Optical tweezers of programmable shape with transverse scattering forces

We propose a non-holographic method to create line traps of arbitrary shape in the sample plane. Setting the phase gradient along theses lines gives control over the transverse forces acting on the confined particles. Phase structures, displayed on a spatial light modulator, are optically processed by a spiral phase filter and imaged onto the object plane of a microscope objective. The resulting bright line structures can be used to trap microparticles. Additionally, they exert transverse scattering forces, which can be exploited for inducing orbital motions or for creating “attracting” or “repelling” points, respectively. We give theoretical and experimental evidence that these scattering forces are proportional to the curvature of the line tweezers.     

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L₂ norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.     

Estimation of curl in paper using a combination of shape measurement and least-squares modelling

Curl is a quality problem that makes paper less suitable for printing. A paper sheet that has structural variations in its thickness such as gradients of fibre orientation, density and filler content, will curl and hence assume a cylindrical shape when its humidity content is changed. We propose a method to measure curl that can be used for automated analysis of the paper quality. The shape of the curled paper is measured using a stereoscopic camera system, which is capable of viewing a random pattern that is projected onto the specimen. The shape of the object is found by calculating the perspective difference in the camera set-up using digital image correlation. The quality parameters that are searched for are the magnitude of curl, which is defined as the inverse of the radius of curvature and the orientation of the curled paper. These parameters are estimated by performing a least-squares fit of a cylindrical shape to the three-dimensional measurement data. The least-squares model is non-linear and an iterative technique based on the Gauss–Newton algorithm is used.     

Surface shape estimation from photometric images

One of the most studied techniques for recovering surface shapes using a computer vision system is the photometric. This method is based on the analysis of one or several images of an object illuminated from a known direction . This kind of images can be considered reflectance maps that give us information of the surface gradient. In computer vision, the problem of recovering 3D information from shaded images is considered an inverse problem. To integrate surface gradient information it is proposed a regularization technique that gives a stable solution of the inverse problem and allows the possibility of reducing errors caused by noise. Results applied on synthetic and real experimental images are presented.     

Level set method for the inverse elliptic problem in nonlinear electromagnetism

An inverse problem of inhomogeneity identification inside a nonlinear magnetic material from the local measurements of the magnetic induction is investigated. The representation of the shape of the inhomogeneity and its evolution during an iterative reconstruction process is achieved by the level set method. The reconstruction is realized by the minimization of a cost function using the steepest descent method. The gradient directions are evaluated using the sensitivity equation and the adjoint variable method. Simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities, convex and non-convex, as well as multiple inhomogeneities.     

Characterization of phase transformation in shape memory alloys by atomic force microscope

An atomic force microscope operated at various temperatures is introduced to evaluate phase transformation temperature and to observe microstructure for a shape memory alloy at same time in this paper. A commercial hot-rolled TiNi shape memory alloy bar is ground, polished and etched. The specimen is then observed by atomic force microscopy at the temperature range of 20–100 C in nitrogen gas. The topographies of a TiNi specimen show twinning martensite with rough surface and smooth austenite at various temperatures. The shape memory effect of the TiNi alloy is analyzed based on the shifts of the topographies obtained at various temperatures, which are used to evaluate the phase transformation temperature between martensite and austenite. The phase transformation temperature is also confirmed in a differential scanning calorimeter (DSC) experiment.     

Stable forms of two-dimensional crystals and graphene

We show that the “two-dimensional” graphene is stable due to transverse short-range displacements of carbon atoms, which may be described in a framework of Ising model with competing interactions. When temperature decreases, two transitions, high temperature disorder into order and order into low-temperature glass, arise. The graphene looks like a microscopic “washboard” with the wavelength of about 2–4 Å. Due to up–down asymmetry of the lattice distortions in graphene on substrate, a mini-bandgap arises. This leads to many new phenomena: a rectification of AC current induced by microwave or infrared radiation, the existence of self-trapping and a new type of fermionic mini-exciton-polaritons.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties

This study is devoted to the derivation of some properties of the von Kármán equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions—with vanishing prescribed forces and displacements—are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called “triple self-adjointness”, is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

力梯度辛方法在圆型限制性三体问题中的应用

旋转坐标系下的圆型限制性三体问 题因含非惯性系所附加的影响部分使得动能不是动量的严格二次型, 可能导致力梯度辛积分算法的应用遇到困难. 从Lie算子运算出发, 严格论证了力梯度算子在这种情形下的物理意义 仍然像质心惯性坐标系下的圆型限制性三体问题那样是引力的梯度, 而不是引力与非惯性力所得合力的梯度, 表明了力梯度辛方法适合求解旋转坐标系下的圆型限制性三体问题. 通过应用四阶力梯度辛方法、最优化四阶力梯度辛方法和Forest-Ruth 辛方法分别求解该问题, 进行了数值对比研究, 结果显示最优化型力梯度算法能够取得最好精度. 还应用最优化型算法计算两邻近轨道的Lyapunov指数和快速Lyapunov指标, 确保高精度辛方法能够贯穿于这些混沌指标计算的全过程, 以便准确刻画此系统的动力学定性性质. 关键词： 辛积分器 圆型限制性三体问题 混沌 Lyapunov指数     

Theory of the line shape of gamma-photon transitions

There is a discussion of the line shape for the nuclear two-quantum transition due to the simultaneous absorption of a Mossbauer quantum and a quantum from a classical photon source. The problem is solved through the use of density-matrix equations, and an equation is derived for the intensity of absorbed gamma rays. In contrast with the case of the ordinary probabilistic approach, the equation for the intensity of gamma absorption reveals a dependence on the relaxation times within the ground and excited states. The line shape for gamma-photon transitions for low intensities of the incident radiation usually differs little from the shape of the Mossbauer line. However, at large irradiation intensities, a shift and a broadening of the resonant lines occur, proportional to the radiation power. These corrections must be taken into account in, e.g., gamma-magnetic resonance.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 12, No. 7, pp. 24–30, July, 1969.     

Damage and fracture evolution in brittle materials by shape optimization methods

This paper is devoted to a numerical implementation of the Francfort–Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.     

Diffusion pore imaging with generalized temporal gradient profiles

In porous material research, one main interest of nuclear magnetic resonance diffusion (NMR) experiments is the determination of the shape of pores. While it has been a longstanding question if this is in principle achievable, it has been shown recently that it is indeed possible to perform NMR-based diffusion pore imaging. In this work we present a generalization of these previous results. We show that the specific temporal gradient profiles that were used so far are not unique as more general temporal diffusion gradient profiles may be used. These temporal gradient profiles may consist of any number of “short” gradient pulses, which fulfil the short-gradient approximation. Additionally, “long” gradient pulses of small amplitude may be present, which can be used to fulfil the rephasing condition for the complete profile. Some exceptions exist. For example, classical q-space gradients consisting of two short gradient pulses of opposite sign cannot be used as the phase information is lost due to the temporal antisymmetry of this profile.     

Alternate stresses and temperature variation as factors of influence of ultrasonic vibration on mechanical and functional properties of shape memory alloys

It is known that the main factors in a variation in the shape memory alloy properties under insonation are heating of the material and alternate stresses action. In the present work the experimental study of the mechanical behaviour and functional properties of shape memory alloy under the action of alternate stresses and varying temperature was carried out. The data obtained had demonstrated that an increase in temperature of the sample resulted in a decrease or increase in deformation stress depending on the structural state of the TiNi sample. It was shown that in the case of the alloy in the martensitic state, a decrease in stress was observed, and on the other hand, in the austenitic state an increase in stress took place. It was found that action of alternate stresses led to appearance of strain jumps on the strain–temperature curves during cooling and heating the sample through the temperature range of martensitic transformation under the constant stress. The value of the strain jumps depended on the amplitude of alternate stresses and the completeness of martensitic transformation. It was shown that the heat action of ultrasonic vibration to the mechanical behaviour of shape memory alloys was due to the non-monotonic dependence of yield stress on the temperature. The force action of ultrasonic vibration to the functional properties was caused by formation of additional oriented martensite.     

New implicitly solvable potential produced by second order shape invariance

The procedure proposed recently by Bougie et al. (2010) to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.     

The Gravitating Perfect Fluid-Scalar Field Equations: Quintessence and Tachyonic Matter

The system consisting of a self gravitating perfect fluid and scalar field is considered in detail. The scalar fields considered are the quintessence and tachyonic forms which have important application in cosmology. Mathematical properties of the general system of equations are studied including the algebraic and differential identities as well as the eigenvalue structure. The Cauchy problem for both quintessence and the tachyon is presented. We discuss the initial constraint equations which must be satisfied by the initial data. A Cauchy evolution scheme is presented in the form of a Taylor series about the Cauchy surface. Finally, a simple numerical example is provided to illustrate this scheme.     

A simple algorithm for spectral line deconvolution

The objective of this work is to develop a numerical procedure to subtract the instrumental function from a measured spectral line profile. The measuring device (for example, a Fabry-Perot Interferometer) distorts the spectral line profile and the experimentally measured one is a convolution of this profile and the instrumental function. Restoring the spectral line profile is strongly affected by numerical instabilities and the problem has been overcome by using the Tikhonov regularization method. The approach is very simple and easy for programming and it is particularly useful for “noisy” experimental data.