| 1. |
- Faouzi, Tarik, et al.
(författare)
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A deep look into the Dagum family of isotropic covariance functions
- 2022
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Ingår i: Journal of Applied Probability. - Cambridge : Cambridge University Press. - 0021-9002 .- 1475-6072. ; 59:4, s. 1026-1041
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Tidskriftsartikel (refereegranskat)abstract
- The Dagum family of isotropic covariance functions has two parameters that allow fordecoupling of the fractal dimension and the Hurst effect for Gaussian random fields thatare stationary and isotropic over Euclidean spaces. Sufficient conditions that allow forpositive definiteness in $R^d$ of the Dagum family have been proposed on the basis ofthe fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a verylimited extent only, and this paper gives insight into this direction. Specifically, we studyfinite and asymptotic properties of the isotropic spectral density (intended as the Hankeltransform) of the Dagum model. Also, we establish some closed-form expressions forthe Dagum spectral density in terms of the Fox–Wright functions. Finally, we provideasymptotic properties for such a class of spectral densities.
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| 2. |
- Zhang, Xian, et al.
(författare)
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Elastodynamic problem on tensor random fields with fractal and Hurst effects
- 2022
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Ingår i: Meccanica (Milano. Print). - : Springer Science+Business Media B.V.. - 0025-6455 .- 1572-9648. ; 57:4, s. 957-970
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Tidskriftsartikel (refereegranskat)abstract
- This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.
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