures. The implicit procedure solves the momentum
equations on the grid for the fluid and the structure at the same time. It does not need to explicitly track the interface.
Since at the interface, the fluid and the structure material points move in the same velocity field, the no-slip
boundary condition is implicitly enforced. It is a big advantage for most of the fluid-structure interaction problems.
The explicit procedure is similar to the traditional FEA-based methods. That is, the fluid and the structure are solved
separately. But there is no force and displacement mapping required. Since the solutions are solved on the same
background grid, instead of mapping the force and the displacement to the interface, the force and the displacement
can directly be interpolated to the background grid nodes when solving equations for the fluid or the structure. This
significantly simplifies the information exchanging process between the fluid and the structure. There is no loss of
accuracy and numerical stability at the interface because the same procedure is followed by both the fluid and the
structure. The explicit procedure is also flexible in applying different boundary conditions to the fluid and the
structure.
For particle method, the most difficult part is the boundary treatment. In our previous study10, wall boundary
condition for fluid flow around the interface is applied. The idea is to use virtual particles to treat the interface by
assigning them into the solid structure with equal distance to the closest real particles outside. Those virtual particles
carry the reflecting boundary conditions of the closest real particles outside of the interface. Wall boundary
condition gives good results in general, but it has serious limitations. One limitation is the ability to treat sharp
corners. If the interface is not smooth, and there are sharp changes in the curvature of the interface, wall boundary
condition might have problem at these sharp corners. Another limitation is the ability to treat thin solid bodies. If the
body is too thin, virtual particles from one side of the solid body might entangle with the virtual particles from the
other side of the body. This will totally destroy the boundary condition. Although fine mesh can be used to alleviate
this problem, the computational cost will also increase dramatically.
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Note that in the MPM formulation, the solutions are solved on the background grid, and then are used to update
the material properties, like velocity, position, and deformation. For those material points at the interface, instead of
using virtual particles to guarantee the boundary condition, we can use the interface and the nearby “normal”
material points to obtain the parameters that are needed for updating the properties. By getting rid of the virtual
particles, the boundary treatment becomes more flexible and robust. Least squares technique can be used for
obtaining the necessary parameters. Actually, least squares technique has already been applied to treat wall
boundary for finite volume method by Koh et al.11 In our treatment, for the material points at the interface, the
velocity, internal energy, and density will be obtained using the least squares technique. The position can be updated
using the velocity, pressure can be calculated using the equation of state, and the deformation can be obtained by
using the velocity gradient. We solve the solution on the material points with least squares technique, instead of
using the background grid and shape function.
Furthermore, the whole system can be solved using the least squares technique. As pointed out above, the
difference between this extension and traditional MPM is that the solutions are solved directly on the particles, not
on the grid. We call this method as pure particle method (PPM). Comparing to traditional MPM, this extension has
several advantages. First, the linear interpolation between the particles and the background grid is replaced by high
order least squares, which significantly increases the accuracy. Second, no limiter is needed to suppress the
numerical oscillations across discontinuities, such as shocks, for high speed fluid flows. Third, it increases the
flexibility in treating the fluid-solid interface. For this extension, the background grid is not used during the
calculation, and it is only used for convenience of tracking the particle position. In addition, it is also needed for
dynamic AMR that designated to refine the particles.
This extension is similar to Smoothed-Particle Dynamics (SPH) in a sense that for the two methods, the solutions
are all solved on the particles, and they all use certain functions to interpolate the properties onto the particles. A big
difference between the two methods is the ability to treat the interface. For SPH, virtual particles are usually used to
guarantee the boundary conditions, due to the kernel functions that are used. This has the same limitations as wall
boundary condition, which was discussed above. For PPM, no virtual particles are needed. The boundary condition
can be easily and accurately applied. We know that for SPH or other particle methods, one of the biggest challenges
is the boundary treatment. The new extension we proposed to current MPM is a breakthrough in advancing particle
methods.
IV.
Preliminary Results and Discussions
Along with other enhancements we have made to MPM 10, dynamic AMR and MPI parallelization have been
successfully implemented in the MPM code. Several test cases have been reported. Here we will show some results
using these methods.
A. Subsonic Flow past a RAE 2822 Airfoil
The RAE2822 airfoil has been used widely in the past for a variety of flow conditions. In our test case, we
studied the subsonic flow at Mach 0.9 past the airfoil. Characteristic boundary conditions are used at the inlet and
outlet, and far-field non-reflective boundary conditions are used at the top and bottom part of the domain. A
schematic of the computational domain along with the boundary conditions used is shown in Fig. 1. The
computations are performed using 4 levels of mesh refinement, or particle refinement as well. A zoom-in schematic
of the particles placed in different AMR levels is shown in Fig. 2.
A plot of the pressure contours is shown in Fig. 3. A compression wave can be observed in front of the airfoil
and there is also a transonic shock wave observed on the surface of the airfoil. A low pressure region is also seen
near the surface of the airfoil as expected. The results indicate that MPM can capture the flow physics of a CFD
problem in an accurate and robust fashion. To have a quantitative comparison, Fig. 4 plots the pressure coefficient
against the experimental values. It can be seen that other than a slight difference at the upper-leading edge, MPM
gives pretty good agreement with the experimental measurements.
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Figure 1. Schematic Showing the Computational DomainFigure 2. Zoom-in Schematic of the Particles Placed in
and the Appropriate Boundary Conditions.
Different AMR Levels.
Figure 3. Pressure Contours after 2500 Time Steps.
Figure 4. Pressure Coefficient Comparison
B. Supersonic Flow past a Bar
A supersonic flow of Mach 2.5 past an elastic bar is tested to validate the MPM unified solver for fluid-solid
interaction problems. In this test, the solid bar is placed in air with one end fixed and one end free. The bar is
simulated as elastic material with young’s modulus E=7.0e8 Pa. It is designed to be soft to ensure large deformation.
The supersonic air flow is from left to right. The upper and lower boundaries of computational domain are assigned
as reflecting boundary conditions. The particle regularization is also applied every 30 steps. The snapshots of the
fluid density field along with the contour of the solid bar as a function of time are shown in Fig. 5.
From Fig. 5 we can see that the reflected λ shock waves by the solid bar in the fluid flows are correctly
captured. With the large structure displacement and deformation, the interface tracking points are working well, and
the fluid-structure interface is precisely captured. The coupling between the fluid and structure dynamics is also
working well. The simulation results clearly validate our algorithm and demonstrate the ability of the code in
treating the fluid and structure coupling problems with large structure displacement and deformation.
C. Steady-State Subsonic Flow past a Sphere
To test the capability of our solver for three-dimensional geometries, a subsonic flow at Mach 0.8 past a sphere
is studied. In this test, a sphere of diameter 1 unit is placed inside a domain of size 10x10x10 units. We use 3 levels
of grid refinement for this case with the coarsest mesh size of 40x40x40, which leads to an initial total number of
material points of 1.4 million. We compared the drag coefficient of the sphere at steady state to experiment. The plot
of drag coefficient is shown in Fig. 6. As can be seen from the figure, reasonable agreement is achieved. The
pressure contours on the surface of the sphere as well as the surrounding flow field for 2 perpendicular planes is
shown in Fig. 7. A close-up view of velocity contours is shown in Fig. 8(a) to demonstrate the particle
distribution/movement near the surface of the sphere. The streamlines in the x-z plane is also depicted in Fig. 8(b).
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Figure 5. Simulation Results about Solid Bar Vibration Induced by a Supersonic Inflow (the Time
History Is from Left to Right and from Upper to Lower).
Figure 6. Drag Coefficient Comparison for Subsonic
Flow past a Sphere at M=0.8.
Figure 7. Pressure Contours on Surface of
Sphere as well as Surrounding Flow Field.
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(a)
(b)
Figure 8. (a) Particle Distribution near Surface of Sphere along with Velocity Contours. (b) Streamlines
in the X-Z Plane.
D. Fluid-Structure Interaction of a Subsonic Sphere
Here, we extend the above case to perform a fluid-structure interaction past a subsonic sphere. The flow
conditions are identical to the previous example, except here an elastic material with young’s modulus E=7.0e6 Pa
and density=2.7 Kg/m3 is assigned to the sphere. The sphere is designed to be soft and light to ensure large
deformation and movement. The sphere can move freely in the simulation domain. Fig. 9 shows the surface mesh of
the sphere at two different time steps. It can be clearly seen that the sphere is moving under the subsonic flow. The
hydrostatic pressure contours, along with the velocity streamlines in the X-Y plane at 500 steps are also shown in
Fig. 10 to demonstrate the fluid-structure coupling effects. It can be seen that in the solid, the hydrostatic pressure at
the front is higher, and at the back is lower, which is consistent with the hydrostatic pressure in the fluid. The
hydrostatic pressure is continuous at the fluid-solid interface, which means that the pressure from the fluid is
correctly transferred to the solid. The velocity streamlines are also continuous at the fluid-solid interface, and
penetrate through the solid, which demonstrates the velocity coordination between the two. The simulation results
clearly demonstrate the capability of the solver in handling three-dimensional fluid-solid interaction problems.
Figure 9. Surface Mesh of the Sphere at Two Different Times.
Red – Initial Surface Mesh; Blue – Surface Mesh at 500 Time Steps.
E. Subsonic Flow past a Model Aircraft
An Aluminum Fighter Aircraft (AFA) is used to test the code’s ability of handling complex geometries. As
shown in Fig. 11, it is a generic aircraft with a wing aspect ratio of 4.8, taper ratio 30%, leading edge sweep 37
degrees, and a weight of 30000 lbs. The total wing span was about 9 units with a length of 18 units. The domain size
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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.
Figure 10. The Hydrostatic Pressure Contours, Along With the Velocity Streamlines in the
X-Y Plane at 500 Time Steps
Figure 11. Schematic of Aluminum Fighter Aircraft (AFA)
We performed two calculations, one corresponding to an angle of attack of 0 degree and one with 5 degrees. The
Mach numbers for both cases are 0.8. Plots pertaining to zero degree angle of attack are shown in Fig. 12 and those
pertaining to 5 degrees AOA are shown in Fig. 13. We can clearly see the two counter-rotating vortices at the tip of
the main wings for the 5 degrees AOA case, which is consistent with the theory. This case shows great promise of
the particle-based methods for dealing with complex geometries.
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Figure 12. Pressure Distribution on the Aircraft Surface and Surrounding Fluid.
Figure 13. Streamlines past AFA at 5 Degrees Angle of Attack
V. Conclusions and Recommendation for Future Work
MPM has been significantly extended and enhanced by ADI to apply to complex fluid-structure interaction and
aerodynamic problems. Test cases have shown that the method can accurately predict subsonic, supersonic, and
hypersonic flows around complex geometries. Its ability in treating nonlinear large structure deformation, and its
straight-forward coupling procedure, makes it a unique and powerful tool for solving complex fluid-structure
interaction and aeroelastic problems. Combined with dynamic adaptive mesh refinement technique and MPI
parallelization technique, it is ready to be used for simulating large dimension, complex geometry fluid-structure
interaction and aerodynamic problems. The next step is to (1) integrate it into a multi-physics numerical toolset,
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ASTE-P, which is being developed by Advanced Dynamics Inc, and (2) apply the integrated code for more complex
fluid-structure interaction and aeroelasticity applications.
Acknowledgement
This work is supported by NASA SBIR Phase I under contract No. NNX07CA39P and Phase II contract No.
NNX08CA39C and Kentucky State SBIR Matching Fund Phase I under contract No. KSTC-184-512-07-018 and
Phase II under contract No. KSTC-184-512-07-032.
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