Atoms finite element

GleadaU et al. (2014a,b) developed an effective cavity theory using the atomic finite element method (AFEM) for the change of Young s modulus during degradation... 

One type of computer simulation which Beeler did not include (it was only just beginning when he wrote in 1970) was finite-element simulation of fabrication and other production processes, such as for instance rolling of metals. This involves exclusively continuum aspects particles , or atoms, do not play a part. 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

These observations are equivalent to a coarse-grained view of the system, which is tantamount to a description in terms of continuum mechanics. [It is clear that "points" of the continuum may not refer to such small collections of atoms that thermal fluctuations of the coordinates of their centers of mass become substantial fractions of their strain displacements.] The elastomer is thus considered to consist of a large number of quasi-finite elements, which interact with one another through dividing surfaces. 

Advances in computational capability have raised our ability to model and simulate materials structure and properties to the level at which computer experiments can sometimes offer significant guidance to experimentation, or at least provide significant insights into experimental design and interpretation. For self-assembled macromolecular structures, these simulations can be approached from the atomic-molecular scale through the use of molecular dynamics or finite element analysis. Chapter 6 discusses opportunities in computational chemical science and computational materials science. 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8,2.9,2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific droplet sizes (forexample,.D0 16,Z>05, andZ)0 84) are to be considered in the calculations of the 2-D motion, cooling and solidification histories. 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

V.B. Shenoy et al An adaptive finite element approach to atomic-scale mechanics-the quasicontinuum method. J. Mech. Phys. Solids 47, 611-642 (1999)... 

R.E. Rudd, J.Q. Broughton Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B 58, R5893-R5896 (1998)... 

Here it suffices to say that the coupling interface between the finite element mesh and the atomic coordinates in the molecular dynamics is accomplished through resolving the near part of the finite element mesh on the atomic coordinates. On either side of the interface, the atoms and the finite element mesh overlap. Finite element cells that intersect the interface and atoms that interact across the interface each contribute to the Hamiltonian at half strength. See Rudd and Broughton (1998) for further discussion of the relationship between molecular dynamics and finite element methods. 

A very different approach is the use of non-atom-centered basis functions such as plane waves. Due to their intrinsic periodic nature, they are mostly employed for electronic structure calculations of periodic solids [10]. A more recent development is the usage of real-space wavefunctions either by discretization on real-space grids or in a finite-element fashion [11], In a non-atom-centered basis, the basis set obviously does not depend on the atomic positions, which makes it ideally suited for ab initio molecular dynamics simulations, since the forces acting on the nuclei can be evaluated much more easily than in an atom-centered basis [10]. 

Rudd, R.E. and Broughton, J.Q. (1998) Coarse-grained Molecular Dynamics and the Atomic Limit of Finite Elements. Phys. Rev. B, 58, R5893-R5896. 

Fig. 12.11. Schematic of treatment of dislocation core in mixed atomistic/continmun setting (adapted from Shenoy et al. (1999)). The left hand figure shows which set of atoms are chosen as representative atoms, and the right hand figure shows the corresponding finite element mesh.

From a geometric perspective, the physics of degree of freedom elimination is based upon the kinematic slavery that emanates from the use of the finite element as the central numerical engine of the method. An alternative view is that of constraint. By virtue of the use of finite element interpolation, vast numbers of the atomic-level degrees of freedom are constrained. The main point is that some subset of the full atomic set of degrees of freedom is targeted as the representative set of atoms, as shown in fig. 12.11. These atoms form the nodes in a finite element mesh, and the positions of any of the remaining atoms are found by finite element interpolation via... 

Shenoy V. B., Miller R., Tadmor E. B., Rodney D, Phillips R. and Ortiz M., An Adaptive Finite Element Approach to Atomic-Scale Mechanics - The Quasicontinuum Method J. Mech. Phys. Solids AT, 611 (1999). 

The programs described so far use basis-set expansions for the one-electron spinors. The fully numerical approach, which is still a challenging task for general molecules in nonrelativistic theory (Andrae 2001), has also been tested for Dirac-Fock calculations on diatomics (DtisterhOft etal. 1994,1998 Kullie etal. 1999 Sundholm 1987,1994 Sundholm et al. 1987 v. Kopylow and Kolb 1998 v. Kopylow et al. 1998 Yang et al. 1992). The finite-element method (FEM) was tested for Dirac-Fock and Kohn—Sham calculations by Kolb and co-workers in the 1990s. However, this approach has not yet been developed into a general method for systems with more than two atoms only test systems, namely few-electron linear molecules at some fixed intemuclear distance, have been studied with the FEM. Nonetheless, these numerical techniques are able to calculate the Dirac-Fock limit and thus yield reference data for comparisons with more approximate basis-set approaches. The limits of the numerical techniques are at hand ... 

Table 4 Energy eigenvalues for different states for the hydrogen atom confined by an impenetrable spherical box of radius tq = 2 au. The reported results were obtained by Killinbeck [32] and Friedman et al. [33] using the series method and the finite element method, respectively...

It is also worth mentioning that numerical solutions of the Schrodinger equation frequently enclose the atom in a spherical box of finite radius for example, discrete variable methods, finite elements methods and variational methods which employ expansions in terms of functions of finite support, such as -splines, all assume that the wave function vanishes for r > R, which is exactly the situation we deal with here. For such solutions to give an accurate description of the unconfined system it is, of course, necessary to choose R sufficiently large that there is negligible difference between the confined and unconfined atoms. 

Finite element three-body studies of bound and resonant states in atoms and molecules. 

The development of a full angular momentum, three dimensional, smooth exterior complex dilated, finite element method for computing bound and resonant states in a wide class of quantum systems is described. Applications to the antiprotonic helium system, doubly excited states in the helium atom and to a model of a molecular van der Waals complex are discussed. 2001 by Academic Press. 

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