Coupled Equations (coupled + equation)

Distribution by Scientific Domains


Selected Abstracts


Chemical reactions in the gas phase and in condensed matter: From wavefunctions to density operators

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 13 2009
David A. Micha
Abstract This contribution generalizes the treatment of chemical reactions in the gas phase based on the reaction channel decomposition of the wavefunction, by introducing a similar channel decomposition of the statistical density operator valid also for condensed phases such as liquid solutions and solid surfaces. Coupled equations for the channel components of the density operator are derived and a brief presentation is given of their partial Wigner transform, which leads to a general treatment for coupling quantum and classical variables. This provides a general approach for reactions involving electronically excited states in many-atom systems. It is pointed out that reactions involving coupled quantal and classical variables can be correctly described provided (a) initial conditions for trajectories are generated from quantal distributions and (b) the bundle of trajectories for the whole initial classical phase space is propagated coupled to the quantal elements of the density matrix and used in the calculation of reaction flux averages. 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]


Coupled HM analysis using zero-thickness interface elements with double nodes.

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 18 2008
Part I: Theoretical model
Abstract In recent years, the authors have proposed a new double-node zero-thickness interface element for diffusion analysis via the finite element method (FEM) (Int. J. Numer. Anal. Meth. Geomech. 2004; 28(9): 947,962). In the present paper, that formulation is combined with an existing mechanical formulation in order to obtain a fully coupled hydro-mechanical (or HM) model applicable to fractured/fracturing geomaterials. Each element (continuum or interface) is formulated in terms of the displacements (u) and the fluid pressure (p) at the nodes. After assembly, a particular expression of the traditional ,u,p' system of coupled equations is obtained, which is highly non-linear due to the strong dependence between the permeability and the aperture of discontinuities. The formulation is valid for both pre-existing and developing discontinuities by using the appropriate constitutive model that relates effective stresses to relative displacements in the interface. The system of coupled equations is solved following two different numerical approaches: staggered and fully coupled. In the latter, the Newton,Raphson method is used, and it is shown that the Jacobian matrix becomes non-symmetric due to the dependence of the discontinuity permeability on the aperture. In the part II companion paper (Int. J. Numer. Anal. Meth. Geomech. 2008; DOI: 10.1002/nag.730), the formulation proposed is verified and illustrated with some application examples. Copyright 2008 John Wiley & Sons, Ltd. [source]


Vertical vibration of an elastic strip footing on saturated soil

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 5 2008
Y. Q. Cai
Abstract Based on Biot's dynamic coupled equations, the vertical vibration of an elastic strip footing on the surface of saturated soil is studied. Utilizing the Fourier transform, the governing dynamic differential equations for saturated poroelastic medium are solved. Considering the mixed boundary value conditions at the bottom of the foundation, a pair of dual integral equations about the vertical vibration of an elastic strip footing is derived, which can be converted to a set of linear equations by means of infinite series of orthogonal functions. The relation between the dynamic compliance coefficients and the dimensionless frequency tends to be gentle with decreasing footing rigidity, while the dimensionless frequency has only small effect on the dynamic compliance coefficients. When the dynamic permeability is large, its effect on the dynamic compliance coefficients should be taken into consideration. Furthermore, the dynamic compliance coefficients are found to be not sensitive to Poisson's ratio of the soil for footing on saturated soil. Copyright 2007 John Wiley & Sons, Ltd. [source]


Kinematic and dynamic analysis of open-loop mechanical systems using non-linear recursive formulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2006
Yunn-Lin Hwang
Abstract In this paper, a non-linear recursive formulation is developed for kinematic and dynamic analysis of open-loop mechanical systems. The non-linear equations of motion are developed for deformable links that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial co-ordinates as well as the assumed displacement field, and these time invariant quantities represent the dynamic coupling between the rigid-body modes and elastic deformations. A new recursive formulation is presented for solving equations of motion for open-loop chains consisting of interconnected rigid and deformable open-loop mechanical systems. This formulation is expressed by the recursive relationships and the generalized non-linear equations for deformable mechanical systems to obtain a large system of loosely coupled equations of motion. The main processor program consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic co-ordinates. Simultaneously, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic co-ordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The solution techniques used in this investigation yield a much smaller operations count and can more efficiently implement in any computer. The algorithms and solutions presented in this paper are illustrated by using an industrial robotic manipulator system. The numerical results using this formulation are also presented and discussed in this paper. Copyright 2006 John Wiley & Sons, Ltd. [source]


Modified method of characteristics for solving population balance equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2003
Laurent Pilon
Abstract This paper presents a new numerical method for solving the population balance equation using the modified method of characteristics. Aggregation and break-up are neglected but the density function variations in the three-dimensional space and its dependence on the external fields are accounted for. The method is an interpretation of the Lagrangian approach. Based on a pre-specified grid, it follows the particles backward in time as opposed to forward in the case of traditional method of characteristics. Unlike the direct marching method, the inverse marching method uses a fixed grid thus, making it compatible with other numerical schemes (e.g. finite-volume, finite elements) that may be used to solve other coupled equations such as the mass, momentum, and energy conservation equations. The numerical solutions are compared with the exact analytical solutions for simple one-dimensional flow cases. Very good agreement between the numerical and the theoretical solutions has been obtained confirming the validity of the numerical procedure and the associated computer program. Copyright 2003 John Wiley & Sons, Ltd. [source]


Phenomenological description of polarization switching in ferroelectric semiconductors with charged defects

PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 4 2005
Anna N. Morozovska
Abstract We have proposed the phenomenological description of polarization switching peculiarities in ferroelectric semiconductors with charged defects and prevailing extrinsic conductivity. More precisely, we have modified the Landau,Ginsburg approach and shown that the macroscopic state of the aforementioned inhomogeneous system can be described by three coupled equations for three order parameters. Both the experimentally observed coercive field values, well below the thermodynamic one, and the various hysteresis loop deformations (minor, constricted and double loops) have been obtained in the framework of our model. The obtained results qualitatively explain the ferroelectric switching in such bulk ferroelectric materials as SBN single crystals doped with Ce, lead zirconate titanate (PZT) films doped with Nd, and La-doped PZT ceramics. ( 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Coupling Techniques for Thermal and Mechanical Fluid-Structure-Interactions in Aeronautics

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
Matthias Haupt
For the coupled thermal and mechanical analysis of spacecraft structures a simulation environment was developed containing the necessary coupling techniques. The numerical concept uses the weak form of the interface conditions on the coupling surface. The iterative solution of the coupled equations is based on the classical Dirichlet-Neumann approach. Transient problems are handled with iterative staggered schemes. A flexible component-based software environment combines existing fluid and structural analysis codes. Aspects of the architecture and its implementation are described. Finally an application to a spacecraft structure is shown. ( 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]