The "equation-free'' approach has been proposed in recent years as a general framework for developing multiscale methods to efficiently capture the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the "equation-free'' umbrella, the projective integrators and the patch dynamics. We discuss some very simple examples in the context of the "equation-free'' approach. These examples seem to indicate that while its general philosophy is quite attractive and indeed similar to many other approaches in concurrent multiscale modeling, there are severe limitations to the specific implementation proposed by the equation-free approach.
In this work, we review the connection between the subjects of homogenization and nonlocal modeling and discuss the relevant computational issues. By further exploring this connection, we hope to promote the cross fertilization of ideas from the different research fronts. We illustrate how homogenization may help characterizing the nature and the form of nonlocal interactions hypothesized in nonlocal models. We also offer some perspective on how studies of nonlocality may help the development of more effective numerical methods for homogenization.
A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and …
Hierarchical multiscale modeling for flows in fractured media using Generalized Multiscale Finite Element Method
Texas A&M University（德克萨斯农工大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Chevron ETC（雪佛龙ETC）；China University of Petroleum (East China)（中国石油大学（华东））
Multiscale techniques have been widely shown to potentially overcome the limitation of homogenization schemes in representing the microscopic failure mechanisms in heterogeneous media as well as their influence on their structural response at the macroscopic level. Such techniques allow the use of fully detailed models to be avoided, thus resulting in a notable decrease in the overall computational cost at fixed numerical accuracy compared to the so-called direct numerical simulations. In the present work, two different multiscale modeling approaches are presented for the analysis of microstru…
We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depend on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean …
Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is $dt=H^2/κ_{max}$, where $κ_{max}$ is the la…
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocit…
In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media using Koopman operators. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale …
The authors have shown in previous contributions that reduced order modeling with optimal cubature applied to finite element square (FE2) techniques results in a reliable and affordable multiscale approach, the HPR-FE2 technique. Such technique is assessed here for an industrial case study of a generic 3D reinforced composite whose microstructure is represented by two general microcells accounting for different deformation mechanisms, microstrucural phases and geometry arrangement. Specifically, in this approach the microstrain modes used for building the reduced order model (ROM) are obtained…
In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection-diffusion-reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and m…
In modern engineering designs, advanced materials (e.g., fiber/particle-reinforced polymers, metallic alloys, laminar composites, etc.) are widely used, where microscale heterogeneities such as grains, inclusions, voids, micro-cracks, and interfaces significantly affect the macroscopic constitutive behaviors. Obviously, an accurate description of the multiscale material behaviors is of great importance to the success of material design and structural analysis. The Representative Volume Element (RVE) analysis method provides a rigorous means to obtain homogenized macroscopic material properties…
Multiscale modeling of granular flows with application to crowd dynamics
University of Salerno（萨莱诺大学）；Istituto per le Applicazioni del Calcolo “M. Picone”, CNR（CNR计算应用研究所（M. Picone））；Department of Mathematics, Politecnico di Torino（都灵理工大学数学系）
In this work we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration…
Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media
Texas A&M University（德克萨斯农工大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Chinese University of Hong Kong（香港中文大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system ma…
In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized-$α$ method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub…
Aluminum alloys are increasingly utilized as lightweight materials in the automobile industry due to their superior capability in withstanding high mechanical loads. A significant challenge impeding the large-scale use of these alloys in high-performance applications is the presence of manufacturing-induced, spatially varying porosity defects. In order to understand the impacts of these defects on the macro-mechanical properties of cast alloys, multiscale simulations are often required. In this paper, we introduce a computationally efficient reduced-order multiscale framework to simulate the b…
