scholarly journals The Roles of Random Boundary Conditions in Spin Systems

2020 ◽  
pp. 371-381
Author(s):  
Eric O. Endo ◽  
Aernout C.  D. van Enter ◽  
Arnaud Le Ny
Keyword(s):  
Boundary Conditions ◽  
Spin Systems ◽  
Random Boundary

Extreme-Value-Based Statistical Bounding of Low, Mid, and High Frequency Responses of a Forced Plate With Random Boundary Conditions

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
A. Seçgin ◽  
J. F. Dunne ◽  
L. Zoghaib
Keyword(s):  
Monte Carlo ◽  
Energy Density ◽  
Boundary Conditions ◽  
Total Energy ◽  
High Frequency ◽  
Extreme Value ◽  
Frequency Range ◽  
Random Boundary ◽  
Total Energy Density ◽  
Dsc Method

The problem of statistically bounding the response of an engineering structure with random boundary conditions is addressed across the entire frequency range: from the low, through the mid, to the high frequency region. Extreme-value-based bounding of both the FRF and the energy density response is examined for a rectangular linear plate with harmonic point forcing. The proposed extreme-value (EV) approach, previously tested only in the low frequency region for uncoupled and acoustically-coupled uncertain structures, is examined here in the mid and high frequency regions in addition to testing at low frequencies. EV-based bounding uses an asymptotic threshold exceedance model of Type-I, to extrapolate the m-observational return period to an arbitrarily-large batch of structures. It does this by repeatedly calibrating the threshold model at discrete frequencies using a small sample of response data generated by Monte Carlo simulation or measurement. Here the discrete singular convolution (DSC) method – a transfrequency computation approach for deterministic vibration - is used to generate Monte Carlo samples. The accuracy of the DSC method is first verified (i) in terms of the spatial distribution of total energy density and (ii) across the frequency range, by comparison with a mode superposition method and Statistical Energy Analysis (SEA). EV-based bound extrapolations of the receptance FRF and total energy density are then compared with: (i) directly-estimated bounds using a full set of Monte Carlo simulations and (ii) with total mean energy levels obtained with SEA. This paper shows that for a rectangular plate structure with random boundary conditions, EV-based statistical bounding of both the FRF and total energy density response is generally applicable across the entire frequency range.

scholarly journals A comparative study of 2d Ising model at different boundary conditions using Cellular Automata

2018 ◽  
Vol 29 (08) ◽  
pp. 1850066
Author(s):  
Jahangir Mohammed ◽  
Swapna Mahapatra
Keyword(s):  
Ising Model ◽  
Cellular Automata ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Spontaneous Magnetization ◽  
Spin Systems ◽  
Square Lattice ◽  
Random Orientation ◽  
Lattice Size ◽  
Different Types

Using Cellular Automata, we simulate spin systems corresponding to [Formula: see text] Ising model with various kinds of boundary conditions (bcs). The appearance of spontaneous magnetization in the absence of magnetic field is studied with a [Formula: see text] square lattice with five different bcs, i.e. periodic, adiabatic, reflexive, fixed ([Formula: see text] or [Formula: see text]) bcs with three initial conditions (all spins up, all spins down and random orientation of spins). In the context of [Formula: see text] Ising model, we have calculated the magnetization, energy, specific heat, susceptibility and entropy with each of the bcs and observed that the phase transition occurs around [Formula: see text] as obtained by Onsager. We compare the behavior of magnetization versus temperature for different types of bcs by calculating the number of points close to the line of zero magnetization after [Formula: see text] at various lattice sizes. We observe that the periodic, adiabatic and reflexive bcs give closer approximation to the value of [Formula: see text] than fixed [Formula: see text] and fixed [Formula: see text] bcs with all three initial conditions for lattice sizes less than [Formula: see text]. However, for lattice size between [Formula: see text] and [Formula: see text], fixed [Formula: see text] bc and fixed [Formula: see text] bc give closer approximation to the [Formula: see text] with initial conditions all spin down configuration and all spin up configuration, respectively.

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