Tensorial Minkowski functionals of triply periodic minimal surfaces

Mickel, Walter; Schröder‐Turk, Gerd E.; Mecke, Klaus

doi:10.1098/rsfs.2012.0007

Cited by 22 publications

(23 citation statements)

References 100 publications

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“…Even applications to non-cellular shapes with arbitrary topology are possible, albeit with altered interpretation. 63,74 Finally, our analysis supports the more frequent use of the low-weight q l , in particular q 2 , that have been largely overlooked in the literature. q 2 carries the same information as the anisotropy index β 0,2 1 of Refs.…”

Section: Resultssupporting

confidence: 61%

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

Mickel¹,

Kapfer²,

Schröder‐Turk³

et al. 2013

The Journal of Chemical Physics

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Local structure characterization with the bond-orientational order parameters q 4 , q 6 ,. . . introduced by Steinhardt et al. has become a standard tool in condensed matter physics, with applications including glass, jamming, melting or crystallization transitions and cluster formation. Here we discuss two fundamental flaws in the definition of these parameters that significantly affect their interpretation for studies of disordered systems, and offer a remedy. First, the definition of the bond-orientational order parameters considers the geometrical arrangement of a set of neighboring spheres NN(p) around a given central particle p; we show that procedure to select the spheres constituting the neighborhood NN(p) can have greater influence on both the numerical values and qualitative trend of q l than a change of the physical parameters, such as packing fraction. Second, the discrete nature of neighborhood implies that NN(p) is not a continuous function of the particle coordinates; this discontinuity, inherited by q l , leads to a lack of robustness of the q l as structure metrics. Both issues can be avoided by a morphometric approach leading to the robust Minkowski structure metrics q ′ l . These q ′ l are of a similar mathematical form as the conventional bond-orientational order parameters and are mathematically equivalent to the recently introduced Minkowski tensors [Europhys. Lett. 90, 34001 (2010); Phys. Rev. E. 85, 030301 (2012)].

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Section: Resultssupporting

confidence: 61%

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

Mickel¹,

Kapfer²,

Schröder‐Turk³

et al. 2013

The Journal of Chemical Physics

Self Cite

250

236

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“… have also used the other Minkowski functionals to characterize the structure of trabecular bone. However, while the scalar functionals are rotation invariant, the Minkowski tensors provide anisotropy indices that are explicitly designed to quantify anisotropy, in a more robust fashion than the MIL analysis.…”

Section: Alternative Anisotropy Measures Based On Minkowski Tensorsmentioning

confidence: 99%

“…57 The Euler characteristic has already been widely applied in medical physics to characterize the connectivity of trabecular bone, 4,28,58,59 and R€ ath et al 60 have also used the other Minkowski functionals to characterize the structure of trabecular bone. However, while the scalar functionals are rotation invariant, the Minkowski tensors provide anisotropy indices that are explicitly designed to quantify anisotropy, 14,15,61 in a more robust fashion than the MIL analysis.…”

Section: A Definition and Applicationsmentioning

confidence: 99%

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

2017

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Our analysis reveals several shortcomings of the mean intercept length tensor analysis that pose conceptual problems and limitations on the information content of this commonly used analysis method. We suggest the Minkowski tensors from integral geometry as alternative sensitive measures of anisotropy. The Minkowski tensors allow for a robust, comprehensive, and systematic approach to quantify various aspects of structural anisotropy. We show the Minkowski tensors to be more sensitive, in the sense, that they can quantify the remnant anisotropy of structures not captured by the mean intercept length analysis. If applied to porous tissue and microstructures, this improved structure characterization can yield new insights into the relationships between geometry and material properties.

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“…Figure 2 shows the unit cells of the primitive-IPC, IWP-IPC, neovius-IPC, gyroid-IPC, and diamond-IPC which have cubic symmetry. Hence, their coefficients of thermal expansion are isotropic [49,50]. On the other hand, the CTE of the CLP-IPC is transversely-isotropic since it has tetragonal symmetry [50,51].…”

Section: Architecturesmentioning

confidence: 99%

Micromechanical finite element predictions of a reduced coefficient of thermal expansion for 3D periodic architectured interpenetrating phase composites

Abueidda

Dalaq

Al‐Rub

et al. 2015

Composite Structures

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Tensorial Minkowski functionals of triply periodic minimal surfaces

Cited by 22 publications

References 100 publications

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

Micromechanical finite element predictions of a reduced coefficient of thermal expansion for 3D periodic architectured interpenetrating phase composites

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