Exponential compact higher order scheme and their stability analysis for unsteady convection-diffusion equations

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Peclet number 0 <= Pe <= 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

On the Erdos-Szekeres n-interior-point problem

The n-interior-point variant of the Erdos Szekeres problem is the following: for every n, n >= 1, does there exist a g(n) such that every point set in the plane with at least g(n) interior points has a convex polygon containing exactly n interior points. The existence of g(n) has been proved only for n <= 3. In this paper, we show that for any fixed r >= 2, and for every n >= 5, every point set having sufficiently large number of interior points and at most r convex layers contains a subset with exactly n interior points. We also consider a relaxation of the notion of convex polygons and show that for every n, n >= 1, any point set with at least n interior points has an almost convex polygon (a simple polygon with at most one concave vertex) that contains exactly n interior points. (C) 2013 Elsevier Ltd. All rights reserved.

Correspondence between neutron depolarization and higher order magnetic susceptibility to investigate ferromagnetic clusters in phase separated systems

It is a tough task to distinguish a short-range ferromagnetically correlated cluster-glass phase from a canonical spin-glass-like phase in many magnetic oxide systems using conventional magnetometry measurements. As a case study, we investigate the magnetic ground state of La0.85Sr0.15CoO3, which is often debated based on phase separation issues. We report the results of two samples of La0.85Sr0.15CoO3 (S-1 and S-2) prepared under different conditions. Neutron depolarization, higher harmonic ac susceptibility and magnetic relaxation studies were carried out along with conventional magnetometry measurements to differentiate subtle changes at the microscopic level. There is no evidence of ferromagnetic correlation in the sample S-2 attributed to a spin-glass phase, and this is compounded by the lack of existence of a second order component of higher harmonic ac susceptibility and neutron depolarization. A magnetic relaxation experiment at different temperatures complements the spin glass characteristic in S-2. All these signal a sharp variance when we consider the cluster-glass-like phase (phase separated) in S-1, especially when prepared from an improper chemical synthesis process. This shows that the nonlinear ac susceptibility is a viable tool to detect ferromagnetic clusters such as those the neutron depolarization study can reveal.

Probing a spin-glass state in SrRuO3 thin films through higher-order statistics of resistance fluctuations

The complex perovskite oxide SrRuO3 shows intriguing transport properties at low temperatures due to the interplay of spin, charge, and orbital degrees of freedom. One of the open questions in this system is regarding the origin and nature of the low-temperature glassy state. In this paper we report on measurements of higher-order statistics of resistance fluctuations performed in epitaxial thin films of SrRuO3 to probe this issue. We observe large low-frequency non-Gaussian resistance fluctuations over a certain temperature range. Our observations are compatible with that of a spin-glass system with properties described by hierarchical dynamics rather than with that of a simple ferromagnet with a large coercivity.

An interior penalty method for distributed optimal control problems governed by the biharmonic operator

In this paper, a C-0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement. (C) 2014 Elsevier Ltd. All rights reserved.

Stability analysis of thick piezoelectric metal based FGM plate using first order and higher order shear deformation theory

This paper presents the stability analysis of functionally graded plate integrated with piezoelectric actuator and sensor at the top and bottom face, subjected to electrical and mechanical loading. The finite element formulation is based on first order and higher order shear deformation theory, degenerated shell element, von-Karman hypothesis and piezoelectric effect. The equation for static analysis is derived by using the minimum energy principle and solutions for critical buckling load is obtained by solving eigenvalue problem. The material properties of the functionally graded plate are assumed to be graded along the thickness direction according to simple power law function. Two types of boundary conditions are used, such as SSSS (simply supported) and CSCS (simply supported along two opposite side perpendicular to the direction of compression and clamped along the other two sides). Sensor voltage is calculated using present analysis for various power law indices and FG (functionally graded) material gradations. The stability analysis of piezoelectric FG plate is carried out to present the effects of power law index, material variations, applied mechanical pressure and piezo effect on buckling and stability characteristics of FG plate.

A New Version of Smoothed Point Method to Simulate Linear Elastic Wave Propagation

By using the kernel function of the smoothed particle hydrodynamics (SPH) and modification of statistical volumes of the boundary points and their kernel functions, a new version of smoothed point method is established for simulating elastic waves in solid. With the simplicity of SPH kept, the method is easy to handle stress boundary conditions, especially for the transmitting boundary condition. A result improving by de-convolution is also proposed to achieve high accuracy under a relatively large smooth length. A numerical example is given and compared favorably with the analytical solution.

Fracture criterion based on the higher-order asymptotic fields

A complete development for the higher-order asymptotic solutions of the crack tip fields and finite element calculations for mode I loading of hardening materials in plane strain are performed. The results show that in the higher-order asymptotic solution (to the twentieth order), only three coefficients are independent. These coefficients are determined by matching with the finite element solutions carried out in the present paper (our attention is focused on the first five terms of the higher-order asymptotic solution). We obtain an analytic characterization of crack tip fields, which conform very well to the finite element solutions over wide range. A modified two parameter criterion based on the asymptotic solution of five terms is presented. The upper bound and lower bound fracture toughness curves predicted by modified two parameter criterion are given. These two curves agree with most of the experimental data and fully capture the proper trend.

Higher-order analysis of crack-tip fields in power-law hardening materials

In this paper, we present an exact higher-order asymptotic analysis on the near-crack-tip fields in elastic-plastic materials under plane strain, Mode I. A four- or five-term asymptotic series of the solutions is derived. It is found that when 1.6 < n less-than-or-equal-to 2.8 (here, n is the hardening exponent), the elastic effect enters the third-order stress field; but when 2.8< n less-than-or-equal-to 3.7 this effect turns to enter the fourth-order field, with the fifth-order field independent. Moreover, if n>3.7, the elasticity only affects the fields whose order is higher than 4. In this case, the fourth-order field remains independent. Our investigation also shows that as long as n is larger than 1.6, the third-order field is always not independent, whose amplitude coefficient K3 depends either on K1 or on both K1 and K2 (K1 and K2 arc the amplitude coefficients of the first- and second-order fields, respectively). Firmly, good agreement is found between our results and O'Dowd and Shih's numerical ones[8] by comparison.

Higher-order analysis of crack tip fields in elastic power-law hardening materials

A HIGHER-ORDER asymptotic analysis of a stationary crack in an elastic power-law hardening material has been carried out for plane strain, Mode 1. The extent to which elasticity affects the near-tip fields is determined by the strain hardening exponent n. Five terms in the asymptotic series for the stresses have been derived for n = 3. However, only three amplitudes can be independently prescribed. These are K1, K2 and K5 corresponding to amplitudes of the first-, second- and fifth-order terms. Four terms in the asymptotic series have been obtained for n = 5, 7 and 10; in these cases, the independent amplitudes are K1, K2 and K4. It is found that appropriate choices of K2 and K4 can reproduce near-tip fields representative of a broad range of crack tip constraints in moderate and low hardening materials. Indeed, fields characterized by distinctly different stress triaxiality levels (established by finite element analysis) have been matched by the asymptotic series. The zone of dominance of the asymptotic series extends over distances of about 10 crack openings ahead of the crack tip encompassing length scales that are microstructurally significant. Furthermore, the higher-order terms collectively describe a spatially uniform hydrostatic stress field (of adjustable magnitude) ahead of the crack. Our results lend support to a suggestion that J and a measure of near-tip stress triaxiality can describe the full range of near-tip states.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher-order dispersions

Spatiotemporal instabilities in nonlinear Kerr media with arbitrary higher-order dispersions are studied by use of standard linear-stability analysis. A generic expression for instability growth rate that unifies and expands on previous results for temporal, spatial, and spatiotemporal instabilities is obtained. It is shown that all odd-order dispersions contribute nothing to instability, whereas all even-order dispersions not only affect the conventional instability regions but may also lead to the appearance of new instability regions. The role of fourth-order dispersion in spatiotemporal instabilities is studied exemplificatively to demonstrate the generic results. Numerical simulations confirm the obtained analytic results. (C) 2002 Optical Society of America.

Higher order vortex gyrotropic modes in circular ferromagnetic nanodots

Magnetic vortex that consists of an in-plane curling magnetization configuration and a needle-like core region with out-of-plane magnetization is known to be the ground state of geometrically confined submicron soft magnetic elements. Here magnetodynamics of relatively thick (50-100 nm) circular Ni80Fe20 dots were probed by broadband ferromagnetic resonance in the absence of external magnetic field. Spin excitation modes related to the thickness dependent vortex core gyrotropic dynamics were detected experimentally in the gigahertz frequency range. Both analytical theory and micromagnetic simulations revealed that these exchange dominated modes are flexure oscillations of the vortex core string with n = 0,1,2 nodes along the dot thickness. The intensity of the mode with n = 1 depends significantly on both dot thickness and diameter and in some cases is higher than the one of the uniform mode with n = 0. This opens promising perspectives in the area of spin transfer torque oscillators.