DEPLOYMENT SIMULATIONS OF COMPLEX ANTENNA STRUCTURES USING AN IMPLICIT NON-LINEAR FINITE ELEMENT SOLVER

Transcription

1 DEPLOYMENT SIMULATIONS OF COMPLEX ANTENNA STRUCTURES USING AN IMPLICIT NON-LINEAR FINITE ELEMENT SOLVER F. Cugnon (1), D. Granville (1), P. Howard (2), M. Milano (3), J. Santiago Prowald (4) (1) Samtech Liège Science Park, Rue des Chasseurs-Ardennais, 8, B-4031 Liège, Belgium (2) EADS Astrium Satellites Anchorage Road, Portsmouth, PO3 5PU, U.K. (3) Thales Alenia Space Italy Via Saccomuro 24, Roma, Italy (4) European Space Agency Keplerlaan 1, 2200 AG Noordwijk, The Netherlands ABSTRACT Lightweight deployable structures are required in many space missions. The numerical analysis of deployable space structures is becoming more and more necessary and a software tool supporting this activity, from the preliminary design to the final verification, has thus a clear interest in space industry. The goal of the DESCAS (DEployment Simulation of Complex Antenna Structures) ESA project was to improve the capabilities of the SAMCEF Mecano [1][2] commercial software in order to demonstrate its ability to provide accurate solutions for those mathematically stiff problems [3]. The main results of this project are presented in this paper, highlighting the capability of this implicit non-linear finite element solver to model efficiently and simulate the dynamic behavior of flexible deployable systems. Two industrial benchmarks proposed by ESA have been treated during the project and are detailed in the paper, demonstrating the capabilities of the Software. The first one is a Large Deployable Reflector Antenna developed by Thales Alenia Space and the Russian-Georgian subcontractor NPO-EGS. The second test case is a deployable antenna based on a Tape-Spring hinge concept, proposed by Astrium UK. 1. INTRODUCTION Scientific, Earth Observation and Telecommunications missions require the deployment of antennas, solar arrays, booms and other appendices that need to be stowed during launch. The need for larger deployable structures forces engineers to properly study more and more complex mechanical systems already at the early phases of the design, as well as during their operational life. This scenario leads to the necessity of simultaneously evaluating, under a rather large number of mechanical parameters, the exact 3D kinematics, the mechanical loads, stresses and vibrations, large amplitude motion and multibody flexible dynamics in the space environment and on-ground testing. This paper addresses the numerical simulation of the deployment of large space structures, including the transient loads generated during the deployment. It is a demanding task due to the need to simultaneously capture widely spread phenomena in terms of their respective time scales, such as slow actuation and low-frequency dynamics in longduration deployments, together with higher-frequency response. This produces what is known in the literature as stiff systems of differential equations, and imposes new challenges on the time integration algorithms in multi-body FEM analysis. The arbitrary increase of physical or numerical damping in the simulation, as the most readily available way to overcome the numerical instability, turns out to be a source of inaccuracy in the overall prediction. In addition, strain energy is released or stored in structural elements during the opening phase. The drawback is that transient loads like shocks, affect the deployment and can represent dangerous load cases.

2 Improvement of existing software tools (e.g. SAMCEF Mecano) is therefore needed in relation to robustness in stiff numerical environments and convergence analysis. In addition, validation with experiments is mandatory for the production of reliable numerical tools. This validation will be carried out on the existing mathematical models of the Large Deployable Antenna, developed under ESA contract by TASI as prime, and the FLATS demonstrator by Astrium UK. 2. TIME INTEGRATION OF STIFF PROBLEMS Time integration of stiff problems is still a challenge. During the DESCAS project many implicit time integrators have been studied and tested. From this theoretical study, we can say that the β-newmark family of integration schemes have second order unconditional stability. This property disappears when solutions are strongly non-linear as for stiff problems. However, in most cases introducing controlled numerical damping (HHT and Chung Hulbert) and using adequate time stepping strategy can control instabilities. An alternative to ensure the stability of the solution is by means of schemes that verify the preservation of the total energy of the system at each time step [4]. There is a class of algorithms (EPS) which have been developed in order to verify the conservation of energy. It seems that the main idea is similar between the different methods and comes from the work of Simo [5]. The main idea is to use a mid-point rule algorithm and to compute the internal forces in a specific way, which depends on the element formulation. Those methods have two major drawbacks which are the necessity to re-write the complete library of elements and the fact that high frequencies remain in the solutions. To overcome the problem of damping high frequencies, several authors [4][6] propose alternative energy dissipative schemes (EDS) based on the time-discontinuous Galerkin method. The last type of method we investigated are implicit Runge-Kutta methods [7]. The most popular version of those methods is the RADAU IIA. Application of this method to multibody systems is presented in [8]. From intensive testing on lightweight deployable structures we came to the following conclusions. Energy conservation is demonstrated on academic tests when EPS is used, but obtained results contain numerical high frequencies oscillations that are not damped if any shocks occur during deployment. When industrial problems are considered, convergence is never reached. EDS and Radau IIA perform well, in some case (structural stiff problems) better than HHT; but in case of impact problems, advantage of higher order integrators is lost. The key point to model efficiently flexible deployable system is the use of time integrators that could damp numerical high frequency as HHT or Chung Hulbert schemes. Accuracy and robustness depend strongly on a proper definition of the mathematical models and structural damping in particular. Using C 1 function to define local stiffness elements brings some robustness to the Newton-Raphson process. Combination of those allows obtaining converged solutions for non-academic problems. This approach is recommended and used for the simulation of the deployment of the both antennas presented below. Fig.1. One deployable cell (open configuration)

3 The simulation of a deployable cell (Fig.1) exhibiting ends of deployment shocks due to stiff locking device in hinges and stiffener cables allow demonstrating convergence of the solution get using HHT scheme and above recommendations on the mathematical model. Fig.2 shows dynamics seen in one hinge for several simulations done with increasing precision requested from the automatic time stepping algorithm. Ones can observe the convergence of obtained solutions. Fig.2. Rotation at hinge 1 (Dt1, Dt2, Dt3, Dt4, Dt5, Dt6) 3. SIMULATION OF DEPLOYABLE SYSTEMS 3.1. Large Deployable Reflector (LDR) This model developed in 2000 by Alenia Spazio with the technical support of SAMTECH is based on the programming capabilities of SAMCEF to build a hierarchical model based on the repetition of basics parameterized sectors. At that time the simulation could not be carried out to the end of the deployment because of end of deployment chocks instabilities. In the DESCAS project, we considered this deployment simulation as a validation test case for SAMCEF Mecano. Starting from the Alenia s model, we investigate the past convergence problem and test several alternatives to improve the convergence. Thanks to the use of SAMCEF version 13, and in particular to the use of the Chung-Hulbert integration scheme with numerical damping on higher frequencies and the possibility to modify the iteration matrix of shell elements in such a way that it is positive in order to avoid negative pivots during the iterations, we could obtain the solution shown on Fig.3.

4 Fig.3. LDR in deployed configuration. The robustness and the accuracy of this model have been successfully investigated performing sensitivity analyses to all modeling parameters. We also demonstrate that the model could be used for both on earth and in orbit simulations [9]. To control the deployment with force actuation, we have implemented an actuation at a prescribed power. This is done by replacing the previous initially applied prescribed displacements by some non-linear force elements connected to the distance sensors used to model the cables. After several tries, we have calibrated the power P = F*v to about Watts by actuator. The imposed non-linear force is thus defined by: F = v 5*10 6 Fig.4 shows that the deployment time is close to the one of the reference prescribed displacement. Looking to Fig.5, we observe that the deployment velocity is now more or less constant.

5 Fig.4. a) Radial displacement of the pantograph - b) Rotation of the pantograph (force actuation & displacement actuation) Fig.5. Angular velocity of the pantograph (force actuation & displacement actuation) Starting from a reference solution, we shown that the LDR model is robust and weakly sensitive to most of the parameters related to damping. Reducing the damping doesn t really modify the solution, it mainly increase the computation costs. The only strong requirement is the use of the Chung-Hulbert time integration scheme to allow the control of local undesired high frequency oscillations in the ribs. We also demonstrated that the model could be driven using more realistic force actuation P-Band FLAT antenna The antenna concept proposed by Astrium UK has been selected as validation breadboard for the DESCAS project. The demonstrator used for the deployment test is a 1/5 th scale model of the P-Band FLATS concept; it has been described in detail in [9]. The flat hinge concept is shown on Fig.6, where we can see an exploded view of the folding area and a folded sample. The hinge line has three main functions. The first one is to assure the kinematical articulation between two panels, the second one is to supply the deployment torque by restituting the energy stored when it is folded, and the last one is to ensure the locking of the hinge line when it is deployed.

6 Fig.6. Exploded view of the structure & folded sample (courtesy of Astrium UK and ESA). Due to the contact between them, the face-skins supply most of the deployment torque, which is augmented by the additional discreet tape-springs placed across the hinge-lines. Those ensure the locking action on deployment and aid in the deployed stiffness. The complete system is shown on Fig.7; its deployment is controlled by a cable. Fig.7. Experimental system (courtesy of Astrium UK and ESA). Starting from an existing mesh, several techniques have been used to reduce the computational cost. First, taking into account the symmetry of this problem, only one half of the structure should be considered. Secondly, most of the deformations occur at the hinge-lines levels where the non-linearities (buckling and contact) are present. A linear behavior of the panels can then be assumed and super-elements used to represent them. To perform transient analyses, contact conditions are defined between all components of the flat hinges, kinematical joints (hinges) are added to each panel to prescribe folding kinematical conditions and hold-on interfaces are introduced to assure that a minimum distance between the panels is imposed. This condition is imposed by defining distance sensors between some points of the panels and associating to them some elastic stop-spring 1 degree-of-freedom elements. The cable is modeled by a set of distance sensor joining the center points of the faces of the neighbor panels. At the localizations of those points, all the nodes along the thickness of the panels are rigidly connected. A kinematical constraint is introduced to define the total length of the cable and to control it during the deployment. The defined kinematical constraint is: d + d 1 d 0 + d 2 + d 3 f ( t) = 0

7 where d i are the lengths measure by the sensor and f(t) is the prescribed value of the total length of the cable. The condition on d says that the cable has no stiffness in compression. Complete model is shown on Fig.8 during folding operations. Fig.8. Model with hold-on interfaces and the control cable After tuning model parameters on a model with a single panel, we could simulate the folding and the complete deployment of the antenna. The results of the first simulation differed slightly from the experimental video provided by Astrium. As some uncertainties exist on the modeling data, mainly on the actuation system, we build a simplified 2D model based on a torque-rotation law corresponding to the detail model of the hinge line. This model allows carrying a complete simulation (folding & deployment) in a few seconds instead of several days for the complete model. It was used to study the sensitivity of the model to many parameters [9]. This study shows that if a friction effect on the cable when it passes through the holes of panels is considered, the deployment kinematics (Fig.9) is slightly modified and results are in good agreement with experimental data.

8 Fig.9. FLATS antenna at times 305, 310, 315, 320 & CONCLUSIONS The DESCAS project, under ESA contract, has allowed improving the capabilities of Samcef Mecano for the integration of systems of stiff equations, resulting from the flexible multibody finite element models of the LDA and FLATS deployable antennas. Energy preserving schemes have been investigated, resulting not always in stable integration. Therefore, energy dissipating schemes with targeted high frequency damping, such as Hilber-Hughes- Taylor and Chung-Hulbert, have been adopted for the transient deployment analyses, together with an adequate time stepping strategy. The convergence has been proven and the end of deployment is reached numerically without instabilities. In general, it has been concluded that the families of implicit integration schemes inserted in Mecano, are suited for the transient deployment analysis of such flexible multibody problems. 5. REFERENCES [1]. SAMCEF Mecano, [2]. M. Géradin & A. Cardona, Flexible multi-body dynamics: a finite element approach", John Willey & Sons, 2001 [3]. F. Cugnon, Summary report of the DESCAS project (ESA/ESTEC Contract N /06/NL/Sfe), 2009 [4]. Lens E., Cardona A, An energy preserving/decaying scheme for nonlinearly constrained multibody systems. Multibody Syst. Dyn., Springer 2007 [5]. Simo J.C., Tarnow N. and Wong K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, in Comput. Methods Appl. Mech.Engrg., 100, , 1992 [6]. Bauchau, O., Joo, T., Computational schemes for nonlinear elasto-dynamics, in Int. J. Numer. Methods Eng. 45, (1999) [7]. Hairer, E. & Wanner, G., Solving ordinary differential equations. II. Stiff and differential algebraic problems, 2nd edn. Springer, [8]. O.Bauchau & A. Epple, Performance evaluation of numerical integration schemes for flexible multibody systems, in Multibody Dynamics 2007, ECCOMAS conference [9]. F. Cugnon, DESCAS technical notes (2009).