Advanced Computational Methods for 3D Simulation ...

T. Tezduyar, V. Kalro and W. Garrard, "Advanced Computational Methods for 3D Simulation of Parafoils", AIAA Paper 99-1712, Proceedings of the 15th CEAS/AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, Toulouse, France (1999), http://dx.doi.org/10.2514/6.1999-1712

Corrected/Updated References 1. S.K. Aliabadi, W.L. Garrard, V. Kalro, S. Mittal and T.E. Tezduyar, "Parallel Finite Element Computation of the Dynamics of Large Ram Air Parachutes", AIAA Paper 95-1581, Proceedings of AIAA 13th Aerodynamic Decelerator Systems Technology Conference and Seminar, Clearwater Beach, Florida (1995). 2. W.L. Garrard, T.E. Tezduyar, S.K. Aliabadi, V. Kalro, J. Luker and S. Mittal, "Inflation Analysis of Ram Air Inflated Gliding Parachutes", AIAA Paper 95-1565, Proceedings of AIAA 13th Aerodynamic Decelerator Systems Technology Conference and Seminar, Clearwater Beach, Florida (1995). 3. V. Kalro, S. Aliabadi, W. Garrard, T. Tezduyar, S. Mittal, and K. Stein, "Parallel Finite Element Simulation of Large Ram-Air Parachutes", International Journal for Numerical Methods in Fluids, 24 (1997) 1353-1369. 4. V. Kalro, W. Garrard and T. Tezduyar, "Parallel Finite Element Simulation of the Flare Maneuver of Large Ram-Air Parachutes", AIAA Paper 97-1427, Proceedings of AIAA 14h Aerodynamic Decelerator Systems Technology Conference and Seminar, San Francisco, California (1997). 5. T. Tezduyar, V. Kalro and W. Garrard, "Parallel Computational Methods for 3D Simulation of a Parafoil with Prescribed Shape Changes", Parallel Computing, 23 (1997) 1349-1363. 6 T.E. Tezduyar, M. Behr and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces -- The Deforming-Spatial-Domain/Space-Time Procedure: I. The Concept and the Preliminary Numerical Tests", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 339-351. 7. T.E. Tezduyar, M. Behr, S. Mittal and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces -- The Deforming-Spatial-Domain/Space-Time Procedure: II. Computation of Free-surface Flows, Two-liquid Flows, and Flows with Drifting Cylinders", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 353-371. 8. S.K. Aliabadi and T.E. Tezduyar, "Space-Time Finite Element Computation of Compressible Flows Involving Moving Boundaries and Interfaces", Computer Methods in Applied Mechanics and Engineering, 107 (1993) 209-223. 12. T.E. Tezduyar and T.J.R. Hughes, "Finite Element Formulations for Convection Dominated Flows with Particular Emphasis on the Compressible Euler Equations", AIAA Paper 83-0125, Proceedings of AIAA 21st Aerospace Sciences Meeting, Reno, Nevada (1983). 13. T.E. Tezduyar, "Stabilized Finite Element Formulations for Incompressible Flow Computations", Advances in Applied Mechanics, 28 (1992) 1-44.

14. A.A. Johnson and T.E. Tezduyar, "Mesh Update Strategies in Parallel Finite Element Computations of Flow Problems with Moving Boundaries and Interfaces", Computer Methods in Applied Mechanics and Engineering, 119 (1994) 73-94. 17. T.E. Tezduyar, M. Behr, S. Mittal and A.A. Johnson, "Computation of Unsteady Incompressible Flows with the Stabilized Finite Element Methods--Space-Time Formulations, Iterative Strategies and Massively Parallel Implementations", New Methods in Transient Analysis, PVP-Vol.246/AMD-Vol.143, ASME, New York (1992) 7-24. 18. T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson and S. Mittal, "Massively Parallel Finite Element Computation of Three-dimensional Flow Problems", Proceedings of the 6th Japan Numerical Fluid Dynamics Symposium, Tokyo, Japan (1992). 19. T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, and C. Waters, "3D Simulation of Flow Problems with Parallel Finite Element Computations on the Cray T3D", Computational Mechanics '95, Proceedings of International Conference on Computational Engineering Science, Mauna Lani, Hawaii (1995). 20. T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro and M. Litke, "Flow Simulation and High Performance Computing", Computational Mechanics, 18 (1996) 397-412. 22. S. Mittal and T.E. Tezduyar, "Massively Parallel Finite Element Computation of Incompressible Flows Involving Fluid-Body Interactions", Computer Methods in Applied Mechanics and Engineering, 112 (1994) 253-282.

ADVANCED COMPUTATIONAL METHODS FOR 3D SIMULATION OF PARAFOILS T. Tezduyar 1 , V. Kalro 2 and W. Garrard 3 1

Mechanical Engineering and Materials Science Army High Performance Computing Research Center Rice University- MS 321, 6100 Main Street Houston, TX 77005-1892, USA 2

Fluent, Inc. 500 Davis Street, Suite 600 Evanston, IL 60201, USA 3

Aerospace Engineering and Mechanics 107 Akerman Hall University of Minnesota Minneapolis, MN 55455, USA

http://www.mems.rice.edu/TAFSM/ J anuary 8, 1999

Abstract In this paper we provide an overview of the computational methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) for 3D simulation of parafoils, particularly large, ram-air parachutes. The model is based on 3D NavierStokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The methods developed include a stabilized space-time finite element formulation that accommodates for the motion and shape changes, special mesh generation and mesh moving strategies, iterative solution techniques for the large, coupled nonlinear equation systems encountered, and parallel implementation of these methods on distributed-memory and shared-memory parallel computing systems such as the Thinking Machines CM-5, CRAY T3E, and multiprocessor SGI systems. This set of methods developed over the past few years gives us new, powerful tools that enable us to carry out parafoil simulations at new levels of sophistication and computational scales.

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1.

Introduction

Simulation of parafoils is known to be an inherently very challenging task in computational fluid dynamics. Various stages involved in the operation of these parafoils, such as the initial stages of inflation, transformation to a fully deployed parafoil, steady gliding, flare maneuver, and landing, makes the task very multi-faceted. Our computational research on this subject is part of a greater effort in simulation and modeling of airdrop systems which has been, for the past several years, one of the "targeted challenges" for the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http: I /www. mems. rice. edu/TAFSM/]. Our work on this subject was originally motivated by the need to use advanced computational strategies to address a technological issue of major interest to the US Army- design, deployment and operation of large, ram-air parachutes. Now our research in this area is being carried out, together with a number of other projects in airdrop systems, in collaboration with the US Army Natick Research, Development and Engineering Center. The ram-air parachutes we are interested in can be as large as 10,800 square-feet (60ft x 180 ft), with a payload capacity of as much as 42,000 lbs. The scale of these parachute systems, combined with the various stages of complex behavior described above, makes wind tunnel tests for such large parachutes impossible. Therefore, simulation and modeling, with the recent advances in computational methods and computing platforms, becomes a credible, promising complement to costly drop tests. The basic strategy in our approach to handle this very complex problem has been to divide the problem into a number of computationally tractable stages, and use for each stage a mathematical model as realistic as we can handle in terms of the capabilities of our methods and limitations of the computing platform available. Different levels of simplifying assumptions are used in this stage. Gradually, the simplifying assumptions are removed. For example, in this paper we assume that the changes in the parafoil shape are prescribed. This assumption relieves us from the additional challenge of solving for the structural deformation of the parachute. However, the methods are currently being extended by the T*AFSM to solution of the membrane equations assumed to be modeling the structural deformation, and simulation of the parafoil behavior in a coupled, fluid-structure interaction approach. As part of our stage-by-stage approach, earlier we carried out simulations for initial stages of inflation, transformation to a fully deployed parafoil, and steady glide (see [1-3]). Later we escalated the level of sophistication to coupled aerodynamics-dynamics interactions, specifically the flare maneuver stage [4, 5]. In the flare maneuver, the parafoil velocity is reduced by pulling down the trailing edges of the parafoil. In this overview paper, we give as an example this flare maneuver simulation. Because the deflection of the trailing edges produces an effect similar to that of flaps on aircraft wings, we will refer to this trailing edge deflections as pulling down the flaps. The mathematical model used in the computations consists of two components: 1) time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the 2

parafoil; and 2) Newton's law of motion governing the dynamics of the parafoil. The aerodynamic forces acting on the parafoil, which are needed in determining the motion of the parafoil, are derived from the computed flow field. For the aerodynamic computations, the position and displacement rate of the internal boundaries of the computational domain (i.e. the canopy ) are derived from the computed motion of the parafoil. These calculations need to be done in a coupled fashion. In our model we assume that the change in the shape of the parafoil is prescribed as a function of time based on shape and size information from design data. The governing equations are reviewed in detail in Section 2. The changes in the shape of the computational domain makes this problem a special case of a more general class of flow problems with moving boundaries and interfaces. The location of the boundaries and interfaces is unknown, and needs to be determined as part of the overall solution. There are a number of alternatives in terms of handling this general class of problems. A stabilized space-time finite element formulation with time-varying spatial domains, developed earlier [6- 8], is the method we use. This method, named DeformableSpatial-Domain/Stabilized Space-Time (DSD/SST) formulation, was developed for solution of flow problems with fluid-object and fluid-structure interactions, moving mechanical components, and free surfaces and two-fluid interfaces. In DSD /SST method we write the variational formulation of the problem in a space-time domain, and thus automatically take into account the time-variation of the spatial domain. Linear, or possibly higher-order, interpolation functions, which are continuous in space but discontinuous in time, are used. Because the functions are discontinuous in time, the computations are carried out one spacetime slab at a time, without adding a real fourth dimension to the computations. Still, the method is costlier compared to semi-discrete finite element methods, and should be used, in our opinion, only for complex problems that require a formulation that can handle moving boundaries and interfaces. For example, the stabilized space-time formulation was used earlier by other researchers [9, 10] to solve problems with fixed spatial domains for the purpose of better numerical stability and accuracy. This purpose, as important as we also believe it is, is not a strong enough justification in our view for using this relatively costly method for flow problems with just fixed domains. We note that, as an additional numerical benefit, the stabilization [11- 13] prevents the numerical instabilities and oscillations that might be encountered in flow computations involving high Reynolds numbers, high Mach numbers, shocks, and very thin boundary layers. The stabilization also assures numerical stability for computations in which the flow variables are approximated with the combination of interpolation functions which would normally not be acceptable. These finite element formulations are described in Section 3. As the spatial domain changes with time, the finite element mesh spanning that domain needs to be updated. This is an important component of methods for moving boundaries and interfaces. If not managed effectively, mesh update cost might overwhelm the cost of the computation. For example, if we remesh (i.e. generate a new set of nodes and elements) too often in problems with complex geometries, the cost of automatic mesh generation could

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become a major part of the overall computational cost. We have developed advanced mesh moving methods [3, 14] which reduce the frequency of remeshing. For simulation of parafoils with prescribed shape changes, we developed a special algebraic mesh generation and mesh moving method which requires no remeshing throughout the entire simulation. The mesh generation and update method used in the simulations are briefly described in Section 4. In 3D computation of flows with moving boundaries and interfaces, the finite element discretizations in space and time generate very large coupled, nonlinear equation systems. These equations need to be solved at every time step of the simulation. For this purpose we use the Newton-Raphson method. At each Newton-Raphson step, we need to solve a linear equation system to compute the increment vector. These linear systems are solved iteratively, where at each iteration the residual of the system is formed by an elementvector-based (matrix-free) method [15]. With this, we use a diagonal preconditioner and the GMRES update technique [16]. The solution method for the nonlinear equation system is summarized in Section 5. We have implemented our methods on distributed-memory parallel computing platforms such as the Thinking Machines CM-5, CRAY T3D and T3E, as well as the shared-memory parallel computing platforms such as the multi-processor SGI systems (see [17-20]). The parallel implementations are described very briefly in Section 6. The problem description and computational results for flare maneuver simulation as a numerical example is given in Section 7. This simulation involves solution of over 3.6 million coupled, nonlinear equations at every time step of the computations. These computations were carried on the CM-5. 2.

Mathematical Model and Governing Equations

Let nt be a 3D spatial domain that changes with respect to time t. The N avier-Stokes equations of incompressible flows can be written as follows:

au + u · Vu -

p( at

f) - V · a = 0

(1)

V ·u = 0

(2)

where pis constant density, u is the velocity, and f is the external force consisting of gravity. The stress tensor is defined as

u(u,p) =-pi+ 2J-Lc(u),

(3)

where pis the pressure, I is the identity tensor, 11 is the viscosity, and c(u) is the strain rate tensor: 1

c(u)

=

2((Vu) + (Vuf). 4

(4)

The Dirichlet- and Neumann-type boundary conditions are represented as u

= g on (ft) 9 ,

n.u

h on ( r t )h'

(5)

where (ft) 9 and (ft)h are complementary subsets of the boundary ft, n is the unit normal vector at the boundary, and g and h are given functions. The initial condition on the velocity is specified as u(x, 0) = Uo on rlo, (6) where u 0 is divergence free. The parafoil is modeled as an object with time-dependent geometry. The time-evolution of this geometry is assumed to be prescribed. The equations of motion for the parafoil system consist of the following:

(~) a,

F+W

(7) (8)

where F and M are the aerodynamic force and moment acting on the parafoil, and W is the gravitational force acting on the parafoiljpayload system with moment of inertia IM (with respect to the mass center of this system). Here a is the linear acceleration of the mass center of the parafoil/payload system, jj is the angular acceleration, and g is the gravitational acceleration. The aerodynamic force and moment acting on the parafoil is determined from the computed flow field. The flow boundary conditions corresponding to the parafoil surface are determined from the computed motion of the parafoil system. Therefore, the equations governing the flow and the equations of motion for the parafoil are solved in a coupled fashion. A zero-equation turbulence model is used in computing the flow past the parafoil. This model is a simplified version of the algebraic Baldwin-Lomax [21] model: the viscosity 1-" is replaced by 1-" + J-lt, where the eddy viscosity 1-lt is defined as

1-"t

l

pl2JwJ,

(9)

K:n(1- exp( -n+ jA+)).

(10)

K: = 0.41 and A+= 25.0 are constants, n is the normal distance to the wall, and n+ is the same distance expressed in wall units. In our computations this distance is measured from the closest node on the surface.

Herew is the vorticity vector,

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3.

Stabilized Space-Time Formulation for Moving Boundaries and Interfaces

Let us partition the time domain (0, T) into subintervals In = (tn, tn+d, where 0 = t 0 < t 1 < · · · < tN = T. Let On = Otn and fn = ftn· We define the space-time slab Qn as the domain enclosed by the surfaces On, On+I, and Pn, where Pn is the surface described by the boundary ft as t traverses In. As it is the case with ft, surface Pn is decomposed into (Pn) 9 and (Pn)h with respect to the type of boundary condition being imposed. For each space-time slab, we define the corresponding finite element function spaces (S~)n, (V~)n, (S;)n, and (V;)n· Over the element domain, this space is formed by using first-order polynomials in space and time. Globally, the interpolation functions are continuous in space but discontinuous in time. The stabilized space-time formulation for deforming domains is then written as follows: given (uh)~, find uh E (S~)n and ph E (S;)n such that Vwh E (V~)n and qh E (V;)n

1 w · + u · Vu + 1 + 1 1 + 2:: -TL(w, [L(u,p)+ L1 Qn

h p ( -8uh 8t

h

h

f ) dO

t:(wh) : u(ph' uh)dQ

Qn

qhV. uhdQ

Qn

1

net

e= l.

Q;'.

q) ·

p

pf) dQ

net

6V · whpV · uhdQ

Q;'.

e=l

+

J (wh)~.p ((uh)~- (uh)~) dO J(Pn)h wh · hhdP !ln

(11)

This process is applied sequentially to all the space-time slabs Q0 , Q1 , Q2 , .•• , QN-l· In the variational formulation given by Eq. (11), the following notation is being used:

L(w,q)

=

p ( ~7

+ u · Vw) + Vq- 2J-LV · t:(w),

(uh)~

lim u(tn ±c)

e-+0

1 (.. .

)dQ

Qn

1Pn (...)dP

], 1 (.. . j 1 (.. . In

-

iln

In rn

6

(12)

(13)

)dOdt,

(14)

)dfdt.

(15)

The computations start with

(16) Explanation of the terms in the variational formulation, and the definitions of the stabilization coefficients T and 6 are given in [3].

4.

Updating the Finite Element Mesh

In flow problems with moving boundaries and interfaces, as the spatial domain changes in time, updating the mesh efficiently becomes one of the most important issues. In our approach to this issue, the underlying philosophy is quite simple: try to update the mesh, as much as you can, by moving and deforming the mesh, with as few remeshings (generating a new set of nodes and elements) in between as possible. If the initial mesh is generated by an automatic mesh generator, the motion of the mesh is determined by solving a set of equations that govern this motion. That would constitute an automatic mesh moving approach. Although the computational cost for solution of the mesh moving equations is much smaller compared to the cost for solution of the flow equations, in general we would like avoid using an automatic mesh moving strategy. Furthermore, as we occasionally need to remesh, we need to again call the automatic mesh generator. This could be a very costly step, and sometimes overwhelming. That would be the second incentive for using, whenever we can, special mesh generators instead of an automatic mesh generator, and special mesh moving techniques instead of an automatic mesh moving approach. In our example simulation of the flare maneuver, the flap motion is described as a function of time by interpolating between the configurations with zero and full flaps. For the entire span of the parafoil, the flap length is 20% of the chord length. For the parafoil shape with full flaps, the flap deflection is constant for the outer 50% of the span, and decreases linearly to zero for the inner half. The duration of the flare maneuver used in the simulation is from flight data. During the flare, the parafoil is allowed to translate and pitch, where the transitional velocity components and the pitch angle are determined as part of the overall solution. To accommodate for pitching, the parafoil is enclosed in a cylindrical mesh zone which absorbs the deformation resulting from this motion (see Figure 1). To accommodate for the translation of the parafoil, the entire mesh, including the cylindrical zone, translates with the parafoil. A similar, 2D scheme was used earlier [22] to investigate the dynamics of 2D airfoils.

5.

Solution of the Nonlinear Equations

Full finite element discretization of the variational formulation given by Eq. (11) leads to a coupled, nonlinear equation system

(17) 7

where dn+l is the vector of unknown nodal values of u and p corresponding to the space-time slab between the time levels n and n +1. This equation system, typically involving millions of unknowns in our computations, needs to be solved at every time step. We solve this system with the Newton-Raphson method: (18) where d~+l is the value of dn+l from iteration k, and 6d~+ 1 is the correction computed for d~+I· In a · tual implementation, ( ~~) is com put d approximately, in the sense that som of the parameters, such a the mun rical stabilization coefficients, are frozen during the differentiation. We represent Eq. (18) as a system

Ax

=

b

(19)

that needs to be solved at every Newton-Raphson step, where A, in general, is a nonsymmetric matrix. For the sizes of systems we are dealing with, it is prohibitively costly to solve Eq. (19) with direct solution techniques. Therefore, we solve Eq. (19) with an iterative method which involves the evaluation of b- Axm at every such iteration, where Xm is the value of x from iteration m. The evaluation of a matrix-vector product Ax does not require the actual formation of the matrix A. This can be achieved by a computation of the form (aexe), where ae and xe are the element-level constituents of A and x, respectively, corresponding to element e. represents the finite element assembly operation to form the global arrays like A Here and x. In element-vector-based (matrix-free) computations, even the element-level matrices a e do not need to be formed or stored. Instead, we recognize that, because A originates from ( ~~), we can write

Ax = lim N ( d

+ t:x) -

e~O

£

N (d) '

(20)

where c is a suitably small numerical limit parameter. In addition to resulting in major savings in memory by storing the element-level vectors instead of matrices, this afproach allows us to circumvent the complex task of calculating the derivatives involved in \ ~~).

6.

Parallel Implementations

We have implemented the methods described in the previous sections on both sharedmemory and distributed-memory parallel platforms. In most of our implementations, the majority of the computational effort consists of operations performed at the element level.

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These operations can be executed fully in parallel because all operations for each element are local to the processor the element is assigned to. The element-level operations need to be accompanied by transferring data between the elements and global data arrays. This task is rather straightforward for shared-memory parallel computing, because all processors are accessing a common memory bank. However, for distributed-memory parallel computing, the data transfer task requires communication between different processors. This involves gather (local/element *- global/node) and scatter (local/element ---+ global/node) data transfer modes. This task has the potential of becoming a bottleneck. Therefore, careful partitioning of the total set of elements to the processors becomes essential for maximizing data locality (see for example [23]) and for minimizing off-processor communication. Our earlier distributed-memory parallel computations were based on a data-parallel paradigm, and were carried out on the CM-5. In our current research, we utilize the message-passing paradigm based on Message Passing Interface (MPI), and the computations are carried out on the CRAY T3E. 7.

Example Simulations: Flare Maneuver

The flare maneuver is used to reduce speed just prior to landing and thus reduce the impact when the payload touches the ground. The maneuver is initiated by pulling down the flaps on either side of the parafoil. This increases the effective camber and creates larger aerodynamic forces, and therefore reduces the speed. Flares may be initiated in a gradual manner by using winches, or by a jerk retraction using a pyrotechnically driven charge [24]. Figure 2 shows the flight path angle (t), pitch angle (8), angle of attack (a), and the rigging angle (fl,. ). The rigging angle plays an important role in the flare maneuver. For the purpose of a qualitative comparison, we first show in Figure 3 the velocity profiles from test data [24] for a winch-driven flare maneuver. The parafoil model used for our example simulations has a 10,800 square-feet (60 jt x 180 jt) area, and the system weighs 42,000 lbs. The parafoil has a Clark Y cross section, with ratio of line length to span (R/b) equal to 0.6. The flap deflection for the outer 50% of the span is "' 9.6ft. In these simulations the flare is modeled as a gradual transition. At the start and end of the flare the acceleration of the flaps during the retraction goes to zero smoothly. This resembles a winch-driven flare. The mesh consists of 469,493 nodes and 455,520 hexahedral elements. In this space-time computation, 3,666,432 coupled nonlinear equations are solved at every time step. The size of the Krylov subspace is set to 10, and this computation requires "' 30 sec per time step on a 512-node CM-5. The initial condition at t = O.Os corresponds to the parafoil in steady gliding descent. The flare is initiated at t = 3. 75s and the parafoil achieves maximum flap deflection at t = 22.5s. Boundary conditions consist of specifying the free-stream velocity at the inflow boundary, zero-normal velocity and zero-shear stress at the side boundaries, and traction-free condition at the outflow boundary. 9

Flare simulations are carried out at three different rigging angles: f.lr = 3°, 6°, and go. Figure 4 shows the time histories of flight angles, velocity components, moments, and forces. The reference point for the moment is the position of the payload. The steady glide speed prior to flare increases with increasing f.lr· During the initial stages of the maneuver, the trailing edge retraction results in increased aerodynamic forces and moments on the parafoil. The increased drag and the much smaller mass of the parafoil ( rv 9% of the total parafoil/payload system) causes it to rotate backwards from the payload. This tendency to rotate backwards is offset by the increased nose down moment. The increased drag also causes the system to slow down. There is a significant decrease in the horizontal speed and some reduction in the vertical speed. Correspondingly, there is an increase in a. As the flare maneuver continues beyond the t = 22.5s, the reduction in speed decreases the lift, and this causes the system to accelerate (under the influence of its own weight), and the vertical speed returns to a level close to that before flare. The horizontal speed maintains its reduced value and the system glides at an a higher than that prior to flare. The reduction in the horizontal speed is rv 20-25 fps.

8.

Concluding Remarks

In recent years, a number of computational methods and tools have been developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) for simulation of parafoils. This paper provides an overview of these methods and includes some example simulations of parafoil flare maneuver. We assume that the changes in the parafoil shape are prescribed. With this assumption, we limit our attention to models where we are not faced with solving for the structural deformation of the parachute. As governing equations, we use the N avierStokes equations of incompressible flows for the flow problem, and equations of motion for the dynamics of the parafoil system. These two sets of governing equations are solved in a coupled fashion. The methods developed include a stabilized space-time finite element formulation that accommodates for the motion and shape changes, special mesh generation and mesh moving strategies, iterative solution techniques for the large, coupled nonlinear equation systems encountered, and parallel implementation of these methods on distributedmemory and shared-memory parallel computing systems such as the Thinking Machines CM-5, CRAY T3E, and SGI systems. As numerical example, we show simulation of flare maneuver of a parafoil for three different rigging angles. We demonstrated that the methods described gives us new, powerful tools that enable us to carry out parafoil simulations at new levels of sophistication and computational scale. We also realize that a lot more needs to be done. Methods are being developed and tested by the T*AFSM for solution of the membrane equations assumed to be governing the structural deformation, and simulation of the parafoil behavior in a coupled, fluid-structure interaction approach. Of course, as any other research group in flow simulation and modeling, we are concerned about turbulence modeling and looking for better models which would allow us to carry out these type of

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computations more accurately but still with practical mesh sizes. Acknowledgment

This research was sponsored by ARO under grant DAAH04-93-G-0514 and by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-0003/contract number DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. References

[1] S.K. Aliabadi, W.L. Garrard, V. Kalro, S. Mittal, T.E. Tezduyar, and K.R. Stein, "Parallel finite element computations of the dynamics of large ram air parachutes", in AIAA Paper 95-1581, 13th Aerodynamic Decelerator and Systems Conference, Clearwater, Fl, (1995) 278- 293. [2] W.L. Garrard, T.E. Tezduyar, S.K. Aliabadi, V. Kalro, J. Luker, and S. Mittal, "Inflation analysis ram air inflated gliding parachutes", in AIAA Paper 95-1565, 13th Aerodynamic Decelerator and Systems Conference, Clearwater, Fl, (1995) 186-198. [3] V. Kalro, S. Aliabadi, W. Garrard, T. Tezduyar, S. Mittal, and K.Stein, "Parallel finite element simulation of large ram-air parachutes", International Journal for Numerical Methods in Fluids, 23 (1997) 1353-1369. [4] V. Kalro, W. Garrard, and T.E. Tezduyar, "Parallel finite element computation of the flare maneuver of a large ram-air parachute", in AIAA Paper 97-1427, 14th Aerodynamic Decelerator and Systems Conference, San Francisco, CA, (1997). [5] T. Tezduyar, V. Kalro and W. Garrard, "Parallel computational methods for 3D simulation of a parafoil with prescribed shape changes", Parallel Computing, 23 (1997) 13491363. [6] T.E. Tezduyar, M. Behr, and J. Liou, "A new strategy for finite element computations involving moving boundaries and interfaces - the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary tests", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 339-351. [7] T.E. Tezduyar, M. Behr, S. Mittal, and J. Liou, "A new strategy for finite element computations involving moving boundaries and interfaces - the deforming-spatialdomain/space-time procedure: II. Computation of free-surface :flows, two-liquid :flows, and flows with drifting cylinders", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 353-371. 11

[8] S.K. Aliabadi and T.E. Tezduyar, "Space-time finite element computation of compressible flows involving moving boundaries and interfaces", Computer Methods in Applied Mechanics and Engineering, 107 (1993) 209-223. [9] T.J.R. Hughes and G.M. Hulbert, "Space-time finite element methods for elastodynamics: formulations and error estimates", Computer Methods in Applied Mechanics and Enginee1·ing, 66 (1988) 339-363. [10] P. Hansbo and A. Szepessy, "A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations", Computer Methods in Applied Mechanics and Engineering, 84 (1990) 175- 192.

[11] A.N. Brooks and T.J.R. Hughes, "Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations", Computer Methods in Applied Mechanics and Engineering, 32 (1982) 199-259. [12] T.E. Tezduyar and T.J.R. Hughes, "Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations", in Proceedings of AIAA 21st Aerospace Sciences Meeting, AIAA Paper 83-0125, Reno, Nevada, (1983). [13] T.E. Tezduyar, "Stabilized finite element formulations for incompressible flow computations", Advances in Applied Mechanics, 28 (1991) 1-44. [14] A.A. Johnson and T.E. Tezduyar, "Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces", Computer Methods in Applied Mechanics and Engineering, 119 (1994) 73-94. [15] Z. Johan, T.J.R. Hughes, and F. Shakib, "A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids", Computer Methods in Applied Mechanics and Engineering, 87 (1991) 281-304. [16] Y. Saad and M. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems", SIAM Journal of Scientific and Statistical Computing, 7 (1986) 856- 869. [17] T.E. Tezduyar, M. Behr, S. Mittal, and A.A. Johnson, "Computation of unsteady incompressible flows with the finite element methods - space-time formulations, iterative strategies and massively parallel implementations", in P. Smolinski, W.K. Liu, G. Hulbert, and K. Tamma, editors, New 1\!Iethods in Transient Analysis, AMD-Vol.143, ASME, New York, (1992) 7-24.

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[18] T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, and S. Mittal, "Massively parallel finite element computation of three-dimensional flow problems", in Proceedings of the 6th Japan Numerical Fluid Dynamics Symposium, Tokyo, Japan, (1992) 15-24. [19] T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, and C. Waters, "3D simulation of flow problems with parallel finite element computations on the CRAY T3D", in Computational Mechanics '95, Proceedings of International Conference on Computational Engineering Science, Mauna Lani, Hawaii, (1995). [20] T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, and M. Litke, "Flow simulation and high performance computing", Computational Mechanics, 18 (1996) 397-412. [21] B. Baldwin and H. Lomax, "Thin layer approximation and algebaric turbulence model for separated turbulent flows", in AIAA Paper 78-257, AIAA 16th Aerospace Sciences Meeting, Huntsville, Alabama, (1978). [22] S. Mittal and T.E. Tezduyar, "Massively parallel finite element computation of incompressible flows involving fluid-body interactions", Computer Methods in Applied Mechanics and Engineering, 112 (1994) 253-282. [23] Z. Johan, Data Parallel Finite Element Techniques for Large-Scale Computational Fluid Dynamics, Ph.D. thesis, Department of Mechanical Engineering, Stanford University, 1992. [24] ARS Group, "Advanced recovery systems phase 2- final report- volume III- advanced control systems", Technical Report NAS8-36631, Pioneer Aerospace Corporation, 1990.

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Figure 1. Updating the Finite Element Mesh. The top image shows the entire mesh. The bottom left and central images illustrate how the pitch and flap motions are handled. The circular region around the parafoil undergoes deformation by a differential rotation. The bottom right image shows a close up view of the mesh near the parafoil.

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·-.

··a··-..,.----...... x-.. . . .

··-·-·-·-·- .... ·-· . c ............ :!"""-----

y

t

~

............ - - - - - : : : : . .....

: I

,' i

' i

i i i

z

i

a=y+8 X p

y

~----------------~x

Figure 2. Example Simulations: Flare Maneuver. Definition of the flight path angle (1), pitch angle (0), angle of attack (a), and the rigging angle (Pr).

100 vy-+-

vx. -·--·

90

80

__._ .._....··-.......-._ . . . __

70

.._

~

"3 8.

"'

__

.

'·

60

so 40 30 20 JOL-----~----~----_J

0

10

______L __ _ _ _~----~

15

Tlme(s)

20

25

30

Figure 3. Example Simulations: Flare Maneuver. Test data for a 15-foot winch-driven flare.

15

30,------r------~----~------~----~------,

gnmma rig-ang

giU1llllB::::r!g-ang gommB_f1g-ang

25

lhet~t_ng-ong

lhela_rig-ong

20

lheLa_rig-OJJg

100,------r----~------~-----r-----.------,

36 ----· 9 ·····

9 . .•..

80

/.1''·~---··~- ··

vx._r!g-ong

3-

vx_rig-ang

9 ---

vx_n g-ong

90

36-- -

vy_rig-ong vy_r!g-Mg

vy_ng-ong

r-- - - -

6-

J-

6- 9 ~-·

70

10 60

50 40

-5

30 -15

- - - · .. .... . __, _ __,~,.,'·-----· ..... _. ··-

··

-·-··------~-----

20

~OOL----4~--~---~--~~---7.~--~-

10

20

tim~«)

40

50

60

50000

f:\Jig-u.ng

3-

fy.J!g-ong ly.J!g-ang lyJlg-ong

3-

f:_,!:=~~ ~ ~---~:·

-----------

-. ~-~~~_..........

6 •·•·• 9 .....

40000

·--~>-- -~~.:,;;:::;;..:.--,-·--·

10 o~--~ . o~-~~2~0---~~:----~40 ~--~50~--~60'

Omc(sec)

-50000 60000

----····-· ""',

.------..-----~------,........----.,.....-----.-------,

mz_r}g-Mg = 3 -

mz_ng-ang = 6 ---··

mz_rig-ong =9 -

·100000

::: -JSOOOO

, /·-......_ ... _ ......-

···--- •.__._ •.-

..·-·-

~;/--. -·-·--------·-----··---·----

= 1/

30000

20000 10000

·400000 - ·----··-"'

j

-450000

1-----.:_~~;:::::;:.:-::--··

' 10000 0~_;_~=10=--...,2:':0---~30:-----c40~--~50:-----}60

iI

______ ../ /

-sooooo o~--~10:----~2':;-o---~3o~--~40~---50-;;;---........;60.

time(sec)

Lime(sec)

Figure 4. Example Simulations: Flare Maneuver. For /-l r 3°, 6° and go, clockwise from top left: time histories of flight angles, velocity components, moments, and forces.

16