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arxiv: 2605.17387 · v1 · pith:H5YW45CUnew · submitted 2026-05-17 · 💻 cs.CE · cs.MS

Spatial Optimization of Interconnected Systems in Non-Convex Design Spaces

S. Westerhof , T. Hofman This is my paper

Pith reviewed 2026-05-19 23:03 UTC · model grok-4.3

classification 💻 cs.CE cs.MS
keywords spatial optimizationnon-convex design spacesinterconnected systemsMaximal Disjoint Ball DecompositionSPI2 frameworkcomponent placementCAD workflowphysics-based optimization
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The pith

A smooth inside-outside function lets optimizers pack interconnected components into arbitrary non-convex geometries while handling routing and physics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the SPI2 framework to handle arbitrary non-convex design boundaries by introducing a differentiable inside-outside check. It uses the Maximal Disjoint Ball Decomposition method to evaluate component placement during optimization. The approach adds direct calculations for center of gravity and moment of inertia and includes an end-to-end CAD workflow for importing parts and rebuilding the assembly. A demonstration on a fictional aircraft auxiliary unit shows the optimizer arranging multiple connected components inside a custom shape while meeting routing and physical objectives. Geometric feasibility holds within numerical tolerance.

Core claim

The paper establishes a spatial optimization methodology that extends the SPI2 framework to arbitrary non-convex design boundaries through a smooth, differentiable inside-outside evaluation supplied by the Maximal Disjoint Ball Decomposition method, incorporates center-of-gravity and moment-of-inertia terms directly into the objective, and supplies an end-to-end CAD workflow; the resulting optimizer places multiple interconnected components inside a custom geometry while simultaneously satisfying routing and physics-based objectives and maintains geometric feasibility within numerical tolerance.

What carries the argument

The Maximal Disjoint Ball Decomposition (MDBD) method, which supplies a smooth and differentiable inside-outside evaluation for components inside arbitrary non-convex boundaries during optimization.

If this is right

  • Multiple interconnected components can be placed and routed inside custom non-convex geometries.
  • Routing constraints and physics-based objectives such as center-of-gravity and moment-of-inertia are optimized simultaneously.
  • An end-to-end CAD workflow allows direct import of components and reconstruction of the final assembly.
  • Geometric feasibility of the placed components is preserved within numerical tolerance.
  • The method demonstrates practical applicability on engineering examples such as an aircraft auxiliary unit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The differentiable boundary handling could support extension to dynamic or time-varying design spaces where boundaries change during optimization.
  • Similar ball-decomposition techniques might apply to packing problems in other domains such as electronics enclosure layout or vehicle interior arrangement.
  • Replacing the fictional demonstration geometry with measured or scanned real-world boundaries would provide a direct test of robustness.
  • The framework's modular structure suggests it could incorporate additional physics models beyond inertia without changing the core placement logic.

Load-bearing premise

The Maximal Disjoint Ball Decomposition method must supply an accurate and differentiable inside-outside evaluation suitable for arbitrary non-convex design boundaries throughout the optimization.

What would settle it

If an optimization run places a component such that it intersects the design boundary yet the MDBD-based inside-outside function reports a feasible value within tolerance, the central feasibility claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.17387 by S. Westerhof, T. Hofman.

Figure 1
Figure 1. Figure 1: Overview of the full framework from CAD drawing [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Depiction of the model with various components: [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example of a design boundary section, where the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overview of the nested optimization framework. Steps [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Overview of the placement–routing–physics outcome [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

This paper presents a spatial optimization methodology that extends the Spatial Packaging of Interconnected Systems with Physical Interaction (SPI2) framework to support arbitrary, non-convex design boundaries. We introduce a smooth, differentiable inside-outside evaluation for components represented using the Maximal Disjoint Ball Decomposition (MDBD) method. The framework also incorporates center-of-gravity and moment-of-inertia calculations directly into the optimization, and provides an end-to-end computer-aided design (CAD) workflow for importing components and reconstructing the optimized assembly. The method is demonstrated on a fictional aircraft auxiliary unit. Results show that the optimizer can place multiple interconnected components within a custom geometry while simultaneously handling routing and physics-based objectives. The approach maintains geometric feasibility within numerical tolerance and illustrates the potential of MDBD-based SPI2 methods for practical engineering design applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper extends the SPI2 framework to arbitrary non-convex design boundaries by introducing a smooth, differentiable inside-outside evaluation based on Maximal Disjoint Ball Decomposition (MDBD). It incorporates center-of-gravity and moment-of-inertia calculations directly into the optimization objective and provides an end-to-end CAD workflow for importing components and reconstructing the optimized assembly. The method is demonstrated on a fictional aircraft auxiliary unit, with the claim that the optimizer successfully places multiple interconnected components while simultaneously handling routing and physics-based objectives and maintaining geometric feasibility within numerical tolerance.

Significance. If the MDBD evaluation is shown to remain accurate and differentiable across non-convex features, the work would enable practical gradient-based spatial optimization of interconnected systems in realistic engineering geometries, extending prior SPI2 methods with new physics integration and CAD reconstruction steps. The approach addresses a relevant gap in handling non-convex boundaries for component placement and routing.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'the approach maintains geometric feasibility within numerical tolerance' is load-bearing for the overall contribution but is unsupported by any quantitative metrics (e.g., maximum constraint violation, Hausdorff distance to the true boundary, or gradient error versus finite differences) from the fictional aircraft demonstration.
  2. [Abstract] Abstract: the assumption that MDBD supplies an accurate and differentiable inside-outside oracle for arbitrary non-convex boundaries is not demonstrated; potential loss of differentiability or uncovered thin regions at sharp concavities or narrow corridors would directly undermine the feasibility of gradient-based placement, routing, and inertia optimization.
minor comments (1)
  1. The abstract refers to a 'fictional aircraft auxiliary unit' without specifying the number of components, the nature of the custom geometry, or the routing constraints; adding these details would improve reproducibility of the demonstration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's constructive comments on our manuscript extending the SPI2 framework. We address each major comment point by point below and commit to revisions that strengthen the quantitative support and demonstrations without altering the core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'the approach maintains geometric feasibility within numerical tolerance' is load-bearing for the overall contribution but is unsupported by any quantitative metrics (e.g., maximum constraint violation, Hausdorff distance to the true boundary, or gradient error versus finite differences) from the fictional aircraft demonstration.

    Authors: We agree that the abstract claim regarding geometric feasibility would benefit from explicit quantitative backing in the demonstration. In the revised version, we will incorporate specific metrics from the aircraft auxiliary unit example, including maximum constraint violation values, Hausdorff distance between the MDBD approximation and the true boundary, and gradient error comparisons against finite differences. These additions will directly support the feasibility statement. revision: yes

  2. Referee: [Abstract] Abstract: the assumption that MDBD supplies an accurate and differentiable inside-outside oracle for arbitrary non-convex boundaries is not demonstrated; potential loss of differentiability or uncovered thin regions at sharp concavities or narrow corridors would directly undermine the feasibility of gradient-based placement, routing, and inertia optimization.

    Authors: The paper describes MDBD as yielding a smooth, differentiable inside-outside function by construction. We acknowledge that the current demonstration does not explicitly verify its accuracy or differentiability at sharp concavities or in narrow corridors. The revision will add targeted analysis and visualizations for the aircraft geometry to illustrate behavior in these regions, including any observed limitations or safeguards. revision: yes

Circularity Check

0 steps flagged

Extends prior SPI2 framework with independent MDBD differentiable evaluation and physics objectives

full rationale

The paper explicitly builds on the SPI2 framework from prior work but introduces new elements: a smooth differentiable inside-outside evaluation via MDBD for non-convex boundaries, direct incorporation of center-of-gravity and moment-of-inertia calculations, and an end-to-end CAD workflow. These additions do not reduce by construction to fitted parameters or self-cited results; the central optimization claims rest on the new MDBD oracle and gradient-based handling rather than re-deriving prior inputs. No self-definitional loops, fitted predictions, or load-bearing uniqueness theorems from overlapping authors are present in the provided derivation chain. This is a standard extension with minor self-citation that is not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; specific free parameters, axioms, and invented entities cannot be extracted. The approach likely depends on standard gradient-based optimization assumptions and the existing SPI2 framework.

pith-pipeline@v0.9.0 · 5662 in / 1121 out tokens · 42924 ms · 2026-05-19T23:03:35.930195+00:00 · methodology

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Reference graph

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