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Gauss' Theorem (gauss + theorem)
Selected AbstractsA Taylor series-based finite volume method for the Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008G. X. Wu Abstract A Taylor series-based finite volume formulation has been developed to solve the Navier,Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible. Copyright © 2008 John Wiley & Sons, Ltd. [source] Investigation of nanoscale electrohydrodynamic transport phenomena in charged porous materialsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005P. Pivonka Abstract Depending on the permeability of porous materials, different mass transport mechanisms have to be distinguished. Whereas mass transport through porous media characterized by low permeabilities is governed by diffusion, mass transport through highly permeable materials is governed by advection. Additionally a large number of porous materials are characterized by the presence of surface charge which affects the permeability of the porous medium. Depending on the ion transport mechanism various phenomena such as co-ion exclusion, development of diffusion,exclusion potentials, and streaming potentials may be encountered. Whereas these various phenomena are commonly described by means of different transport models, a unified description of these phenomena can be made within the framework of electrohydrodynamics. In this paper the fundamental equations describing nanoscale multi-ion transport are given. These equations comprise the generalized Nernst,Planck equation, Gauss' theorem of electrostatics, and the Navier,Stokes equation. Various phenomena such as the development of exclusion potentials, diffusion,exclusion potentials, and streaming potentials are investigated by means of finite element analyses. Furthermore, the influence of the surface charge on permeability and ion transport are studied in detail for transient and steady-state problems. The nanoscale findings provide insight into events observed at larger scales in charged porous materials. Copyright © 2005 John Wiley Sons, Ltd. [source] An accurate gradient and Hessian reconstruction method for cell-centered finite volume discretizations on general unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010Lee J. Betchen Abstract In this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell-centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second-order accurate estimate of the gradient, along with a first-order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods. Copyright © 2009 John Wiley & Sons, Ltd. [source] Proof of a decomposition theorem for symmetric tensors on spaces with constant curvatureANNALEN DER PHYSIK, Issue 8 2008N. Straumann Abstract In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses , beside the Hodge decomposition for one-forms , an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations. [source] Exact transformation of a wide variety of domain integrals into boundary integrals in boundary element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008M. R. Hematiyan Abstract In this paper, a sufficient condition for transforming domain integrals into boundary integral is described. The transformation is accomplished by Green's and Gauss' theorems. It is shown that a wide range of domain integrals including some integrals in boundary element method satisfy this sufficient condition and can be simply transformed into boundary. Although emphasis is made on potential and elastostatic problems, this method can also be used for many other applications. Using the present method, a wide range of 2D and 3D domain integrals over simply or multiply connected regions can be transformed exactly into the boundary. The resultant boundary integrals are numerically evaluated using an adaptive version of the Simpson integration method. Several examples are provided to show the efficiency and accuracy of the present method. Copyright © 2007 John Wiley & Sons, Ltd. [source] | ||||||||||