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Directional Derivatives (directional + derivative)
Selected AbstractsOn the differentiation of the Rodrigues formula and its significance for the vector-like parameterization of Reissner,Simo beam theoryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2002M. Ritto-Corręa Abstract In this paper we present a systematic way of differentiating, up to the second directional derivative, (i) the Rodrigues formula and (ii) the spin-rotation vector variation relationship. To achieve this goal, several trigonometric functions are grouped into a family of scalar quantities, which can be expressed in terms of a single power series. These results are then applied to the vector-like parameterization of Reissner,Simo beam theory, enabling a straightforward derivation and leading to a clearer formulation. In particular, and in contrast with previous formulations, a relatively compact and obviously symmetric form of the tangent operator is obtained. The paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples. Copyright © 2002 John Wiley & Sons, Ltd. [source] Traces of Sobolev functions with one square integrable directional derivativeMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2006M. Gregoratti Abstract We consider the Sobolev spaces of square integrable functions v, from ,n or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ,,,,v. In these spaces we define closed trace operators on the boundaries ,Q and on the hyperplanes {r,, = z}, z , ,\{0}, which turn out to be possibly unbounded with respect to the usual L2 -norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L2, we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ,n and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd. [source] 2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2010Akpofure E. Taigbenu Abstract This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six-node quadratic triangular and the four and eight-node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] Multiscalet basis in Galerkin's method for solving three-dimensional electromagnetic integral equationsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 4 2008M. S. Tong Abstract Multiscalets in the multiwavelet family are used as the basis and testing functions in Galerkin's method. Since the multiscalets are orthogonal to their translations under the Sobolev inner product, the resulting Galerkin's method behaves like a collocation method but possesses the ability of derivative tracking for unknown functions in solving integral equations. The former makes the method simple in implementation and the latter allows to use coarse meshes in discretization. These robust features have been demonstrated in solving two-dimensional (2D) electromagnetic (EM) problems, but have not been exploited in three-dimensional (3D) scenarios. For 3D problems, the unknown functions in the integral equations are dependent on two coordinate variables. In order to preserve the use of coarse meshes for 3D cases, we realize the omnidirectional derivative tracking by tracking the directional derivatives along two orthogonal directions, or equivalently tracking the gradient. This process yields a nonsquare matrix equation and we use the least-squares method (LSM) to solve it. Numerical examples show that the multiscalet-based Galerkin's method is also robust in solving for 3D EM integral equations with a minor cost increase from LSM. Copyright © 2007 John Wiley & Sons, Ltd. [source] Differentiability of Lieb functional in electronic density functional theoryINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 10 2007Paul E. Lammert Abstract A solid understanding of the Lieb functional FL is important because of its centrality in the foundations of electronic density functional theory. A basic question is whether directional derivatives of FL at an ensemble-V-representable density are given by (minus) the potential. A widely accepted purported proof that FL is Gâteaux differentiable at EV-representable densities would say, "yes." But that proof is fallacious, as shown here. FL is not Gâteaux differentiable in the normal sense, nor is it continuous. By means of a constructive approach, however, we are able to show that the derivative of FL at an EV-representable density ,0 in the direction of ,1 is given by the potential if ,0 and ,1 are everywhere strictly greater than zero, and they and the ground state wave function have square integrable derivatives through second order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source] The particular solutions for thin plates resting on Pasternak foundations under arbitrary loadingsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010Chia-Cheng Tsai Abstract Analytical particular solutions of splines and monomials are obtained for problems of thin plate resting on Pasternak foundation under arbitrary loadings, which are governed by a fourth-order partial differential equation (PDEs). These analytical particular solutions are valuable when the arbitrary loadings are approximated by augmented polyharmonic splines (APS) constructed by splines and monomials. In our derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators whose particular solutions are known in literature. Then, we use the difference trick to recover the analytical particular solutions of the original operator. In addition, we show that the derived particular solution of spline with its first few directional derivatives are bounded as r , 0. This solution procedure may have the potential in obtaining analytical particular solutions of higher order PDEs constructed by products of Helmholtz-type operators. Furthermore, we demonstrate the usages of these analytical particular solutions by few numerical cases in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] Image analysis using p -Laplacian and geometrical PDEsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007A. KuijperArticle first published online: 29 FEB 200 Minimizing the integral ,,1/p |,L |pd , for an image L under suitable boundary conditions gives PDEs that are well-known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p -Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||