A non-intrusive geometrically nonlinear augmentation to generic linear aeroelastic models

Abstract

A new approach to build geometrically-nonlinear dynamic aeroelastic models is proposed that only uses information typically available in linear aeroelastic analyses, namely a generic (linear) finite-element model and frequency-domain aerodynamic influence coefficient matrices (AICs). Good computational efficiency is achieved through a two-step process: Firstly, a geometric reduction of the structure is carried out through static or dynamic condensation on nodes along the main load paths of the vehicle. Secondly, manipulation of the resulting linear normal modes (LNMs), the condensed stiffness and mass matrices, and the nodal coordinates provides the modal coefficients of the intrinsic beam equations along these load paths. This preserves the LNMs of the original problem and augments them with the geometrically nonlinear terms of beam theory. The structural description is in material coordinates and modal AICs are thus naturally included as follower forces. Numerical examples include cantilever wings built using detailed models, for which effects such as nonlinear aeroelastic equilibrium, nonlinear dynamics and structural-driven limit-cycle oscillations are shown. Results demonstrate the ability of the methodology to seamlessly and efficiently incorporate critical nonlinear effects to (linear) arbitrarily large aeroelastic models of high aspect ratio wings.

Introduction

Next-generation passenger aircraft will need to meet much stricter emissions and fuel efficiency targets. This will undoubtedly accelerate the push towards increased used of advanced lighter materials to reduce operating weight and configurations with very high aspect ratio wings to reduce induced drag. Aeroelastic effects will thus dominate many aspects of wing design, although higher flexibility may also have positive effects in damping some external disturbances, such as gusts. As a result, airframe design processes may need to be revised (Palacios et al., 2014), and, in particular, geometrical-nonlinear effects may need to be accounted for in stages of the analysis where they are currently neglected. This has already been the case in more radical configurations, such as large solar-powered aircraft (Noll et al., 2007), where linear analysis yielded non-conservative results as they missed critical nonlinear aeroelastic couplings (Patil and Hodges, 2004). While fully-coupled nonlinear aeroelastic simulation is now available (Qiao et al., 2018), certification of a new air vehicle currently requires 100,000s of load case simulations (Kier, 2017), as it considers maneuvers and gust loads at different velocities and altitudes, and for a range of mass cases and configurations. Statistical methods may help reduce the total simulation burden, yet there is still a need for a computationally-efficient models that provide an acceptable level of fidelity to describe the vehicle dynamics. Furthermore, design of control strategies also requires reduced models, and standard approaches with linear models can fail under large wing deformations (Wang et al., 2018).

Two main approaches have been considered so far to construct computationally-efficient aeroelastic models with geometrical nonlinearity, namely, descriptions based on beam theory, and model-order reduction methods. Geometrically nonlinear beams have been extensively used to study these problems (see, for instance, the review by Afonso et al., 2017). While they have been very useful to understand key physics and explore the design space, they impose substantial limitations when detailed finite-element (FE) models are already available. Extraction of the equivalent beam properties is needed through either a homogenization process (Dizy et al., 2013) or a stick modeling approach (Riso et al., 2020). For large structures, it has been shown that either approach can lead to significant errors when compared to nonlinear simulations using built-up finite-element models (Palacios and Cea, 2019, Medeiros et al., 2020). Alternatively, system identification approaches have been proposed (Mignolet et al., 2013) that capture the nonlinear response of structure to construct reduced-order models (ROMs) for fast computations in nonlinear dynamic aeroelasticity (Medeiros et al., 2020). The main issue here lies in the prior calculation of a training database made of a large number of static nonlinear computations, which then acts as the model internal physics in the actual simulations. This approach also needs the full model to be suitable for nonlinear simulations, which could be a real constrain, e.g. on linear models calibrated against Ground Vibration Tests (GVT).

The goal of this work is to develop an efficient computational framework to study the geometrically nonlinear effects on already existing (linear) industrial-scale aeroelastic models, building on an alternative approach (Wang et al., 2015, Palacios and Cea, 2019) that does not require prior nonlinear computations on a full 3D finite-element model. Instead, the geometric layout of the structure and the coupling between modes in the intrinsic beam equations (Hodges, 2003) – that, by construction, take sectional velocities and internal forces as main variables – are employed. A feature of this approach is that mass and stiffness properties of the initial configuration require no updating as the model deforms. Furthermore, only quadratic terms of the main variables are needed to capture the complete space of geometric nonlinearities. As for the modeling of the aerodynamic forces, although 2-D airfoil unsteady aerodynamics are still often used for high-aspect-ratio wing aeroelasticity, 3-D effects have been shown to play an important role (Modaress-Aval et al., 2019) and will be considered here. The aerodynamic forces are obtained here from the Doublet-Lattice Method (DLM), which solves a linearization of the compressible, inviscid, unsteady flow equations in the 3-dimensional domain, and it is commonly used in linear aircraft aeroelastic analysis (Kim, 2019). It will be shown how key geometrically nonlinear effects (wing inextensionality, follower force effects), can be taken into account despite the linear – i.e. not updating with geometry – aerodynamic matrices obtained from the DLM. The Unsteady Vortex Lattice Method (UVLM) (Murua et al., 2012) is another potential method for geometrically-nonlinear aeroelastic simulations, as it fully accounts for large deformations both in the structure and the wake. However, it is substantially more computationally demanding and it is limited to incompressible conditions, therefore not appropriate for the typical flow regimes of commercial aircraft. The current description can also accommodate higher fidelity aerodynamic models, such as 3D panel methods (Kier, 2017) that include thickness effects, CFD-generated aerodynamic influence coefficient matrices (Güner et al., 2019), and it will be shown that it is suitable for an industrial loads and aeroelastic environment.

The structure of this paper is as follows. Section 2 presents the methodological approach proposed in this work; Section 3 shows static and dynamic aeroelastic results with three different models, all of which were originally built as linear MSC Nastran aeroelastic models; and Section 4 presents the main conclusions of the work.