Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale system at a desired macroscale system level. In this scheme, the given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate simulations, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve translational invariance, rotational invariance, and self-adjoint symmetry, thus guaranteeing that conservation laws associated with these symmetries are preserved in the macroscale simulation. Spectral and algebraic analyses of the proposed scheme in both one and two dimensions reveal mechanisms for further improving the accuracy of the simulations. Consistency of the patch scheme's macroscale dynamics with the original microscale model is proved. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation in a wide range of multiscale scenarios of interest to scientists and engineers

Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale complex system at a desired macroscale system level. In this scheme, a given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate predictions, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve self-adjoint symmetry, thus guaranteeing that the macroscale model maintains some important conservation laws of the original microscale model. Consistency of the patch scheme’s macroscale dynamics with the original microscale model is proved for systems in 1D and 2D space, and these proofs immediately extend to higher dimensions. Expanding from a system with a single configuration to an ensemble of configurations establishes that the proven consistency also holds for cases where the microscale periodicity does not integrally fill the patches. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation, as demonstrated here with canonical examples in 1D and 2D space based on heterogenous diffusion, and is applicable to a wide range of multiscale scenarios of interest to scientists and engineers.J. E. Bunder, I. G. Kevrekidis and A. J. Robert

Thermodynamic Properties of the One-Dimensional Extended Quantum Compass Model in the Presence of a Transverse Field

The presence of a quantum critical point can significantly affect the thermodynamic properties of a material at finite temperatures. This is reflected, e.g., in the entropy landscape S(T; c) in the vicinity of a quantum critical point, yielding particularly strong variations for varying the tuning parameter c such as magnetic field. In this work we have studied the thermodynamic properties of the quantum compass model in the presence of a transverse field. The specific heat, entropy and cooling rate under an adiabatic demagnetization process have been calculated. During an adiabatic (de)magnetization process temperature drops in the vicinity of a field-induced zero-temperature quantum phase transitions. However close to field-induced quantum phase transitions we observe a large magnetocaloric effect

Study of Loschmidt Echo for a qubit coupled to an XY-spin chain environment

We study the temporal evolution of a central spin-1/2 (qubit) coupled to the environment which is chosen to be a spin-1/2 transverse XY spin chain. We explore the entire phase diagram of the spin-Hamiltonian and investigate the behavior of Loschmidt echo(LE) close to critical and multicritical point(MCP). To achieve this, the qubit is coupled to the spin chain through the anisotropy term as well as one of the interaction terms. Our study reveals that the echo has a faster decay with the system size (in the short time limit) close to a MCP and also the scaling obeyed by the quasiperiod of the collapse and revival of the LE is different in comparison to that close to a QCP. We also show that even when approached along the gapless critical line, the scaling of the LE is determined by the MCP where the energy gap shows a faster decay with the system size. This claim is verified by studying the short-time and also the collapse and revival behavior of the LE at a quasicritical point on the ferromagnetic side of the MCP. We also connect our observation to the decoherence of the central spin.Comment: Accepted for publication in EPJ

Quantum Correlation in One-dimensional Extend Quantum Compass Model

We study the correlations in the one-dimensional extended quantum compass model in a transverse magnetic field. By exactly solving the Hamiltonian, we find that the quantum correlation of the ground state of one-dimensional quantum compass model is vanishing. We show that quantum discord can not only locate the quantum critical points, but also discern the orders of phase transitions. Furthermore, entanglement quantified by concurrence is also compared.Comment: 8 pages, 14 figures, to appear in Eur. Phys. J.

A toolbox of equation-free functions in Matlab/Octave for efficient system level simulation

Published online: 30 October 2020The ‘equation-free toolbox’ empowers the computer-assisted analysis of complex, multiscale systems. Its aim is to enable scientists and engineers to immediately use microscopic simulators to perform macro-scale system level tasks and analysis, because micro-scale simulations are often the best available description of a system. The methodology bypasses the derivation of macroscopic evolution equations by computing the micro-scale simulator only over short bursts in time on small patches in space, with bursts and patches well-separated in time and space respectively. We introduce the suite of coded equation-free functions in an accessible way, link to more detailed descriptions, discuss their mathematical support, and introduce a novel and efficient algorithm for Projective Integration. Some facets of toolbox development of equation-free functions are then detailed. Download the toolbox functions and use to empower efficient and accurate simulation in a wide range of science and engineering problems.John Maclean, J. E. Bunder and A. J. Robert

Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity

Advance Access Publication on 19 March 2018Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially resulting in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problemsChen Chen, A. J. Roberts and J. E. Bunde

Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation

The equation-free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction---the patch dynamics scheme. A recent novel patch scheme empowered the equation-free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the equation-free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming, and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the subpatch dynamics.John Maclean, J. E. Bunder, I. G. Kevrekidis, and A. J. Robert

Lyapunov exponents of the Kuramoto-Sivashinksy PDE

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.Russell A. Edson, J.E. Bunder, Trent W. Mattner and A.J. Robert

An Equation Free algorithm accurately simulates macroscale shocks arising from heterogeneous microscale systems

Scientists and engineers often create accurate, trustworthy, computational simulation schemes—but all too often these are too computationally expensive to execute over the time or spatial domain of interest. The equation-free approach is to marry such trusted simulations to a framework for numerical macroscale reduction—the patch dynamics scheme. This article extends the patch scheme to scenarios in which the trusted simulation resolves abrupt state changes on the microscale that appear as shocks on the macroscale. Accurate simulation for problems in these scenarios requires capturing the shock within a novel patch, and also modifying the patch coupling rules in the vicinity in order to maintain accuracy. With these two extensions to the patch scheme, straightforward arguments derive consistency conditions that match the usual order of accuracy for patch schemes. The new scheme is successfully tested to simulate a heterogeneous microscale partial differential equation.This technique willempower scientists and engineers to accurately and efficiently simulate, over large spatial domains, multiscale multiphysics systems that have rapid transition layers on the microscale.John Maclean, Judith E. Bunder, Ioannis G. Kevrekidis, and Anthony J. Robert