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

• Morphing aircraft must preserve compatibility among structural motion, aerodynamic geometry, surface velocity, and generalized loading during configuration changes.Source: §I · p. 3
• 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.Source: §I · p. 3
• 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.Source: §I · p. 3
• 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.Source: Nomenclature · p. 2, §I · p. 3
• Verification combines exact-reference convergence data with a direct comparison of quadratic and cubic static-manifold corrections.Source: §I · p. 3

• 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.Source: A. Twists, wrenches, and the component… · p. 4
• 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.Source: B. Structural and attachment hypotheses · p. 5
• 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.Source: B. Structural and attachment hypotheses · p. 5
• 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.Source: C. Dependency chain · p. 5
• 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.Source: A. Exact relative log and elastic… · p. 6

• 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.Source: B. Section interpolation and the role… · p. 7
• Ordinary Taylor–BCH truncation can misclassify the finite reference relative log d_0 as an elastic perturbation.Source: C. Reference-anchored finite-order… · p. 7
• 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.Source: C. Reference-anchored finite-order… · p. 7
• 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.Source: C. Reference-anchored finite-order… · p. 8
• Finite-order claims refer to endpoint perturbation degree, not powers of reference curvature or reference element rotation.Source: C. Reference-anchored finite-order… · p. 8

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).Source: A. Body fields, compatibility, and the… · p. 8

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.Source: B. Component-local elastic potential · p. 9
• 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.Source: B. Component-local elastic potential · p. 9

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.Source: C. Mechanics-informed static… · p. 10
• 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.Source: C. Mechanics-informed static… · pp. 10–11

• Common-source compatibility requires the relative-log and section expansions to use the same endpoint chart.Source: A. Reference anchoring and current… · p. 11, B. Tangents, metric, normal, and… · p. 13
• The central abstraction establishes material offsets from reference geometry and transports them through the deformed section field.Source: A. Reference anchoring and current… · p. 11
• Using shared attachment derivatives makes the rate map and force pullback dual, preserving virtual work by construction.Source: B. Tangents, metric, normal, and… · pp. 12–13
• The parameter-space chain rule generates covariant bases, metrics, dual bases, and oriented unit normals from the attachment map.Source: B. Tangents, metric, normal, and… · p. 12
• At fixed material coordinates, section variables have no explicit time derivative, so surface motion is inherited through section kinematics.Source: B. Tangents, metric, normal, and… · p. 12
• The declared attachment-coordinate orientation fixes the normal sign, while a negative parameter Jacobian can reverse it.Source: B. Tangents, metric, normal, and… · pp. 12–13

C. Discrete sites and their error forms

• Separating collocation, pressure, equivalent-load, and structural-station sites makes their discretization policy and associated errors explicit.Source: C. Discrete sites and their error forms · p. 13
• Omitting the residual couple is a force-equivalence approximation distinct from pressure quadrature.Source: C. Discrete sites and their error forms · p. 13
• Using a right-hand-side velocity at a site different from the reference point introduces a kinematic defect.Source: C. Discrete sites and their error forms · p. 13

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.Source: A. No penetration, circulation, and… · p. 14
• Linearized unsteady wake realizations depend jointly on the declared reference geometry, wake-convection law, state definition, and Kutta map.Source: A. No penetration, circulation, and… · p. 14
• The linearized wake equation does not represent a general nonlinear free wake on a moving geometry.Source: A. No penetration, circulation, and… · p. 14

B. Moving-surface pressure equation

• Moving-surface pressure uses the difference of unsteady Bernoulli relations at fixed material coordinates, including surface metric effects.Source: B. Moving-surface pressure equation · pp. 14–15
• Attachment-generated normals, surface velocities, and Jacobians are reused in both the aerodynamic boundary condition and load map.Source: B. Moving-surface pressure equation · p. 15

• Graph assembly sums local inertial, elastic, and aerodynamic one-forms into the global d’Alembert one-form.Source: A. Local one-forms and graph pullback · p. 15
• Backward recursion accumulates every descendant wrench at its parent, giving proximal joints the wrench of their downstream subtree.Source: A. Local one-forms and graph pullback · p. 15
• The semidiscrete power identity follows when local load one-forms use the transpose of the tangent map that generates local velocities.Source: A. Local one-forms and graph pullback · p. 16
• Internal graph-port wrenches cancel because connected components share admissible port velocities and carry opposite wrenches.Source: A. Local one-forms and graph pullback · p. 16
• With no explicit time dependence, assembled d’Alembert equations satisfy the power relation up to explicitly declared dissipative or nonconservative work.Source: A. Local one-forms and graph pullback · p. 16
• The wrench lever arm changes its reference point within a component, whereas graph adjoints pull the resulting covector through generalized coordinates.Source: A. Local one-forms and graph pullback · p. 16
• Wake reduction selects generalized-force channels after local pressure loads are pulled to structural coordinates, avoiding reduction against poorly representative pressure coordinates.Source: B. Mechanism-relevant aerodynamic… · pp. 16–17
• Balanced truncation is applied only to the wake state, while direct channels remain; selected output norms are sensitivity indicators, not observability Gramians.Source: B. Mechanism-relevant aerodynamic… · p. 17

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.Source: A. Exact-reference relative-log… · p. 17
• Rigid-reference tests report zero error, while directional-action discrepancies remain between 6.6 × 10^-13 and 6.0 × 10^-12.Source: A. Exact-reference relative-log… · p. 17

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.Source: A. Exact-reference relative-log… · p. 18, C. Geometrically nonlinear cantilever… · p. 20

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.Source: C. Geometrically nonlinear cantilever… · p. 19, Table 2 · p. 19
• 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%.Source: C. Geometrically nonlinear cantilever… · p. 20

• 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.Source: §X · p. 21
• 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.Source: §X · pp. 21–22
• The formulation combines Euler–Poincaré beam balance, graph cotangent pullback, moving-surface relations, and explicit discretization proxies for structural and multibody assembly.Source: §X · p. 22
• The exact-reference relative-log construction exhibits the expected N_d+1 convergence order, supporting consistency of the structural expansion.Source: §X · p. 22
• 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.Source: §X · p. 22

• The construction introduces an exact-reference increment map and uses homogeneous series reversion after establishing nonsingularity of d_0 in the selected chart.Source: §B.1 · pp. 23–24
• The relative logarithm d supplies deformation data, while the section coordinate σ provides an exact split for internal geometry used in the attached surface.Source: §B.2 · p. 24
• Finite-order expansions retain finite reference geometry exactly and truncate endpoint perturbations through separately specified small–small BCH and final endpoint degrees.Source: §B.2 · p. 24
• Final endpoint projection is necessary because BCH slot degree can understate endpoint degree when an anchored increment is quadratic.Source: §B.2 · p. 24
• The logarithmic expansions require a continuous local branch, and endpoint perturbations must remain within the reference chart; value convergence alone is insufficient for dynamics.Source: §B.3 · p. 24
• Potential-energy order determines the required deformation order: fourth-order potential needs d through degree three, whereas sixth-order potential needs degree five.Source: C. Static-Manifold and Potential… · p. 25
• 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.Source: C. Static-Manifold and Potential… · p. 25

• Section rotation and translation enter attachment derivatives, and point Jacobians pull surface forces and couples back to elastic coordinates.Source: §D.1 · p. 26, §D.2 · p. 26
• 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.Source: §D.2 · p. 26, §D.3 · p. 26
• 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.Source: §D.3 · p. 26
• 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.Source: §E.1 · p. 27
• Graph assembly accumulates local one-forms through subtree covector pullback, so proximal joints collect downstream contributions without a separately prescribed moment rule.Source: §E.1 · p. 27
• The implementation sequence defines reference geometry, component reduction, attached-surface kinematics, aerodynamic one-forms, graph assembly, and dynamic reduction before verification.Source: §E.2 · pp. 27–28
• Verification checks attachment identities, normal orientation, value and action convergence, virtual-work duality, static-manifold closure, equivalent-point residuals, and reduced-state diagnostics.Source: §E.2 · p. 28