A parallel 3D computational method for fluid–structure interactions in parachute systems

Abstract

We present a parallel finite element computational method for 3D simulation of fluid–structure interactions (FSI) in parachute systems. The flow solver is based on a stabilized finite element formulation applicable to problems involving moving boundaries and governed by the Navier–Stokes equations of incompressible flows. The structural dynamics (SD) solver is based on the total Lagrangian description of motion, with cable and membrane elements. The nonlinear equation system is solved iteratively, with a segregated treatment of the fluid and SD equations. The large linear equation systems that need to be solved at every nonlinear iteration are also solved iteratively. The parallel implementation is accomplished using a message-passing programming environment. As a test case, the method is applied to computation of the equilibrium configuration of an anchored ram-air parachute placed in an air stream.

Introduction

The parallel 3D computational method presented in this paper has been developed for modeling fluid–structure interactions (FSI) encountered in airdrop systems. The airdrop systems we focus on here include conventional personnel round parachutes, cross parachutes which have limited glide capability, and large gliding ram-air parachutes (parafoils) which can carry payloads up to 21 tons. The design emphasis for these parachute systems is precision delivery under demanding deployment conditions such as strong wind gusts and large offsets. Since conventional design techniques are time-consuming, expensive, and semi-empirical at best, our objective is to build a reliable and cost effective design tool based on the advanced flow simulation and modeling methods we have been developing.

Parachute systems present very complex dynamics arising from interactions between the canopy, suspension lines, payload, and the surrounding air. Parachutes can experience significant canopy deformations and changes in orientation at any stage of their deployment and operation. To correctly represent the actual behavior, modeling of these parachute systems has to involve solution of the governing equations over computational domains which change their shapes in time.

In the earlier models we developed [1], [2], [3], the parachute canopy was represented as a structure with prescribed shape changes, and its dynamics was determined as part of the overall solution of the coupled flow and dynamics equations. In a complementary effort, Benney et al. [4] presented detailed structural analysis of parachute systems with assumed air pressure distributions. Stein et al. [5] presented an axisymmetric model where the parachute structure was represented by cable and axisymmetric membrane elements, and its response was determined by solution of the equation system which took into account the coupling between this structural model and the fluid dynamics of the air surrounding it.

In the computational modeling presented in this paper, the fluid dynamics is governed by the Navier–Stokes equations of incompressible flows. The structural dynamics (SD) is governed by the membrane and cable equations with the Lagrangian description of motion.

In general, for flow problems involving moving boundaries and interfaces, including FSI, we employ the deformable-spatial-domain/stabilized space-time (DSD/SST) method introduced by Tezduyar et al. [6], [7]. This method takes automatically into account the changes in the shape of the spatial domain. The stabilized finite element formulations prevent numerical oscillations and other instabilities in solving problems with high Reynolds numbers and strong boundary layers. They also allow us to use equal-order interpolation functions for velocity and pressure. Some of the most established stabilized formulations for incompressible flows are the streamline-upwind/Petrov–Galerkin (SUPG) formulation [8], Galerkin/least-squares (GLS) formulation [9]), and pressure-stabilizing/Petrov–Galerkin (PSPG) formulation [10]. These formulations stabilize the method without introducing excessive numerical dissipation. 2D FSI methods, based on the DSD/SST formulation and developed for flow problems with moving mechanical components, were reported by Wren et al. [11]. 3D FSI simulations of a round parachute canopy with a flow solver based on the DSD/SST method were reported by Stein et al. [12]. In this paper, we focus on computing the equilibrium configuration of the parachute. Since the temporal accuracy is not an important issue for this simulation, we carry out the computations with the arbitrary Lagrangian–Eulerian (ALE) version of our stabilized formulation.

The SD solver is also based on a finite element formulation, and uses cable and membrane elements. The coupled, nonlinear equations system arising from this FSI model is solved iteratively. This is accomplished with a segregated treatment of the fluid and SD equation systems. Each of these nonlinear equation systems is solved with the Newton–Raphson method. The linear equation systems that need to be solved at each Newton–Raphson step are also solved iteratively. The flow and SD solvers have both been implemented for parallel computation within a message-passing programming environment. The complete simulation tool includes an algebraic mesh mover developed for updating the fluid mesh based on the motion of the fluid–structure interface.

The flow solver is described in Section 2, and the SD solver in Section 3. The FSI coupling technique is described in Section 4. Section 5 is a brief summary of the parallel implementation. In Section 6, we report, as a test case, computation of the equilibrium configuration of an anchored ram-air parachute placed in an air stream. The concluding remarks are given in Section 7.