Computational speed

A common acronym is MFLOPS, millions of floating-point operations per second. Because most scientific computations are limited by the speed at which floating point operations can be performed, this is a common measure of peak computing speed. Supercomputers of 1991 offered peak speeds of 1000 MFLOPS (1 GFLOP) and higher. 

Likewise, efficient interface reconstruction algorithms and mixed cell thermodynamics routines have been developed to make three-dimensional Eulerian calculations much more affordable. In general, however, computer speed and memory limitations still prevent the analyst from doing routine three-dimensional calculations with the resolution required to be assured of numerically converged solutions. As an example. Fig. 9.29 shows the setup for a test involving the oblique impact of a copper ball on a hardened steel target... 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

Beeler defined the broad scope of computer experiments as follows Any conceptual model whose definition can be represented as a unique branching sequence of arithmetical and logical decision steps can be analysed in a computer experiment... The utility of the computer... springs mainly from its computational speed. But that utility goes further as Beeler says, conventional analytical treatments of many-body aspects of materials problems run into awkward mathematical problems computer experiments bypass these problems. 

There is a trade-off between the accuracy of the calculation and the amount of computation required. In general, the more severe the approximations, the more limited is the range of applicability of the particular calculation. An organic chemist who wishes to make use of the results of MO calclulations must therefore make a judgment about the suitability of the various methods to the particular problem. The rapid increases that have occurred in computer speed and power have made the application of sophisticated methods practical for increasingly larger molecules. 

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

Molecular dynamics (MD) permits the nature of contact formation, indentation, and adhesion to be examined on the nanometer scale. These are computer experiments in which the equations of motion of each constituent particle are considered. The evolution of the system of interacting particles can thus be tracked with high spatial and temporal resolution. As computer speeds increase, so do the number of constituent particles that can be considered within realistic time frames. To enable experimental comparison, many MD simulations take the form of a tip-substrate geometry correspoudiug to scauniug probe methods of iuvestigatiug siugle-asperity coutacts (see Sectiou III.A). 

Traditionally, trajectory calculations were only performed on previously calculated (or empirically estimated) potential energy surfaces. With the increased computational speed of modern computers, it has also become possible to employ direct dynamics trajectory calculations [34, 35]. In this method, a global potential energy surface is not needed. Instead, from some... 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

The behavior of CA is linked to the geometry of the lattice, though the difference between running a simulation on a lattice of one geometry and a different geometry may be computational speed, rather than effectiveness. There has been some work on CA of dimensionality greater than two, but the behavior of three-dimensional CA is difficult to visualize because of the need for semitransparency in the display of the cells. The problem is, understandably, even more severe in four dimensions. If we concentrate on rectangular lattices, the factors that determine the way that the system evolves are the permissible states for the cells and the transition rules between those states. 

In the early 1970s, the Intel 8080 microprocessor was dazzling engineers around the world with its blazing computational speed of 2MHz Digital became the anthem for a new... 

Like the climate system described in Chapter 7, this diagenetic system consists of a chain of identical reservoirs that are coupled only to adjacent reservoirs. Elements of the sleq array are nonzero close to the diagonal only. Unnecessary work can be avoided and computational speed increased by limiting the calculation to the nonzero elements. The climate system, however, has only one dependent variable, temperature, to be calculated in each reservoir. The band of nonzero elements in the sleq array is only three elements wide, corresponding to the connection between temperatures in the reservoir being calculated and in the two adjacent reservoirs. The diagenetic system here contains two dependent variables, total dissolved carbon and calcium ions, in each reservoir. The species are coupled to one another in each reservoir by carbonate dissolution and its dependence on the saturation state. They also are coupled by diffusion to their own concentrations in adjacent reservoirs. The method of solution that I shall develop in this section can be applied to any number of interacting species in a one-dimensional chain of identical reservoirs. 

Density functional theory (DFT),32 also a semi-empirical method, is capable of handling medium-sized systems of biological interest, and it is not limited to the second row of the periodic table. DFT has been used in the study of some small protein and peptide surfaces. Nevertheless, it is still limited by computer speed and memory. DFT offers a quantum mechanically based approach from a fundamentally different perspective, using electron density with an accuracy equivalent to post Hartree-Fock theory. The ideas have been around for many years,33 34 but only in the last ten years have numerous studies been published. DFT, compared to ab initio... 

Dielectric constant is directly proportional to the capacitance of a material. Present computer operations are limited by the coupling capacitance between circuit paths and integrated circuits on multilayer boards since the computing speed between integrated circuits is reduced by this capacitance and the power required to operate is increased.11 If the dielectric constant is reduced a thinner dielectric provides equivalent capacitance, and the ground plane can be moved closer to the line, so that additional lines can be accommodated for the same cross-talk. Thus, the effect of a low dielectric constant will be to increase the speed ofthe signal and improve the density of the packaging, and this will result in improved system performance.2... 

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