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Compositional Aeroelastic Operators for Morphing Flexible Multibody Aircraft: A Geometric Framework with Structural Verification

Gelin Chen, Chen Song, Chao Yang

arXiv:2609.05464v1cs.CEmath.NAphysics.flu-dyn

TL;DR

Morphing aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible during joint and component configuration changes. The paper uses a compositional formulation with fixed material attachment between lifting surfaces and geometrically exact beams, separating relative-log strain coordinates from section-field geometry. Exact-reference relative-log data exhibit the expected N_d+1 convergence order, and adding the cubic static-manifold term reduces full-record displacement error in every completed load case relative to the quadratic manifold.

  • Problem

    Morphing aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible during joint and component configuration changes.

  • Method

    The paper uses a compositional formulation with fixed material attachment between lifting surfaces and geometrically exact beams, separating relative-log strain coordinates from section-field geometry.

  • Results

    Exact-reference relative-log data exhibit the expected N_d+1 convergence order, and adding the cubic static-manifold term reduces full-record displacement error in every completed load case relative to the quadratic manifold.

  • Takeaways & Limitations

    The results support the consistency of the structural construction and provide structural and interface-level evidence for the framework.

  • Takeaways & Limitations

    The attachment map is assumed locally invertible and fixed in time on each surface patch.

Abstract

from arXiv · show

Morphing flexible multibody aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible as joints and flexible components change configuration. A compositional formulation is developed around an assumed material attachment between each lifting surface and a geometrically exact beam. Separating the component root pose from the section field shows that the body strain and elastic potential of a component depend on its own elastic coordinates, while upstream motion enters kinetic terms and external-load pullbacks. At element level, an exact relative logarithm $d$ supplies strain and potential energy, whereas a reference-anchored section coordinate $σ$ supplies deformed section geometry. Finite-order expansions retain the finite reference geometry exactly and truncate only endpoint perturbations. The attachment map then generates surface points, tangents, normals, velocities, and force Jacobians from common section kinematics. Euler--Poincare beam balance, moving-surface potential-flow relations, graph cotangent assembly, and the associated semidiscrete power identity are stated in a common twist--wrench convention. Collocation, pressure, equivalent-load, and structural-station sites are distinguished to expose their approximation errors. Verification gives the expected $N_d+1$ convergence order for degree-$N_d$ relative-log expansions. In a geometrically nonlinear cantilever comparison, a cubic static-manifold correction reduces mean full-record displacement error from $0.479$ to $0.255$ over four completed load cases. These results provide structural and interface-level evidence rather than validation of a complete aircraft aeroelastic prediction.

Nomenclature

The paper frames morphing aeroelasticity as an interface-consistency problem and introduces a compositional attachment-based formulation with structural, aerodynamic, and graph-level operators.

  • Morphing aircraft must preserve compatibility among structural motion, aerodynamic geometry, surface velocity, and generalized loading during configuration changes.
  • A fixed material attachment assigns each aerodynamic point to a beam section and reference-frame offset, generating differentiable surface geometry, velocity, and force maps.This common source provides geometric consistency within the assumed attachment model.
  • The formulation separates the relative logarithm d for structural strain energy from the section coordinate σ for internal surface geometry.Both are expanded about finite reference geometry without treating that reference as a small elastic quantity.
  • The structural field equation, aerodynamic relations, and multibody graph assembly use a common twist–wrench pairing.The paper also distinguishes operator, pressure, equivalent-load, and structural station-proxy sites to expose approximation errors.
  • Verification combines exact-reference convergence data with a direct comparison of quadratic and cubic static-manifold corrections.

II. Geometric Setting and Compositional Hypotheses

The geometric setting represents the aircraft as a directed acyclic component graph and assumes a locally invertible, fixed material attachment between each beam section and aerodynamic surface patch.

  • Each flexible component is connected through a parent port pose in a directed acyclic graph, with transpose maps providing compatible backward wrench recursion.No separate moment-transfer convention is required.
  • The theory assumes elastic constitutive data are material-coordinate based, attachment charts are locally invertible, and physical loads assemble through cotangent maps.Beam discretization, aerodynamic paneling, and state reduction are introduced afterward.
  • The attachment map assigns surface parameters to material section coordinates and a single-valued reference-frame offset on each nondegenerate patch.If a global chart is unavailable, compatible patches are used; the principal development assumes the map is fixed in time.
  • Endpoint coordinates generate separate structural and geometric branches, preventing section interpolation from being used implicitly as a strain measure.Their shared endpoint source keeps structural energy and surface geometry tied to the same deformed component.
  • The exact relative logarithm supplies compatible relative deformation for strain and potential calculations, while section interpolation reconstructs the geometric field.For exponential interpolation, the left-trivialized strain is constant on the element.

B. Section interpolation and the role of 𝜎

The section coordinate σ supplies endpoint-consistent section pose and attached geometry, while reference-anchored expansions retain finite reference quantities exactly and truncate endpoint perturbations.

  • σ interpolates the section field between endpoint coordinates, satisfying σ(0)=q_a and σ(1)=q_b in the exact chart.Differentiation of σ provides the operators used for section pose, velocity, virtual motion, and attached geometry.
  • Ordinary Taylor–BCH truncation can misclassify the finite reference relative log d_0 as an elastic perturbation.
  • The anchored construction keeps the reference anchor finite while each Magnus term has degree k only in the endpoint perturbation variables.Dependence on the finite anchor remains exact.
  • A degree-N card projects the anchored increment and small–small BCH expansion, with residual O(∥z_e∥^(N+1)) in a valid local chart.
  • Finite-order claims refer to endpoint perturbation degree, not powers of reference curvature or reference element rotation.

IV. Euler–Poincaré Structure and Nonlinear Reduction

The structural formulation combines Euler–Poincaré beam balance, component-local strain energy, and a variationally complete nonlinear static-manifold reduction.

  • Euler–Poincaré beam balance: Equation (29) fixes the signs of inertia, elastic stress, external work, and boundary ports in the Euler–Poincaré beam field equation.The outward endpoint internal ports are −n(0) and n(L).
  • Component-local potential: Under the stated factorization and material-coordinate-independent root pose, component strain energy depends only on the component’s elastic coordinates.Its direct derivative with respect to upstream root or joint coordinates is zero.
  • Component-local potential: Upstream coordinates still enter kinetic terms, external work, graph pullbacks, local stress resultants, and configuration-dependent additional potentials.Joint springs, gravity, and contact need not satisfy the component-local potential result.
  • Nonlinear reduction: The static manifold y=h(q) is constructed from homogeneous terms satisfying degree-by-degree homological equations, with h(0)=0 and Dh(0)=0 under the stated basis conditions.
  • Nonlinear reduction: The reduced Lagrangian must include the configuration-dependent tangent pullback, because changing only the potential would not produce a complete manifold reduction.Section VIII comparisons use the complete tangent pullback.

V. Surface Geometry from an Assumed Material Attachment

The attachment formulation separates reference geometry from the deformed section field, generating compatible surface geometry, kinematics, and load maps from common section coordinates.

  • Common-source compatibility requires the relative-log and section expansions to use the same endpoint chart.
  • The central abstraction establishes material offsets from reference geometry and transports them through the deformed section field.
  • Using shared attachment derivatives makes the rate map and force pullback dual, preserving virtual work by construction.
  • The parameter-space chain rule generates covariant bases, metrics, dual bases, and oriented unit normals from the attachment map.
  • At fixed material coordinates, section variables have no explicit time derivative, so surface motion is inherited through section kinematics.
  • The declared attachment-coordinate orientation fixes the normal sign, while a negative parameter Jacobian can reverse it.

C. Discrete sites and their error forms

The formulation distinguishes aerodynamic and structural evaluation sites so their separate discretization choices expose pressure, load-transfer, station, and velocity errors.

  • C. Discrete sites and their error forms: Separating collocation, pressure, equivalent-load, and structural-station sites makes their discretization policy and associated errors explicit.
  • C. Discrete sites and their error forms: Omitting the residual couple is a force-equivalence approximation distinct from pressure quadrature.
  • C. Discrete sites and their error forms: Using a right-hand-side velocity at a site different from the reference point introduces a kinematic defect.
  • A. No penetration, circulation, and wake state: The equations do not require classical quarter-chord vortex and three-quarter-chord collocation placement; the operator site is declared by the panel model.
  • A. No penetration, circulation, and wake state: Linearized unsteady wake realizations depend jointly on the declared reference geometry, wake-convection law, state definition, and Kutta map.
  • A. No penetration, circulation, and wake state: The linearized wake equation does not represent a general nonlinear free wake on a moving geometry.
  • B. Moving-surface pressure equation: Moving-surface pressure uses the difference of unsteady Bernoulli relations at fixed material coordinates, including surface metric effects.
  • B. Moving-surface pressure equation: Attachment-generated normals, surface velocities, and Jacobians are reused in both the aerodynamic boundary condition and load map.

VII. Graph Cotangent Assembly and Wake Reduction

Graph cotangent assembly pulls local aerodynamic, inertial, and elastic contributions into global generalized work, while wake reduction preserves selected mechanical channels.

  • Graph assembly sums local inertial, elastic, and aerodynamic one-forms into the global d’Alembert one-form.
  • Backward recursion accumulates every descendant wrench at its parent, giving proximal joints the wrench of their downstream subtree.
  • The semidiscrete power identity follows when local load one-forms use the transpose of the tangent map that generates local velocities.
  • Internal graph-port wrenches cancel because connected components share admissible port velocities and carry opposite wrenches.
  • With no explicit time dependence, assembled d’Alembert equations satisfy the power relation up to explicitly declared dissipative or nonconservative work.
  • The wrench lever arm changes its reference point within a component, whereas graph adjoints pull the resulting covector through generalized coordinates.
  • Wake reduction selects generalized-force channels after local pressure loads are pulled to structural coordinates, avoiding reduction against poorly representative pressure coordinates.
  • Balanced truncation is applied only to the wake state, while direct channels remain; selected output norms are sensitivity indicators, not observability Gramians.

A. Exact-reference relative-log convergence

Exact-reference tests show the relative-log expansion achieves the expected order while preserving finite reference geometry, and the cantilever comparison shows consistent benefit from cubic static correction within a limited structural check.

  • A. Exact-reference relative-log convergence: The relative-log expansion achieves observed N_d+1 convergence for degrees N_d=1,...,5, with slopes from 2.001 to 6.000.The finite reference d_0 remains fixed as h varies, supporting the perturbation-degree interpretation.
  • A. Exact-reference relative-log convergence: Rigid-reference tests report zero error, while directional-action discrepancies remain between 6.6 × 10^-13 and 6.0 × 10^-12.
  • Scope and limitations: The verification is local and structural/interface-level rather than evidence of complete nonlinear aircraft aeroelastic predictive accuracy.Broader claims require additional reference-solver, geometry, loading, damping, and experimental or independent verification studies.
  • C. Geometrically nonlinear cantilever comparison: The H2 + H3 route reduces full-record displacement error in every completed cantilever case, lowering the mean from 0.479 to 0.255.This corresponds to a 46.7% reduction over four completed load cases.
  • C. Geometrically nonlinear cantilever comparison: The cubic route lowers mean peak error from 4.73% to 3.25%, although the linear model has the smallest peak-amplitude error in these four cases.Full-record trajectory error remains distinct from peak agreement, and cubic-route errors remain 15–36%.

X. Conclusions

The paper presents a compositional aeroelastic framework that separates component elastic fields from upstream motion and derives shared structural–aerodynamic interfaces. Verification shows expected relative-log convergence and improved nonlinear cantilever reduction, while remaining evidence is structural and interface-level rather than complete aircraft validation.

  • The compositional formulation separates each component’s elastic section field from root and upstream motion, assigning strain energy locally while retaining upstream effects in inertia, external work, and graph ports.
  • A common attachment map generates surface geometry, velocities, and force Jacobians, allowing aerodynamic boundary conditions and pressure one-forms to remain paired under virtual work.
  • The formulation combines Euler–Poincaré beam balance, graph cotangent pullback, moving-surface relations, and explicit discretization proxies for structural and multibody assembly.
  • The exact-reference relative-log construction exhibits the expected N_d+1 convergence order, supporting consistency of the structural expansion.
  • The results provide structural and interface-level evidence, with further validation requiring aerodynamic refinement, attachment comparisons, and independently verified or experimental responses across morphing configurations.

B.1. Exact-reference construction for 𝑑

The exact-reference construction expands relative deformation around a finite reference while preserving reference geometry and controlling endpoint perturbation order. Consistent truncation couples chart reversion, section-coordinate reconstruction, and potential-energy order.

  • The construction introduces an exact-reference increment map and uses homogeneous series reversion after establishing nonsingularity of d_0 in the selected chart.
  • The relative logarithm d supplies deformation data, while the section coordinate σ provides an exact split for internal geometry used in the attached surface.
  • Finite-order expansions retain finite reference geometry exactly and truncate endpoint perturbations through separately specified small–small BCH and final endpoint degrees.
  • Final endpoint projection is necessary because BCH slot degree can understate endpoint degree when an anchored increment is quadratic.
  • The logarithmic expansions require a continuous local branch, and endpoint perturbations must remain within the reference chart; value convergence alone is insufficient for dynamics.
  • Potential-energy order determines the required deformation order: fourth-order potential needs d through degree three, whereas sixth-order potential needs degree five.
  • Static-manifold corrections and relative-log expansions are matched as one reduced construction, with H2 and H2+H3 requiring progressively higher reconstructed d and potential orders.

D. Attachment Differentials and Equivalent-Point Errors

The attachment-differential construction propagates section motion into surface and load quantities, while equivalent-point analysis identifies the limits of force-only load relocation. A staged workflow and verification suite expose these approximation boundaries.

  • Section rotation and translation enter attachment derivatives, and point Jacobians pull surface forces and couples back to elastic coordinates.
  • Moving a force to a station without updating its moment produces a residual-couple error; force relocation can reproduce only the target-moment component perpendicular to the force.
  • A moment component parallel to a nonzero force cannot be represented by relocating that force and must remain as a free couple; when force is zero, any nonzero panel moment is purely a couple.
  • In serial morphing segments, each component retains its own elastic variables and attachment map, while downstream inertia and aerodynamic loading still contribute to upstream coordinates through graph pullback.
  • Graph assembly accumulates local one-forms through subtree covector pullback, so proximal joints collect downstream contributions without a separately prescribed moment rule.
  • The implementation sequence defines reference geometry, component reduction, attached-surface kinematics, aerodynamic one-forms, graph assembly, and dynamic reduction before verification.
  • Verification checks attachment identities, normal orientation, value and action convergence, virtual-work duality, static-manifold closure, equivalent-point residuals, and reduced-state diagnostics.
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