Numerical Simulations of Fluid-Structure Interaction Problems using MpCCI

Transcription

1 Numerical Simulations of Fluid-Structure Interaction Problems using MpCCI François Thirifay and Philippe Geuzaine CENAERO, Avenue Jean Mermoz 30, B-6041 Gosselies, Belgium Abstract. This paper reports on the development of a computationally efficient methodology for the simulation of nonlinear fluid-structure interaction problems. Preliminary results are presented for the steady and unsteady aeroelastic problems associated with the AGARD Wing Objectives A large class of fluid-structure interaction (FSI) problems require the simultaneous application of computational fluid dynamics (CFD), computational structural dynamics (CSD) and computational thermodynamics (CTD) codes. Since each discipline has developed powerful specialized tools, a partitioned procedure is preferred for solving coupled field nonlinear FSI problems. In the present research a in-house three-dimensional unstructured CFD solver is coupled to a commercial finite element structural solver. The CFD solver developed at CENAERO is a domain decomposition based parallel threedimensional Euler and Navier-Stokes solver which features a combination of finite volume and finite element discretizations on unstructured tetrahedral meshes. The solver uses the AOMD (Algorithm Oriented Mesh Database) library [1] for the management of the topological mesh entities across the processors. Pseudo time-integration is performed for steady state flows with the backward Euler scheme. The ALE version [2] of the three-point backward difference scheme is used for unsteady computations on moving meshes. Since these schemes are implicit, the flow solver must solve at each time-step a system of nonlinear equations. For this purpose, it relies on an inexact Newton method based on a finite difference Newton-GMRES algorithm. The flow solver is coupled through the MpCCI (Mesh based parallel Code Coupling Interface) software [3] to the SAMCEF Mecano code [4] in order to perform aeroelastic simulations. This code can perform linear, nonlinear, steady state or transient analyses. MpCCI is a standardized and portable library which allows the coupling of simulation codes. It provides conservative data interpolation across non-matching fluid and structure mesh interfaces and data communication between different codes through MPI software channels. The interface to SAMCEF Mecano is obtained through the modification of the MCUSER user-functions. An iterative partitioned procedure, where the fluid and structure subsystems are solved by different schemes that are tailored to their different mathematical models, is used to solve FSI problems. A typical iteration of such a procedure applied to aeroelastic problems can be summarized as follows: (1) compute the fluid force and transfer it to the structure, (2) solve the 1

2 (a) Undeformed model (b) f 1 = 9.8 Hz (f 1,exp = 9.6 Hz) (c) f 2 = 39.3 Hz (f 2,exp = 38.1 Hz) (d) f 3 = 49.7 Hz (f 3,exp = 48.3 Hz) Figure 1: CSD model for the AGARD Wing structure subsystem, (3) transfer the displacement of the wet boundary of the structure to the fluid subsystem and update the position of the fluid mesh accordingly (with a structure analogy method in the present case), and (4) advance the fluid subsystem to the next iteration. In the steady case, an under-relaxation factor is used for the transfer of the displacements of the wet boundary of the structure, while in the unsteady case carefully designed structure displacement predictor and structure force corrector are used for bypassing the inner iterations encountered in strongly coupled solution procedures while preserving the second-order time-accurate of the individual fluid and structure time-integrators [5]. 2 Numerical results The inviscid simulation of the steady and unsteady aeroelastic responses of the AGARD Wing are considered. This wing is an AGARD standard aeroelastic configuration with a 45 degrees quarter-chord sweep angle, a panel aspect ratio of 1.65, a taper ratio of 0.66, and a NACA 65A004 airfoil section. The wing is clamped at its root. The structural model selected here is the so-called 2.5-ft weakened model 3 whose measured modal frequencies and windtunnel flutter test results are reported in [6]. A finite element structural model of this wing (see Fig. 1(a)) is constructed with 800 triangular shell elements (element T028 in SAMCEF Mecano) using the information given in [6]. This model yields natural mode shapes and frequencies that are similar to those derived experimentally (see Fig. 1(b) 1(d)). A three-dimensional fluid mesh that contains 178,938 points and 942,782 (see Fig. 2(a)) is also constructed. First, the steady aeroelastic response of the wing is computed with the freestream conditions set to M = 0.901, α = 2, ρ = slugs/in 3 and p = 11 slugs/(in s 2 ). A steady state solution is first computed around the undeformed configuration of the wing. This solution is then used as an initial condition, and the steady aeroelastic response of the wing is computed with the procedure described previously. Figure 2(b) displays the final deformed shape of the 2

3 wing, while Fig. 3(a) reports the fast convergence of the computed vertical force for two values of the under-relaxation parameter. Second, the unsteady aeroelastic response of the wing is computed for the same freestream conditions except for the angle of attack α set to 0. The finite element structural model is perturbed along its first bending mode (see Fig. 1(b)), and a steady state solution is computed around the deformed configuration of the wing (see Fig. 2(b)). This perturbation is then used as an initial condition, and the aeroelastic response of the wing is computed with a coupling timestep t = 10 3 s. This time-step corresponds to sampling the period of the first torsional mode of the dry wing in 25 points, as usually done for any second-order implicit time-integration scheme. Figure 3(b) shows that the computed aeroelastic response matches nicely a reference solution obtained with a time-step t = 10 4 s. It also shows that for the given freestream conditions the AGARD Wing does not flutter. This is consistent with the experimental data published in [6] and the computational results found in [7]. (a) Mesh (b) Undeformed (solid) and deformed (colored with Mach number contours) shapes Figure 2: CFD model for the AGARD Wing Third, the flutter analysis of the AGARD Wing is performed with the same coupling time-step. In order to observe the onset of flutter, the freestream pressure is varied, while the freestream density is set as in [6]. Fig. 3(c) shows the flutter index as a function of the Mach number. A good agreement with the experiment is observed in the subsonic and transonic regimes, while the comparison is less favorable in the supersonic regime. These results are similar to those presented in [8] with a much coarser mesh containing 22,014 nodes, and those obtained in [7] but with a much smaller coupling time-step. 3

4 42 40 ω=0.5 ω= F z (lb) Iteration (a) Vertical force convergence t = 10-3 s Reference 10 F z (lb) Time (s) (b) Vertical force history k mesh 179k mesh Experiment Flutter index Mach number (c) Flutter index Figure 3: Steady and unsteady aeroelastic responses of the AGARD Wing

5 Acknowledgment The authors acknowledge the support by the Walloon Region and the European funds ERDF and ESF under contract N EP1A References [1] J.-F. Remacle and M. S. Shephard. An algorithm oriented mesh database. International Journal for Numerical Methods in Engineering, 58: , [2] P. Geuzaine, C. Grandmont, and C. Farhat. Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations. Journal of Computational Physics, 191(1): , [3] K. Wolf. MpCCI - a general coupling library for multidisciplinary simulation. In Workshop on Scalable Solver Software Multiscale Coupling and Computational Earth Science, [4] SAMTECH. SAMCEF User Manual 11.0, [5] P. Geuzaine, K. van der Zee, and C. Farhat. Second-order time-accurate loosely-coupled solution algorithms for nonlinear FSI problems. In P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau, editors, Proceedings of the ECCOMAS 2004 Conference, Jyväskylä, Finland, July [6] E. C. Yates. AGARD standard aeroelastic configuration for dynamic response, candidate configuration I. Wing Technical Report TM , NASA, [7] K. K. Gupta. Development of a finite element aeroelastic analysis capability. Journal of Aircraft, 33(5): , [8] C. Farhat and M. Lesoinne. Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Computer Methods in Applied Mechanics and Engineering, 182: ,