Partitioned procedures for the transient solution of coupled aeroelastic problems – Part II: energy transfer analysis and three-dimensional applications

Abstract

We consider the problem of solving large-scale nonlinear dynamic aeroelasticity problems in the time-domain using a fluid/structure partitioned procedure. We present a mathematical framework for assessing some important numerical properties of the chosen partitioned procedure, and predicting its performance for realistic applications. Our analysis framework is based on the estimation of the energy that is artificially introduced at the fluid/structure interface by the staggering process that is inherent to most partitioned solution methods. This framework also suggests alternative approaches for time-discretizing the transfer of aerodynamic data from the fluid subsystem to the structure subsystem that improves the accuracy and the stability properties of the underlying partitioned method. We apply this framework to the analysis of several partitioned procedures that have been previously proposed for the solution of nonlinear transient aeroelastic problems. Using two- and three-dimensional, transonic and supersonic, wing and panel aeroelastic applications, we validate this framework and highlight its impact on the design and selection of a staggering algorithm for the solution of coupled fluid/structure equations.

Introduction

Wing flutter, fighter tail buffeting, flow-induced pipe vibrations, and atrial flutter are examples of fluid/structure interaction phenomena that are of great concern to aerospace, mechanical, civil, and biomedical engineering. These as well as several other fluid/structure interaction problems can be described as aeroelastic problems. When the underlying fluid system is represented by a nonlinear model (i.e., the Euler or Navier–Stokes equations), and the transient response of the given structure is of interest, these problems are referred to as nonlinear transient aeroelastic problems.

A nonlinear transient aeroelastic problem where the fluid domain boundaries undergo a motion with a large amplitude can be formulated as a three-field problem – the fluid, the structure, and the dynamic mesh that is often represented by a pseudo-structural system – governed by the following coupled semidiscrete equations [1], [2], [3].

$$\text{d}\text{d}\text{t}\text{(}\text{AW}\text{)+}\text{F}{}^{\mathrm{<mtext>c</mtext>}}\text{(}\text{W}\text{,}\text{x}\text{,}\text{x}\text{̇}\text{)=}\text{R}\text{(}\text{W}\text{,}\text{x}\text{),}$$$$\text{M}\text{d}{}^2\text{u}\text{d}\text{t}{}^2\text{+}\text{D}\text{d}\text{u}\text{d}\text{t}\text{+}\text{Ku}\text{=}\text{f}{}^{\mathrm{<mtext>ext</mtext>}}\text{(}\text{W}\text{(}\text{x}\text{,t),}\text{x}\text{),}$$$$\text{M}\text{d}{}^2\text{x}\text{d}\text{t}{}^2\text{+}\text{D}\text{d}\text{x}\text{d}\text{t}\text{+}\text{K}\text{x}\text{=0,}$$

where a dot denotes a time-derivative, $\text{x}$ the displacement or position vector of the moving fluid grid points (depending on the context), $\text{W}$ the fluid state vector, $\text{A}$ results from the finite element/volume discretization of the fluid equations, $\text{F}{}^{\mathrm{<mtext>c</mtext>}}\text{=}\text{F}\text{−}\text{x}\text{̇}\text{W}$ the vector of arbitrary Lagrangian Eulerian (ALE) convective fluxes, $\text{F}$ the vector of convective fluxes, $\text{R}$ the vector of diffusive fluxes, $\text{u}$ the structural displacement vector, $\text{M}\text{,}\text{D}$, and $\text{K}$, respectively, the finite element mass, damping, and stiffness matrices of the structure, $\text{f}{}^{\mathrm{<mtext>ext</mtext>}}$ the vector of external forces acting on the structure, and $\text{M}\text{,}\text{D}$, and $\text{K}$ are fictitious mass, damping, and stiffness matrices associated with the moving fluid grid and constructed to control its motion. , express the equilibrium of the fluid and structure subsystems, respectively. Eq. (3) is a mathematical representation of a fluid dynamic mesh. It is usually driven by Dirichlet boundary conditions at the fluid/structure interface that enforce the compatibility of the displacement and velocity fields of the structure and the fluid mesh. Note that $\text{M}\text{=}\text{D}\text{=0}$ includes as particular cases the spring analogy and continuum mechanics-based mesh motion schemes advocated by many investigators (for example, see [4], [5]).

For simple and small-scale structural problems – for example, for an airfoil with one or two vibrational degrees of freedom – Eq. (2) can be efficiently recast in first-order form so that the fluid and the structural equations of motion can be combined into a single formulation (for example, see [6]). In such a case, a “monolithic” fully explicit or fully implicit treatment of the coupled fluid/structure equations of motion is possible. However, for more complex structural systems, the simultaneous solution of , , by a monolithic scheme is in general computationally challenging, mathematically and economically suboptimal, and software-wise unmanageable.

Alternatively, , , can be solved by a partitioned procedure where the fluid and structure subproblems are time-discretized by different methods tailored to their different mathematical models, and the resulting discrete equations can be solved by a “staggered”, or “segregated”, or “time-lagged” algorithm [2], [3], [7], [8], [9]. Such a strategy simplifies explicit/implicit treatment, subcycling, load balancing, software modularity, and replacements as better mathematical models and methods emerge in the fluid and/or structure disciplines. The basic and most popular staggered algorithm, referred to in this paper as the conventional serial staggered (CSS) procedure, goes as follows: (a) transfer the motion of the wet boundary of the structure to the fluid system, (b) update the position of the moving fluid mesh accordingly, (c) advance the fluid system using a given flow time-integrator and compute new pressure and fluid stress fields, (d) convert the new fluid pressure and stress fields into a structural load, and (e) advance the structural system under the flow-induced load using a given structure time-integrator. Such a staggered procedure, which can be described as a loosely coupled solution algorithm, can also be equipped with a subcycling strategy where the fluid and structure subsystems are advanced using different time-steps Δt_F and Δt_S [3]. Usually, one has Δt_F⩽Δt_S.

Unfortunately, it is well-known that the time-accuracy of the CSS procedure is in general at least one order lower than that of its underlying flow and structure time-integrators, and its stability limit can be much more restrictive than that of the flow and/or structure solvers. For this reason, several ad-hoc strategies have been published in the literature for improving the time-accuracy and stability properties of the CSS procedure. Most of them consist essentially in inserting some type of predictor/corrector iterations within each cycle of the CSS procedure, in order to compensate for the time-lag between the fluid and structure solvers [7], [10].

Previously, we have considered the mathematical analysis of the time-accuracy and numerical stability of partitioned procedures constructed for the solution of , , , in an attempt to improve our understanding of their behavior, and design better alternatives to the CSS method [3]. However, because of the dependence of the structure equations of equilibrium on the motion of the fluid dynamic mesh is implicit rather than explicit, and the fluid equations of motion can be strongly nonlinear, we had to confine ourselves to the mathematical investigation of a one-dimensional aeroelastic model problem. This model problem was obtained by linearizing , , around a position of aeroelastic equilibrium. Furthermore, Eq. (3) was replaced by “transpiration” fluxes at the fluid/structure interface, and therefore the model problem was formulated as a two-field and two-way coupled fluid/structure interaction problem. We have constructed a family of staggered algorithms for solving this model problem, and proved that an unconditionally stable partitioned procedure that furthermore retains the order of time-accuracy of its underlying flow and structure time-integrators can be constructed, by superposing a subiteration-free but carefully constructed corrector scheme to the basic CSS method. Based on our mathematical analysis, we have also established guidelines for exchanging aerodynamic and elastodynamic data in the presence of subcycling, in a manner that preserves the unconditional stability and order of time-accuracy of a given partitioned procedure. We were able to apply some but not all of these ideas to the realistic problem represented by , , (see also [11]). However, the formal analysis of staggered algorithms applied to the solution of , , remains a formidable challenge, because of the same reasons that previously incited us to consider a representative model problem.

For this reason, we complement in this paper our previous work by an additional criterion for assessing the suitability of a given partitioned procedure for the solution of the nonlinear transient aerolastic , , . This criterion is essentially based on the evaluation of the energy that is numerically – and hence, artificially – created at the fluid/structure interface by the staggering process. We show that this criterion typically dictates the choice of the predictor, and/or the time-discretization of the transfer of the fluid pressure and stress fields to the structure subsystem. We apply this criterion to the analysis of several partitioned procedures that have been previously proposed for solving , , . We validate it, and illustrate its effectiveness at discriminating between staggered algorithms as well as improving them, by considering the solution of two- and three-dimensional, transonic and supersonic, wing and panel flutter problems.