## The Heterogeneous Multi-Scale Methods (2002)

Venue: | Comm. Math. Sci |

Citations: | 120 - 13 self |

### BibTeX

@ARTICLE{E02theheterogeneous,

author = {Weinan E and Björn Engquist},

title = {The Heterogeneous Multi-Scale Methods},

journal = {Comm. Math. Sci},

year = {2002},

volume = {1},

pages = {87--132}

}

### Years of Citing Articles

### OpenURL

### Abstract

The heterogeneous multi-scale method (HMM) is presented as a general methodology for the ecient numerical computation of problems with multi-scales and multiphysics, on multi-grids. Both variational and dynamic problems are considered. The method relies on an ecient coupling between the macroscopic and microscopic models.

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Citation Context ...rm we are dealing with, ~ A(U; U ), is still uniformly elliptic if a " (u; u) is. Accuracy can be estimated using thesrst Strang lemma on the approximation of stiness matrix forsnite element meth=-=ods [13] although-=- for the present problem it is simpler to proceed directly. Lemma 6.1 Assume that a " (u; u) is uniformly V -elliptic in the sense that a " (u; u) C 0 Z D jruj 2 dx: 35 for u 2 H 1 0 (D). T... |

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Citation Context ... A(U; V ) by ~ A(U; V ). Finally, the macroscale problem is ~ A(U; V ) = (f; V ) for all V 2 V . This method is closely related to the analytic tool of two-scale convergence in homogenization theory [=-=3, 18, 50]. Th-=-ere the idea is to use oscillatory test functions of the form ' x; x " in order to probe both the macroscopic behavior and the oscillations in the unknowns. Here we split this into two steps: we... |

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Citation Context ... A(U; V ) by ~ A(U; V ). Finally, the macroscale problem is ~ A(U; V ) = (f; V ) for all V 2 V . This method is closely related to the analytic tool of two-scale convergence in homogenization theory [=-=3, 18, 50]. Th-=-ere the idea is to use oscillatory test functions of the form ' x; x " in order to probe both the macroscopic behavior and the oscillations in the unknowns. Here we split this into two steps: we... |

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Citation Context ...ssibility of reducing the temporal and state-space complexity of the microscopic model, by exploiting scale separation. Such ideas can already be found in the literature, for example for sti ODEs in [=-=33, 43, 5, 35, 28]-=-, for kinetic schemes in gas dynamics in [72]. Our current work draws inspiration from the recent work of Kevrekidis and co-workers [28, 29]. Closely related ideas are also found in [71]. This paper i... |

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Citation Context ...model. Scale separation can be exploited to considerably reduce the complexity of the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics =-=[11]-=-, quasi-continuum methods [66, 62, 61] and projective methods for systems with multiscales [28, 29], HMM also provides a methodology for designing new methods for a large variety of multiscale problem... |

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Citation Context ...ticularly dicult to extend such methods to nonlinear problems, or problems for which the microstructure evolves in time. [39] extends the method of [6] to high dimensions. For other related work, see =-=[40, 60]-=-. The quasi-continuum method can be formulated in the same way. Here our problem is to simulate the nonlinear elastic behavior of crystalline solids without using a stored-energy functional, and we ba... |

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Citation Context ...es in the next section use the standard piecewise linear finite element method. For dynamic problems that are conservative, we may use the methods developed for nonlinear conservation laws (see, e.g. =-=[44]-=-). Examples include the Godunov scheme, Lax-Friedrichs scheme, and the discontinuous Galerkin method. For dynamic problems that are non-conservative, one could simply use a standard ODE solver, such a... |

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Citation Context ...lly one should apply the absorbing boundary conditions that aim at eliminating the spurious eect of the articial boundary. Such boundary conditions were proposed in the context of wave equations in [1=-=4, 26]-=-, and for molecular dynamics in [21, 22]. However, extending them to general situations does not seem to be a simple task, and we will take it up in future work. In summary, we can express the F -esti... |

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Citation Context ...ude defects in crystals where atomistic descriptions have to be used near the defects and continuum theories are valid away from the defects [66], contact line dynamics [17, 55], turbulentsame fronts =-=[53]-=-, and chemical systems with localized chemical reactions where quantum mechanics has to be used in the chemically active regions and classical mechanics can be used elsewhere. Problems of type B are f... |

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Citation Context ...lly one should apply the absorbing boundary conditions that aim at eliminating the spurious eect of the articial boundary. Such boundary conditions were proposed in the context of wave equations in [1=-=4, 26]-=-, and for molecular dynamics in [21, 22]. However, extending them to general situations does not seem to be a simple task, and we will take it up in future work. In summary, we can express the F -esti... |

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Citation Context ...control. The dimension of the state space is reduced when the reduced model is derived. The numerical techniques involved may, for example, be singular value decomposition or Krylov subspace methods, =-=[9]-=-. The reduced state space will in our case typically correspond to the macro scale. The other technique is based on wavelet compression. Such compression is common in signal and image processing but h... |

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Citation Context ...ic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [11], quasi-continuum methods [66, 62, 61] and projective methods for systems with multiscales =-=[28, 29]-=-, HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications are... |

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Citation Context ...rities, instead of been converted to the macrostates at the end of each macro time step. This is already done in algorithms involving multi-levels of physical models that are coupled together locally =-=[1, 2, 21, 22, 59]-=-. 8 Conclusion The heterogeneous multi-scale method (HMM) is a general methodology that allows us to eciently move between the macroscopic and microscopic models, and to best exploit scale separation ... |

25 |
Calculation of viscoelastic flow using molecular models: the CONNFFESSIT approach
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Citation Context ...as, including coupling quantum mechanics with molecular dynamics [12, 73, 75], coupling atomistics with continuum theory [66, 1, 55, 53, 11, 24, 45, 37], coupling kinetic theory with continuum theory =-=[8, 30, 42, 72]-=-, coupling kinetic Monte Carlo methods with continuum theory [58], homogenization theory [5, 28, 40, 60, 41], and coarse-grained bifurcation analysis [67, 57]. From the point of view of numerical anal... |

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Citation Context ... A(U; V ) by ~ A(U; V ). Finally, the macroscale problem is ~ A(U; V ) = (f; V ) for all V 2 V . This method is closely related to the analytic tool of two-scale convergence in homogenization theory [=-=3, 18, 50]. Th-=-ere the idea is to use oscillatory test functions of the form ' x; x " in order to probe both the macroscopic behavior and the oscillations in the unknowns. Here we split this into two steps: we... |

22 | The Gaussian moment closure for gas dynamics
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Citation Context ...s that an expanded set of macroscopic variables might decrease t R . Such an expectation is embodied in approaches such as Grad's 13-moment closures and the entropy-based closure schemes of Levermore =-=[44]-=-. If this is the case, the procedure described in section 7 can be used tosnd this expanded set. HMM can then be used based on this set, and eciency will be gained as a result of decreasing t R . Anot... |

20 | Origin of crack tip instabilities - Marder, Gross |

19 | Multigrid methods in lattice field computations - Brandt - 1992 |

16 |
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Citation Context ...uids in which constitutive relations are replaced by kinetic models for the conformation of macromolecules [27, 42, 73]. Such a coupled multi-scale, multi-physics approach is discussed in many papers =-=[1, 2, 12, 21, 22, 66, 62, 57, 56, 10, 29, 15]-=-, and is a central theme of the present paper. 3 In what follows we will concentrate on two types of multi-scale problems: A. A macroscopic description is known but cease to be valid in a localized re... |

16 | Fracture as a phase transition - Truskinovsky - 1996 |

15 |
Coarse-Grained Molecular Dynamics and the Atomic limit of Finite Elements
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Citation Context ...uids in which constitutive relations are replaced by kinetic models for the conformation of macromolecules [27, 42, 73]. Such a coupled multi-scale, multi-physics approach is discussed in many papers =-=[1, 2, 12, 21, 22, 66, 62, 57, 56, 10, 29, 15]-=-, and is a central theme of the present paper. 3 In what follows we will concentrate on two types of multi-scale problems: A. A macroscopic description is known but cease to be valid in a localized re... |

15 | Multiscale modeling and computation - E, Engquist |

14 |
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Citation Context ... model. Problems of type A include defects in crystals where atomistic descriptions have to be used near the defects and continuum theories are valid away from the defects [66], contact line dynamics =-=[17, 55]-=-, turbulentsame fronts [53], and chemical systems with localized chemical reactions where quantum mechanics has to be used in the chemically active regions and classical mechanics can be used elsewher... |

14 |
Front Tracking Applied to Rayleigh-Taylor Instability", in preparation
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Citation Context ...f the full model that resolves the detailed microscopic structure of the interface. A variety of powerful techniques have been developed for thesrst class of methods, including front tracking methods =-=[31-=-] [69], level set methods [51], immersed boundary methods [52]. Algorithmic development for the second class of methods have so far centered on adaptive mesh renement in the interfacial region. Even t... |

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Citation Context ...hosen to be the unit cell. 3. Dynamic Problems of Nonconservative Type. Examples include 1. @ t u " = X i;j a " i;j (x; u) @ 2 u @x i @x j where fa " (x; u)g is as discussed before. 2. =-=Spinsip models [30]-=- that leads to Ginzburg-Landau type of equations. In this case, we write the macroscale model as U t = F (U) where F (U) can be a nonlinear operator acting on U . For the macroscale scheme, we choose ... |

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Citation Context ...as, including coupling quantum mechanics with molecular dynamics [12, 73, 75], coupling atomistics with continuum theory [66, 1, 55, 53, 11, 24, 45, 37], coupling kinetic theory with continuum theory =-=[8, 30, 42, 72]-=-, coupling kinetic Monte Carlo methods with continuum theory [58], homogenization theory [5, 28, 40, 60, 41], and coarse-grained bifurcation analysis [67, 57]. From the point of view of numerical anal... |