Heterogeneous Multiscale Methods: A Review

By E. Weinan, Björn Engquist, Xiantao Li, Weiqing Ren, Eric Vanden-Eijnden. Communications in Computational Physics, 2007

828 citations recorded by the source corpus.

This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be ...

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/9772bc56141d1466e2381cee8325f81b2cd326fd

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Citations in the corpus

Partial differential equations for self-organization in cellular and developmental biology (2008, methodology): “We note that such techniques are common in other fields, such as the modelling of dislocations and shocks in physics [138] and theoretical electrophysiology [137, 139], but have yet to become standard in biological self-organization modelling.”
Reduction of the resonance error---Part 1: Approximation of homogenized coefficients (2011, methodology): “In essence, numerical homogenization techniques aim at computing solutions to partial differential equations whose coefficients depend on a small parameter denoted by ε, without resolving all the details at the scale ε (see for instance [15, 16] for the multiscale finite element method, [6, 5] for t”
Multi-Scale Surface Texturing in Tribology—Current Knowledge and Future Perspectives (2019, background): “A more in-depth understanding of numerical multi-scale methods for micro-meso-macroscopic scales coupling can be found in [143,144,224,225].”
Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High‐Resolution Simulations (2017, methodology): “However, multiscale modeling relies on a scale separation between the global-model mesh size and the domain size of the nested high-resolution simulation [E et al., 2007].”
A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods (2009, background): “There is a vast literature surrounding these methods, as summarized in several recent reviews [9, 26, 34, 13, 31].”
An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs (2013, background): “[13, 2, 1] for selective applications to highly oscillatory problems), which is a very general framework for efficiently computing approximations to problems that exhibit multiple spatial or temporal scales; a review of HMM can be found in [35].”
A Micro-Macro Parareal Algorithm: Application to Singularly Perturbed Ordinary Differential Equations (2012, background): “We refer, for instance, to the work on equation-free [22, 23] or heterogeneous multiscale methods [8, 9], and references therein.”
Nonintrusive and Structure Preserving Multiscale Integration of Stiff ODEs, SDEs, and Hamiltonian Systems with Hidden Slow Dynamics via Flow Averaging (2009, methodology): “By using Verlet/leapfrog macro-solvers, methods that are symplectic on slow variables (when those variables can be identified) have been proposed in the framework of HMM (the Heterogeneous Multiscale Method) in [102, 24].”
A Unified Interpretation of Stress in Molecular Systems (2010, methodology): “s are quasicontinuum [53,59], coupling of lengthscales [51], cluster quasicontinuum [31], bridging domain [66], coupled atomistics and discrete dislocations [54], and heterogeneous multiscale methods [15], to name just a few. Refer to [58] for a comparison of some prominentatomistic/continuumcouplin”