Extended Lagrangian technique

The extended Lagrangian technique on which the Car-Parrinello method is based can be used also in other contexts. Whenever the forces on some atoms... 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

This article is organized as follows. In Section 2 ab initio molecular dynamics methods are described. Specifically, in Section 2.1 we discuss the extended Lagrangian atom-centered density matrix (ADMP) technique for simultaneous dynamics of electrons and nuclei in large clusters, and in Section 2.2 we discuss the quantum wavepacket ab initio molecular dynamics (QWAIMD) method. Simulations conducted and new insights obtained from using these approaches are discussed in Section 3 and the concluding remarks are given in Section 4. 

Waldvogel and Poulikakos1501 extended the model and numerical techniques of Zhao et al.13681 by incorporating solidification and droplet-substrate contact resistance in the heat transfer model. They conducted both theoretical and experimental studies on the impact and solidification of molten solder droplets on a multilayer substrate. The theoretical model was based on the Lagrangian formulation, and accounted for a host of thermal-fluid phenomena,... 

A similar approach exists for performing constant temperature MD calculations. The Lagrangian (Eq. [48]) is extended by terms for additional (fictitious) degrees of freedom that couple the system to an external heat reser-voir. A combination of the Parrinello-Raman and Nose schemes is possible to sample an isobaric-isothermic (NPT) ensemble. These methods as well as other techniques for performing calculations in different statistical ensembles are reviewed by Allen and Tildesley in detail. 

Before we finish this subsection, we would like to discuss the practical limitations of the Poincare sections, which require Lagrangian particle tracking for extended times. In reality. Fig. 2 presents stroboscopic images of the same four particles passing through thousands of mixing block boundaries. This has two basic implications. First, numerical calculation of the Poincare sections requires either analytical solutions or high-order accurate discretizations of the velocity field. Otherwise, the results may suffer from numerical diffusion and dispersion errors, and the KAM boundaries may not be identified accurately. Second, it is experimentally difficult, if not impossible, to track particles (in three-dimensions) beyond a certain distance allowed by the field of view of the microscopy technique. Despite these... 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

FeUceUi et al. [57] extended the model to three-dimensional problems and applied it to solidification of a Pb-Sn alloy in a parallelepiped and in a cylinder. The transport equations were discretized and integrated in time using a FEM based on the bilinear Lagrangian isoparametric element. A Petrov-Galerkin technique was used to... 

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