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

Bunder, J. E.; Kevrekidis, Ioannis G.; Roberts, Anthony John

doi:10.1007/s00211-021-01232-5

Cited by 7 publications

(5 citation statements)

References 42 publications

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“…These results and general proofs were first done for homogeneous systems (e.g., Roberts & Kevrekidis 2007, Roberts et al 2014. They were subsequently extended to heterogeneous microscales (Bunder et al 2017), and recently extended to alternative inter-patch coupling that preserves self-adjointness (Bunder et al 2021). Interestingly, the extension of the theoretical support to heterogeneous cases invokes the ensemble of all phase-shifts of the heterogeneity.…”

Section: Mathematical Analysis Proves Consistencymentioning

confidence: 91%

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Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches

Roberts¹,

Tran‐Duc²,

Bunder³

et al. 2023

Preprint

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Modern 'smart' materials have complex microscale structure, often with unknown macroscale closure. The Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately simulate over large scales through computations on only small well-separated patches of the microscale system. Here the microscale system is a solid beam of random heterogeneous elasticity. The continuing challenge is to compute the given physics on just the microscale patches, and couple the patches across un-simulated macroscale space, in order to establish efficiency, accuracy, consistency, and stability on the macroscale. Dynamical systems theory supports the scheme. This research program is to develop a systematic non-intrusive approach, both computationally and analytically proven, to model and compute accurately macroscale system levels of general complex physical and engineering systems. Contents

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Section: Mathematical Analysis Proves Consistencymentioning

confidence: 91%

“…Mostly, the published proofs explicitly address dissipative (nonlinear) systems. However, as discussed by Bunder et al (2021), the patch scheme in space only recasts spatial interactions, so whether the time derivative is ∂/∂t of dissipation or ∂ 2 /∂t 2 of waves makes little difference.…”

Section: Mathematical Analysis Proves Consistencymentioning

confidence: 99%

Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches

Roberts¹,

Tran‐Duc²,

Bunder³

et al. 2023

Preprint

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“…In the equation-free multiscale framework, a patch scheme performs detailed microscale simulations within small, widely separated, patches of space (e.g., the small violet squares enclosing green grid in Figure 3a), and couples the patches (patch coupling) via interpolation over the macroscale space between the patches (Hyman 2005;Kevrekidis et al 2004;Kevrekidis and Samaey 2009). One can achieve arbitrarily high orders of macroscale consistency for patch schemes via appropriate high order interpolation for the patch coupling (Bunder et al 2021;Roberts andKevrekidis 2005, 2007). For development and analysis of the staggered patch grids, throughout this article we use spectral interpolation (Bunder et al 2020, §2.2.3), via Fourier transforms, to couple patches.…”

Section: Staggered Patch Grids For Equation-free Multiscale Modelling...mentioning

confidence: 99%

Staggered grids for multidimensional multiscale modelling

Divahar¹,

Roberts²,

Mattner³

et al. 2022

Preprint

Self Cite

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Numerical schemes for wave-like systems with small dissipation are often inaccurate and unstable due to truncation errors and numerical roundoff errors. Hence, numerical simulations of wave-like systems lacking proper handling of these numerical issues often fail to represent the physical characteristics of wave phenomena. This challenge gets even more intricate for multiscale modelling, especially in multiple dimensions. When using the usual collocated grid, about two-thirds of the resolved wave modes are incorrect with significant dispersion. But, numerical schemes on staggered grids (with alternating variable arrangement) are significantly less dispersive and preserve much of the wave characteristics. Also, the group velocity of the energy propagation in the numerical waves on a staggered grid is in the correct direction, in contrast to the collocated grid. For high accuracy and to preserve much of the wave characteristics, this article extends the concept of staggered grids in full-domain modelling to multidimensional multiscale modelling. Specifically, this article develops

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“…Based on the assumption of the appropriateness of such a macro-scale field behavior, artificial boundary conditions for each tooth have been devised for the meaningfully coupled simulation of the micro-scale problem across all teeth [e.g. 4,40]. These teeth-edge conditions constitute an important ingredient of the gap-tooth scheme, as they couple the dynamics of adjacent teeth and allow the global pattern of the solution to emerge from the simulations in (seemingly) separate teeth.…”

Section: Equation-free Approach For Homogenizationmentioning

confidence: 99%

“…In this example [7,4], we consider diffusion on a two-dimensional periodic domain with latticetype micro-scale heterogeneity. We define a two-dimensional lattice, indexed by (i, j), with the same spacing h in both directions on the bi-periodic domain [0, 2π) 2 .…”

Section: Diffusion In Two Dimensions With Lattice Heterogeneitymentioning

confidence: 99%

Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations

Arbabi,

Bunder,

Samaey

et al. 2020

Preprint

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The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential operators requires extensive spatiotemporal data. For learning coarse-scale PDEs from computational fine-scale simulation data, the training data collection process can be prohibitively expensive. We propose to transformatively facilitate this training data collection process by linking machine learning (here, neural networks) with modern multiscale scientific computation (here, equation-free numerics). These equation-free techniques operate over sparse collections of small, appropriately coupled, space-time subdomains ("patches"), parsimoniously producing the required macro-scale training data. Our illustrative example involves the discovery of effective homogenized equations in one and two dimensions, for problems with fine-scale material property variations. The approach holds promise towards making the discovery of accurate, macro-scale effective materials PDE models possible by efficiently summarizing the physics embodied in "the best" fine-scale simulation models available.

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

Cited by 7 publications

References 42 publications

Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches

Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches

Staggered grids for multidimensional multiscale modelling

Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations

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