Computational Methods and Applications in Parachute Maneuvers

Transcription

1 COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM 04, Sept. 5-10, 2004, Beijing, China c 2004 Tsinghua University Press & Springer-Verlag Computational Methods and Applications in Parachute Maneuvers Keith R. Stein 1, Tayfun E. Tezduyar 2, Sunil V. Sathe 2, Masayoshi Senga 2, Richard J. Benney 3 and Richard D. Charles 3 1 Department of Physics, Bethel University, St. Paul, Minnesota, U.S.A. 2 Mechanical Engineering, Rice University - MS 321, Houston, Texas, U.S.A. 3 U.S. Army Soldier Systems Center, Natick, Massachusetts, U.S.A. k-stein@bethel.edu, tezduyar@rice.edu, sathe@rice.edu, msenga@rice.edu, richard.benney@natick.army.mil, richard.charles@natick.army.mil Abstract: Fluid structure interactions (FSI) are experienced during all stages of parachute operations. During special parachute maneuvers, and even during terminal descent, parachute canopies experience significant dynamics and undergo large shape changes. The FSI involved in the parachute operations introduce a variety of challenges that must be properly addressed in computational modeling efforts [1]. These challenges provide motivation for using methods that are based on interface-tracking techniques. Accordingly, we use the Deforming-Spatial- Domain/Stabilized Space Time (DSD/SST) [2] formulation for our flow computations and a Lagrangian finite element formulation for the parachute deformation computations [3], with advanced mesh update methods. Here, we present our methods for simulation of parachute FSI behavior and address some of the modeling challenges involved. Results are presented for round parachutes, with focus on maneuvers of the steerable kind. Key words: airdrop systems, fluid structure interaction INTRODUCTION New parachute systems are being developed for aerial delivery of supplies with greater precision than has been achieved in the past. One type of parachute system that offers increased precision is the steerable round parachute, which allows for limited maneuverability and glide to be achieved using a traditional round parachute system. When equipped with a guidance system, this type of parachute can offer automated maneuvering, can increase drop zone accuracy, and can allow for in-flight corrections during parachute descent to overcome wind-induced error. Non-steerable round parachutes are very much under the control of the winds after leaving the aircraft and, for this reason, are often referred to as dumb parachutes. Additionally, non-steerable round parachutes have no capability for limiting drop zone dispersion between the first and last parachutes exiting the aircraft. For this reason, there is interest in developing a class of precision parachute systems, with steerable round parachutes being an economical solution for many airdrop situations. In this paper we focus on the behavior of a steerable round parachute during maneuvers. Specifically, we study the limited glide that can be achieved with a round parachute by deflecting a portion of the canopy through riser control. Each riser connects the payload to multiple suspension lines, which are attached to a section of the canopy skirt. Thus, each riser

2 controls some section of the parachute canopy. Here, initial gliding performance of a G 12 cargo parachute is simulated through the deflection of one half of the canopy by retraction of two risers. Simulation of parachute maneuvers involves a variety of challenges that must be addressed in accurate modeling. Parachute canopies have a strong response to the surrounding flow fields, experiencing significant dynamics and undergoing large shape changes, even during uncontrolled descent. During maneuver (i.e., riser controls), canopy dynamics in fluid structure interactions (FSI) is increased due to the forced deflections in the canopy and the subsequent response of the parachute. Following the initiation of the riser control, the parachute pitches towards (or away from) the retracted (or extended) section of the canopy and achieves a limited amount of glide. The pitching of the parachute leads into gliding descent, which continues as long as the riser control is held. For the simulations described in this paper, the parachute operates at sufficiently low speed and, hence, the aerodynamics is governed by the Navier Stokes equations of incompressible flows. The parachute is treated as a cable membrane tension structure, governed by the linear momentum balance equation. Parachute FSI during control line maneuvers involve canopies that undergo shape changes, with the spatial domain occupied by the fluid varying (i.e. deforming) with respect to time. To accomodate canopy shape changes, we use methods which are based on interface-tracking techniques. We use the Deforming-Spatial- Domain/Stabilized Space Time (DSD/SST) [2] formulation for the flow field computations and a Lagrangian finite element formulation [3] for the parachute deformation computations. In addition to the fluid and structural computations, we use advanced mesh update methods that we have developed to handle parachute deformations within the fluid mesh. FSI behavior is modeled with the quasi-direct coupling method introduced in [4]. We use a six degree-of-freedom (6-dof) modeling approach to simulate the glide behavior of the parachute after the parachute has had some time to respond to the riser control. Fluid object coupling for the 6-dof approach is achieved in a block-iterative fashion, similar to the approach we have widely used in earlier FSI computations [5, 6]. The 6-dof approach is more economical than a coupled FSI computation and allows for faster computation of the glide performance of the parachute with deflected canopy. We assume here that glide performance is driven primarily by the large canopy deflections imposed by the riser controls, and less by the smaller canopy motions due to the FSI that continue throughout the descent. In the following sections we describe our computational methods and demonstrate these methods for a steerable round cargo parachute. We begin in the next section by describing the governing equations and finite element methods, including the DSD/SST finite element formulation. A brief description of our mesh update methods is included. Following that, we describe the quasi-direct coupling used for FSI computations and the block-iterative approach used for 6-dof computations. After this, we present results from the FSI and 6-dof computation involving a steerable round cargo parachute system. Finally, we end with some concluding remarks. FINITE ELEMENT MODEL Fluid Dynamics Let Ω t IR n sd be the spatial fluid mechanics domain with boundary Γt at time t (0, T), where the subscript t indicates the time-dependence of the spatial domain and its boundary.

3 The Navier Stokes equations of incompressible flows can be written on Ω t and t (0, T) as ( ) u ρ t + u u f σ = 0, (1) u = 0, (2) where ρ, u and f are the density, velocity and external force, respectively. The stress tensor σ is defined as σ(p,u) = pi + 2µε(u). Here p, I and µ are the pressure, identity tensor and viscosity, respectively. The strain rate tensor is defined as ε(u) = 1 ( ) ( u) + ( u) T. (4) 2 Both Dirichlet- and Neumann-type boundary conditions are accounted for: u = g on (Γ t ) g, n σ = h on (Γ t ) h. (5) Here (Γ t ) g and (Γ t ) h are complementary subsets of the boundary Γ t, n is the unit normal vector at the boundary, and g and h are given functions. A divergence-free velocity field is specified as the initial condition. The DSD/SST finite element method is suitable for handling spatial domains for the fluid that change in time. In this method, the finite element formulation of the governing equations is written over a sequence of N space time slabs Q n, where Q n is the slice of the space time domain between the time levels t n and t n+1. At each time step, the integrations involved in the finite element formulation are performed over Q n. The space time finite element interpolation functions are continuous within a space time slab, but discontinuous from one space time slab to another. The notation ( ) n and ( ) + n denotes the function values at t n as approached from below and above. Each Q n is decomposed into space time elements Q e n, where e = 1, 2,..., (n el ) n. The subscript n used with n el is to account for the general case in which the number of space time elements may change from one space time slab to another. The Dirichlet- and Neumann-type boundary conditions are enforced over (P n ) g and (P n ) h, the complementary subsets of the lateral boundary of the space time slab. The finite element trial function spaces (Su) h n for velocity and (Sp h ) n for pressure, and the test function spaces (Vu h) n and (Vp h) n = (Sp h) n are defined by using, over Q n, first-order polynomials in both space and time. The DSD/SST formulation is written as follows: given (u h ) n, find u h (Su) h n and p h (Sp h) n such that w h (Vu h) n and q h (Vp h) n: ( ) u w h h ρ Q n t + uh u h f h dq + ε(w h ) : σ(p h,u h )dq Q n w h h h dp + q h u h dq + (w h ) + n ρ ( ) (u h ) + n (u h ) n dω (P n) h Q n Ω n + + [ ( ) ] 1 w h τ SUPG ρ + u h w h + τ PSPG q h Q e n ρ t ( ) ] u h ρ t + uh u h σ(p h,u h ) ρf h dq ν LSIC w h ρ u h dq = 0, (6) (n el ) n e=1 [ (n el ) n e=1 Q e n (3)

4 where τ SUPG (streamline-upwind/petrov-galerkin), τ PSPG (pressure-stabilizing/petrov-galerkin), and ν LSIC (least-squares on incompressibility constraint) are the stabilization parameters (see [7]). This formulation is applied to all space time slabs Q 0, Q 1, Q 2,...,Q N 1, starting with (u h ) 0 = u 0. For an earlier, detailed reference on this stabilized formulation see [2]. For the FSI computations in this paper, w h / t has been dropped from the SUPG stabilization term in Eqn. (6). Structural Dynamics Let Ω s t IRn xd be the spatial domain bounded by Γ s t, where n xd = 2 for membranes and n xd = 1 for cables. The boundary Γ s t is composed of (Γ s t) g and (Γ s t) h. Here, the superscript s corresponds to the structure. The equations of motion for the structural system are: ( d ρ s 2 ) y dt + ηdy 2 dt fs σ s = 0, (7) where, y is the displacement, ρ s is the material density, f s is the external body force, σ s is the Cauchy stress tensor, and η is the mass-proportional damping coefficient. The damping provides additional stability and is used in inflating (i.e., prestressing) the parachute, but not during the FSI computations presented in this paper. Our numerical method uses total Lagrangian formulation of the problem. Thus, stresses are expressed in terms of the 2nd Piola Kirchoff stress tensor S, which is related to the Cauchy stress tensor through a kinematic transformation. Under the assumption of large displacements and rotations, small strains, and no material damping, the membranes and cables are treated as Hookean materials with linear elastic properties. For membranes, under the assumption of plane stress, S becomes S ij = ( λm G ij G kl + µ m [ G il G jk + G ik G jl]) E kl, (8) where λ m = 2λ mµ m (λ m + 2µ m ). (9) Here, E kl are the components of the Cauchy-Green strain tensor, G ij are the components of the contravariant metric tensor in the original configuration, and λ m and µ m are Lamé constants. For cables, under the assumption of uniaxial tension, S becomes S 11 = E c G 11 G 11 E 11, (10) where E c is the cable Young s modulus. The semi-discrete finite element formulation for the structural dynamics is based on the principle of virtual work: Ω s 0 w h ρ sd2 y h dt 2 dωs + w h ηρ sdyh Ω s dt dωs + δe : S h dω s = w h (t + ρ s f s )dω s. (11) 0 Ω s 0 Ω s t Here the weighting function w h is also the virtual displacement. The air pressure force on the canopy surface is represented by vector t, a geometrically nonlinear term that increases the overall nonlinearity of the formulation. The left-hand-side terms of Eq. (11) are referred to in the original configuration and the right-hand-side terms for the deformed configuration at time t.

5 Mesh Update Appropriate mesh update methods are needed to handle spatial domains for the fluid that change in time. Selection of these methods depends on several factors, such as the complexity of the moving boundary or interface, how unsteady the moving boundary or interface is, and how the initial mesh is generated. In general, the mesh update consists of moving the mesh in a way that limits element distortion, with full or partial remeshing when the element distortion becomes excessive. The only condition that mesh motion must satisfy is that at the moving boundary or interface the normal velocity of the mesh has to match the normal velocity of the fluid. Beyond that, we have flexibility in how to move the mesh, with the main objective being to reduce the frequency of remeshing. In computation of parachute FSI reported here we use an automatic mesh moving scheme [8] based on the equations of linear elasticity, with selective treatment of mesh elements and also the deformation modes in terms of shape and volume changes so that smaller elements enjoy more protection from mesh deformation. The motion of the internal nodes is determined by solving these additional equations, with the boundary conditions for these mesh motion equations specified in such a way that they match the normal velocity of the fluid at the interface. Several additional features have been developed and added to our central mesh moving strategy to reduce the frequency of remeshing [9]. These features have proven effective for a variety of test cases [10, 11]. COUPLING Fluid Structure Interactions Parachute FSI involve situations where the structure is very light, with structural response very sensitive to small changes in the fluid dynamics. In these types of problems, achieving acceptable levels of convergence in the nonlinear equation systems for the FSI can be difficult. We use the quasi-direct coupling method, which has been shown to be effective for FSI involving light structures [4]. Spatial and temporal discretizations of the finite element formulations for the fluid and structure (see Eqs. (6) and (11)) lead to a fully-coupled, nonlinear system of equations that can be symbolically written as N 1 (d 1,d 2 ) = F 1, N 2 (d 1,d 2 ) = F 2, (12) (13) where d 1 and d 2 are the vectors of nodal unknowns associated with the fluid unknown function (u 1 ) and structure unknown function (u 2 ), respectively. Solution of these equations with Newton Raphson method would require the solution in each iteration of a linearized system of the form A 11 x 1 + A 12 x 2 = b 1, A 21 x 1 + A 22 x 2 = b 2, (14) where b 1 and b 2 are the residuals of the nonlinear system, x 1 and x 2 are incremental updates for d 1 and d 2, and A βγ = N β d γ. (15) In our situation, the off-diagonal blocks A 12 and A 21 represent the fluid structure coupling matrices. Retaining these coupling matrices in the current form would require accounting

6 for the dependence of A 12 on mesh motion. In this paper, we use the quasi-direct coupling approach, which partially includes the coupling blocks, without accounting for the dependence of A 12 on mesh motion. This approach provides a much tighter fluid structure coupling than the block-iterative coupling and, therefore, delivers significant improvement in convergence. Fluid Object Interactions In certain situations it may be desirable to model a parachute as a rigid body with the dynamics governed by the force and moment balance equations. This 6-dof approach is more ecomonical than a coupled FSI computation and can be used to simulate parachute glide after the canopy is deflected with a riser control. For the 6-dof simulations presented in the following section we use a block-iterative method for the fluid-object coupling. Here, the parachute is treated as a rigid object with the deformed parachute geometry resulting from a preceding FSI computation. The block-iterative coupling technique can be viewed as an approximate Newton Raphson method, where the fluid object coupling matrices are ignored in the solution of the linear equation system each Newton Raphson iteration. This is equivalent to dropping A 12 and A 21 from Eq. (14), where b 2, x 2, and d 2 now come from the system of equations for the 6-dof motion. MANEUVER OF A G 12 PARACHUTE WITH RISER CONTROLS Computations are carried out to study the response of a G 12 cargo parachute to a riser control involving the retraction of two risers, deflecting one half of the canopy. The G 12 is a 64 ft diameter cargo parachute designed to deliver payloads of 2200 lb at descent speeds of 28 ft/s. The G 12 is constructed with 64 suspension lines each 51.2 ft that extend from the canopy to four risers each ft. The risers are each connected to 16 suspension lines and merge to a single confluence point, which in turn is connected to four cables each ft that hold the payload. A schematic with dimensions for the G 12 model is shown in Figure 1 (left). The structural model is composed of membranes, cables, and concentrated masses. The canopy is modeled with triangular membrane elements. Linear cable elements are used to model the suspension lines, radial reinforcements along the canopy, risers, and payload support cables. The payload is modeled with eight concentrated masses that are interconnected with a set of truss elements. Material properties are selected to be representative of the G 12. Our model for the G 12 in its constructed configuration is shown in Figure 1 (middle). Several measures are taken in preparations for the G 12 riser control simulations. First, a stand alone structural deformation computation is carried out to determine the inflated (i.e., prestressed) shape of the G 12 (See Figure 1, right). Secondly, a stand alone flow field computation is carried out to obtain a developed flow about the inflated G 12. Finally, a FSI computation is carried out prior to the riser control to remove any mismatch between the fluid forces and the parachute shape. The simulations presented in the following two subsections are carried out in two stages. In the first stage, canopy shape changes and the onset of glide that results from riser retraction is simulated with a FSI computation. In the second stage, glide behavior is simulated with a 6-dof computation. Here, we assume that glide performance is driven primarily by the significant canopy shape changes that result from the retracted risers, and less by smaller canopy motions from continued FSI behavior.

7 Figure 1: G 12 Parachute: Dimensions (left); Constructed geometry (middle); Prestressed geometry (right). Fluid Structure Interaction of G 12 Parachute with Retraction of Two Risers The response of the G 12 to a two-riser retraction is simulated using the quasi-direct method described. The FSI for the riser control is carried out in two computations. The first computation is carried out to model the dynamic retraction of two risers by 3.84 f t in 0.29 s. Riser retraction is modeled by prescribing that the cables defining the retractable risers shrink in time. This is accomplished by decreasing the natural length of the cables over the duration of the retraction. These reductions of cable length are accounted for in the calculation of cable stress terms, whereas inertia terms are always based on the initial cable lengths. In the second computation the retracted risers are held, allowing the G 12 to respond to the retraction for an additional 0.22 s and to begin to enter into a state of glide. Results for the FSI of the two-riser retraction are shown in Figures 2-3. The G 12 geometry is shown in Figure 2 prior to the retraction (left) and at the end of the FSI computations (right). The shortening of the two retracted risers and the deflection of the right half of the canopy are evident. Although the canopy has undergone a significant change in shape, the resulting pitch and glide of the parachute are just being initiated at this point in time. The pitching and gliding become more apparent during the following 6-dof computation. The flow field surrounding the G 12 at four stages of the FSI computation are shown in Figure 3, with colors representing the pressure field surrounding the canopy. The pressure field from the FSI results in a horizontal component of aerodynamic force in the direction of anticipated glide. As expected, retraction of the two risers results in a horizontal force which will tend to drive the parachute towards the retracted risers (i.e., the +x-direction). The transition to glide is simulated in the 6-dof computations, as described in the following subsection. Fluid Object Interaction for 6-Dof Behavior of G 12 Parachute with Two Retracted Risers The solution at the end of the FSI computation is used as an initial condition for an ensuing 6-dof simulation to study the gliding performance of the G 12 with the deflected canopy due to the retracted risers. Here, the G 12 parachute is treated as a rigid body and the dynamics is determined based on the force and moment balance equations, with forces

8 Figure 2: G 12 geometry prior to (left) and after (right) the two-riser retraction. and moments arising from the unsteady fluid pressure distribution on the canopy surface. The simulation is carried out for a physical time of approximately 5.4 s, during which the canopy pitches towards the deflected section of the canopy and transitions to glide. During transition, the canopy pitches from a nearly vertical orientation due to a moment induced by the deflected canopy geometry. Eventually, the canopy pitches beyond equilibrium and the moment becomes negative. In time, the G 12 will approach an equilibrium glide orientation, although some oscillations will continue due to the unsteady nature of the flow field. To handle the gliding parachute, we specify boundary conditions for the mesh motion equations such that the outer boundaries of the mesh move with the center of mass of the parachute canopy. Implementation of this mesh boundary condition assures that the parachute canopy interface remains centered in the fluid mesh. Results from the 6-dof computations for the gliding performance of the G 12 with deflected risers are shown in Figures 4-6. The trajectory of the payload and the velocity for the center of mass are shown in Figure 4. For our simulations, the two retracted risers are on the +x side of the parachute and will induce a preferred glide in the x z plane. Hence, the trajectory plot is displayed in the x z plane, with the understanding that induced motion in the y- direction is minimal. The trajectory indicates a transition from vertical descent to glide in the +x-direction and is shown along with the anticipated trajectory without the maneuver. The components of velocity in the G 12 glide plane are shown for the center of mass (Figure 4, right). As expected, induced glide is primarily in the +x-direction, with the glide speed building up as the simulation progresses. It is apparent from the figure that the glide speed and descent speed are of the same order of magnitude by the end of the computation. The time history for the horizontal force in the +x-direction is shown in Figure 5. This is the glide-inducing component the total aerodynamic force, and is shown along with the weight of the G 12 parachute payload system. The build up of the x-component of force, which initiates the pitching motion of the G 12 about the y-axis, is evident. The time history

9 Figure 3: Sequence of G-12 pictures (left to right and top to bottom) showing the velocity vectors colored by pressure, during and immediately after two-riser retraction. for the glide-inducing component of the moment (i.e., about the y-axis) acting on the G 12 canopy is shown in Figure 6. This component of the total moment is dominant, and remains positive initially and becomes negative at about 3.25 s. The moment causes the G 12 to begin pitching forward around the y-axis with increasing angular speed for the first 3.25 s and then to continue pitching forward with decreasing angular speed. Eventually, the G 12 will pitch back and settle into an equilibrium glide orientation. CONCLUDING REMARKS We described our computational methods for the simulation of parachute fluid structure interaction that are applicable to situations involving maneuvers for round parachutes. These problems involve parachute canopies that interact strongly with the surrounding air flow and undergo large shape changes in time. Hence, our methods are based on interfacetracking techniques. We have used the Deforming-Spatial-Domain/Stabilized Space Time (DSD/SST) formulation, with advanced mesh update methods, which is effective in solving the fluid dynamics for this class of problems. The DSD/SST formulation and a Lagrangian finite element formulation for parachute deformations were coupled in a quasi-direct fashion for the FSI studies presented in this paper. For the 6-dof studies, we coupled the DSD/SST method to the parachute force and moment balance equations in a block-iterative fashion. Simulations were carried out to study maneuvers for a round cargo parachute. Specifically, we focused on the glide performance of a G 12 parachute due to the retraction of risers controlling one half of the canopy. Results were presented for the dynamic response resulting from the retraction of two risers, the transition to glide, and the gliding performance. FSI computations based on the quasi-direct coupling approach were presented to study the initial behaviors, for which FSI play a most dominant role. These computations were carried out to study the dynamic response of a G 12 parachute to the surrounding flow field during the

10 Trajectory of payload Trajectory of payload without maneuver Glide speed Descent speed Vertical Position (ft) Speed (ft/s) Horizontal Position (ft) Time (s) Figure 4: Trajectory of G 12 payload (left) and velocity of center of mass (right) during 6-dof motion Force in glide direction Weight of parachute system Force (lb) Time (s) Figure 5: Horizontal force in the glide direction and G 12 parachute system weight. retraction of two risers. Follow on 6-dof computations were carried out for the G 12 with the deflected canopy to study the transition to glide for the parachute. The simulations presented here have demonstrated how the quasi-direct coupling method for FSI can be used along with a more efficient 6-dof fluid-solid interaction model to study relevant problems involving parachute systems. Acknowledgments: This work was supported by the Army Natick Soldier Center.

11 Moment (lb-ft) Time (s) Figure 6: Moment acting on a G-12 about an anticipated pitching axis during 6-dof motion. REFERENCES [1] K. Stein, T. Tezduyar, and R. Benney, Applications in airdrop systems: Fluid-structure interaction modeling, in Proceedings of the Fifth World Congress on Computational Mechanics (Web Site), On-line publication: Paper-ID: 81545, Vienna, Austria, (2002). [2] T.E. Tezduyar, Stabilized finite element formulations for incompressible flow computations, Advances in Applied Mechanics, 28 (1992) [3] M.L. Accorsi, J.W. Leonard, R. Benney, and K. Stein, Structural modeling of parachute dynamics, AIAA Journal, 38 (2000) [4] T.E. Tezduyar, S. Sathe, R. Keedy, and K. Stein, Space time techniques for finite element computation of flows with moving boundaries and interfaces, in S. Gallegos, I. Herrera, S. Botello, F. Zarate, and G. Ayala, editors, Proceedings of the III International Congress on Numerical Methods in Engineering and Applied Science, CD-ROM, [5] K. Stein, R. Benney, V. Kalro, T.E. Tezduyar, J. Leonard, and M. Accorsi, Parachute fluid structure interactions: 3-D Computation, Computer Methods in Applied Mechanics and Engineering, 190 (2000) [6] K.R. Stein, R.J. Benney, T.E. Tezduyar, J.W. Leonard, and M.L. Accorsi, Fluid structure interactions of a round parachute: Modeling and simulation techniques, Journal of Aircraft, 38 (2001) [7] T. Tezduyar and Y. Osawa, Methods for parallel computation of complex flow problems, Parallel Computing, 25 (1999) [8] 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 New Methods in Transient Analysis, PVP-Vol.246/AMD-Vol.143, ASME, New York, (1992) 7 24.

12 [9] T. Tezduyar, Finite element interface-tracking and interface-capturing techniques for flows with moving boundaries and interfaces, in Proceedings of the ASME Symposium on Fluid-Physics and Heat Transfer for Macro- and Micro-Scale Gas-Liquid and Phase- Change Flows (CD-ROM), ASME Paper IMECE2001/HTD-24206, ASME, New York, New York, (2001). [10] K. Stein, T. Tezduyar, and R. Benney, Mesh moving techniques for fluid structure interactions with large displacements, Journal of Applied Mechanics, 70 (2003) [11] K. Stein and T. Tezduyar, Advanced mesh update techniques for problems involving large displacements, in Proceedings of the Fifth World Congress on Computational Mechanics, On-line publication: Paper-ID: 81489, Vienna, Austria, (2002).