Geometry Parameterisation via a Structural Modal Basis for Aerodynamic Shape Optimisation

• C denotes the pseudo-structure’s linear constraint operator, while E(x), ν(x), and ρ(x) are its Young’s modulus, Poisson’s ratio, and density fields.Source: §1 · p. 2
• K and M are the pseudo-structure stiffness and mass matrices; λ_i and ω_i are its ith eigenvalue and natural angular frequency.Source: §1 · p. 2
• Φ_s collects the selected mode shapes, s is their count, and a contains their modal amplitudes; x_0 and x(a) denote the baseline and updated geometries.Source: §1 · p. 2

• Payload, manufacturing, integration, certification, and low-observability constraints often leave aerodynamic optimisation with only small, highly constrained geometry changes.Source: §2 · p. 2
• Too many controls make search costly and shapes hard to constrain or interpret, while too few can exclude aerodynamically important changes.Source: §2 · p. 2
• MPM builds a tunable pseudo-structure and selects its modes, using boundary conditions and coupling to set admissibility and material-property edits to prioritise motions.Source: §2 · p. 2
• The resulting coordinates are smooth, coupled, and engineer-interpretable, positioning the basis between direct controls, deformation fields, and data-driven reduced bases.Source: §2 · p. 2

• Parameterisation methods are assessed by trade-offs among conciseness, robustness, and flexibility, especially for tightly constrained three-dimensional aircraft geometry.Source: §3 · pp. 2–3
• Analytic and direct-geometry methods are interpretable and CAD-compatible, but can restrict strongly three-dimensional edits or incur costly variable coupling and constraint management.Source: §3 · p. 3
• Morphing methods offer broad shape reach, whereas reduced-order and learned bases compress variation; their use can depend on separate admissibility controls or available representative data [14–30].Source: §3 · p. 3
• Figure 1 identifies conventional MEIRD edits through NURBS control points, OpenVSP sections, and CAD guide and loft curves.Source: Figure 1 · p. 3, §3 · p. 4
• The proposed pseudo-structure basis embeds admissibility through supports and coupling, can be retuned without a shape library, and keeps solver-agnostic, coupled, interpretable modes.Source: §3 · p. 4

4.1 Define the Pseudo-Structure and Solve the Eigenvalue Problem

• The pseudo-structure’s topology and properties generate candidate modes through a generalised eigenvalue problem, and it need not represent the aircraft’s physical structure.Boundary conditions, coupling, stiffness, density, thickness, and added masses shape the admissible and prioritised motions before optimisation.Source: §4.1 · p. 4
• Boundary conditions and coupling rules impose linear restrictions on admissible modes, while property edits prioritise motions but do not replace those constraints.Source: §4.1 · pp. 4–5
• A selected set from one mass-normalised eigenproblem provides independent coordinates, but its modes are orthogonal in the pseudo-structural mass metric, not necessarily a later geometry-space metric.Modes combined across different pseudo-structures or settings should be checked for linear dependence.Source: §4.1 · p. 5

4.2 Inspect, Tune, and Select the Subset of Modes

• Inspecting candidate modes before optimisation lets engineers tune supports and structural properties toward desired global and local deformation; stiffness smooths short-wavelength motion, while mass changes inertial weighting.Reducing local stiffness can bring more localised motion earlier in the modal ordering; increasing local mass can bias regional participation in lower-order modes.Source: §4.2 · p. 5
• Mode selection is an engineering choice: lower-order modes often capture global behaviour, higher-order modes often capture local behaviour, and the selected count defines the basis.Source: §4.2 · p. 5

4.3 Selected Basis and Optimisation Coordinates

• Selected modal amplitudes parameterise geometry directly; optional pivoted QR supplies Euclidean-orthonormal coordinates without changing the admissible deformation family.The rotated directions may reduce coordinate coupling and improve numerical efficiency for some optimisers, but are generally less directly interpretable.Source: §4.3 · pp. 5–6, §4.2 · p. 5

5.1 Physics-derived Basis Functions: 2-Element Cantilever Beam

• On desired dihedral shapes, the raw two-vector MPM basis covered 196 of 1,000 samples versus 161 of 1,000 for same-family SVD; the third principal angle against full-family SVD was 72.4°.MPM performed worse on the broader arbitrary family, which included sagging and overshoot shapes unlikely for transport-aircraft dihedral. Raw MPM modes remained smooth and interpretable, while rotated QR directions were less stable to interpret and SVD directions depended strongly on the training library.Source: Table 2 · p. 8, §5.1 · p. 10

5.2 Basis-Design Tuning: 3D Thin Airfoil Wing

• At e ≥0.999, the untuned wing required seven modes, stiffness tuning three with a kink at y = 45, and tip-mass tuning three with a smoother twist.The baseline twist was visibly noisy; the outboard soft patch smoothed inboard and outboard regions but introduced a kink, while a tip mass was simpler to tune than a distributed stiffness patch.Source: §5.2 · pp. 10–11, Figure 7 · p. 11, Figure 8 · p. 11
• Increasing tip mass shifted selected modes from global to localized outboard response, while a composite case combined stiffness, mass, and hinge-line constraints and still reached e ≥0.999 with three modes.Two MPCs allowed rigid-body translation and rotation of hinge lines while permitting leading- and trailing-edge deformation; the selected modes included bending, torsion, and sweep patterns.Source: §5.2 · pp. 11–12, Figure 10 · p. 12

5.3 Constraint Editing: Military Engine Intake Research Duct (MEIRD)

• For MEIRD revisions expressible through supports, releases, or local attachments, MPM updates admissibility by editing the pseudo-structure and re-solving while keeping geometry mapping fixed; stay-out zones and engine-diameter changes require downstream checks or basis rebuilding.Cases 1–5 fit the support-defined workflow; stay-out-zone compliance is not embedded in the basis, and an engine-face diameter change requires rebuilding around the new interface condition.Source: §5.3 · pp. 13–15

5.4 Complex Geometry: Advanced Military Airborne Platforms

• The advanced-platform studies used a constrained optimization loop coupling modal deformation, mesh morphing, surrogate-assisted PSO, and high-fidelity CFD.Structural modes were computed with ANSYS Mechanical, aerodynamic analyses with ANSYS Fluent, and modal motion was projected to the CFD outer mould line using MPR.Source: §5.4 · p. 15

5.4.1 Case 1: Bifurcated Serpentine Intake

• The bifurcated serpentine intake case balanced mass-flow and fan-face-distortion requirements against payload- and signature-driven exclusion zones, using three reference conditions selected from 12 mission-relevant conditions.The reference conditions represented critical static thrust (ID 1B), high-angle-of-attack low-altitude departure (ID 7), and low-angle-of-attack high-altitude performance (ID 11).Source: §5.4.1 · pp. 15–16
• The intake used seven low-order bending, torsional, and breathing modes as design coordinates, with payload and low-observable constraints enforced through variable-range limits.The optimisation maintained at least 95% operating efficiency at conditions ID 7 and ID 11, targeting unity AIP recovery and reduced forward loading.Source: §5.4.1 · p. 16
• The selected intake design improved total relative pressure recovery by about 12% and reduced system head loss by about 7%, while respecting exclusion zones.Convergence required three MPM update iterations to impose additional constraints and progressively reduce free-form deformation.Source: §5.4.1 · p. 16, Figure 15 · p. 17

5.4.2 Case 2: Nose Pitch (Trim / Longitudinal Stability Reconciliation)

• The nose case addressed a longitudinal stability risk from an adverse centre-of-gravity shift by optimising the nose as a local control region with objectives for elevator trim and drag.Hard geometric limits maintained internal volume, and a manufacturability filter enforced minimum radii on skin edges.Source: §5.4.2 · pp. 16, 18
• For nose control, modes 1 and 2 plus a localised component derived from mode 5 formed the basis, while higher-frequency modes were generally excluded.The pseudo-structure was fixed at the rear frame, constrained against torsional rotation, and tuned with a rigid lower-skin reference point; practical parameter bounds supported manufacturability and solver robustness.Source: §5.4.2 · p. 18
• The nose modification reduced required elevator trim by an average of about 0.5 degrees across analysed conditions, with a net drag change of less than 1%.Convergence took one MPM iteration; small modal amplitudes kept mesh quality acceptable with smoothing and dynamic mesh updates only.Source: §5.4.2 · p. 18, Figure 17 · p. 18

6 Discussion

• Selected modes provide concise coordinates, with lower-order modes often producing recognisable global changes and higher-order modes providing local refinement.The selected basis produces smooth deformation fields rather than pointwise geometric edits.Source: §6 · p. 18
• For tightly constrained designs, basis alignment with the admissible family can matter more than unrestricted shape reach.Stiffness edits, attached masses, and protected hinge constraints shifted the thin-wing modal family; support-defined revisions for MEIRD cases could be absorbed by pseudo-structural edits and a new eigenvalue solve, while geometry transfer stayed fixed.Source: §6 · p. 18
• The approach remained viable on complex aerospace geometry when required motion was smooth, coupled, and strongly constrained by program requirements.Source: §6 · p. 19
• Optional QR conditioning can reduce coordinate coupling for an optimisation algorithm, but does not change the admissible deformation family.It changes only the coordinates describing the selected subspace.Source: §6 · p. 19
• Sharp features and large planform edits may require many modes, reducing conciseness; the method is also sensitive to pseudo-structural choices and mode and amplitude bounds.The beam comparison found trained reduction can outperform the pseudo-structure-derived basis after the desired family has already been generated; thin-wing admissibility can trade hinge-line coherence against whole-field flexibility.Source: §6 · p. 19

7 Conclusion

• The method constructs a tunable pseudo-structure, solves its eigenvalue problem, and uses selected modes as geometry coordinates.Boundary conditions, coupling assumptions, stiffness, density, thickness, and added masses are basis-design choices made before optimisation.Source: §7 · p. 19
• The beam, thin-wing, MEIRD, and advanced-platform studies show that basis-design choices influence modal-coordinate quality and that pseudo-structural edits can accommodate complex constrained geometries.The thin-wing study reports that tuning stiffness, attached masses, and hinge constraints shifted the modal family toward the required response while reducing selected modes.Source: §7 · p. 19
• The resulting coordinates are solver-agnostic, concise, smooth, and engineer-interpretable for constrained aerodynamic design.The admissible deformation family can be shaped and inspected before optimisation.Source: §7 · p. 19
• Sharp local features, inequality volume constraints, and discrete interface redefinitions can make smooth modal bases less concise or require admissibility logic outside basis definition.Performance also depends on pseudo-structural choices, selected mode count, and amplitude bounds.Source: §7 · p. 19

Contact Author

• The listed organisation is Boeing Aerostructures Australia.Source: Contact Author · p. 19
• The listed address is 226 Lorimer Street, Port Melbourne, VIC 3207, Australia.Source: Contact Author · p. 19

Copyright Statement

• The authors affirm copyright ownership of original material, permission for third-party material, and permission for ICAS proceedings or individual off-print publication and distribution.Source: Copyright Statement · p. 20