Nonlinear Model Order Reduction for Coupled Aeroelastic-Flight Dynamic Systems

Abbreviations

• Abbreviations identify the paper’s recurring terms, including FOM, NMOR, ROM, and UVLM.The list also defines DOF, HALE, LCO, NFOM, NROM, UAV, and related abbreviations.Source: Abbreviations · p. 2

Roman Symbols

• Roman symbols cover system quantities such as the Jacobian A, interaction coefficients D_kij and E_kijl, and state vectors w and z.The list also names aerodynamic, structural, and input quantities.Source: Roman Symbols · p. 2

Greek Symbols

• Greek symbols identify quantities including angle of attack α, flexibility parameters σ, and Jacobian eigenvalue λ_i.The list also defines perturbation, attitude, and modal quantities.Source: Greek Symbols · p. 3

Operators

• The operators B and C denote the symmetric bilinear second-order and trilinear third-order operators, respectively.Source: Operators · p. 3

• Large wing deformations in HALE and solar-powered aircraft couple flexibility, unsteady aerodynamics, and rigid-body flight dynamics.The Helios accident is cited as evidence that separating flight-dynamics and aeroelastic analyses is inappropriate for such configurations.Source: §1 · p. 3
• Coupled full-order models can have hundreds to thousands of degrees of freedom, making parametric studies and control design computationally prohibitive.A single nonlinear time-domain simulation can take hours, while certification-relevant parameter sweeps require hundreds of simulations.Source: §1 · p. 3
• Existing MOR approaches have limits: POD needs excitation-dependent training data, harmonic balance targets periodic or quasi-periodic regimes, and linear identification misses essential nonlinearities.Source: §1 · p. 3
• Prior nonlinear aeroelastic studies examined large-deflection behavior, adaptive parametric ROMs, and Taylor-derivative interpolation for high-amplitude limit-cycle oscillations.Riso and Cesnik [2023a] reported comparable flutter and post-flutter accuracy for strip-theory and higher-fidelity UVLM models.Source: §1 · p. 4
• The paper uses Taylor-series NMOR to project coupled-system dynamics onto Jacobian eigenvectors, retaining nonlinear terms through second- and third-order coefficients.The resulting models are excitation-independent once generated and support rapid parametric studies.Source: §1 · p. 4
• ∼1,600 DOF to 9 with up to 600× computational speedup are demonstrated across three test cases of increasing complexity.Source: §1 · p. 4
• The contributions include a systematic eigenvector-basis strategy combining rigid-body and structural modes, alongside evidence of formulation independence.The framework is presented and validated across three increasingly complex test cases.Source: §1 · p. 4

2.1 General Framework

• At trim, the state vector separates augmented aerodynamic, structural, and rigid-body states; very flexible aircraft may require nonlinear trim for deformations exceeding 10–25% of wingspan.The coupled model combines three subsystems, with aerodynamic loads resolved in frames attached to the deformed beam cross-section.Source: §2.1 · p. 4, §2 · p. 4, Figure 1 · p. 5

2.2 Unsteady Aerodynamic Model

• Strip-theory aerodynamic coefficients divide lift, pitching moment, and hinge moment into section-motion, flap, and gust contributions.Each spanwise section uses local aerodynamic frames attached to the deformed beam cross-section.Source: §2.2 · p. 5, Figure 2 · p. 5

2.2.1 Section motion contribution

• Theodorsen’s frequency-domain section-motion model is replaced for time-domain simulation by a two-exponential Wagner approximation with two augmented states per strip.The approximation uses Ψ1 = 0.165, Ψ2 = 0.335, ε1 = 0.0455, and ε2 = 0.3.Source: §2.2.1 · p. 6

2.2.2 Atmospheric gust contribution

• The Küssner gust approximation adds two augmented states per strip, giving four aerodynamic states per strip without flap and eight with flap.Gust penetration represents unsteady lift build-up as a thin aerofoil enters a sharp-edged gust.Source: §2.2.2 · p. 6

2.3 Structural Dynamics

• Geometrically-exact beam dynamics use two-noded finite elements with six translational and rotational degrees of freedom per node and frame-invariant strains that accommodate large displacements and rotations.The nodal rotation tensor is obtained from the Cartesian rotation vector using the Rodrigues formula.Source: §2.3 · p. 6, Figure 3 · p. 7
• The virtual-work finite-element equations include configuration-dependent tangent mass, gyroscopic and Coriolis contributions, elastic and geometric-stiffening forces, and aerodynamic forces.Aerodynamic generalized forces come from the strip-theory model.Source: §2.3 · pp. 6–7

2.4 Flight Dynamics

• Rigid-body flight dynamics use body-frame Newton–Euler equations for translational and angular motion, while quaternion attitude propagation avoids Euler-angle singularities and enforces unit norm.The translational equation includes aerodynamic, gravitational, and thrust forces; the rotational equation includes aerodynamic and gravitational moments.Source: §2.4 · p. 7

2.5 Coupled System in First-Order Form

• Combining aerodynamic, structural, flight, and quaternion equations yields a first-order system whose trim-state Jacobian separates linear dynamics from control, gust, and nonlinear residual terms.The Jacobian’s off-diagonal blocks encode coupling among fluid, structural, and rigid-body partitions.Source: §2.5 · p. 7

3.1 Taylor Series Expansion

• The residual is expanded around equilibrium through third order, defining symmetric bilinear and trilinear operators for its second and third derivatives.The expansion uses the perturbation from the equilibrium state.Source: §3.1 · p. 8

3.2 Eigenvector Basis Construction

• The basis retains near-origin real modes for rigid-body and gust-coupling dynamics and lightly damped complex modes for dominant structural oscillations.Biorthonormal left and right eigenvectors make the linear modal equations decoupled, leaving nonlinear interactions to couple the modes.Source: §3.2 · p. 8

3.3 Reduced-Order Equations

• Galerkin projection onto the selected eigenvectors yields a reduced system, typically with 4–9 modes instead of 500–2,000 full-order states.The reduced equations retain physical nonlinear coupling and can be excited by arbitrary gust profiles, control inputs, or initial conditions without regeneration.Source: §3.3 · pp. 8–9

3.4 Matrix-Free Computation of Higher-Order Terms

• Matrix-free finite differences compute higher-order operator actions on selected eigenvector directions, avoiding explicit formation of prohibitively large bilinear and trilinear operators.The perturbation parameter balances truncation and round-off errors.Source: §3.4 · p. 9
• For 9 modes, coefficient construction requires about 81 second-order and 729 third-order residual evaluations, while subsequent ROM time marching costs negligibly compared with the full-order model.Construction cost is incurred once and is a modest multiple of the trim-solution cost.Source: §3.4 · p. 9

4 Results

• The validation covers three test cases of progressively increasing complexity, reproduced from previously published studies for a self-contained presentation.Table 1 summarizes the cases.Source: §4 · p. 9, Table 1 · p. 9

4.1 Three-Degree-of-Freedom Aerofoil

• The aerofoil has pitch, plunge, and trailing-edge flap degrees of freedom, with cubic hardening in pitch and plunge; its 14-state full-order model is reduced using four retained eigenvalues.The retained set includes structural-mode pairs and a real eigenvalue associated with Küssner gust coupling.Source: §4.1 · pp. 9–10

4.1.1 Sub-critical response

• At sub-critical speed, the system decays to the trivial equilibrium and the nonlinear ROM reproduces pitch, plunge, and flap histories with negligible error.The cubic hardening nonlinearity plays a minor role in this regime, where the linear ROM also performs well.Source: §4.1.1 · p. 10

4.1.2 Post-flutter limit-cycle oscillation

• The nonlinear ROM captures the post-flutter limit cycle and closely matches the full-order model, whereas the linear ROM cannot predict bounded post-flutter oscillations.The cubic hardening nonlinearity limits the unstable oscillation to a stable limit cycle; a second-order Taylor expansion is sufficient to reproduce this cubic structural behavior.Source: §4.1.2 · pp. 10, 12

4.2 HALE Aircraft Configuration

• The 2,016-state HALE model represents a 32-m-span aircraft and is reduced using nine retained eigenvalues.The model includes structural, rigid-body, attitude, and augmented aerodynamic states.Source: §4.2 · pp. 12–13, Figure 8 · p. 12, §4.2.4 · p. 15

4.2.1 Structural validation

• The strip-theory flutter speeds are closer to Patil et al.’s results using two-dimensional Peters aerodynamics, while minor differences from UVLM are attributed to three-dimensional aerodynamic effects.The structural natural-frequency comparison is reported as excellent agreement with published results.Source: §4.2.1 · p. 13, Table 2 · p. 13, Table 3 · p. 13

4.2.2 Static aeroelastic deformation

• At α = 2°, the strip-theory model agrees well with the higher-fidelity Euler CFD and UVLM deformation results; at α = 4°, predictions diverge because strip theory omits three-dimensional aerodynamic effects.The divergence identifies a fidelity boundary for the two-dimensional aerodynamic model.Source: §4.2.2 · p. 13, Figure 9 · p. 14

4.2.4 Reduced-order model

• At the design trim condition, significant wing-tip deflection makes geometric nonlinearities relevant to the reduced-order model.The 9 retained eigenvalues represent rigid-body and gust-coupling dynamics alongside bending and torsion modes.Source: §4.2.2 · p. 13, §4.2.4 · p. 15

4.2.3 Vertical equilibrium trimming

• Large wing bending requires a larger angle of attack for vertical force equilibrium because lift no longer acts vertically; two-dimensional aerodynamics reproduce the described physics of flexible aircraft with large deformations and rigid-body motion.The reported trim uses a payload positioned 2 m from the reference location.Source: §4.2.3 · pp. 13, 15

4.3 Very Flexible Flying-Wing

• The case is a 32-m-span, symmetric, tail-less free-flying wing with 1,616 states, including rigid-body, attitude, structural, and aerodynamic states.It is a challenging NMOR case because of large deformations, coupling to flight dynamics, and absent inherent pitch stiffness from a conventional tail.Source: §4.3 · p. 15, Table 4 · p. 16, Table 5 · p. 16

4.3.1 Eigenvector basis selection

• The coupled Jacobian basis retains 9 eigenvalues: 4 real eigenvalues near the origin and 5 complex conjugate pairs.The complex eigenvalues correspond to the first through fourth bending modes and first torsion, modified by aerodynamic coupling.Source: §4.3.1 · p. 15, Figure 12 · p. 16, Table 6 · p. 17

4.3.2 ROM convergence study

• The 9-mode ROM overlaps with the nonlinear full-order model for stochastic-gust wing-tip response; adding the first bending mode recovers the dominant oscillation frequency and amplitude.One mode captures the rigid-body angle-of-attack response but not structural aeroelastic dynamics; four modes reproduce the slow phugoid-like oscillation but overpredict its amplitude.Source: §4.3.2 · p. 17, Figure 13 · p. 17

4.3.3 Computational speedup

• 600× speedup: the 9-mode nonlinear ROM completed 37 gust cases in 22 minutes, versus 222 hours for the full-order simulations.The 80-element mesh used in the ROM study was converged, with negligible difference from the 160-element mesh.Source: §4.3.3 · p. 18, Figure 14 · p. 18

4.3.4 Linear versus nonlinear ROM accuracy

• The linear ROM accurately predicts full-order response below approximately 10% wingspan deformation, but misses geometric stiffening at larger deformations and overestimates response amplitude.The stiffening occurs as the wing bends upward and the lift-vector direction changes.Source: §4.3.4 · p. 18, Figure 15 · p. 19
• The second-order nonlinear ROM tracks the full-order response at wing-tip deformations around 10% of the wingspan, while third-order terms provide negligible improvement for the gust cases considered.Quadratic modal interactions generate effective cubic nonlinearities that reproduce geometric stiffening; third-order terms require 729 additional residual evaluations, versus 81 for second-order terms.Source: §4.3.4 · p. 18, Figure 15 · p. 19

4.3.5 Flexibility effect on flight dynamics

• Increasing flexibility increases wing-tip deformation and marginally affects flight-dynamic response through changes in aerodynamic forces.The study varies stiffness scaling from a very stiff wing (σ = 0.001) to four times nominal flexibility (σ = 4).Source: §4.3.5 · p. 19

4.3.6 Summary of NMOR performance

• Table 7 summarizes NMOR performance across the three test cases.The supplied passage identifies the table's scope but does not provide its cell values.Source: §4.3.6 · p. 19, Table 7 · p. 19

Sufficiency of second-order Taylor expansion

• Across the test cases, a second-order Taylor expansion provides accuracy comparable to the full nonlinear model, including for cubic structural nonlinearities.Interactions between distinct modal amplitudes produce effective cubic behavior through the projected quadratic operator.Source: §5 · pp. 19–20

Formulation independence

• NMOR requires residual evaluation at arbitrary states and solution of the coupled Jacobian eigenproblem, not access to the full-order model's internal structure.The stated scope includes CFD-coupled systems, UVLM-based models, and commercial finite-element codes.Source: §5 · p. 20

Basis selection strategy

• The two-tier basis strategy selects real eigenvalues for rigid-body and gust modes, then lightly damped complex eigenvalues for structural modes; 9 modes suffice for the flying wing.The first bending mode is the most critical addition beyond rigid-body dynamics.Source: §5 · p. 20

Limitations

• A single-equilibrium Taylor expansion may need updating for flight regimes with large trim-state changes, while strip theory limits accuracy at high angles of attack or with strong three-dimensional flow.The discussion identifies multipoint expansion as a potential remedy for changing trim states.Source: §5 · p. 20

6 Conclusions

• The NMOR technique reduces systems of approximately 1,600 degrees of freedom to 9 and achieves speedups up to 600× for parametric gust studies.The conclusion reports 222 hours reduced to 22 minutes.Source: §6 · p. 21
• The second-order Taylor expansion captures cubic structural nonlinearities through quadratic modal interactions, avoiding computationally expensive third-order terms.The conclusion also states that the approach is independent of the full-order model formulation and uses a systematic eigenvector-basis strategy.Source: §6 · p. 21
• The reduced models support rapid parametric studies and longer-duration simulations for aeroelastic stability verification and design evaluations of HALE and solar-powered aircraft.These applications are stated as practical uses of the framework.Source: §6 · p. 21