scholarly journals Stress-weighted spatial averaging of random fields in geotechnical risk assessment

2021 ◽  
Vol 43 (4) ◽  
pp. 465-478
Author(s):  
Włodzimierz Brząkała
Keyword(s):  
Random Fields ◽  
Variance Reduction ◽  
Random Variable ◽  
Volume Averaging ◽  
Stress Fields ◽  
Soil Parameters ◽  
Spatial Averaging ◽  
Uniform Stress ◽  
Frictional Material ◽  
Spatial Fluctuations

Abstract Effects of spatial fluctuations of soil parameters are considered in a new context – considering variability of soil parameters in conjunction with non-uniform stress fields, which can locally amplify (or suppress) subsoil inhomogeneities. In this way, several design situations for the Coulomb frictional material with random tan(φ(x)) reveal a reduction of variance, which is less significant than for the standard volume averaging. When looking for an ‘effective’ random variable [tan(φ)]a – that is, a random variable, which is equivalent to the random field tan(φ( x )) – the Vanmarcke averaging by simple volume integrals is insufficient; it systematically overestimates effects of variance reduction, thus causing potentially unsafe situations. The new proposed approach is coherent, formally defined and more realistic.

Displacement threshold energy change in uniform stress fields

1986 ◽  
Vol 135 (2) ◽  
pp. K113-K117
Author(s):  
V. V. Kirsanov ◽  
E. M. Kislitsina
Keyword(s):  
Threshold Energy ◽  
Energy Change ◽  
Stress Fields ◽  
Uniform Stress ◽  
Displacement Threshold

scholarly journals Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200095
Author(s):  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Conformal Mapping ◽  
Screw Dislocation ◽  
Lamellar Structure ◽  
Stress Fields ◽  
Uniform Stress ◽  
Dislocation Dipole ◽  
Lamellar Structures ◽  
Eshelby Inclusion ◽  
Mapping Techniques ◽  
Concentrated Couple

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

scholarly journals A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

2019 ◽  
Vol 24 (1) ◽  
pp. 311-331 ◽  
Author(s):  
Prashant Kumar ◽  
Carmen Rodrigo ◽  
Francisco J. Gaspar ◽  
Cornelis W. Oosterlee
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Nonlinear Problems ◽  
Picard Iteration ◽  
Richards Equation ◽  
Porous Media Flow ◽  
Soil Parameters ◽  
Multilevel Monte Carlo ◽  
Highly Nonlinear ◽  
Non Gaussian

AbstractWe present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190407
Author(s):  
Bob Svendsen
Keyword(s):  
Angular Momentum ◽  
Constitutive Relations ◽  
Volume Averaging ◽  
Mass Point ◽  
Momentum Balance ◽  
Spatial Averaging ◽  
Material Frame Indifference ◽  
Phase Space Transport ◽  
Polar Continua ◽  
Material Frame

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

scholarly journals Extremes of Homogeneous Gaussian Random Fields

2015 ◽  
Vol 52 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Natalia Soja-Kukieła
Keyword(s):  
Correlation Function ◽  
Asymptotic Expansion ◽  
Random Fields ◽  
Survival Function ◽  
Random Variable ◽  
Gaussian Random Fields ◽  
Extremal Index ◽  
Gaussian Field ◽  
Gaussian Fields ◽  
Sample Paths

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn1(u),tn2(u)) ∈[0,x]∙[0,y]X(s, t) ≤ u), where n1(u)n2(u) = u2/α1+2/α2Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

Dislocation distributions in non-uniform stress fields

1990 ◽  
Vol 130 (2) ◽  
pp. 139-150 ◽  
Author(s):  
Xiaoyu Hu ◽  
H. Margolin
Keyword(s):  
Stress Fields ◽  
Uniform Stress
Sign in / Sign up

Export Citation Format

Share Document