Plastic Flow of Random Media: Micromechanics, Markov Property and Slip-Lines

1992 ◽  
Vol 45 (3S) ◽  
pp. S75-S81 ◽  
Author(s):  
Martin Ostoja-Starzewski
Keyword(s):  
Random Media ◽  
Classical Method ◽  
Mean Field ◽  
Markov Property ◽  
Continuum Approximation ◽  
Field Equations ◽  
Slip Lines ◽  
Perfectly Plastic ◽  
Stochastic Solution ◽  
Difference Form

The classical method of slip-lines (characteristics) of planar flow of perfectly-plastic media is generalized to a stochastic setting. The media are charaterized by space-homogeneous statistics of the yield limit k, whose derivation is outlined on the basis of micromechanics. The field equations of the random continuum approximation lead to a stochastic hyperbolic system. This system, when stated in a finite difference form, displays a Markov property for the forward evolution. On that basis, two methods of solution of boundary value problems - an exact one and a mean-field one - are outlined through an example of a Cauchy problem. The principal observation is that even for a weak material randomness the stochastic solution may differ qualitatively from that of a homogeneous deterministic medium and have a strong scatter.

Micromechanics as a Basis of Stochastic Finite Elements and Differences: An Overview

1993 ◽  
Vol 46 (11S) ◽  
pp. S136-S147 ◽  
Author(s):  
M. Ostoja-Starzewski
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Classical Method ◽  
Perfectly Plastic ◽  
Plastic Materials ◽  
Solution Methods ◽  
Stochastic Solution ◽  
Stochastic Finite Elements

A generalization of conventional deterministic finite element and difference methods to deal with spatial material fluctuations hinges on the problem of determination of stochastic constitutive laws. This problem is analyzed here through a paradigm of micromechanics of elastic polycrystals and matrix-inclusion composites. Passage to a sought-for random meso-continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that the microstructure cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields may be introduced to bound the material response from above and from below. Since the RVE corresponds to a single finite element, or finite difference cell, not infinitely larger than the crystal size, these two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The window-based random continuum formulation is also employed in analysis of rigid perfectly-plastic materials, whereby the classical method of slip-lines is generalized to a stochastic finite difference scheme. The present paper is complemented by a comparison of this methodology to other existing stochastic solution methods.

Precipitation in small systems—II. Mean field equations more effective than population balance

1996 ◽  
Vol 51 (19) ◽  
pp. 4423-4436 ◽  
Author(s):  
S. Manjunath ◽  
K.S. Gandhi ◽  
R. Kumar ◽  
Doraiswami Ramkrishna
Keyword(s):  
Mean Field ◽  
Population Balance ◽  
Field Equations ◽  
Small Systems ◽  
Mean Field Equations

Solution of the mean field equations for spontaneous fission

1987 ◽  
Vol 35 (3) ◽  
pp. 1007-1027 ◽  
Author(s):  
G. Puddu ◽  
J. W. Negele
Keyword(s):  
Spontaneous Fission ◽  
Mean Field ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean

scholarly journals Population dynamics with spatial structure and an Allee effect

2020 ◽  
Author(s):  
Anudeep Surendran ◽  
Michael Plank ◽  
Matthew Simpson
Keyword(s):  
Population Dynamics ◽  
Spatial Structure ◽  
Short Range ◽  
Allee Effect ◽  
Mean Field ◽  
Mean Field Approximation ◽  
Continuum Approximation ◽  
Spatial Moment ◽  
Mean Field Models ◽  
The Mean

AbstractAllee effects describe populations in which long-term survival is only possible if the population density is above some threshold level. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to population extinction, whereas initial densities above the threshold eventually asymptote to some positive carrying capacity density. Mean field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as short-range competition and dispersal. The influence of such non mean-field effects has not been studied in the presence of an Allee effect. To address this we develop an individual-based model (IBM) that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework accurately recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure that mean-field models neglect. For example, we show that there are cases where the mean-field model predicts extinction but the population actually survives and vice versa. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

scholarly journals OLD PROBLEMS, CLASSICAL METHODS, NEW SOLUTIONS

2020 ◽  
Vol 23 (04) ◽  
pp. 2050024
Author(s):  
ALEXANDER LIPTON
Keyword(s):  
Classical Method ◽  
Mean Field ◽  
Trading Strategies ◽  
Financial Mathematics ◽  
Uhlenbeck Process ◽  
Put Options ◽  
Default Intensity ◽  
American Put Options ◽  
The Mean ◽  
American Put

We use a powerful extension of the classical method of heat potentials, recently developed by the present author and his collaborators, to solve several significant problems of financial mathematics. We consider the following problems in detail: (a) calibrating the default boundary in the structural default framework to a constant default intensity; (b) calculating default probability for a representative bank in the mean-field framework; and (c) finding the hitting time probability density of an Ornstein–Uhlenbeck process. Several other problems, including pricing American put options and finding optimal mean-reverting trading strategies, are mentioned in passing. Besides, two nonfinancial applications — the supercooled Stefan problem and the integrate-and-fire neuroscience problem — are briefly discussed as well.

Mean field equations with probability measure in 2D-turbulence

2014 ◽  
Vol 63 (S1) ◽  
pp. 255-264 ◽  
Author(s):  
Tonia Ricciardi ◽  
Gabriella Zecca
Keyword(s):  
Probability Measure ◽  
Mean Field ◽  
Field Equations ◽  
2D Turbulence ◽  
Mean Field Equations

scholarly journals Distributed Control of Clustered Populations of Thermostatic Loads in Multi-Area Systems: A Mean Field Game Approach

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6483
Author(s):  
Vincenzo Trovato ◽  
Antonio De Paola ◽  
Goran Strbac
Keyword(s):  
Power System ◽  
Frequency Response ◽  
Mean Field ◽  
Cost Effective ◽  
Support Network ◽  
Field Equations ◽  
Mean Field Game ◽  
Flexible Resources ◽  
Mean Field Equations ◽  
The Impact

Thermostatically controlled loads (TCLs) can effectively support network operation through their intrinsic flexibility and play a pivotal role in delivering cost effective decarbonization. This paper proposes a scalable distributed solution for the operation of large populations of TCLs providing frequency response and performing energy arbitrage. Each TCL is described as a price-responsive rational agent that participates in an integrated energy/frequency response market and schedules its operation in order to minimize its energy costs and maximize the revenues from frequency response provision. A mean field game formulation is used to implement a compact description of the interactions between typical power system characteristics and TCLs flexibility properties. In order to accommodate the heterogeneity of the thermostatic loads into the mean field equations, the whole population of TCLs is clustered into smaller subsets of devices with similar properties, using k-means clustering techniques. This framework is applied to a multi-area power system to study the impact of network congestions and of spatial variation of flexible resources in grids with large penetration of renewable generation sources. Numerical simulations on relevant case studies allow to explicitly quantify the effect of these factors on the value of TCLs flexibility and on the overall efficiency of the power system.

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