McLaren’s improved snub cube and other new spherical designs in three dimensions

Hardin, R. H.; Sloane, N. J. A.

doi:10.1007/bf02711518

Cited by 289 publications

(245 citation statements)

References 24 publications

(47 reference statements)

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“…(A.1) is equal to zero. By means of a computer search, Hardin and Sloane (1996) found spherical designs for all t 13 with a minimal number of points; for instance, they produced a (94, 13)-design. A variety of algorithms have also been proposed for explicitly constructing asymptotically uniform distributions of points on S 2 .…”

Section: Resultsmentioning

confidence: 99%

Earth magnetic field modeling from Oersted and Champ data

2010

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We present a method to model geomagnetic field requiring only a restricted number of measurements on magnetic survey satellite orbits. These points are chosen in an optimal-or close to optimal-manner relying on recent developments in the problem of numerical integration over spheres. The method allows us to compute a series of models at short time intervals, namely 10 days in the present study. At each of these close dates several models are computed from independent sets of data; their redundancy in turn provides a control of results thanks to which the selection of data-for example, as a function of magnetic activity or latitude-may be reduced. We find that the internal low degree Gauss coefficients derived from Oersted and Champ data, respectively, differ from one another by 1 or 2 nT. We then take as a second example of the method application a brief study of the so-called external field. We compare the first-degree axisymmetric field with the D st index.

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Section: Resultsmentioning

confidence: 99%

Earth magnetic field modeling from Oersted and Champ data

2010

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“…It is a 2-design in RP 2 but not a 3-design. However, a 16-point 3-design exists: it is equivalent to an antipodal 32-point spherical 7-design in S 2 , and such a design is constructed in Section 4 of [HS96]. It follows that the 16-line configuration is not universally optimal, and hence the list in Theorem 2 is complete.…”

Section: Appendix a The Orthoplex Boundmentioning

confidence: 99%

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2012

J. Amer. Math. Soc.

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“…Therefore, an increase of the integration directions leads to more accurate results. Another possible choice would be to consider another integration technique such as that used by Hardin and Sloane (1996) or the non-linear transformation, proposed by Alastrué et al (2009b), in order to adjust the distribution of the integration directions to the statistical distribution function. The discretization and associated peaks may be considered as lacking accuracy.…”

Section: Micro Mechanics Of the Tissuementioning

confidence: 99%

Anisotropic microsphere-based approach to damage in soft fibered tissue

Saéz

Alastrué²,

Peña

et al. 2011

Biomech Model Mechanobiol

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An anisotropic damage model for soft fibered tissue is presented in this paper, using a multi scale scheme and focusing on the directionally-dependent behavior of these materials. For this purpose, a microstructural or, more precisely, a microsphere-based approach is used to model the contribution of the fibers. The link between micro-structural contribution and macroscopic response is achieved by means of computational homogenization, involving numerical integration over the surface of the unit sphere. In order to deal with the distribution of the fibrils within the fiber, a von Mises probability function is incorporated, and the mechanical (phenomenological) behavior of the fibrils is defined by an exponential-type model. We will restrict ourselves to affine deformations of the network, neglecting any cross-link between fibrils and sliding between fibers and the surrounding ground matrix. Damage in the fiber bundles is introduced through a thermodynamic formulation, which is directly included in the hyperelastic model. When the fibers are stretched far from their natural state, they become damaged. The damage increases gradually due to the progressive failure of the fibrils that make up such a structure. This model has been implemented in a finite element code, and different boundary value problems are solved and

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McLaren’s improved snub cube and other new spherical designs in three dimensions

Abstract: Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N = 24, 26, ≥ 28; 7-designs for N = 24, 30, 32, 34, ≥ 36; 8-

Cited by 289 publications

References 24 publications

Earth magnetic field modeling from Oersted and Champ data

Earth magnetic field modeling from Oersted and Champ data

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Anisotropic microsphere-based approach to damage in soft fibered tissue

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