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Structural Affinity: An Algorithm for Coherent Cage-Based Shape Design

Aviraj Newatia, Shengan Yang, Hantao Zhong, Rika Antonova

arXiv:2610.23173v1cs.LGcs.GRcs.RO

TL;DR

Arbitrary shape optimization has many degrees of freedom, while CBD-based optimization can still produce distorted shapes and relies on loss regularizers that compete with task objectives. Structural Affinity couples cage-handle motion through an enclosing cage and precomputed affinities. Across deformation transfer, aerodynamic optimization, and robot morphology co-design, it yields more regular shapes with reduced reliance on tuned losses.

  • Problem

    Arbitrary shape search has many degrees of freedom and can include irregular or degenerate shapes, while CBD remains sensitive to handle movement and loss regularizers compete with task objectives.

    Source: §1 · p. 1

  • Method

    Structural Affinity derives a precomputed affinity matrix from outer-cage mean-value coordinates to propagate motion between inner-cage handles.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    Across deformation transfer, aerodynamic optimization, and robot morphology co-design, Structural Affinity improves shape regularity; in deformation transfer it retains 85–98% of the final cage-level regularity gain without two explicit loss regularizers.

    Source: §7 · p. 9

  • Takeaways & Limitations

    The dominant cage-level regularity effect comes from the parameterization itself, while original losses remain complementary when maximum surface regularity is preferred.

    Source: §7 · p. 9

    The affinity formulation is derived for mean-value coordinates; deriving it for alternative weighting schemes or free-form deformation remains future work.

    Source: §7 · p. 9

Abstract

from arXiv · show

Cage-based deformation (CBD) parametrises 3D geometry through a small set of control handles, effectively enabling a variety of tasks from animation and interactive modelling to tool-based co-design, neural deformation transfer, and aerodynamic shape optimization. Many optimization approaches relying on CBD employ regularisation strategies to reduce the distortion of the deformed mesh by encouraging smooth, regular cage deformations. These techniques offer soft guidance to an optimization algorithm towards desirable deformations, but compete with the task objective rather than changing the deformation parameterization itself. In this work we introduce Structural Affinity for CBD, an explicit regularisation method which enables smooth deformations through control cages by exploiting a two-cage structure to couple cage-handle motion directly. Structural Affinity precomputes an affinity matrix using mean-value coordinates, binding the inner control-cage to a coarser outer-cage. This matrix defines how much a cage handle displacement propagates to neighbouring handles which substantially decreases and often eliminates cage self-intersections, reduces mesh distortion, and adds linear overhead at deformation time. Across robotic control--morphology co-design, source-to-target neural deformation transfer, and aerodynamic wing shape optimization, Structural Affinity produces more regular deformations and substantially reduces reliance on hand-designed structural loss terms. We report consistent gains on three complementary diagnostics: the cotangent Laplacian, a proxy for surface smoothness; cage distortion, capturing whether control handles move coherently; and surface roughness, which measures undesirable oscillation in wing geometry.

1 INTRODUCTION

Shape optimization over arbitrary 3D geometry has many degrees of freedom and includes irregular or degenerate shapes. Structural Affinity couples cage-handle motion through an enclosing outer cage, reducing reliance on task-specific loss regularization.

1 INTRODUCTION

  • Searching arbitrary shapes creates a combinatorial space with many degrees of freedom and regions of irregular, distorted, or degenerate shapes.Source: §1 · p. 1
  • CBD reduces the shape search space, but poorly controlled handle motion can still produce distorted, spiky, or degenerate shapes.Source: §1 · p. 1
  • Structural Affinity uses an enclosing outer cage to propagate handle displacements across the control cage, reducing reliance on loss regularizers.With N inner-cage handles and L retained off-diagonal affinities, propagation costs O(N + L) at deformation time.Source: §1 · p. 2

2 BACKGROUND AND PRIOR WORK

Prior work uses cage-based and related parameterizations to shape deformation behavior and reduce the search space. The paper situates its comparisons in neural deformation transfer, aerodynamic design, and robot morphology co-design.

2 BACKGROUND AND PRIOR WORK

  • CBD represents each mesh vertex as a fixed linear combination of cage handles, with the weighting scheme determining deformation behavior.MVC gives smooth closed-form weights for closed cages, Green coordinates give quasi-conformal deformations, and Somigliana coordinates allow more elastic behavior.Source: §2 · p. 2
  • Cage handles can move independently, so modeling tools use soft selection, morphology mappings, or least-squares direct manipulation to couple their motion.Source: §2 · p. 2
  • Neural Cages predicts a cage and handle displacements for target-shape transfer, trained with Chamfer, MVC, and two Laplacian-inspired regularity losses.The losses penalize changes in point-to-plane distance over a vertex’s 1-ring and changes in local PCA normals.Source: §2 · p. 2
  • Aerodynamic design commonly uses B-splines or FFD lattices, while CBD is rarely used despite its ability to follow underlying geometry more closely.Source: §2 · p. 2
  • Robot morphology co-design optimizes morphology and controller jointly; this work uses gradient-free morphology optimization with an RL policy to compare CBD and CBD-SA.Source: §2 · pp. 2–3

3 CAGE-BASED DEFORMATION WITH STRUCTURAL AFFINITY

Structural Affinity binds the inner control cage to an enclosing outer cage and derives affinities from the inner vertices’ mean-value-coordinate distributions. A precomputed affinity matrix propagates handle displacements while preserving reachability of inner-cage motions under the stated dense-matrix conditions.

3 CAGE-BASED DEFORMATION WITH STRUCTURAL AFFINITY

  • The mesh is bound to an inner control cage, which is itself bound to an enclosing outer cage using mean-value coordinates.The outer-cage weights describe each inner-cage vertex’s position relative to the enclosing structure.Source: §3 · p. 3
  • Inner-cage vertices are structurally related when their outer-cage weight distributions are similar, defining affinities that respect the outer cage without an online solve.Source: §3 · p. 3

3.1 DERIVING STRUCTURAL AFFINITY

  • A naive weighted average of outer-handle motion ratios makes every vertex appear to follow every other vertex completely, so the method caps ratios at one before averaging.Source: §3.1 · p. 3
  • The capped affinity lies in [0, 1], is symmetric with a unit diagonal, and is small for vertices attached mainly to different outer handles.The matrix is sparsified by retaining the diagonal and selected off-diagonal entries, or by removing entries below 10^-4; it is not row-normalized.Source: §3.1 · p. 3

3.2 DEFORMATION PROPAGATION

  • At deformation time, each displaced handle propagates motion in proportion to its affinity, and simultaneous handle displacements combine through a matrix product.The affinity strength α controls propagation; for α ≤ 1, no handle moves further than the displaced handle.Source: §3.2 · p. 4
  • For a dense affinity matrix and 0 ≤ α < 1, every inner-cage motion remains reachable; with bounded neighbors per handle, propagation costs O(N + L).All outer cages in this work are convex; nonconvex cages can produce negative mean-value-coordinate weights.Source: §3.2 · p. 4

4 PREDICTING CAGE DEFORMATIONS WITH NEURAL NETWORKS

In Neural Cages deformation transfer, Structural Affinity provides cage regularity largely through its parameterization, while reconstruction fidelity remains close to Vanilla NC. Explicit shape-regularity losses add surface-smoothness gains, but the affinity mechanism retains substantial cage-regularity improvement without them.

4 PREDICTING CAGE DEFORMATIONS WITH NEURAL NETWORKS

  • The evaluation trains on Chair, Table, and Car source–target pairs, using the Neural Cages split and 5,000 surface-sampled points per shape.Metrics include reconstruction, deformed-mesh Cot-Laplacian distortion, and cage Laplacian, edge, and self-intersection measures.Source: §4 · p. 4
  • CBD-SA is compared with full-loss Vanilla NC under NoReg and restored-loss settings, using a shared fixed 12-vertex icosahedral outer cage per category.NoReg removes Lp2f and Lnormal; the restored-loss variant uses the full published loss, with architecture and training schedule shared across methods.Source: §4 · p. 5
  • Without either regularizer, CBD-SA NoReg retains 93–98% of cage-Laplacian and 85–98% of cage-edge improvement over Vanilla NC; CBD-SA NC Loss reduces cage-Laplacian distortion by 81–84%.The latter also reduces Cot-Laplacian distortion by 26–69%; NoReg retains 35–74% of that gain.Source: Figure 2 · p. 4, §4 · p. 5
  • Mean reconstruction error remains within 1% of Vanilla NC, while NoReg and NC Loss differ by only 0.03% and 0.05% under the complementary reconstruction metrics.The reported comparison uses DCD and SAUCD; the section notes that Chamfer Distance is a poor measure of mesh alignment.Source: §4 · p. 5
  • Projecting motion onto the outer-cage MVC subspace does not reproduce the result, and Euclidean Gaussian propagation remains a strong alternative when the outer cage adds little structure.The outer cage acts as soft coupling rather than a hard motion constraint.Source: §4 · pp. 5–6

5 AERODYNAMIC SHAPE OPTIMIZATION

Under the low-fidelity wing-optimization model, CBD-SA improves drag and deformation roughness over the evaluated parameterizations at matched design-variable counts. Its lower modeled drag persists across matched valid runs from perturbed starts and under a coarser outer-cage discretization.

5 AERODYNAMIC SHAPE OPTIMIZATION

  • CBD and CBD-SA are compared at 12, 48, and 192 design variables under a low-fidelity vortex-lattice CRM drag-minimization setup; results compare parameterizations, not absolute benchmark optima.The problem minimizes drag at fixed lift with angle of attack as a trim variable and a bending-energy constraint.Source: §5 · p. 6
  • At matched design-variable counts, CBD-SA strictly outperforms on drag and roughness; at medium and large scales it beats both FFD settings and vanilla CBD on both measures.The comparison uses final drag coefficient and deformation roughness, which captures spanwise waviness of final mesh displacement.Source: §5 · pp. 6–7
  • From identical perturbed starts, CBD-SA reaches lower final modeled drag in all 8 matched valid pairs at 5% perturbation and all 9 at 10%.Final CD falls by 1.53% and 0.75%, respectively; affinity adds 20.2% and 7.14% relative to CBD's progress from the common initialization.Source: §5 · p. 7, Figure 6 · p. 7
  • With coarser convex-hull outer cages, CBD-SA again has lower modeled drag in every matched valid run and reduces deformation roughness by 18.2% and 13.0%.Drag reductions are 2.06% and 0.83%; the lower-drag result does not depend on finer discretization of the convex outer cage.Source: §5 · p. 7

6 ROBOT MORPHOLOGY CO-DESIGN

Structural Affinity was evaluated in robot morphology co-design for object-picking fingertips, with quantitative simulation comparisons and real-robot trials. Compared with CBD, CBD-SA produced higher t-nut success and shapes described as broader and more physically plausible.

Simulation Experiments

  • The team co-optimized fingertip shapes and an RL policy using gradient-free global optimization after 20 minutes of policy pre-training.During global optimization, the policy was conditioned on the cage shape parameterization.Source: §6 · p. 7
  • CBD-SA raises mean t-nut success from 35.60% ± 43.49% to 87.35% ± 11.05% nominally, and from 7.25% ± 10.25% to 32.58% ± 20.35% under randomization.The randomized condition perturbs object and contact parameters; both conditions use four training seeds.Source: Figure 7 · p. 8, §6 · p. 8
  • CBD-SA produced broader, more manufacturable fingertips, while vanilla CBD could yield spiky shapes that exploit simulation inaccuracies by piercing or knocking objects.The comparison includes t-nut and low-profile spatula examples.Source: §6 · p. 8, Figure 8 · p. 8, Figure 9 · p. 9

Real Robot Experiments

  • Real-robot trials demonstrated physical feasibility of selected CBD-SA designs, whereas corresponding CBD fingertips could not be cast and would not survive object contact.The tested objects included tiny, long or unbalanced, and round flat objects.Source: §6 · pp. 8–9

7 CONCLUSION AND FUTURE WORK

The conclusion reports benefits across deformation transfer, aerodynamic optimization, and robot morphology co-design. It also identifies MVC as the basis of the current affinity formulation and points to alternative weighting schemes and FFD as future directions.

7 CONCLUSION AND FUTURE WORK

  • Structural Affinity benefits deformation transfer, aerodynamic shape optimization, and robot morphology co-design, with smoother, more regular shapes and reduced reliance on tuned loss regularizers.For deformation transfer, it retains 85–98% of the final cage-level regularity gain without two explicit loss regularizers while reconstruction fidelity remains within 1% of Vanilla NC.Source: §7 · p. 9
  • Under the shared low-fidelity aerodynamic model, geometry-informed coupling yields repeatable modeled-drag improvements across perturbed initial geometries.The conclusion motivates evaluation under higher-fidelity aerodynamic models.Source: §7 · p. 9
  • The current affinity formulation is derived from MVC; alternative weighting formulations and applying outer-cage binding to FFD remain future directions.Outer cages must avoid pathological or degenerate constructions.Source: §7 · p. 9

A.1 PROPERTIES

The section characterizes affinity as shared mass between MVC rows, then discusses invertibility, propagation limitations, computational cost, evaluation, and Neural Cages settings. It reports substantial regularity gains and sensitivity results alongside fidelity constraints.

A.1 PROPERTIES

  • C_ij is one minus the total-variation distance between MVC rows, equivalently their largest shared mass; dense propagation is invertible for 0 ≤ α < 1, but sparsification needs a separate guarantee.The sparse operator is invertible whenever α∥C − I∥2 < 1; nonconvex outer cages can produce negative affinities, and accumulated handle contributions need not prevent intersections.Source: §A.1 · p. 15

A.2 COMPUTATIONAL COMPLEXITY

  • With N inner-cage vertices and L retained off-diagonal affinities, multi-handle propagation costs O(N + L); it is O(N) for fixed average retained neighbors and O(N^2) in the dense worst case.Linear scaling requires the average number of retained neighbors to remain bounded as N grows.Source: §A.2 · pp. 15–16

B.1 SHAPE-REGULARITY LOSSES

  • Yifan et al. (2020) use Lp2f and Lnormal shape-regularity losses: one penalizes changed point-to-plane distances, and the other measures changes in PCA normals.These losses are described as Laplacian-inspired regularizers.Source: §B.1 · p. 16

B.2 EVALUATION METRICS AND CHAMFER DISTANCE

  • Chamfer is a pointwise cost, so its magnitude alone cannot establish consequential geometric-quality changes; mesh regularity and cage validity provide complementary indicators.The evaluation reports six metrics, all with lower values indicating better performance.Source: §B.2 · p. 16
  • Fidelity metrics remain within 1% of Vanilla NC, while CBD-SA NC Loss reduces Cot-Laplacian distortion by 26%–69% and cage Laplacian distortion by 81%–84%.The authors report DCD and SAUCD alongside regularity metrics because Chamfer alone cannot establish the magnitude of perceptual or structural degradation.Source: §B.2 · p. 17

B.3 HYPERPARAMETERS

  • Neural Cages hyperparameters are grid-searched by category, held across six evaluation seeds, and transferred from CBD-SA NoReg to CBD-SA NC Loss.The selected category settings are reported in Table 2.Source: §B.3 · p. 17, Table 2 · p. 17

B.4 NEURAL CAGES SENSITIVITY AND OUTER-CAGE CONSTRUCTION

  • The outer cage should enclose the inner cage and encode broad structural relationships; alternative Chair supports retain low cage-Laplacian distortion, but unsuitable or degenerate cages can propagate unsuitable motion.The canonical support is fixed within category, while the compared alternatives use three seeds and the canonical result uses six.Source: §B.4 · pp. 17–18, Table 3 · p. 17, Table 4 · p. 18
  • 21 of 22 Chair configurations improve cage Laplacian distortion over Vanilla NC, and 15 eliminate cage self-intersections; the strongest result under +5% mean-Chamfer compatibility is outside the usual α range.At α = 0.6, d = 0.2, cage Laplacian distortion is 0.001194 with +3.89% Chamfer.Source: §B.4 · p. 18

B.5 ABLATIONS: DIRECT PROJECTION AND GAUSSIAN PROPAGATION

The ablations show that Structural Affinity’s soft handle coupling yields more regular cages than hard outer-coordinate projection or Euclidean propagation in the evaluated settings. Its advantage is conditional on the outer cage’s encoded structure, with comparisons spanning neural deformation transfer and aerodynamic setup details.

B.5 ABLATIONS: DIRECT PROJECTION AND GAUSSIAN PROPAGATION

  • CBD-SA has lower effective-cage Laplacian and edge distortion than direct outer-MVC projection across Chair, Table, and Car, while pooled DCD is 0.20% higher and valid-pair SAUCD is 0.63% lower.The largest separation in cage regularity is on Table; CBD-SA preserves full inner-handle control while biasing how handles interact.Source: §B.5 · p. 18, Table 5 · p. 19

B.5.1 GAUSSIAN NOREG COMPARISON

  • 32.8% lower Cot-Laplacian error, 16.9% lower predicted-cage Laplacian, and 4.34% lower edge distortion distinguish CBD-SA from Gaussian NoReg; pooled DCD and SAUCD improve by 0.069% and 0.46%.These are descriptive comparisons across evaluated categories, not a claim that every category or seed improves; Figure 11 reports lower-is-better metrics with positive values favoring CBD-SA.Source: Figure 11 · p. 19, §B.5.1 · p. 19

B.5.2 BASELINE DETAILS

  • With an informative coarse outer cage, CBD-SA has 4.48% lower pooled effective cage-edge distortion than Gaussian, while category Chamfer increases are 4.45% on Chair, 3.83% on Table, and 0.82% on Car.The frozen comparison uses matched seeds; the methods share 87.5–94.2% of selected handle relationships with canonical support and 82.2–83.7% with informative support.Source: §B.5.2 · p. 20

B.6 ADDITIONAL RESULTS

  • CBD-SA shifts Chair, Table, and Car means toward lower Cot-Laplacian distortion while SAUCD remains tightly clustered; Figure 12 shows both metrics on logarithmic axes.Lower is better on both axes, and the shaded region marks the nearest method mean’s joint trade-off.Source: Figure 12 · p. 20

C.1 PROBLEM SETUP

  • The CRM study uses the OAS lifting-surface setup rather than reproducing ADODG Case 4.1 benchmark values, and optimizes wing angle of attack and geometry under lift, bending-energy, drag, and bound constraints.It analyzes a symmetric half-wing at one aerodynamic condition, with a 3-by-9 mesh of chordwise and spanwise stations.Source: §C.1 · p. 21

C.2 HYPERPARAMETERS AND PERTURBATION PROTOCOL

  • For held-out seeds 10–19 at 5% and 10% perturbation, CBD and CBD-SA start from matched perturbed cages and wings and share models, constraints, bounds, derivatives, optimizer settings, and iteration budgets.Affinity parameters were selected using exploratory seeds 0–9 and then frozen; the large cage has 192 active vertical-displacement variables.Source: §C.2 · p. 22

C.3 OPTIMIZER SETTINGS

  • The CRM multi-optimizer comparison uses the settings summarized in Table 10.The supplied passage identifies the settings table but does not specify its values.Source: Table 10 · p. 23

C.4 ADDITIONAL RESULTS

The CRM additional results report successful lift-constraint convergence and smoother deformations under CBD-SA in perturbed-geometry studies. They also compare cage methods against FFD baselines and test support discretization.

C.4 ADDITIONAL RESULTS

  • The maximum final CL residual is 10^-8 for CBD at the medium scale, and the passage reports successful convergence for all optimizers, methods, and scales.The authors state that 10^-8 is within the typical optimizer tolerance for these problems.Source: §C.4 · p. 23, Figure 14 · p. 24
  • CBD-SA reduces deformation roughness by 18.2% at 5% perturbation and 13.0% at 10%, with lower roughness in 7/9 matched pairs at each magnitude.The compared FFD baselines use 6 and 30 design variables, versus 192 for CBD and CBD-SA; under the 38-vertex construction, both cage methods reach lower modeled drag than both FFD baselines in every matched valid comparison.Source: §C.4 · p. 23, Figure 15 · p. 24
  • Table 11 separates paired aerodynamic results for directly constructed 19–27-vertex convex supports and a 38-vertex support-discretization sensitivity study.Its caption reports that CBD-SA reaches lower final modeled drag in every listed matched pair.Source: §C.4 · p. 23, Table 11 · p. 24

D.1 SETUP AND HYPERPARAMETERS

The robot co-design setup uses a convex box outer cage and shared optimizer settings. In spatula lifting, CBD-SA produces smoother fingertips than CBD, and the learned policy lifts the object.

D.1 SETUP AND HYPERPARAMETERS

  • The robot co-design experiments use an axis-aligned convex outer box with 8 vertices and 12 faces, padded by 5% of the inner-cage bounding-box diagonal.Affinity strength α = 0.35 and target density d = 0.15 were selected through limited random search and used across experiments.Source: §D.1 · p. 24

D.2 OPTIMIZER SETTINGS

  • The pretraining and combined RL+GO stages use the same optimizer settings, with unspecified settings taken from the FastTD3 repository.The parameters are listed in Table 12.Source: §D.2 · p. 25, Table 12 · p. 25

D.3 SPATULA LIFT FRAMES

  • RL+GO with CBD-SA produces smooth, wide fingertips that let the LEAP hand slide under the spatula and reorient it stably, while CBD produces jagged fingertips and cannot grasp it.Figure 16 compares the CBD and CBD-SA spatula lift under RL+GO optimization.Source: §D.3 · p. 26, Figure 16 · p. 26

E COMPUTE RESOURCES

The appendix reports the computing resources and run times for the experiments, followed by licenses and usage terms for datasets, code, software, and reference assets.

E COMPUTE RESOURCES

  • Neural Cages and robot co-design experiments ran on an 8-GPU RTX PRO 6000 Blackwell machine with 1.5 TB RAM and dual 64-core AMD EPYC processors; each run took under 40 minutes and 2.5 hours, respectively.Aerodynamic optimization ran on a 384-vCPU, 768-GB-RAM machine, with each run under 60 minutes.Source: E COMPUTE RESOURCES · pp. 26–27

F LICENSES

  • The appendix identifies ShapeNet’s non-commercial research and educational terms and reports MIT licensing for Neural Cages and FastTD3, Apache 2.0 for OpenAeroStruct and OpenMDAO, and other licenses for tools and robot assets.The NASA CRM is credited as public reference geometry, with no specific software-style license identified on its page.Source: F LICENSES · p. 27
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