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Analytical Form (analytical + form)
Selected AbstractsLarge displacement behaviour of a structural model with foundation uplift under impulsive and earthquake excitationsEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 3 2003Giuseppe Oliveto Abstract This paper considers the dynamical behaviour of a structural model with foundation uplift. The equations of motion of the system considered are derived for large displacements thus allowing for the eventual overturning of the system. The transition conditions between successive phases of motion, derived in terms of the specific Lagrangian co-ordinates used in the formulation of the equations of motion, present innovative aspects which resolve some previously inexplicable behaviour in the structural response reported in the literature. The dynamical behaviour of the model is considered under impulsive and long-duration ground motions. The minimum horizontal acceleration impulses for the uplift and the overturning of the system are evaluated in analytical form. The sensitivity of the model to uplifting and to overturning under impulsive excitations is established as a function of few significant structural parameters. Numerical applications have been performed changing either the structural parameters or the loading parameter, in order to analyse several dynamical behaviours and also to validate the analytical results. For earthquake ground motions the results, reported in the form of response spectra, show that linearized models generally underestimate, sometimes significantly, the structural response. Copyright © 2003 John Wiley & Sons, Ltd. [source] Link between the kinetic- and exchange-energy functionals in the generalized gradient approximationINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2002Fabien Tran Abstract An approximate kinetic-energy functional of the generalized gradient approximation form was derived following the "conjointness conjecture" of Lee, Lee, and Parr. The functional shares the analytical form of its gradient dependency with the exchange-energy functionals of Becke and Perdew, Burke, and Ernzerhof. The two free parameters of this functional were determined using the exact values of the kinetic energy of He and Xe atoms. A set of 12 closed-shell atoms was used to test the accuracy of the proposed functional and more than 30 others taken from the literature. It is shown that the conjointness conjecture leads to a very good class of kinetic-energy functionals. Moreover, the functional developed in this work is shown to be one of the most accurate despite its simple analytical form. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source] Application of the frozen atom approximation to the GB/SA continuum model for solvation free energyJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 2 2002Olgun Guvench Abstract The generalized Born/surface area (GB/SA) continuum model for solvation free energy is a fast and accurate alternative to using discrete water molecules in molecular simulations of solvated systems. However, computational studies of large solvated molecular systems such as enzyme,ligand complexes can still be computationally expensive even with continuum solvation methods simply because of the large number of atoms in the solute molecules. Because in such systems often only a relatively small portion of the system such as the ligand binding site is under study, it becomes less attractive to calculate energies and derivatives for all atoms in the system. To curtail computation while still maintaining high energetic accuracy, atoms distant from the site of interest are often frozen; that is, their coordinates are made invariant. Such frozen atoms do not require energetic and derivative updates during the course of a simulation. Herein we describe methodology and results for applying the frozen atom approach to both the generalized Born (GB) and the solvent accessible surface area (SASA) parts of the GB/SA continuum model for solvation free energy. For strictly pairwise energetic terms, such as the Coulombic and van-der-Waals energies, contributions from pairs of frozen atoms can be ignored. This leaves energetic differences unaffected for conformations that vary only in the positions of nonfrozen atoms. Due to the nonlocal nature of the GB analytical form, however, excluding such pairs from a GB calculation leads to unacceptable inaccuracies. To apply a frozen-atom scheme to GB calculations, a buffer region within the frozen-atom zone is generated based on a user-definable cutoff distance from the nonfrozen atoms. Certain pairwise interactions between frozen atoms in the buffer region are retained in the GB computation. This allows high accuracy in conformational GB comparisons to be maintained while achieving significant savings in computational time compared to the full (nonfrozen) calculation. A similar approach for using a buffer region of frozen atoms is taken for the SASA calculation. The SASA calculation is local in nature, and thus exact SASA energies are maintained. With a buffer region of 8 Å for the frozen-atom cases, excellent agreement in differences in energies for three different conformations of cytochrome P450 with a bound camphor ligand are obtained with respect to the nonfrozen cases. For various minimization protocols, simulations run 2 to 10.5 times faster and memory usage is reduced by a factor of 1.5 to 5. Application of the frozen atom method for GB/SA calculations thus can render computationally tractable biologically and medically important simulations such as those used to study ligand,receptor binding conformations and energies in a solvated environment. © 2002 Wiley Periodicals, Inc. J Comput Chem 23: 214,221, 2002 [source] Form factors for core electrons useful for the application of quantum crystallography (QCr) to organic moleculesACTA CRYSTALLOGRAPHICA SECTION A, Issue 4 2002Lulu Huang Form factors are calculated for the core electrons of the first-row atoms B, C, N, O and F. The form factors are presented in an analytical form, as appears in International Tables for X-ray Crystallography [Ibers & Hamilton (1974), Vol. IV, pp. 103,145. Birmingham: Kynoch Press; present distributor Kluwer Academic Publishers, Dordrecht]. Having such form factors for the core electrons reduces the number of parameters to be determined by the methods of quantum crystallography (QCr). It is shown that the form factors obtained are quite accurate. Thus, when they are used in QCr, they are expected to be consistent with accurate charge densities. [source] The Hubbard model extended by nearest-neighbor Coulomb and exchange interaction on a cubic cluster , rigorous and exact resultsANNALEN DER PHYSIK, Issue 6 2010R. Schumann The Hubbard model on a cube was revisited and extended by both nearest-neighbor Coulomb correlation W and nearest-neighbor Heisenberg exchange J. The complete eigensystem was computed exactly for all electron occupancies and all model parameters ranging from minus infinity to plus infinity. For two electrons on the cluster the eigensystem is given in analytical form. For six electrons and infinite on-site correlation U we determinded the groundstate and the groundstate energy of the pure Hubbard model analytically. For fixed electron numbers we found a multitude of ground state level crossings depending on the various model parameters. Furthermore the groundstates of the pure Hubbard model in dependence on a magnetic field h coupled to the spins are shown for the complete U-h plane. The critical magnetic field, where the zero spin groundstate breaks down is given for four and six electrons. Suprisingly we found parameter regions, where the ground state spin does not depend monotonously on J in the extended model. For the cubic cluster gas, i.e. an ensemble of clusters coupled to an electron bath, we calculated the density n (,, T, h) and the thermodynamical density of states from the grand potential. The ground states and the various spin-spin correlation functions are studied for both attractive and repulsive values of the three interaction constants. We determined the various anomalous degeneration lines, where n (,, T = 0, h = 0) shows steps higher than one, since in this parameter regions exotic phenomena as phase separation are to expect in extended models. For the cases where these lines end in triple points, i.e. groundstates of three different occupation numbers are degenerated, we give the related parameter values. Regarding the influence of the nn-exchange and the nn-Coulomb correlation onto the anomalous degeneration we find both lifting and inducing of degeneracies depending on the parameter values. [source] | ||||||||||