In recentyears, Brandt has proposed to extend the multi-grid method to caseswhen the effective problems solved at different levels correspond tovery different kinds of models (Brandt, 2002). For example, themodels used at the finest level might be molecular dynamics or MonteCarlo models whereas the effective models used at the coarse levelscorrespond to some continuum models. Brandt noted that there is noneed to have closed form macroscopic models at the coarse scale sincecoupling to the models used at the fine scale grids automaticallyprovides effective models at the coarse scale.
Quantum mechanics – molecular mechanics (QM-MM) methods
- Forexample, if the microscale model is the NVT ensemble of moleculardynamics, \(d\) might be the temperature.
- In sequential multiscalemodeling, one has a macroscale model in which some details of theconstitutive relations are precomputed using microscale models.
- Typically,near defects such as dislocations, all the atoms are selected.
- The other extreme is to work with a microscale model, such as the first principle of quantum mechanics.
- Multiscale ideas have also been used extensively in contexts where nomulti-physics models are involved.
- The firstis that the implementation of CPMD is based on an extended Lagrangianframework by considering the wavefunctions for electrons in the samesetting as the positions of the nuclei.
Homogenization methods can be applied to many other problems of thistype, in which a heterogeneous behavior is approximated at the largescale by a slowly varying or homogeneous behavior. This is a strategy for choosing thenumerical grid or mesh adaptively based on what is known about thecurrent approximation to the numerical solution. Usually one finds alocal error indicator from the available numerical solution based onwhich one modifies the mesh in order to find a better numericalsolution. This is a way of summing up longrange interaction potentials for a large set of particles. Thecontribution to the interaction potential is decomposed intocomponents with different scales and these different contributions areevaluated at different levels in a hierarchy of grids. Multiscale ideas have also been used extensively in contexts where nomulti-physics models are involved.
Examples of multiscale methods
These slowly varying quantities aretypically the Goldstone modes of the system. For example, the densities ofconserved quantities such as mass, momentum and energy densities areGoldstone modes. The equilibrium states of macroscopicallyhomogeneous systems are parametrized by the values of thesequantities. When the system varies on a macroscopic scale, theseconserved densities also vary, and their dynamics is described by aset of hydrodynamic equations (Spohn, 1991). In this case, locally,the microscopic state of the system is close to some local equilibriumstates parametrized by the local values multi-scale analysis of the conserved densities.
Statistical Techniques
When studying chemical reactions involving large molecules, it oftenhappens that the active areas of the molecules involved in thereaction are rather small. The rest of the molecules just serves toprovide the environment for the reaction. In this case, it is naturalto only treat the reaction zone quantum mechanically, and treat therest using classical description. This is a type A problem.Such a methodology is called theQM-MM (quantum mechanics-molecular mechanics) method (Warshel and Levitt, 1976).
We refer to thefirst type as type A problems and the second type as type B problems. In the multiscale approach, one uses a variety of models at differentlevels of resolution Coding and complexity to study one system. Thedifferent models are linked together either analytically ornumerically. For example, one may study the mechanical behavior ofsolids using both the atomistic and continuum models at the same time,with the constitutive relations needed in the continuum model computedfrom the atomistic model. Quasicontinuum method (Tadmor, Ortiz and Phillips, 1996; Knap and Ortiz, 2001)is a finite element type of method for analyzing the mechanicalbehavior of crystalline solids based on atomistic models. Atriangulation of the physical domain is formed using a subset of theatoms, the representative atoms (or rep-atoms).
Key Objectives of Multiple-Scale Analysis:
Therefore tryingto capture the macroscale behavior without any knowledge about themacroscale model is quite difficult. Of course, the usefulness of HMMdepends on how much prior knowledge one has about the macroscalemodel. In particular, guessing the wrong form of the macroscale modelis likely going to lead to wrong results using HMM. Here the macroscale variable \(U\) may enter the system via some constraints,\(d\) is the data needed in order to set up the microscale model. Forexample, if the microscale model is the NVT ensemble of moleculardynamics, \(d\) might be the temperature. Classically this is a way ofsolving the system of algebraic equations that arise from discretizingdifferential equations by simultaneously using different levels ofgrids.
Macro-micro formulations for polymer fluids
- The equilibrium states of macroscopicallyhomogeneous systems are parametrized by the values of thesequantities.
- We simply have to input the atomic numbers of all the participating atoms, then we have a complete model which is sufficient for chemistry, much of physics, material science, biology, etc.
- In recentyears, Brandt has proposed to extend the multi-grid method to caseswhen the effective problems solved at different levels correspond tovery different kinds of models (Brandt, 2002).
- The basic idea is to use microscalesimulations on patches (which are local spatial-temporal domains) to mimicthe macroscale behavior of a system through interpolation inspace and extrapolation in time.
- For example, themodels used at the finest level might be molecular dynamics or MonteCarlo models whereas the effective models used at the coarse levelscorrespond to some continuum models.
The idea is to decompose the wholecomputational domain into several overlapping or non-overlappingsubdomains and to obtain the numerical solution over the whole domainby iterating over the solutions on these subdomains. The domaindecomposition method is not limited to multiscale problems, but it canbe used for multiscale problems. The firstis that the implementation of CPMD is based on an extended Lagrangianframework by considering the wavefunctions for electrons in the samesetting as the positions of the nuclei. In this extended phase space,one can write down a Lagrangian which incorporates both theHamiltonian for the nuclei and the wavefunctions. This makes the system stiffsince the time scales of the electrons and the nuclei are quitedisparate. However, since we are only interested in the dynamics ofthe nuclei, not the electrons, we can choose a value which is muchlarger than the electron mass, so long as it still gives ussatisfactory accuracy for the nuclear dynamics.
Mathematics > Analysis of PDEs
Despite the fact that there are already so many different multiscalealgorithms, potentially many more will be proposed since multiscalemodeling is relevant to so many different applications. Therefore itis natural to ask whether one can develop some general methodologiesor guidelines. An analogy can be made with the general methodologiesdeveloped for numerically solving differential equations, for example,the finite difference, finite element, finite volume, and spectral methods. These different but also closely related methodologies serveas guidelines for designing numerical methods for specificapplications.
BibTeX formatted citation
Precomputing the inter-atomic forces asfunctions of the positions of all the atoms in the system is notpractical since there are too many independent variables. On the otherhand, in a typical simulation, one only probes an extremely smallportion of the potential energy surface. Concurrent coupling allowsone to evaluate these forces at the locations where they are needed. Traditional multi-grid method is a way of efficiently solving a largesystem of algebraic equations, which may arise from the discretizationof some partial differential equations. For this reason, theeffective operators used at each level can all be regarded as anapproximation to the original operator at that level.