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arxiv: 2605.06011 · v1 · submitted 2026-05-07 · 🧮 math.OC

Distortion-minimized de-homogenization for optimization of cell-size distribution in TPMS structures

Hiroki Kawabe , Kaito Ohtani , Yusibo Yang , Musaddiq Al Ali , Kentaro Yaji This is my paper

Pith reviewed 2026-05-08 08:24 UTC · model grok-4.3

classification 🧮 math.OC
keywords TPMS structuresde-homogenizationtopology optimizationcell-size distributionwavenumber matchingdistortion minimizationstiffness maximizationPoisson equation
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The pith

A wavenumber-matching de-homogenization method lets topology-optimized TPMS cell-size distributions retain nearly all their predicted stiffness after realization.

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

The paper develops a de-homogenization method for triply periodic minimal surface structures whose cell sizes have been optimized at the homogenized level. Instead of conventional periodic modulation, it solves a minimization problem that forces the actual local wavenumbers to match the target distribution as closely as possible. This minimization is recast as a Poisson equation and solved efficiently with the discrete cosine transform. When the method is used inside a stiffness-maximization problem, the final de-homogenized structures show strain-energy errors of only 0.8 percent relative to the homogenized model, versus 63.6 percent for the older approach. The realized designs also deliver more than 50 percent higher stiffness than a uniform lattice in both simulation and experiment.

Core claim

The central claim is that directly minimizing the difference between target and realized wavenumbers, formulated as a Poisson equation and solved via discrete cosine transform, produces de-homogenized TPMS geometries whose mechanical response deviates by less than one percent from the homogenized prediction, in contrast to the large distortions introduced by periodic modulation.

What carries the argument

The Poisson-equation formulation of wavenumber-difference minimization, solved with the discrete cosine transform, that directly prescribes local cell size in the de-homogenized TPMS lattice.

Load-bearing premise

That forcing the local wavenumbers of the realized structure to match those implied by the target cell-size field is enough to keep the mechanical properties close to the homogenized prediction without other geometric or connectivity errors dominating.

What would settle it

A high-resolution finite-element analysis or physical compression test on the de-homogenized geometry that shows the strain energy deviates by more than a few percent from the homogenized value even when the wavenumber mismatch is small.

Figures

Figures reproduced from arXiv: 2605.06011 by Hiroki Kawabe, Kaito Ohtani, Kentaro Yaji, Musaddiq Al Ali, Yusibo Yang.

Figure 1
Figure 1. Figure 1: Overall framework of the proposed method. view at source ↗
Figure 2
Figure 2. Figure 2: RUC with periodic boundary conditions applied to the opposite faces. view at source ↗
Figure 3
Figure 3. Figure 3: Interpolation of the effective properties of the TPMS structure based on the RUC analysis: Young’s modulus (top), Poisson’s ratio (center), and den￾sity (bottom) as functions of the design variable. wavenumbers. These optimized phase distributions can then be used in Eq. (1) to generate the TPMS structure that accurately reflects the spatially varying size P(r). 2.3. Effective property estimation by homoge… view at source ↗
Figure 4
Figure 4. Figure 4: Simulation setup: the analysis domain and boundary conditions for the FEA of the homogenized model (left) and the de-homogenized model (right). view at source ↗
Figure 5
Figure 5. Figure 5: De-homogenization examples: a) 1D-like case with smooth and discrete size distributions, b) simple 3D case, c) complex 3D case with a 3D-scanned view at source ↗
Figure 6
Figure 6. Figure 6: The smoothed size distributions (left) and the corresponding view at source ↗
Figure 8
Figure 8. Figure 8: De-homogenized structures and residual distributions for the 1D case: proposed method (first row), PM method with Gaussian smoothing with view at source ↗
Figure 10
Figure 10. Figure 10: The distributions of the gradients of the phase distributions, show view at source ↗
Figure 11
Figure 11. Figure 11: Dehomogenized structures and residual distributions for the 1D case with discrete size distribution: conventional PM method without Gaussian smoothing view at source ↗
Figure 12
Figure 12. Figure 12: Dehomogenized structures and residual distributions for the 3D case: conventional PM method without Gaussian smoothing (top) and proposed method view at source ↗
Figure 13
Figure 13. Figure 13: Size distributions and residual distributions for the bunny case: conventional PM method with Gaussian smoothing with view at source ↗
Figure 14
Figure 14. Figure 14: Size distributions and residual distributions for the bone case: con view at source ↗
Figure 15
Figure 15. Figure 15: De-homogenized structure for the bunny case: a) Gyroid structure view at source ↗
Figure 17
Figure 17. Figure 17: Displacement distribution for the uniform size case with the sectional view at the center and two opposite side views of the de-homogenized model (top) view at source ↗
Figure 18
Figure 18. Figure 18: Stress distribution for the uniform size case with the sectional view at the center and two opposite side views of the de-homogenized model (top) and view at source ↗
Figure 19
Figure 19. Figure 19: Optimization history: negative strain energy on the left y-axis and view at source ↗
Figure 20
Figure 20. Figure 20: Upsampling results and illustration of the upsampling process: the size optimization results from TO on the coarse mesh (left); the upsampled size view at source ↗
Figure 21
Figure 21. Figure 21: Phase optimization results and marching cube illustration: the phase distributions view at source ↗
Figure 22
Figure 22. Figure 22: Illustration of the stitching and remeshing process. view at source ↗
Figure 23
Figure 23. Figure 23: Displacement distribution of the models with the optimized size distribution obtained from the homogenized TO, showing the de-homogenized models view at source ↗
Figure 25
Figure 25. Figure 25: The test specimens were placed between the base plate view at source ↗
Figure 24
Figure 24. Figure 24: Stress distribution of the models with the optimized size distribution obtained from the homogenized TO, showing the de-homogenized models from the view at source ↗
Figure 25
Figure 25. Figure 25: Experimental setup for the compression tests. view at source ↗
Figure 26
Figure 26. Figure 26: Load-displacement curves for the three results: uniform, optimized view at source ↗
Figure 27
Figure 27. Figure 27: Side view of the specimens from the proposed method during the compression test. The bottom length value indicates the applied displacement for each view at source ↗
Figure 28
Figure 28. Figure 28: Side view of the specimens from the conventional PM method during the compression test. The bottom length value indicates the applied displacement view at source ↗
Figure 29
Figure 29. Figure 29: Side view of the specimens from the uniform case during the compression test. The bottom length value indicates the applied displacement for each view at source ↗
Figure 30
Figure 30. Figure 30: Top view of the test specimens before and 20 days after the compression test. view at source ↗
read the original abstract

This paper presents a homogenized topology optimization (TO) method for spatially optimizing cell-size distribution of triply-periodic minimal surface (TPMS) structures, with high accuracy in the optimized structural response after de-homogenization. To achieve this, we introduce a novel de-homogenization technique that directly minimizes the difference between the wavenumbers obtained from the target and actual size distributions. This minimization problem is efficiently solved as a typical Poisson's equation utilizing the discrete cosine transform. We first verify the proposed de-homogenization method through numerical examples, showcasing its capability in significantly reducing the known distortion of the de-homogenized TPMS structures from the conventional periodic modulation (PM) method. Then, we apply the proposed method to a stiffness maximization problem, to demonstrate its effectiveness in improving the structural response compared to the PM method. The proposed method successfully reduced the distortion of the de-homogenized structures compared to the PM method, leading to 0.8% difference in the strain energy compared to the homogenized model, as opposed to 63.6% difference in the PM method. The optimized structure from the proposed method shows a significant improvement in the strain energy by 50.1% compared to the uniform case in the FE analysis on the de-homogenized models, while the PM method results in a significant decrease of 45.8%. The experimental validation shows that the effective stiffness of the optimized structure from the proposed method is 54.2% higher than that of the uniform case, while the PM method results in a significant decrease by 77.3%. These results exhibit the proposed method effectively increases the accuracy of the de-homogenization, thereby maximizing the potential of the homogenized TO for the spatial cell-size optimization of TPMS structures.

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 / 2 minor

Summary. The paper proposes a de-homogenization method for TPMS structures in homogenized topology optimization that minimizes the difference between target and realized cell-size wavenumbers by recasting the problem as a Poisson equation solved via the discrete cosine transform. Numerical verification and a stiffness-maximization example are used to show that the approach reduces distortion relative to the periodic modulation (PM) baseline, yielding a strain-energy difference of only 0.8% from the homogenized prediction (versus 63.6% for PM), a 50.1% strain-energy gain over the uniform case in FE analysis, and a 54.2% experimental stiffness increase.

Significance. If the reported mechanical fidelity holds, the method would meaningfully improve the reliability of cell-size optimization for TPMS lattices, closing the gap between homogenized predictions and realizable geometries. The DCT-based Poisson solver provides an efficient, parameter-light implementation, and the concrete error reductions together with both FE and experimental validation constitute a tangible advance for practical use of homogenization-based design of cellular structures.

major comments (2)
  1. [§3.2] §3.2 (Poisson formulation and DCT solve): the wavenumber-minimization objective does not explicitly constrain local surface curvature, wall-thickness continuity, or triple-line connectivity; because TPMS effective stiffness depends on these geometric features in addition to local period, the manuscript must demonstrate (via a direct comparison or error map) that the resulting implicit surfaces remain valid minimal surfaces without introducing necks or intersections that would otherwise affect global strain energy.
  2. [§4.3] §4.3 (FE analysis on de-homogenized models): the central 0.8% strain-energy agreement is load-bearing for the claim of improved accuracy, yet the text provides no mesh-convergence study or description of how the implicit surface is discretized into a volume mesh; without these, it remains possible that the reported agreement is partly an artifact of discretization rather than a consequence of wavenumber matching.
minor comments (2)
  1. [§3] The definition of wavenumber and its relation to the cell-size field should be stated explicitly with an equation in the method section to prevent ambiguity when comparing target and realized distributions.
  2. [Figures 4-7] Figure captions in the numerical and experimental sections would benefit from explicit mention of the error metric (strain-energy difference) and the number of realizations used for the PM baseline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify important validation aspects of the de-homogenization approach. We address each major comment below and will incorporate the requested demonstrations and studies in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Poisson formulation and DCT solve): the wavenumber-minimization objective does not explicitly constrain local surface curvature, wall-thickness continuity, or triple-line connectivity; because TPMS effective stiffness depends on these geometric features in addition to local period, the manuscript must demonstrate (via a direct comparison or error map) that the resulting implicit surfaces remain valid minimal surfaces without introducing necks or intersections that would otherwise affect global strain energy.

    Authors: We agree that the wavenumber objective does not explicitly enforce curvature or connectivity constraints, and that these geometric features influence TPMS stiffness. Our numerical examples already indicate that the resulting implicit surfaces preserve minimal-surface character without necks or intersections, as reflected in the 0.8% strain-energy agreement. To provide the requested direct evidence, we will add to §3.2 a side-by-side comparison of mean-curvature distributions, wall-thickness continuity metrics, and triple-line connectivity checks between the proposed method and the PM baseline, together with localized error maps. These additions will confirm that wavenumber matching also yields geometrically valid surfaces. revision: yes

  2. Referee: [§4.3] §4.3 (FE analysis on de-homogenized models): the central 0.8% strain-energy agreement is load-bearing for the claim of improved accuracy, yet the text provides no mesh-convergence study or description of how the implicit surface is discretized into a volume mesh; without these, it remains possible that the reported agreement is partly an artifact of discretization rather than a consequence of wavenumber matching.

    Authors: We concur that mesh convergence and a clear discretization description are necessary to rule out numerical artifacts. The implicit surfaces are converted to volume meshes via marching-cubes isosurfacing followed by tetrahedral refinement; however, these steps are not detailed in the current text. In the revision we will expand §4.3 with an explicit description of the meshing pipeline and a mesh-convergence study (reporting strain-energy values at successively refined element sizes) for both the proposed and PM models. The study will demonstrate that the 0.8% difference remains stable across converged meshes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; de-homogenization objective is independent of reported mechanical metrics

full rationale

The paper defines its de-homogenization step as the solution of a Poisson equation (via DCT) that minimizes the L2 difference between target and realized wavenumber fields. This objective is not mathematically identical to the strain-energy or stiffness values later measured by separate FE analysis and physical experiments on the de-homogenized geometries. The reported performance deltas (0.8 % vs. 63.6 % strain-energy error, 50.1 % improvement vs. uniform) are therefore external benchmarks, not quantities forced by the wavenumber objective itself. No self-citation is invoked to justify uniqueness or to close the derivation loop, and no fitted parameter is relabeled as a prediction. The chain from homogenized TO to de-homogenized FE validation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard homogenization assumptions for periodic media and on the premise that wavenumber mismatch is an adequate surrogate for geometric distortion; no new free parameters, ad-hoc constants, or invented physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Homogenization theory remains valid for TPMS structures whose cell size varies smoothly in space
    Invoked when the topology optimization is performed on the homogenized model before de-homogenization

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