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Register a new userBoundary conditions for macroscale waves in an elastic system with microscale heterogeneity
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Multiscale 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 problems.
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Multiscale modelling methodologies build macroscale models of materials with complicated fine microscale structure. We propose a methodology to derive boundary conditions for the macroscale model of a prototypical non-linear heat exchanger. The derived macroscale boundary conditions improve the accuracy of macroscale model. We verify the new boundary conditions by numerical methods. The techniques developed here can be adapted to a wide range of multiscale reaction-diffusion-advection systems.
We consider one dimensional lattice diffusion model on a microscale grid with many discrete diffusivity values which repeat periodicially. Computer algebra explores how the dynamics of small coupled `patches predict the slow emergent macroscale dynamics. We optimise the geometry and coupling of patches by comparing the macroscale predictions of the patch solutions with the macroscale solution on the infinite domain, which is derived for a general diffusivity period. The results indicate that patch dynamics is a viable method for numerical macroscale modelling of microscale systems with fine scale roughness. Moreover, the minimal error on the macroscale is generally obtained by coupling patches via `buffers that are as large as half of each patch.
Many physical systems are formulated on domains which are relatively large in some directions but relatively thin in other directions. We expect such systems to have emergent structures that vary slowly over the large dimensions. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the `large and `thin dimensions becomes nonphysically infinite. Here we extend to nonlinear dynamics a new approach [IMA J. Appl. Maths, DOI: 10.1093/imamat/hxx021] which analyses the dynamics at each cross-section of the domain via a rigorous multivariate Taylor series. Then centre manifold theory supports the global modelling of the systems emergent dynamics in the large but finite domain. Interactions between the cross-section coupling and both fast and slow dynamics determines quantitative error bounds for the nonlinear modelling. We illustrate the methodology by deriving the large-scale dynamics of a thin liquid film, where the film is subject to a Coriolis force induced by a rotating substrate. The approach developed here quantifies the accuracy of known approximations, extends such approximations to mixed order modelling, and may open previously intractable modelling issues to new tools and insights.
Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario we typically expect the system to have emergent structures that vary slowly over the large dimensions. For practical mathematical modelling of such systems we require efficient and accurate methodologies for reducing the dimension of the original system and extracting the emergent dynamics. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the large and thin dimensions becomes unphysically infinite. Here we build on a new approach, previously establish for systems which are large in only one dimension, which analyses the dynamics at each cross-section of the domain with a rigorous multivariate Taylor series. Then centre manifold theory supports the local modelling of the systems emergent dynamics with coupling to neighbouring cross-sections treated as a non-autonomous forcing. The union over all cross-sections then provides powerful support for the existence and emergence of a centre manifold model global in the large finite domain. Quantitative error estimates are determined from the interactions between the cross-section coupling and both fast and slow dynamics. Two examples provide practical details of our methodology. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling, and to open previously intractable modelling issues to new tools and insights.
Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given as the relaxation parameter goes to zero. Finally, we also establish the existence of exponential attractors.
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