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REVIEW 4 major objections 5 minor 46 references

An Improved Finite Element Modeling Method for Triply Periodic Minimal Surface Structures Based on Element Size and Minimum Jacobian

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-parameter voxel method—jointly setting element size and minimum Jacobian—makes TPMS finite element models converge faster, more accurately, and at about half the solve time of conventional voxel meshes.

desk verdict A practical, incremental voxel-meshing improvement for TPMS structures that deserves a serious referee, but needs an element-validity check and a few clarifications before I'd trust the accuracy claim fully. read the letter →

arxiv 2506.04028 v1 pith:45PXP5ID submitted 2025-06-04 eess.SY cs.SY

Siqi Wang , Chuangyu Jiang , Xiaodong Zhang , Yilong Zhang , Baoqiang Zhang , Huageng Luo This is my paper
classification eess.SYcs.SY
keywords triplyperiodicminimalsurfacefiniteelementmethodvoxelmodelingminimumJacobianmeshconvergencegridindexeffectiveYoung'smodulusgradedTPMSstructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the standard way of turning triply periodic minimal surface (TPMS) lattices into finite element meshes—perfectly cubic voxel elements whose Jacobian is held at 1—is inefficient because the cubic elements cannot follow the curved TPMS boundary, overshoot the geometry, and overestimate stiffness until the mesh is extremely fine. It proposes a two-parameter voxel method that jointly controls element size and minimum Jacobian (MJ), letting boundary cells distort so the mesh hugs the true surface. Using the grid convergence index (GCI) based on Richardson extrapolation, it reports that numerical results stabilize once MJ reaches 0.3, and that for a uniform Gyroid at relative density 0.45 the two-parameter model reaches an asymptotic Young's modulus of 19.97 GPa with a higher convergence order, a lower relative error, and roughly half the solve time of a finer traditional voxel model. If right, this gives engineers a cheap and systematic way to compute the mechanical response of TPMS and graded TPMS structures without hand-built conforming meshes.

What carries the argument

The central object is the minimum Jacobian (MJ) of a hexahedral mesh: the smallest value of the Jacobian determinant among the elements, which measures how much an element is distorted when mapped from its natural coordinate system to global coordinates. Traditional voxel modeling keeps every Jacobian at 1, forcing perfectly regular cubes that cannot conform to curved TPMS boundaries. The paper instead treats MJ as a tunable second parameter alongside element size, allowing boundary voxels to become distorted hexahedra that hug the surface. The assessment machinery is the grid convergence index (GCI) built on Richardson extrapolation, which estimates the asymptotic solution, the order of convergence, and whether the simulations lie in the asymptotic range through the ratio $R_a$, which is approximately 1.

What would settle it

A direct test would be to build a high-quality conforming tetrahedral or CAD-reconstructed hexahedral mesh of the same Gyroid at relative density 0.45 and compare its converged Young's modulus with the claimed 19.97 GPa asymptotic value; a significant discrepancy would show that distortion error outweighs the geometric-fit gain. A second test is to inspect the distorted boundary elements for inverted or near-singular Jacobians and to monitor internal strain-energy convergence as MJ is lowered from 0.3 to 0.2: if the modulus keeps shifting or the solution worsens, the 0.3 threshold is not a universal stabilization point.

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Extended reading notes

Core claim

The central claim is that mesh convergence of voxel-based TPMS models is controlled by two parameters, not one: element size and minimum Jacobian. In traditional voxel models every element is required to have a Jacobian of 1, so boundary voxels protrude beyond the implicit surface; the finite element model ends up with a higher effective relative density than designed and overestimates Young's modulus. By lowering the minimum Jacobian to 0.3 in the meshing tool, boundary hexahedra are allowed to deform and track the curved surface, so many fewer elements are needed to approach the true geometry. The paper shows numerically that the effective Young's modulus stabilizes for MJ values at or below 0.3, and for the Gyroid structure the two-parameter method gives a convergence order of $p=2.756$ versus $p=1.358$ for the voxel model, a $GCI_{12}$ of 1.031% versus 15.932%, and an asymptotic modulus of 19.97 GPa versus 21.817 GPa; the higher voxel asymptote is interpreted as the geometric overshoot of the traditional mesh. The same method applied to a graded Gyroid converges even faster, and the predicted relative moduli fit the Gibson–Ashby relation and land closer to experimental data than the voxel model.

Load-bearing premise

The method's results stand or fall on the assumption that letting boundary cube cells distort down to a minimum Jacobian of 0.3 makes them fit the curved TPMS surface tightly without introducing numerical errors, and that this same 0.3 setting transfers to other TPMS types, relative densities, and graded designs.

Editorial extensions

If this is right

  • Engineers can reach converged stiffness for a uniform Gyroid at relative density 0.45 with a 0.2 mm MJ=0.3 model that solves in about 58 seconds, while a 0.15 mm traditional voxel model takes about 121 seconds for comparable accuracy.
  • The GCI-based framework gives a systematic rule for when to stop refining a voxel TPMS mesh, replacing ad hoc element-count targets that vary across the literature.
  • The MJ=0.3 stabilization threshold offers a practical starting guideline: fix the minimum Jacobian at 0.3, then refine element size until the relative error falls below the chosen target.
  • For graded TPMS structures, the two-parameter method shows even faster convergence than for uniform ones—relative error drops to 0.62% at 0.1 mm with an asymptotic modulus of 19.24 GPa.
  • At matched element sizes, the two-parameter model gives effective moduli closer to experimental measurements than the traditional voxel model, because the distorted boundary elements reduce the spurious stiffness from geometric overshoot.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is whether the distortion-controlled boundary fit transfers to other TPMS families such as Diamond, Primitive, or sheet-network Double Gyroid; the 0.3 threshold may depend on local curvature and relative density.
  • Given the paper's stated limitation that MJ=0.3 elements sharply cut the stable time step in explicit analyses, an extension would relax boundary distortion for impact simulations while keeping it for quasi-static accuracy.
  • A natural refinement would localize the GCI check to the lowest-relative-density region of a graded structure, where prior work expects convergence to be slowest.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a two-parameter voxel method for finite element modeling of triply periodic minimal surface (TPMS) structures, in which both element size and minimum Jacobian (MJ) are controlled. Using Hypermesh's Shrink Wrap module, the authors generate voxel meshes of a Gyroid structure with MJ=0.3 and element sizes from 0.4 to 0.1 mm, and compare the proposed method against the traditional voxel method (MJ=1) via relative error, Richardson extrapolation, and grid convergence index (GCI). They report that the two-parameter method gives a higher order of convergence (p=2.756 vs 1.358), lower GCI, and an asymptotic Young's modulus of 19.97 GPa for a uniform Gyroid at RD=0.45, at about half the solving time of a finer voxel model. The method is also applied to a graded Gyroid, where the authors claim superior convergence. The paper concludes that the two-parameter voxel method improves mesh convergence, solution accuracy, and computational efficiency for TPMS structures.

Significance. If the claims are validated, the two-parameter voxel method could substantially reduce the computational cost of TPMS finite element analyses while maintaining or improving accuracy, and the use of GCI is a positive step toward objective mesh-convergence assessment. The paper is clearly written and provides a useful comparison of convergence metrics. However, the core validation is incomplete: the MJ=0.3 threshold is selected from the same stabilization curve used to demonstrate the method, no element-validity check is reported, and the only external comparison is to scattered literature data through fitted Gibson-Ashby curves. The observed convergence advantage could in principle be an artifact of distorted elements rather than improved geometric fidelity, so the central claim is not yet conclusively established.

major comments (4)
  1. [§2.2.2, Fig. 5] The choice of MJ=0.3 is made because the effective Young's modulus stabilizes for MJ≤0.3 in Fig. 5, which is the same data set used to demonstrate the method. No independent check is provided that the elements generated with MJ=0.3 are valid (e.g., no inverted elements, acceptable Jacobian distribution) or that the resulting finite element model's relative density matches the intended design value (e.g., by summing element volumes). Without these checks, the apparent stabilization and improved convergence could be due to a trade-off between geometric fidelity and distortion-induced numerical error rather than to a genuine improvement in accuracy. This is a load-bearing gap for the paper's central claim.
  2. [§3.2.2, Table 2] The Richardson extrapolation and GCI analysis assumes a systematically refined mesh sequence with a known refinement ratio r. However, the element counts in Table 2 do not scale as h^{-3}: for the voxel model, the number of elements per unit cell at h=0.4 mm is 1328.8 and at h=0.2 mm it is 8836.7, a factor of only 6.65 rather than 8. This indicates that the mesh topology changes with element size in a way that is not a pure uniform subdivision, so the assumptions behind Eqs. (9)-(14) may be violated. The asymptotic solution and the order p could therefore depend on the specific boundary-voxel classification rather than on h alone. The paper should justify that the mesh sequence satisfies the asymptotic-range condition, for example by reporting the actual distribution of element sizes or by using a nested refinement strategy.
  3. [§3.3, Fig. 8] The external validation of solution accuracy is not sufficient. The comparison in Fig. 8 uses Gibson-Ashby curves fitted with two free parameters (C1 and m) to only four relative densities, and the literature data are scattered. A direct comparison against an independent high-fidelity reference, such as a body-fitted tetrahedral or conforming hexahedral mesh, is needed to verify that the two-parameter voxel model converges to the correct value. Without such a reference, the reported asymptotic modulus of 19.97 GPa (Table 3) cannot be certified as free of distortion bias. Additionally, the solving-time comparison (121 s vs 58 s) is not a controlled accuracy-per-cost comparison unless both models are shown to produce the same error level relative to a reference solution.
  4. [§4.2, Table 5] The reported order of convergence p=5.986 for the graded structure is unrealistically high for linear hexahedral elements and much larger than the p=2.756 reported for the uniform case. This strongly suggests that the mesh sequence is not in the asymptotic range or that the solution is not smooth. Furthermore, the relative errors in Table 4 for the MJ=0.3 model are highly non-monotonic (39.28% at 0.35 mm, then 7.40% at 0.3 mm), which contradicts the claim of progressively decreasing error and undermines the significance of the low GCI values (0.029% and 1.815%). The assertion that the two-parameter method is even better for graded structures is therefore not convincingly supported by the data as presented.
minor comments (5)
  1. [§3.2.2, Eq. (10)] The definitions of f1, f2, and f3 and the sign convention in the formula for p should be stated unambiguously. The text says f1, f2, f3 correspond to element sizes 0.1, 0.2, and 0.4 mm, so f1 is the finest mesh; the formula as printed uses (f3-f2)/(f2-f1) without absolute values, which is valid only when the solution is monotone. The paper should clarify this assumption.
  2. [Table 2] The 'number of elements per unit cell' values are non-integers (e.g., 1328.8, 1883.2, 2947.0). This is likely an average over the 64 unit cells in the 4×4×4 structure, but the fractional values are confusing; please state whether these are averages and report the total element count for the full structure.
  3. [Fig. 5] The range and spacing of the MJ values used to generate Fig. 5 are not specified in the text or figure caption. Without knowing which MJ values were tested, the reader cannot assess how well the stabilization at MJ≤0.3 is resolved.
  4. [§5, Conclusions] Minor typographical issues: 'CGI' in conclusion (1) should be 'GCI'; 'Gyriod' in the Fig. 3 caption should be 'Gyroid'; and 'V oxel' in the abstract should be 'Voxel'.
  5. [§4.2] The asymptotic solutions for the graded and uniform structures (19.24 GPa vs 19.97 GPa) are compared without confidence intervals or an uncertainty estimate for the GCI values. Given the small GCI values reported, the 3.7% difference should be discussed in the context of the possible distortion bias discussed in the major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the method is calibrated empirically and benchmarked against external data; remaining issues are correctness and reproducibility concerns, not self-referential derivation.

full rationale

I find no circular step in the paper's derivation chain. The proposed two-parameter voxel method is defined by joint control of element size and minimum Jacobian; the MJ=0.3 threshold is an empirical calibration from Fig. 5, not a quantity derived from the claimed outcomes. The claimed convergence superiority is assessed by relative errors and GCI computed from mesh sequences of both methods; although the asymptotic solution is estimated from the same data via Richardson extrapolation, this is a standard verification metric and is not used as the sole evidence of accuracy. The 'solution accuracy' claim is independently benchmarked against external experimental and numerical data in Fig. 8 (Peng et al., Yang et al., Yan et al.), so it is not self-confirmation. The Gibson-Ashby constants in Eqs. (16)-(17) are fitted to the authors' simulations, but comparing the resulting curves to independent literature data is an external check, not circular. Material parameters from the authors' previous work [36] are a minor self-citation, but they are standard TC4 properties and not load-bearing for the central claim. The limitation about small characteristic lengths at MJ=0.3 for explicit analyses is a practical/correctness note, not circularity. The unverified assumption that Hypermesh's Shrink Wrap with MJ=0.3 yields valid non-inverted hexahedra is a reproducibility and validation risk, but it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard FEM assumptions plus several domain-specific and tool-specific premises. The most important free parameters are the MJ threshold (0.3) chosen from the authors' own stabilization curve and the Gibson-Ashby fitting constants. No new physical entities are introduced.

free parameters (3)
  • Minimum Jacobian threshold MJ = 0.3
    Adopted because Fig. 5 shows effective Young's modulus stabilizes when MJ is below 0.3; no a priori derivation or cross-validation on other TPMS types.
  • Gibson-Ashby coefficient C1 and exponent m (voxel model) = C1=1.11, m=1.96
    Fitted to the authors' own voxel simulation points in Fig. 8, Eq. (16).
  • Gibson-Ashby coefficient C1 and exponent m (two-parameter voxel model) = C1=1.06, m=2.24
    Fitted to the authors' own two-parameter simulation points, Eq. (17); used to claim agreement with experimental data.
assumptions (7)
  • standard math Isoparametric hexahedral element shape functions and Jacobian mapping (Eqs. 3-6)
    Standard FEM background used to define element distortion; not independently verified in this paper.
  • domain assumption Gyroid implicit function with offset parameter C defines the solid geometry (Eq. 1)
    The geometric model is generated from this representation; no validation of the STL generation against physical specimens.
  • ad hoc to paper Hypermesh Shrink Wrap with minimum Jacobian control produces valid voxel hexahedra approximating the STL boundary
    The core tooling premise; the paper does not document the exact meshing settings or prove that MJ=0.3 meshes are non-inverted and geometry-conforming.
  • standard math Richardson extrapolation error model E = C h^p + HOT and asymptotic range condition (Eqs. 9-14)
    Standard numerical analysis assumption; its validity here is checked only by the R_a ratio in Tables 3 and 5.
  • domain assumption Gibson-Ashby power law describes the relative modulus of TPMS structures (Eq. 15)
    Assumed for fitting and comparison; the fitted exponents differ between the two methods, so it is not a single verified law.
  • domain assumption 4x4x4 unit-cell model with rigid plates represents bulk TPMS behavior (Section 2.3)
    Based on cited prior work [16,17], not re-verified for the graded case.
  • domain assumption Effective Young's modulus is extracted from an elasto-plastic bilinear compression simulation (Table 1, Eq. 7)
    The paper does not state the strain range or modulus extraction rule; if the compression is not in the linear regime, the reported modulus is not a Young's modulus.

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Cite this review

Pith. "Pith review of An Improved Finite Element Modeling Method for Triply Periodic Minimal Surface Structures Based on Element Size and Minimum Jacobian." pith.science (2026). https://pith.science/paper/45PXP5ID

@misc{pith2026250604028,
  author       = {Pith},
  title        = {Pith review of: An Improved Finite Element Modeling Method for Triply Periodic Minimal Surface Structures Based on Element Size and Minimum Jacobian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45PXP5ID}},
  note         = {Machine review of arXiv:2506.04028}
}
read the original abstract

Triply periodic minimal surface (TPMS) structures, a type of lattice structure, have garnered significant attention due to their lightweight nature, controllability, and excellent mechanical properties. Voxel-based modeling is a widely used method for investigating the mechanical behavior of such lattice structures through finite element simulations. This study proposes a two-parameter voxel method that incorporates joint control of element size and minimum Jacobian (MJ). Numerical results indicate that the simulation outcomes tend to stabilize when the MJ reaches 0.3. The grid convergence index (GCI), based on Richardson extrapolation, is introduced to systematically assess the numerical convergence behavior of both voxel models and the proposed two-parameter voxel models. This provides a systematic and objective framework for evaluating discretization errors and mesh convergence in TPMS modeling. Compared with traditional voxel method, the proposed method exhibits superior mesh convergence, solution accuracy, and computational efficiency. Furthermore, the two-parameter voxel method also shows excellent applicability in the analysis of graded TPMS structures, exhibiting even better convergence behavior than in uniform structures.

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    H. Yin, W. Zhang, L. Zhu, F. Meng, J. Liu, G . Wen, Review on lattice structures for energy absorption properties, Compos. Struct. 304 (2023) 116397

    work page 2023

  2. [2]

    N. Qiu, Y . Wan, Y . Shen, J. Fang, Experimental and numerical studies on mechanical properties of TPMS structures, Int. J. Mech. Sci. 261 (2024) 108657

    work page 2024

  3. [3]

    C. Peng, K. Fox, M. Qian, H. Nguyen -Xuan, P. Tran, 3D printed sandwich beams with bioinspired cores: Mechanical perfor- mance and modelling, Thin-Walled Struct. 161 (2021) 107471

    work page 2021

  4. [4]

    Al-Ketan, R

    O. Al-Ketan, R. Rowshan, R.K.A. Al -Rub, Topology–mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials, Addit. Manuf. 19 (2018) 167–183

    work page 2018

  5. [5]

    H. Yin, Z. Liu, J. Dai, G . Wen, C. Zhang, Crushing behavior and op timization of sheet-based 3D periodic cellular structures, Compos. Part B Eng. 182 (2020) 107565

    work page 2020

  6. [6]

    S. Li, M. Hu, L. Xiao, W. Song, Compressive properties and collapse behavior of additively -manufactured layered -hybrid lattice structures under static and dynamic loadings, Thin-Walled Struct. 157 (2020) 107153

    work page 2020

  7. [7]

    L. Xiao, W. Song, C. Wang, H. Tang, N. Liu, J. Wang, Yield behavior of open-cell rhombic dodecahedron Ti–6Al–4V lattice at elevated temperatures, Int. J. Mech. Sci. 115 (2016) 310–317

    work page 2016

  8. [8]

    Berger, H.N

    J.B. Berger, H.N. Wadley, R.M. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543 (2017) 533–537

    work page 2017

Show all 46 references
  1. [9]

    Novak, S

    N. Novak, S. Tanaka, K. Hokamoto, A. Mauko, Y .E. Yilmaz, O. Al -Ketan, Z. Ren, High strain rate mechanical behaviour of uniform and hybrid metallic TPMS cellular structures, Thin-Walled Struct. 191 (2023) 111109

    2023

  2. [10]

    L. Wan, D. Hu, M. Wan, Z. Yang, H. Zhang, B. Pi, Lateral crushing behavior of tubular lattice structures with triply periodic minimal surface architectures, Thin-Walled Struct. 189 (2023) 110905

    2023

  3. [11]

    Zhang, Y

    Y . Zhang, Y . Chen, J. Li, J. Wu, L. Qian, Y . Tan, G . Zeng, Protective performance of hybrid triply periodic minimal surface lattice structure, Thin-Walled Struct. 194 (2024) 111288

    2024

  4. [12]

    Santiago, H

    R. Santiago, H. Ramos, S. AlMahri, O. Banabila, H. Alabdouli, D.W. Lee, A. Aziz, N. Rajput, M. Alves, Z. Guan, Modelling and optimisation of TPMS-based lattices subjected to high strain-rate impact loadings, Int. J. Impact Eng. 177 (2023) 104592

    2023

  5. [13]

    Z. Luo, Q. Tang, Q. Feng, S. Ma, J. Song, R. Setchi, F. Guo, Y . Zhang, Finite element analysis of the mechanical properties of sheet-and skeleton-gyroid Ti6Al4V structures produced by laser powder bed fusion, Thin-Walled Struct. 192 (2023) 111098

    2023

  6. [14]

    X. Li, L. Xiao, W. Song, Compressive behavior of selective la ser melting printed Gyroid structures under dynamic loading, Addit. Manuf. 46 (2021) 102054

    2021

  7. [15]

    S. Li, E. Xu, X. Zhan, W. Zheng, L. Li, Stress -driven nonlocal homogenization method for cellular structures, Aerosp. Sci. Technol. 155 (2024) 109632

    2024

  8. [16]

    Maskery, A.O

    I. Maskery, A.O. Aremu, L. Parry, R.D. Wildman, C.J. Tuck, I.A. Ashcroft, Effective design and simulation of surface -based lattice structures featuring volume fraction and cell type grading, Mater. Des. 155 (2018) 220–232

    2018

  9. [17]

    Maskery, L

    I. Maskery, L. Sturm, A.O. Aremu, A. Panesar, C.B. Williams, C.J. Tuck, R.D. Wildman, I.A. Ashcroft, R.J.M. Hague, In- sights into the mechanical properties of several triply periodic minimal surface lattice structures made by polymer additive manu- facturing, Polym. 152 (2018) 62–71

    2018

  10. [18]

    Montazerian, E

    H. Montazerian, E. Davoodi, M. Asadi-Eydivand, J. Kadkhodapour, M. Solati -Hashjin, Porous scaffold internal architecture design based on minimal surfaces: a compromise between permeability and elastic properties, Mater. Des. 126 (2017) 98–114

    2017

  11. [19]

    Aremu, I

    A.O. Aremu, I. Maskery, C. Tuck, I.A. Ashcroft, R.D. Wildman, R.J.M. Hague, A comparative finite element study of cubic unit cells for selective laser melting, in: Proc. Int. Solid Freeform Fabr. Symp., Univ. Texas at Austin, 2014

    2014

  12. [20]

    Y . Wan, N. Qiu, M. Xiao, Y . Xu, J. Fang, Energy dissipation of 3D-printed TPMS lattices under cyclic loading, Int. J. Mech. Sci. 294 (2025) 110245

    2025

  13. [21]

    Afshar, A.P

    M. Afshar, A.P. Anaraki, H. Montazerian, J. Kadkhodapour, Additive manufacturing and mechanical characterization of graded porosity scaffolds designed based on tri ply periodic minimal surface architectures, J. Mech. Behav. Biomed. Mater. 62 (2016) 481–494

    2016

  14. [22]

    Kadkhodapour, H

    J. Kadkhodapour, H. Montazerian, A.C. Darabi, A. Zargarian, S. Schmauder, The relationships between deformation mecha- nisms and mechanical properties of additively manufactured porous biomaterials, J. Mech. Behav. Biomed. Mater. 70 (2017) 28–42

    2017

  15. [23]

    Jiang, W

    W. Jiang, W. Liao, T. Liu, X. Shi, C. Wang, J. Qi, Y . Chen, Z. Wang, C. Zhang, A voxel-based method of multiscale mechan- ical property optimization for the design of graded TPMS structures, Mater. Des. 204 (2021) 109655

    2021

  16. [24]

    HomoGenius: a Foundation Model of Homogenization for Rapid Prediction of Effective Mechanical Properties using Neural Operators

    Wang Y , Li X, Yan Z, Du Y , Bai J, Liu B, Rabczuk T, Liu Y . HomoGenius: a Foundation Model of Homogenization for Rapid Prediction of Effective Mechanical Properties using Neural Operators. arXiv preprint arXiv:2404.07943. 2024

    2024 arXiv

  17. [25]

    Simsek, A

    U. Simsek, A. Akbulut, C.E. Gayir, C. Basaran, P. Sendur, Modal characterization of additively manufactured TPMS struc- tures: comparison between different modeling methods, Int. J. Adv. Manuf. Technol. 115 (2021) 657–674

    2021

  18. [26]

    Schwer, Guide for Verification and Validation in Computational Solid Mechanics, American Society of Mechanical En- gineers Virginia, (2006)

    L.E. Schwer, Guide for Verification and Validation in Computational Solid Mechanics, American Society of Mechanical En- gineers Virginia, (2006)

    2006

  19. [27]

    Aremu, J.P.J

    A.O. Aremu, J.P.J. Brennan -Craddock, A. Panesar, I.A. Ashcroft, R.J.M. Hague, R.D. Wildman, C. Tuck, A voxel -based method of cons tructing and skinning conformal and functionally graded lattice structures suitable for additive manufacturing, Addit. Manuf. 13 (2017) 1–13

    2017

  20. [28]

    Sarkari, S

    M. Sarkari, S. Rahmati, M. Nikkhoo, Computer -aided porous scaffold design for tissue engineering using triply peri odic minimal surfaces, Int. J. Precis. Eng. Manuf. 12 (1) (2011) 61–71

    2011

  21. [29]

    Y . Sun, Q. Li, Dynamic compressive behaviour of cellular materials: A review of phenomenon, mechanism and modelling, Int. J. Impact Eng. 112 (2018) 74–115

    2018

  22. [30]

    Zhang, S

    J. Zhang, S. Xie, T. Li, Z. Liu, S. Zheng, H. Zhou, A study of multi -stage energy absorption characteristics of hybrid sheet TPMS lattices, Thin-Walled Struct. 190 (2023) 110989. · 13 ·

    2023

  23. [31]

    Hoang, T.H

    T.-D. Hoang, T.H. Ngo, K.Q. Tran, S. Li, H. Nguyen -Xuan, A stochastic multiscale asymptotic homogenizatio n approach to 3D printed biodegradable resin TPMS bio-inspired structures, Thin-Walled Struct. 212 (2025) 113100

    2025

  24. [32]

    Maskery, N.T

    I. Maskery, N.T. Aboulkhair, A.O. Aremu, C.J. Tuck, I.A. Ashcroft, Compressive failure modes and energy absorption in additively manufactured double Gyroid lattices, Addit. Manuf. 16 (2017) 24–29

    2017

  25. [33]

    Sé bastien, M

    S.B.G .B. Sé bastien, M. Werner, M. Hannula, S. Sharifi, G .P.R.L. Lajoinie, D. Eglin, J. Hyttinen, A.A. Poot, D.W. Grijpma, Surface curvature in triply -periodic minimal surface architectures as a distinc t design parameter in preparing advanced tissue en- gineering scaffolds...

    2017

  26. [34]

    S. Li, M. Hu, L. Xiao, W. Song, Compressive properties and collapse behavior of additively -manufactured layered-hybrid lattice structures under static and dynamic loadings, Thin-Walled Struct. 157 (2020) 107153

    2020

  27. [35]

    LS-Dyna Keyword User’s Manual, Livermore, n.a

  28. [36]

    S. Wang, C. Jiang, C. Wang, B. Zhang, H. Luo, W. Feng, Bird strike resistance analysis for engine fan blade filled with triply periodic minimal surface, Aerosp. Sci. Technol. 161 (2025) 110109

    2025

  29. [37]

    Aycan, A

    O. Aycan, A. Topuz, L. Kadem, Evaluating uncertainties in CFD simulations of patient -specific aorta models using Grid Convergence Index method, Mech. Res. Commun. 133 (2023) 104188

    2023

  30. [38]

    Gibson, Mechanical behavior of metallic foams, Annu

    L.J. Gibson, Mechanical behavior of metallic foams, Annu. Rev. Mater. Sci. 30 (2000) 191–227

    2000

  31. [39]

    X. Peng, Q. Huang, Y . Zhang, X. Zhang, T. Shen, H. Shu, Z. Jin, Elastic response of anisotropic Gyroid cellular structures under compression: Parametric analysis, Mater. Des. 205 (2021) 109706

    2021

  32. [40]

    L. Yang, C. Yan, C. Han, P. Chen, S. Yang, Y . Shi, Mechanical response of a triply periodic minimal surface cellular struc- tures manufactured by selective laser melting, Int. J. Mech. Sci. 148 (2018) 149–157

    2018

  33. [41]

    C. Yan, L. Hao, A. Hussein, P. Y oung, Ti–6Al–4V triply periodic minimal surface structures for bone implants fabricated via selective laser melting, J. Mech. Behav. Biomed. Mater. 51 (2015) 61–73

    2015

  34. [42]

    Z. Zou, M. Simonelli, J. Katrib, G . Dimitrakis, R. Hague, Microstructure an d tensile properties of additive manufactured Ti–6Al–4V with refined prior-β grain structure obtained by rapid heat treatment, Mater. Sci. Eng. A 814 (2021) 141271

    2021

  35. [43]

    L. Yang, C. Yan, W. Cao, Z. Liu, B. Song, S. Wen, C. Zhang, Y . Shi, S. Yang, Compression–compression fatigue behaviour of Gyroid-type triply periodic minimal surface porous structures fabricated by selective laser melting, Acta Mater. 181 (2019) 49–66

    2019

  36. [44]

    Zhang, X

    J. Zhang, X. Chen, Y . Sun, J. Yang, R. Chen, Y . Xiong, W. Hou, L. Bai, Design of a biomimetic graded TPMS scaffold with quantitatively adjustable pore size, Mater. Des. 218 (2022) 110665

    2022

  37. [45]

    Nguyen-Xuan, K.Q

    H. Nguyen-Xuan, K.Q. Tran, C.H. Thai, J. Lee, Modelling of functionally graded triply periodic minimal surface (FG -TPMS) plates, Compos. Struct. 315 (2023) 116981

    2023

  38. [46]

    Novak, M

    N. Novak, M. Borovinšek, O. Al -Ketan, Z. Ren, M. Vesenjak, Impact and blast resistance of uniform and graded sandwich panels with TPMS cellular structures, Compos. Struct. 300 (2022) 116174

    2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.