35 research outputs found
Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations
No full textCoarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial dif- ferential/difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements empowers the approach to re- solve significant subgrid microscale structures and interactions between neighbouring ele- ments. The crucial aspect of this approach is that centre manifold theory organises the res- olution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements. The techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential/difference equations, such as the forced Burgers’ equation, adopted here as an illustrative example.J.E. Bunder, A.J. Robert
Staggered grids for multidimensional multiscale modelling
No full textAvailable online 10 January 2024For high accuracy and to improve simulated wave characteristics, this article extends the concept of staggered grids to novel multidimensional multiscale modelling enabling efficient computation on sparse patches. Computational schemes for wave-like systems with small dissipation are often inaccurate and unstable due to truncation errors and numerical roundoff errors. Hence 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. But numerical schemes on staggered grids are significantly less dispersive, better model the group velocity, and preserve much of the wave characteristics. This article develops and exhaustively studies all 167 040 possible 2D multiscale staggered grids. Our catalog (Divahar, 2023) interactively plots all of them. Only 120 multiscale staggered patch grids give stable and accurate multiscale schemes. Specifically, this article develops these 120 multiscale staggered grids and demonstrates their stability, accuracy, and wave-preserving characteristic for equation-free multiscale modelling of weakly damped linear waves. These characteristics of the developed multiscale staggered grids must also hold in general for multiscale modelling of many complex spatio-temporal physical phenomena such as the general computational fluid dynamics.J. Divahar, A.J. Roberts, Trent W. Mattner, J.E. Bunder, Ioannis G. Kevrekidi
Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves
No full textLink to a related website: https://unpaywall.org/10.1016/j.cma.2023.116133, Open Access via UnpaywallMany multiscale wave systems exhibit macroscale emergent behaviour, for example, the fluid dynamics of floods and tsunamis. Resolving a large range of spatial scales typically requires prohibitively high computational costs. The small dissipation in wave systems poses a significant challenge to further developing multiscale modelling methods in multiple dimensions. This article develops and evaluates two families of equation-free multiscale methods on novel 2D staggered patch schemes, and demonstrates the power and utility of these multiscale schemes for weakly damped linear waves. A detailed study of sensitivity to numerical roundoff errors establishes the robustness of developed staggered patch schemes. Comprehensive eigenvalue analysis over a wide range of parameters establishes the stability, accuracy, and consistency of the multiscale schemes. Analysis of the computational complexity shows that the measured compute times of the multiscale schemes may be 100,000 times smaller than the compute time for the corresponding (same resolution) full-domain computation. This work provides the essential foundation for efficient large-scale simulation of challenging nonlinear multiscale waves.J. Divahar, A.J. Roberts, Trent W. Mattner, J.E. Bunder, Ioannis G. Kevrekidi
A toolbox of equation-free functions in Matlab/Octave for efficient system level simulation
No full textPublished 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
Nonlinear emergent macroscale PDEs, with error bound, for nonlinear microscale systems
No full textAbstract Many multiscale physical scenarios have a spatial domain which is large in some dimensions but relatively thin in other dimensions. These scenarios includes homogenization problems where microscale heterogeneity is effectively a ‘thin dimension’. In such scenarios, slowly varying, pattern forming, emergent structures typically dominate the large dimensions. Common modelling approximations of the emergent dynamics usually rely on self-consistency arguments or on a nonphysical mathematical limit of an infinite aspect ratio of the large and thin dimensions. Instead, here we extend to nonlinear dynamics a new modelling approach which analyses the dynamics at each cross-section of the domain via a multivariate Taylor series (Roberts and Bunder in IMA J Appl Math 82(5):971–1012, 2017. https://doi.org/10.1093/imamat/hxx021 ). Centre manifold theory extends the analysis at individual cross-sections to a rigorous global model of the system’s emergent dynamics in the large but finite domain. A new remainder term quantifies the error of the nonlinear modelling and is expressed in terms of the interaction between cross-sections and the fast and slow dynamics. We illustrate the rigorous approach by deriving the large-scale nonlinear dynamics of a thin liquid film on a rotating substrate. The approach developed here empowers new mathematical and physical insight and new computational simulations of previously intractable nonlinear multiscale problems
Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling
No full textEquation-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
Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches
No full textModern `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.A. J. Roberts, Thein Tran-Duc, J. E. Bunder, I. G. Kevrekidi
Efficient prediction of static and dynamical responses of functional graded beams using sparse multiscale patches
No full textPublished online: 19 March 2025We develop a multiscale patch scheme for studying the system level characteristics of heterogeneous functional graded beams in 3D via accurate computational homogenisation. The algorithm is an extension of our previous work for 2D beams (Tran-Duc et al. in Int. J. Solids Struct. 292:112719, 2024) to explore out-of-plane dynamics of 3D beams of functional graded materials. The scheme computes the detailed microscale elastic equations only in sparsely spaced, small patches of the domain (akin to FE²), and via symmetry-preserving interpolation between these patches. We develop new applications of the scheme to two classes of functionally graded beams, namely cross-sectionally graded and axially graded. Our approach accurately and provably predicts the macroscale system-wide behaviour. Beam deflection and natural frequencies from the patch computations agree very well with both existing experimental data and the full-domain computations, which provides a new validation of the approach and a new characterisation of the interaction between bending and twisting in graduated beams. The scheme is stable and robust, with errors consistently small and controllable by varying the number of patches. The reduction in the spatial domain of computation substantially improves the computational efficiency, with the computational time reducing by a factor of up to 17 when the patches cover 27% of the beam. The scheme also accurately predicts the homogenised dynamics of periodic micro-structured materials, such as metamaterials, by simply ensuring patches are a multiple of the micro-period. Localised phenomena, such as material failures or cracks or boundary layers, may also be accurately encompassed by fully resolving them within a patch.Thien Tran-Duc, J. E. Bunder, A. J. Robert
Good coupling for the multiscale patch scheme on systems with microscale heterogeneity
No full textAvailable online 16 February 2017Computational simulation of microscale detailed systems is frequently only feasible over spatial domains much smaller than the macroscale of interest. The ‘equation-free’ methodology couples many small patches of microscale computations across space to empower efficient computational simulation over macroscale domains of interest. Motivated by molecular or agent simulations, we analyse the performance of various coupling schemes for patches when the microscale is inherently ‘rough’. As a canonical problem in this universality class, we systematically analyse the case of heterogeneous diffusion on a lattice. Computer algebra explores how the dynamics of coupled patches predict the large scale emergent macroscale dynamics of the computational scheme. We determine good design for the coupling of patches by comparing the macroscale predictions from patch dynamics with the emergent macroscale on the entire domain, thus minimising the computational error of the multiscale modelling. The minimal error on the macroscale is obtained when the coupling utilises averaging regions which are between a third and a half of the patch. Moreover, when the symmetry of the inter-patch coupling matches that of the underlying microscale structure, patch dynamics predicts the desired macroscale dynamics to any specified order of error. The results confirm that the patch scheme is useful for macroscale computational simulation of a range of systems with microscale heterogeneity.J.E. Bunder, A.J. Roberts, I.G. Kevrekidi
Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation
No full textThe 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
