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

Aviraj Newatia; Shengan Yang; Hantao Zhong; Rika Antonova

cs.LGOct 16, 2026

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

Authors: Aviraj Newatia, Shengan Yang, Hantao Zhong, Rika Antonova

Organizations: University of Cambridge

Abstract

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.

Figures & tables

Panel of Figure 2 — Figure 2: Cross-category reductions in Cot-Laplacian and effective-cage Laplacian distortion. Percentages compare the six-seed means with Vanilla NC; positive values indicate lower distortion. The left panel isolates Structural Affinity; the right restores the explicit shape-regularity losses.

Panel of Figure 3 — Figure 3: Qualitative Table example. (a) Deformed cages for Vanilla NC and CBD-SA NoReg. (b) Deformed meshes for Vanilla NC, CBD-SA NoReg and CBD-SA NC Loss.

Category	Method	Cage Lap ( $\times 10^{3}$ )	Cage Edge	CotLap	Self-Intersect ( $\times 10^{4}$ )	DCD	SAUCD
Chair	Vanilla NC	$9.69_{\pm 1.70}$	$0.285_{\pm 0.018}$	$10.32_{\pm 3.53}$	$0.53_{\pm 0.52}$	$0.9100_{\pm 0.0056}$	$1.5450_{\pm 0.0128}$
	CBD-SA NoReg	$1.70_{\pm 0.35}$	$0.110_{\pm 0.007}$	$9.38_{\pm 4.13}$	$0$	$0.9153_{\pm 0.0050}$	$1.5393_{\pm 0.0131}$
	CBD-SA NC Loss	$1.55_{\pm 0.17}$	$0.106_{\pm 0.006}$	$7.62_{\pm 2.85}$	$0$	$0.9144_{\pm 0.0051}$	$1.5447_{\pm 0.0183}$
Table	Vanilla NC	$14.65_{\pm 1.53}$	$0.346_{\pm 0.024}$	$145.75_{\pm 188.79}$	$11.19_{\pm 3.46}$	$0.8716_{\pm 0.0101}$	$1.4316_{\pm 0.0455}$
	CBD-SA NoReg	$3.36_{\pm 0.54}$	$0.180_{\pm 0.013}$	$70.46_{\pm 84.08}$	$0.29_{\pm 0.50}$	$0.8790_{\pm 0.0090}$	$1.4381_{\pm 0.0429}$
	CBD-SA NC Loss	$2.75_{\pm 0.61}$	$0.164_{\pm 0.006}$	$44.64_{\pm 46.89}$	$0$	$0.8774_{\pm 0.0116}$	$1.4394_{\pm 0.0318}$

Table 1: Mean

\pm

sample SD across six training seeds. Lower is better; the lowest mean per category and metric is bolded. DCD uses the same 5,000 surface samples per prediction as CD; SAUCD uses the corresponding meshes with deterministic reduction when they exceed 10,000 vertices. Chamfer results are retained in Table 6 .

Figure 4: Reconstruction (DCD) against surface distortion (Cot-Laplacian) for Chair, Table and Car, on logarithmic axes; lower is better on both. DCD remains tightly clustered while CBD-SA shifts the means toward lower distortion. The method on the shaded region’s boundary, nearest the lower-left corner, has the best joint trade-off. SAUCD shows the same pattern: reconstruction stays tightly clustered while CBD-SA lowers distortion (Figure 12 , Appendix B.2 ).

Figure 5: Final drag coefficient and deformation roughness of optimized CRM geometry.

Figure 6: Aerodynamic shape-optimization results using the 38-vertex convex outer cages and identical perturbed starts. Left: each thin line is one matched valid run and the thick line connects the mean final drag for CBD and CBD-SA (lower is better). Right: the reduction from the common initialization achieved by CBD is indexed to 100, showing the additional gain from structural affinity.

Figure 7: Quantitative t-nut co-design at 60,000 training steps. Nominal uses the original softened-contact simulator and evaluation distribution; Randomized additionally perturbs object pose, mass, friction, and contact parameters. (a) Mean task success across four independent training seeds; orange shows CBD and the blue segment shows the observed increase with Structural Affinity. (b) Points show means and error bars show sample SD across training seeds. Each learned policy is evaluated over 1,000 episodes per condition, comprising ten evaluation seeds of 100 episodes.

Figure 8: RL+GO optimization with CBD yields spiky fingertips with that exploit simulation by stabbing the object to cause it to stick to fingertips in non-physical ways (a,b). In contrast, using CBD-SA yields fingertips well-suited for pushing the object onto a broad surface provided by one of the fingertips without ‘stabbing’ (c,d).

Figure 9: RL+GO optimization with CBD yields jagged fingertips. In contrast, CBD-SA produces fingertips with broader surfaces with physically plausible behavior; Appendix D.3 shows the lift frame by frame.

Figure 10: Photos of experiments for picking up small, flat, or unbalanced objects from a flat surface.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Affinity $\alpha$	Affinity Density $d$
Chair	0.3	0.4
Table	0.2	0.4
Car	0.4	0.4

Appendix

Table 2: Selected hyperparameters per category:

\alpha

controls affinity propagation strength;

d

controls the target graph density. Each category uses a fixed convex 12-vertex icosahedral support in its normalized category frame, shared across samples and seeds. Affinity weights use a min-weight aggregation scheme.

Category	Training shapes used for bounds	Outer cage	Scale/radii
Chair	6,440	12V/20F	scale 2.25, margin 0.15
Table	8,015	12V/20F	$(2.40,2.15,2.22)$
Car	3,339	12V/20F	$(2.20,2.15,2.40)$

Appendix

Table 3: Canonical Neural Cages outer supports. Each cage is fixed within a category and shared across samples, seeds, and loss profiles.

Outer cage	Seeds	Cage Laplacian $\downarrow$	Chamfer $\downarrow$
Canonical 12-vertex support	6	$0.001548\pm 0.000169$	$0.02869\pm 0.00168$
Coarser enclosing support	3	$0.001061\pm 0.000301$	$0.03066\pm 0.00097$
20%-perturbed support	3	$0.001336\pm 0.000067$	$0.02940\pm 0.00132$

Appendix

Table 4: Chair outer-cage sensitivity, reported as mean

\pm

sample SD. Chamfer is included only for compatibility with the original Neural Cages evaluation.

Category	Method	Effective cage Laplacian ( $\times 10^{3}$ )	Effective cage edge
Chair	Direct projection	8.92	0.254
	CBD-SA	5.18	0.232
Table	Direct projection	20.79	0.350
	CBD-SA	7.26	0.285
Car	Direct projection	0.644	0.0716
	CBD-SA	0.630	0.0667

Appendix

Table 5: Direct outer-MVC projection and CBD-SA with the published Neural Cages losses. Values are means across six matched seeds; lower is better. Cage metrics are measured on the effective cage used for deformation, after each method’s forward-pass map.

Figure 11: CBD-SA NoReg compared with Gaussian NoReg using the canonical outer cage. Positive values favor CBD-SA; the panels use separate horizontal scales to show the smaller reconstruction differences. Relative improvements are reported for Cot-Laplacian error, predicted-cage Laplacian and edge distortion, DCD, and SAUCD. Lower is better for every metric.

Figure 12: Reconstruction (SAUCD) against surface distortion (Cot-Laplacian) for Chair, Table and Car, on logarithmic axes; lower is better on both. SAUCD remains tightly clustered while CBD-SA shifts the means toward lower distortion. The shaded region extends from the lower-left corner to the nearest method mean; the method on its boundary has the best joint trade-off.

Category	Method	CD ( $\times 10^{2}$ )	DCD	SAUCD
Chair	Vanilla NC	$2.658_{\pm 0.157}$	$0.9100_{\pm 0.0056}$	$1.5450_{\pm 0.0128}$
	CBD-SA NoReg	$2.876_{\pm 0.195}$	$0.9153_{\pm 0.0050}$	$1.5393_{\pm 0.0131}$
	CBD-SA NC Loss	$2.869_{\pm 0.168}$	$0.9144_{\pm 0.0051}$	$1.5447_{\pm 0.0183}$
Table	Vanilla NC	$3.166_{\pm 0.219}$	$0.8716_{\pm 0.0101}$	$1.4316_{\pm 0.0455}$
	CBD-SA NoReg	$3.301_{\pm 0.268}$	$0.8790_{\pm 0.0090}$	$1.4381_{\pm 0.0429}$
	CBD-SA NC Loss	$3.516_{\pm 0.394}$	$0.8774_{\pm 0.0116}$	$1.4394_{\pm 0.0318}$

Appendix

Table 6: Reconstruction metrics from the same evaluation as Table 1 , retaining Chamfer Distance (CD) for comparison with the original training objective. Mean

\pm

sample SD across six training seeds. Lower is better; the lowest mean is bolded. CD is multiplied by

10^{2}

for readability.

Resolution	FFD Base Grid	FFD Base DVs	FFD Match DVs	CBD-SA DVs
Small	$2\times 3$	$6$	$12$	$12$
Medium	$4\times 6$	$24$	$48$	$48$
Large	$8\times 12$	$96$	$192$	$192$

Appendix

Table 7: Geometry-control resolutions used in the CRM parameterization study. The aerodynamic mesh is fixed for all cases; the shape-control space varies.

Quantity	Value
Geometry	NASA Common Research Model (CRM) wing
Symmetry	Half-wing symmetry
Aerodynamic mesh	$N_{x}=3,\;N_{y}=17$
Baseline twist	CRM twist baked into the baseline mesh
Mach number	$M_{\infty}=0.84$
Freestream density	$\rho_{\infty}=0.38~\mathrm{kg\,m^{-3}}$

Appendix

Table 8: Flow, aerodynamic model, and surface settings used for the CRM experiments.

Scale	Affinity $\alpha$	Affinity Density $d$
Small	0.1	0.05
Medium	0.7	0.15
Large	0.7	0.1

Appendix

Table 9: Selected hyperparameters per scale.

\alpha

controls affinity propagation strength;

d

controls the target graph density. The outer cage is computed once per scale. Affinity weights use a min-weight aggregation scheme.

Optimizer	Driver	Main limits	Additional settings
SLSQP	SciPy	tol = 1e-9 maxiter = 200 small/medium maxiter = 2000 large	objective_scaler = 1e4 debug_print = false print_results = false opt_settings = {}
IPOPT	pyOptSparse	tol = 1e-6 maxiter = 3000 max_iter = 3000	objective_scaler = 1e4 gradient_method = openmdao acceptable_tol = 1e-5 acceptable_iter = 15 hessian_approximation = limited-memory limited_memory_max_history = 1000 mu_strategy = monotone mu_init = 1e-5 nlp_scaling_method = none start_with_resto = yes max_resto_iter = 200

Appendix

Table 10: Optimizer settings used for the CRM multi-optimizer comparison.

Figure 13: Final bending energy of optimized CRM geometry.

Figure 14: Final CL residual of optimized CRM geometry.

Figure 15: Outer-cage discretization ablation using the 19–27-vertex convex hulls and identical perturbed starts. Left: mean final modeled drag for CBD and CBD-SA, together with the paired effect interval and the number of matched runs favoring CBD-SA (lower is better). Right: the reduction from the common initialization achieved by CBD is indexed to 100, showing the additional optimization gain from structural affinity.

Outer cage	Perturb.	Matched	Mean $\Delta C_{D}$	95% CI	Reduction
19–27 vertices	5%	9	$-0.000569$	$[-0.000825,-0.000316]$	2.06%
19–27 vertices	10%	9	$-0.000228$	$[-0.000263,-0.000198]$	0.83%
38 vertices	5%	8	$-0.000422$	$[-0.000710,-0.000210]$	1.53%
38 vertices	10%	9	$-0.000208$	$[-0.000233,-0.000190]$	0.75%

Appendix

Table 11: Paired aerodynamic shape-optimization results. Matched denotes matched valid runs; CBD-SA reaches lower final modeled drag in every listed pair.

Parameter	Value
actor_hidden_dim	128
critic_hidden_dim	256
num_steps	8
v_min	$-10.0$
v_max	$30.0$
policy_noise	$0.0$

Appendix

Table 12: FastTD3 parameters.

Panel of Figure 16 — Figure 16: Spatula lift comparison between CBD and CBD-SA with RL+GO optimization.

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