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David Luckhaus - One of the best experts on this subject based on the ideXlab platform.

  • 6d vibrational quantum dynamics Generalized Coordinate discrete variable representation and a diabatic contraction
    Journal of Chemical Physics, 2000
    Co-Authors: David Luckhaus
    Abstract:

    A new discrete variable representation (DVR) in Generalized vibrational Coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence Coordinates as well as nonlinear maps of Coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the Generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usually are hampered by the typically large number of (avoided) crossings in dense spectra. This problem is particularly severe for discrete variable representations. A solution is provided by the new technique of diabatic rotations leading to a systematic construction of locally diabatic channels. This allows the treatment of very dense spectra where conventional truncation techniques fail. Applying the new approach to the vibrational problem of tetratomic molecules demonstrates its flexibility and efficiency. The examples of formaldehyde, ammonia, and hydrogen peroxide cover the whole range from semirigid (CH2O) to large amplitude inversion (NH3) and torsional tunnelling dynamics (H2O2). In solving the full six-dimensional vibrational eigenvalue problems for CH2O and NH3 the Z-matrix DVR shows at least comparable if not superior numerical efficiency compared with specialized techniques. In the case of H2O2 the technique of diabatic rotations and adiabatic contraction for the first time allows the treatment of the tunneling dynamics significantly above the dissociation threshold up to the fifth OH stretch overtone. The calculated decrease of the tunneling rate by about one order of magnitude agrees well with experimental observations.A new discrete variable representation (DVR) in Generalized vibrational Coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence Coordinates as well as nonlinear maps of Coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the Generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usual...

  • 6d vibrational quantum dynamics Generalized Coordinate discrete variable representation and a diabatic contraction
    Journal of Chemical Physics, 2000
    Co-Authors: David Luckhaus
    Abstract:

    A new discrete variable representation (DVR) in Generalized vibrational Coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence Coordinates as well as nonlinear maps of Coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the Generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usual...

Zhendong Sun - One of the best experts on this subject based on the ideXlab platform.

Tomonaga Okabe - One of the best experts on this subject based on the ideXlab platform.

  • smoothed particle hydrodynamics in a Generalized Coordinate system with a finite deformation constitutive model
    International Journal for Numerical Methods in Engineering, 2015
    Co-Authors: Shigeki Yashiro, Tomonaga Okabe
    Abstract:

    SUMMARY This study proposes smoothed particle hydrodynamics (SPH) in a Generalized Coordinate system. The present approach allocates particles inhomogeneously in the Cartesian Coordinate system and arranges them via mapping in a Generalized Coordinate system in which the particles are aligned at a uniform spacing. This characteristic enables us to employ fine division only in the direction required, for example, in the through-thickness direction for a thin-plate problem and thus to reduce computation cost. This study provides the formulation of SPH in a Generalized Coordinate system with a finite-deformation constitutive model and then verifies it by analyzing quasi-static and dynamic problems of solids. High-velocity impact test was also performed with an aluminum target and a steel sphere, and the predicted crater shape agreed well with the experiment. Furthermore, the numerical study demonstrated that the present approach successfully reduced the computation cost with marginal degradation of accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

  • Smoothed particle hydrodynamics in a Generalized Coordinate system with a finite‐deformation constitutive model
    International Journal for Numerical Methods in Engineering, 2015
    Co-Authors: Shigeki Yashiro, Tomonaga Okabe
    Abstract:

    SUMMARY This study proposes smoothed particle hydrodynamics (SPH) in a Generalized Coordinate system. The present approach allocates particles inhomogeneously in the Cartesian Coordinate system and arranges them via mapping in a Generalized Coordinate system in which the particles are aligned at a uniform spacing. This characteristic enables us to employ fine division only in the direction required, for example, in the through-thickness direction for a thin-plate problem and thus to reduce computation cost. This study provides the formulation of SPH in a Generalized Coordinate system with a finite-deformation constitutive model and then verifies it by analyzing quasi-static and dynamic problems of solids. High-velocity impact test was also performed with an aluminum target and a steel sphere, and the predicted crater shape agreed well with the experiment. Furthermore, the numerical study demonstrated that the present approach successfully reduced the computation cost with marginal degradation of accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Fahhad H Alharbi - One of the best experts on this subject based on the ideXlab platform.

  • efficient high order method for differential equations in unbounded domains using Generalized Coordinate transformation
    Journal of Computational Physics, 2019
    Co-Authors: Faisal Mumtaz, Hamed Ben Mohamed Saidaoui, Fahhad H Alharbi
    Abstract:

    Abstract Many physical phenomena are localized in space, where physical quantities of interest extend to large distances (effectively to infinity) where they vanish smoothly. Thus, its modeling necessitates the imposition of vanishing boundary conditions (BCs) to the formulated differential equations (DEs). Numerical solutions for such DEs can be complicated as they involve operations on physical quantities expanding infinitely in space. Currently, many approaches have been implemented in this context, where they mostly use techniques like domain truncation, using functions intrinsic to unbounded domains, and domain transformation. However, since the actual solutions have a decaying nature, solving such problems with high accuracy and efficiency while taking into consideration a wide range of decaying rates and oscillation frequencies is a major challenge. In this work, we present an efficient high-order method, based on Galerkin method, to solve DEs in unbounded domains using properly designed domain transformations where the physical unbounded domain ( − ∞ , ∞ ) is mapped into a computational bounded domain ( 0 , 1 ) . The mapped sine series is then used to solve the differential equation in the computational domain ( 0 , 1 ) where the vanishing quantities at infinities are mapped to 0 (corresponding to physical −∞) and 1 (corresponding to physical ∞). The designed transformations maintain the orthogonality in both physical and computational spaces. Moreover, using sine series basis allows – in most cases – analytical integrations. For verification, the proposed method is applied to solve 1) Helmholtz equation in one-dimension, where the exact solutions for various cases (decaying exponential or algebraic function) are known, and 2) three-dimensional Schrodinger wave equation with harmonic oscillator. For the Helmholtz problem, the error in the approximate solution was less than 10 − 40 using 100 basis functions, while for the second problem, the error is less than 10 − 25 using the same number of bases. All results and the analyses demonstrate how the properly designed mapped basis sets can be used to develop efficient high-order methods to solve DEs with vanishing boundary conditions and with wide range of decaying rates and oscillation frequencies simultaneously.

Meili Lin - One of the best experts on this subject based on the ideXlab platform.