We consider special basic difference equations which are related to discretizations of Schrödinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schrödinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.
lunes, 28 de diciembre de 2009
Some Basic Difference Equations of Schrödinger Boundary Value Problems
We consider special basic difference equations which are related to discretizations of Schrödinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schrödinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.
A single-node characteristic collocation method for unsteady-state convection-diffusion equations in three-dimensional spaces
We develop a nonconventional single-node characteristic collocation method with piecewise-cubic Hermite polynomials for the numerical simulation to unsteady-state advection-diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The reduction of number of nodes has great potential for problems defined on high space dimensions, which appears in such problems as quantification of uncertainties in subsurface porous media. The method developed here is easy to formulate. Numerical experiments are presented to show the strong potential of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
Explicit integration of bounding surface model for the analysis of earthquake soil liquefaction
This paper presents a new plasticity model developed for the simulation of monotonic and cyclic loading of non-cohesive soils and its implementation to the commercial finite-difference code FLAC, using its User-Defined-Model (UDM) capability. The new model incorporates the framework of Critical State Soil Mechanics, while it relies upon bounding surface plasticity with a vanished elastic region to simulate the non-linear soil response. Stress integration of constitutive relations is performed using a recently proposed explicit scheme with automatic error control and substepping, which so far has been employed in the literature only for constitutive models aiming at monotonic loading. The overall accuracy of this scheme is evaluated at element level by simulating cyclic loading along complex stress paths and by using iso-error maps for paths involving change of the Lode angle. The performance of the new constitutive model and its stress integration scheme in complex boundary value problems involving earthquake-induced liquefaction is evaluated, in terms of accuracy and computational cost, via a number of parametric analyses inspired by the successful simulation of the VELACS centrifuge Model Test No. 2 studying the lateral spreading response of a liquefied sand layer. Copyright © 2009 John Wiley & Sons, Ltd.
A partition of unity-based 'FE-Meshfree' QUAD4 element for geometric non-linear analysis
The recently published 'FE-Meshfree' QUAD4 element is extended to geometrical non-linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh-distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non-linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well-known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh-distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.
Reduction in drag and vortex shedding frequency through porous sheath around a circular cylinder
A numerical study on the laminar vortex shedding and wake flow due to a porous-wrapped solid circular cylinder has been made in this paper. The cylinder is horizontally placed, and is subjected to a uniform cross flow. The aim is to control the vortex shedding and drag force through a thin porous wrapper around a solid cylinder. The flow field is investigated for a wide range of Reynolds number in the laminar regime. The flow in the porous zone is governed by the Darcy-Brinkman-Forchheimer extended model and the Navier-Stokes equations in the fluid region. A control volume approach is adopted for computation of the governing equations along with a second-order upwind scheme, which is used to discretize the convective terms inside the fluid region. The inclusion of a thin porous wrapper produces a significant reduction in drag and damps the oscillation compared with a solid cylinder. Dependence of Strouhal number and drag coefficient on porous layer thickness at different Reynolds number is analyzed. The dependence of Strouhal number and drag on the permeability of the medium is also examined. Copyright © 2009 John Wiley & Sons, Ltd.
Discretizing two-dimensional complex fractured fields for incompressible two-phase flow
A method is introduced to discretize irregular and complex two-dimensional fractured media. The geometry of the fractured media is first analysed by searching and treating the complex configurations. Based on that, the method generated a good mesh quality and allows for including finer grids. An incompressible two-phase flow problem is solved to compare the developed method and a public method based on the approximation of a 1D fracture by the edges of a 2D finite element grid of the porous media. The comparison showed that the developed method (i) represents better the fractured domain by maintaining the geometric integrity of input surfaces and geologic data, (ii) provides, for sample and complex fractured domains, excellent and more accurate results, and (iii) is much less sensitive to the grid sizes. Furthermore, the method has to be more efficient than the other methods for transport problems and has to provide better predictable results; this is mainly based on point (ii) and because the method produces optimal triangular grids. Copyright © 2009 John Wiley & Sons, Ltd.
The characteristic-based split (CBS) meshfree method for free surface flow problems in ALE formulation
A semi-implicit characteristic-based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.
Cell face velocity alternatives in a structured colocated grid for the unsteady Navier-Stokes equations
The use of a colocated variable arrangement for the numerical solution of fluid flow is becoming more and more popular due to its coding simplicity. The inherent decoupling of the pressure and velocity fields in this arrangement can be handled via a special interpolation procedure for the calculation of the cell face velocity named pressure-weighted interpolation method (PWIM) (AIAA J. 1983; 21(11):1525-1532). In this paper a discussion on the alternatives to extend PWIM to unsteady flows is presented along with a very simple criterion to ascertain if a given interpolation practice will produce steady results that are relaxation dependent or time step dependent. Following this criterion it will be shown that some prior schemes presented as time step independent are actually not, although by using special interpolations can be readily adapted to be. A systematic way of deriving different cell face velocity expressions will be presented and new formulae free of [Delta]t dependence will be derived. Several computational exercises will accompany the theoretical discussion to support our claims. Copyright © 2009 John Wiley & Sons, Ltd.
Numerical study of electromagnetic wave propagation through layered structures with chiral media
In this paper, the characteristics of layered structures (photonic or electromagnetic bandgaps), including chiral media, are studied by means of two different numerical methods, one in the time domain (finite differences in the time domain, FDTD) and the other in the frequency domain (coupled-mode method, CMM). The results (reflection and transmission coefficients for a plane wave normally incident over a layered structure) obtained by means of both well different techniques are practically identical. Copyright © 2009 John Wiley & Sons, Ltd.
A new algorithm for the general quadratic programming problems with box constraints
Abstract In this paper, we propose a new branch-and-bound algorithm for the general quadratic problems with box constraints. We, first,
transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix.
Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular
bisection. Finally, preliminary computational results are reported.
transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix.
Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular
bisection. Finally, preliminary computational results are reported.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9358-0
- Authors
- Jianling Li, Guangxi University College of Mathematics and Information Science Nanning Guangxi 530004 China
- Peng Wang, Haikou College of Economics Department of Basic Course Wentan Road 2 Haikou Hainan 570203 China
- Lin Ma, Guangxi University College of Mathematics and Information Science Nanning Guangxi 530004 China
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
Compactly supported positive definite radial functions
Abstract We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce
a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest
ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC
k
smoothness, there is a function inC
k
(ℝ
n
), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean
distance. Another example is derived from odd-degreeB-splines.
a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest
ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC
k
smoothness, there is a function inC
k
(ℝ
n
), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean
distance. Another example is derived from odd-degreeB-splines.
- Content Type Journal Article
- DOI 10.1007/BF03177517
- Authors
- Zongmin Wu, Fudan University Department of Mathematics 200433 Shanghai China
- Journal Advances in Computational Mathematics
- Online ISSN 1572-9044
- Print ISSN 1019-7168
- Journal Volume Volume 4
- Journal Issue Volume 4, Number 1 / December, 1995
On multivariate Hermite interpolation
Abstract We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation
is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and
some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced
in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral
remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.
is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and
some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced
in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral
remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.
- Content Type Journal Article
- DOI 10.1007/BF03177515
- Authors
- Thomas Sauer, University Erlangen-Nuremberg Mathematical Institute D-91054 Erlangen Germany
- Yuan Xu, University of Oregon Department of Mathematics 97403 Eugene OR USA
- Journal Advances in Computational Mathematics
- Online ISSN 1572-9044
- Print ISSN 1019-7168
- Journal Volume Volume 4
- Journal Issue Volume 4, Number 1 / December, 1995
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