## The Heterogeneous Multi-Scale Methods (2002)

Venue: | Comm. Math. Sci |

Citations: | 126 - 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.

### Citations

1490 |
The finite element method for elliptic problems
- Ciarlet
- 1987
(Show Context)
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... |

991 |
Mathematical methods of classical mechanics
- Arnold
- 1978
(Show Context)
Citation Context ...oblem. These equations are the basis for computer simulations. Indeed an impressive array of techniques have been developed to accomplish this task, including averaging methods in classical mechanics =-=[4]-=-, homogenization theory [7], equilibrium statistical mechanics [41], WKB methods, nonequilibrium thermodynamics [16], kinetic theory [41], transition state theory [36], turbulence models, etc. This ap... |

973 |
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
- OSHER, SETHIAN
- 1988
(Show Context)
Citation Context ...s 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 though these methods tend to b... |

605 |
The theory of polymer dynamics
- Doi, Edwards
- 1986
(Show Context)
Citation Context ... u(x, t) = R 3 f(v, x, t)vdv, T(x, t) = 1 ∫ 3ρ R3 f(v, x, t)|v−u| 2 dv 2. For our second example, we will take the microscopic model to be the kinetic models of rod-like molecules in liquid crystals, =-=[18]-=- where the microscopic variable is the orientation-position distribution function f(x, m, t), (x, t) ∈ D, m ∈ S 2 , the unit sphere. Here D = D × S 2 , Dx = S 2 . The macroscopic model is the Landau-d... |

403 |
Asymptotic Analysis for Periodic Structures
- BENSOUSSAN, LIONS, et al.
- 1978
(Show Context)
Citation Context ...the basis for computer simulations. Indeed an impressive array of techniques have been developed to accomplish this task, including averaging methods in classical mechanics [4], homogenization theory =-=[7]-=-, equilibrium statistical mechanics [41], WKB methods, nonequilibrium thermodynamics [16], kinetic theory [41], transition state theory [36], turbulence models, etc. This approach has been both succes... |

359 | Multi-level adaptive solutions to boundary-value problems - Brandt - 1977 |

275 |
Non-equilibrium Thermodynamics
- GROOT, MAZUR
- 1964
(Show Context)
Citation Context ...eveloped to accomplish this task, including averaging methods in classical mechanics [4], homogenization theory [7], equilibrium statistical mechanics [41], WKB methods, nonequilibrium thermodynamics =-=[16]-=-, kinetic theory [41], transition state theory [36], turbulence models, etc. This approach has been both successful, in fact it has so far dominated much of science and engineering, and necessary beca... |

223 | Unified analysis of discontinuous Galerkin methods for elliptic problems - Arnold, Brezzi, et al. |

184 | Homogenization and two-scale convergence
- Allaire
- 1992
(Show Context)
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... |

160 | A multiscale finite element method for elliptic problems in composite materials and porous media
- Hou, Wu
- 1997
(Show Context)
Citation Context ...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 analysis, it is natural to ask whether these seemingly different applications can be put into a common framework... |

139 | Dynamical theory of crystal lattices - Born, Huang - 1954 |

125 |
A general convergence result for a functional related to the theory of homogenization
- Nguetseng
- 1989
(Show Context)
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... |

114 | Functional Integration and Partial Differential Equations - Freidlin - 1985 |

97 |
Solving Ordinary
- Hairer, Wanner
- 1996
(Show Context)
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... |

96 |
Unified approach for molecular dynamics and density-functional theory
- Car, Parrinello
- 1985
(Show Context)
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... |

89 | Boundary value problems with rapidly oscillating random coefficients - Papanicolaou, Varadhan - 1979 |

89 | Quasicontinuum analysis of defects in solids - Tadmor, Ortiz, et al. - 1996 |

88 |
Multiscale phenomena; Green’s functions, the Dirichletto-Neumann formulation, subgrid scale models, bubbles and the origins of the stabilized methods
- Hughes
- 1995
(Show Context)
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... |

79 |
Numerical Methods for Conservation Laws”, Birkhäuser
- LeVeque
- 1990
(Show Context)
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... |

77 |
Radiation boundary conditions for acoustic and elastic calculations
- Engquist, Majda
- 1979
(Show Context)
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... |

71 | Proposed experimental tests of a theory of fine microstructure and the two-well problem - Ball, James - 1992 |

68 |
Turbulent Combustion
- Peters
- 2000
(Show Context)
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... |

58 | Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum - Kevrekidis, Gear |

52 |
Absorbing boundary condition for acoustic and elastic wave equations
- Clayton, Enquist
- 1977
(Show Context)
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... |

52 | The Quasicontinuum Method: Overview, applications and current directions - Miller, Tadmor - 2002 |

47 | Convergence of a nonconforming multiscale finite element method - Efendiev, Hou, et al. |

46 | An Analysis of the Quasicontinuum Method - Knap, Ortiz - 2001 |

45 | Analysis of the heterogeneous multiscale method for ordinary differential equations - E |

43 | Krylov space methods on state-space control models
- Boley
- 1994
(Show Context)
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... |

43 | Coarse” integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods
- Gear, Kevrekidis, et al.
(Show Context)
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... |

40 | Multiscale Scientific Computation: Review 2001”. In - Brandt - 2002 |

38 | From molecular models to continuum mechanics - Blanc, Bris, et al. |

36 | Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case, Archive for Rational Mechanics and Analysis - Braides, Maso, et al. - 1999 |

32 | Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model - Lin - 2003 |

29 | Optimal prediction of underresolved dynamics - Chorin, Kast, et al. - 1998 |

28 | Coarse” stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples - Makeev, Maroudas, et al. |

26 |
Concurrent coupling of length scales: methodology and application, Phys
- Broughton, Abraham, et al.
- 1999
(Show Context)
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 ... |

26 |
Calculation of viscoelastic flow using molecular models
- Laso, Öttinger
- 1993
(Show Context)
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... |

24 |
Homogenization of linear and nonlinear transport equations
- E
- 1992
(Show Context)
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
- Levermore, Morokoff
- 1998
(Show Context)
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 |

17 | Multiscale modeling and computation - E, Engquist - 2003 |

16 |
Spanning the Continuum to Quantum Length Scales in a Dynamic Simulation of Brittle Fracture, Europhysic Letters
- Abraham, Broughton, et al.
- 1998
(Show Context)
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 |
Coarse-grained molecular dynamics and the atomic limit of finite elements
- Rudd, Broughton
- 1998
(Show Context)
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 |
Front tracking applied to RayleighTaylor instability
- Glimm, McBryan, et al.
- 1986
(Show Context)
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... |

14 |
On the spreading of liquids on solid surfaces: static and dynamic contact lines, Ann. Rev. Fluid Mechanics 11
- V
- 1979
(Show Context)
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... |

13 | Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems
- Giacomin, Lebowitz, et al.
- 1999
(Show Context)
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 ... |

13 | Coupling Boltzmann and Navier-Stokes Equations by Friction
- Bourgat, Tallec, et al.
- 1996
(Show Context)
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... |