Small Number Of Discretization Points Research Articles

A numerical scheme based on the collocation and optimization methods for accurate solution of sensitive boundary value problems

Despite the significant advances in the numerical solution of nonlinear boundary value problems, most of the existing methods still encounter with a high sensitivity to the initial guess. The aim of this paper is to propose a less sensitive robust numerical scheme for accurate solution of sensitive boundary value problems. For this purpose, an orthogonal collocation approach for discretization of the problem is utilized. Thereby, the problem is converted to the solution of nonlinear algebraic equations. However, due to the difficulties of solving the obtained system of nonlinear equations, particularly in providing the proper initial guess, the obtained system of equations is transferred to an optimization problem in which the values of the solution at collocation points are considered as decision parameters. The method finds good results even using not good initial guess for decision parameters and even using a small number of discretization points. Numerical results of five benchmark examples are presented and the efficiency of the method is reported.

Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Robin boundary conditions. For the reconstruction, the partial differential equation is solved by a semi-discrete method that combines a pseudospectral collocation method for spatial variables and Crank–Nicolson for time. The conductivity tensor is reconstructed through a non-linear least-squares problem solved by Levenberg–Marquardt method (LMM), along with Morozov's discrepancy principle as stopping rule to cope with noise in the data. Unlike classic LMM implementations that mitigate poor conditioning in calculating iterates using nonsingular diagonal scaling matrices, in this paper, singular regularization matrices are used. The impact of such a modification is illustrated with numerical experiments using discrete differential operators as scaling matrices. Numerical results show that accurate conductivity values can be obtained using a fairly small number of discretization points at a very low computational cost.

Multiscale modeling of embedded graphene sheets based on the higher-order Cauchy-Born rule: Nonlinear static analysis

The nonlinear flexural response of single-layer graphene sheets (SLGSs) resting on elastic matrix is studied using an atomistic-based second gradient continuum model. The higher-order Cauchy-Born rule is used to link the interatomic potential to the strain energy induced in the continuum without any parameter fitting. The graphene is modeled by a hyperelastic membrane whose elastic potential energy is exclusively written in terms of the interatomic potential. This results in a constitutive model independent of any additional phenomenological input and thickness. Moreover, through this linkage, both the material and geometrical nonlinearities are exactly reflected in the constitutive model. To solve the continuum boundary value problem, the differential quadrature (DQ) approach is employed in the context of a variational formulation, and the discretized weak form of the equilibrium equation is obtained. The static response of SLGSs under a uniformly distributed load is evaluated. It is found that the present multiscale model can reproduce the results of other coupled atomistic-continuum and full atomistic approaches with a small number of discrete points. Also, the effect of the second-order deformation gradient is found to be significant on the bending deflection of SLGS specifically on the one with high flexural stiffness.

Rapid Path Planning for Zero-Propellant Maneuvers

AbstractThe zero-propellant maneuver (ZPM) is an advanced space station large-angle attitude maneuver technique using control momentum gyroscopes (CMGs), and rapid path planning (RPP) enhances its performance. This paper proposes a novel trajectory optimization technique—the RPP method—to generate the ZPM path. The core of the RPP method is the reconstruction of a continuous-time feasible solution upon the rapid computation of the nonlinear programming (NLP) problem. The method maintains the sparsity of the transcribed NLP and achieves the continuous-time feasibility of the solution with a small number of discretization points. Thus, rapid acquisition of a suboptimal solution may be achieved. The performance of the RPP method is demonstrated by showing that continuous-time feasible ZPM paths with high performance index may be obtained rapidly.

An efficient pseudospectral method for numerical solution of nonlinear singular initial and boundary value problems arising in astrophysics

In the manuscript, a pseudospectral method is developed for approximate and efficient solution of nonlinear singular Lane–Emden–Fowler initial and boundary value problems arising in astrophysics. In the proposed method, the Gauss pseudospectral method is utilized to reduce the problem to the solution of a system of algebraic equations. Furthermore, the Gauss pseudospectral method is developed for finding the first zero of the solution of this equation that gives the radius of the star, in which the numerous properties of the star such as mass, central pressure, and binding energy can be computed through their relations to this solution. The main advantage of the proposed method is that good results are obtained even by using a small number of discretization points and the rate of convergence is high. The accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.

RETRACTED: Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory

In this study, we propose an efficient pseudospectral method for solving two-point boundary value problems in optimal control theory. In our proposed approach, the Gauss pseudospectral method is utilized to reduce a two-point boundary value problem into the solution of a system of algebraic equations. However, the convergence to the solution of the system of equations obtained may be slow, or it can even fail, if a very good initial estimate of the optimal solution is not available. To overcome this drawback, we employ a numerical continuation method, which resolves the sensitivity of the proposed method to the initial estimate. The main advantages of the present combined method are that good results are obtained even when using a small number of discretization points, while the sensitivity to the initial estimate when solving the final system of algebraic equations is resolved successfully. The proposed method is especially useful when shooting methods fail due to the sensitivity or stiffness of the problem. We present numerical results for two examples to demonstrate the efficiency of the combined method.

Pseudospectral modal method for conical diffraction of gratings

For lamellar gratings and other layered periodic structures, the modal methods (including both analytic and numerical ones) are often the most efficient, since they avoid the discretization of one spatial variable. The pseudospectral modal method (PSMM) previously developed for in-plane diffraction problems of one-dimensional gratings achieves high accuracy for a small number of discretization points, and it outperforms most other modal methods. In this paper, an extension of the PSMM to conical diffraction problems is presented and implemented. Numerical examples are used to demonstrate the high accuracy and excellent convergence property of this method for both dielectric and metallic gratings.

Model based precision structural measurements on barely resolved objects

A model based method for the accurate quantification of the 3D structure of fluorescently labelled cellular objects similar in size to the optical resolution limit is presented. This method is applied to both simulated confocal images of chromatin structures and to real confocal data obtained on a Fluorescence in situ Hybridization (FISH) labelled gene domain. The model assumes that the object is composed of a small number of discrete points which are convolved with the microscope point spread function to give the image. Fitting this model to image data results in a method to assess object structure which is accurate, shows a low bias, and does not require user intervention or the potentially subjective setting of a threshold.

Efficient method for analyzing leaky cavities in two-dimensional photonic crystals

An efficient numerical method is developed for analyzing leaky cavities in two-dimensional photonic crystals, assuming that the cavities are located in the vicinity of a photonic crystal waveguide or close to a homogeneous medium. A leaky cavity mode is characterized by its complex eigenfrequency and by the outgoing wave behavior of the field in the nearby waveguide or homogeneous medium. In our method, the eigenfrequency of a leaky cavity mode is determined from a condition formulated on one single edge of a defect unit cell. Such a condition is obtained by using rigorous boundary conditions to terminate photonic crystal waveguides and homogeneous media, Dirichlet-to-Neumann maps of the unit cells to reduce the problem to the cell edges, and standard Gaussian elimination to further reduce the problem to one single edge. The reduction process can be performed efficiently. Accurate solutions can be obtained when a small number of discrete points are used on each edge of the unit cells.

High Frequency Quantitative Photoelasticity Applied to Jet Engine Components

For sometime, reflection photoelasticity has been used in a qualitative manner as an aid to the placement of strain gauges in vibration tests on turbine and compressor blades. Often, the motivation for such tests is the validation of numerical models used in life-time predictions. Strain gauges supply data at a small number of discrete points, whereas the photoelastic fringe patterns provide information over the whole field of view. Digital photoelasticity based on phase-shifting allows these fringe patterns to be processed into maps of surface strain data that can be used to verify the computational results. Digital photoelasticity has been developed over the last two decades utilising spectral or phase approaches sometimes combined with Fourier analysis; and in most reported applications only idealised case studies are considered. Recently, the application of digital photoelasticity in high frequency blade tests has been developed as a robust methodology that can be applied using standard equipment or using solid-state polariscopes. The paper includes descriptions of the methodology and exemplar results to demonstrate the efficacy of this advanced application of photoelasticity for full-field validations of numerical modelling.

Beyond mean field study of excited states: Analysis within the Lipkin model

We compare the generator coordinate method (GCM) and the random-phase approximation (RPA) in the framework of the exactly solvable Lipkin-Meshkov-Glick model. We show that the discretized GCM works quite well and permits to obtain results close to the exact results with a small number of discretization points.

A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag

We introduce a modified Milstein scheme for pathwise approximation of scalar stochastic delay differential equations with constant time lag on a fixed finite time interval. Our algorithm is based on equidistant evaluation of the driving Brownian motion and is simply obtained by replacing iterated Itô-integrals by products of appropriate Brownian increments in the definition of the Milstein scheme. We prove that the piecewise linear interpolation of the modified Milstein scheme is asymptotically optimal with respect to the mean square L 2 -error within the class of all pathwise approximations that use observations of the driving Brownian motion at equidistant points. Moreover, for a large class of equations our scheme is also asymptotically optimal for mean square approximation of the solution at the final time point. Our asymptotic optimality results are complemented by a comparison with the Euler scheme based on exact error formulas for a linear test equation. This comparison demonstrates the superiority of the modified Milstein scheme even for a very small number of discretization points. Finally, we provide a generalization of our approach to the case of a system of SDDEs with an arbitrary finite number of constant delays. We conjecture that the above optimality results carry over to the generalized scheme.

Discrete Finger and Palmar Feature Extraction for Personal Authentication

A practical method for a noncontacting and real-time feature extraction for personal authentication is proposed. The finger geometry and feature extraction of the palmar flexion creases are integrated into a small number of discrete points based on the anatomical observation. For a video image of either palm, a palm placed freely facing toward a video camera is acquired. The fingers are brought together, and the palm is straightened out to eliminate any constraints. The discrete feature points for the fingers involve intersection points of the three finger (digital) flexion creases on the four finger skeletal lines. The feature points for the palm involve intersection points of the major palmar flexion creases and/or prominent creases of the palm on the extended finger skeletal lines. The orientations of the creases at the intersection points are also extracted features to be matched. The matching results are perfect for about 500 palm samples from 50 subjects so far. This discrete point processing, requiring no time-consumptive palmprint image analysis and requiring less than one second processing time, will contribute to a noncontacting, real-time and reliable feature extraction, easily combinable with other traits, for the personal authentication.

The Influence of Charge Variation on the Adsorbed Configuration of a Model Cationic Oligomer onto Colloidal Silica

The adsorbed amounts and interfacial conformations of a low-molecular-weight oligomer of a weak cationic polyelectrolyte (15-unit chain of dimethylaminoethyl methacrylate, DMAEMA) on colloidal silica were examined, with the ultimate intent of providing perspective on the adsorption of high-molecular-weight weak polyelectrolytes. At all but the lowest ionic strengths or highest pHs, over the full range of adsorbed oligomer amounts on each isotherm, the interfacial conformation was relatively insensitive to coverage with train fractions from 0.8 to 1. This occurs because short oligomeric chains are not capable of forming long loops or tails found in higher-molecular-weight homopolymer layers. Variations in pH altered the backbone and surface charge densities, changing the density of contact points for adsorption. This was apparent in the NMR solvent relaxation behavior, which suggested the tightest binding near pH 6, and a sharp drop in the train fraction at high pH, just as the adsorption began to diminish with reduced backbone protonation. Variations in ionic strength screened repulsions among adsorbed oligomers, an effect most apparent at low ionic strengths. NMR solvent relaxation data provided substantial evidence for differences in the anchoring of DMAEMA oligomer and adsorbed trains of a nonionic homopolymer such as polyethylene oxide (T. Cosgrove, M. A. Cohen Stuart, and G. P. van der Beek, Langmuir7, 327 (1991)). DMAEMA adsorbs at a small number of discrete points on side chains, while with polyethylene oxide the literature suggested every monomer on the main backbone can potentially adhere to the surface, giving a less mobile interface from the perspective of the solvent.

Higher-order variational finite difference schemes for solving 3-D paraxial wave equations using splitting techniques

Numerical schemes for solving 3-D paraxial equations are constructed using splitting techniques. The solution can be reduced to a series of 2-D paraxial equations in each direction of splitting. The discretization along the depth is based on higher-order conservative schemes. The discretization along the transverse variables is based on higher-order finite difference variational schemes. Numerical experiments illustrate the advantages of higher-order schemes, which are much less dispersive, even for a small number of discretization points per wavelength.

Local parabolic reference approximation of thermal Feynman path integrals in quantum Monte Carlo simulations

We have developed a new propagator, called the local parabolic reference (LPR), for use in the numerical evaluation of discretized Feynman path integrals by Metropolis Monte Carlo simulations. The form of the propagator is motivated by fitting a local quadratic reference potential (with positive, negative or zero curvature) to the potential energy surface of interest, and constructing the exact propagator for this reference potential. The final form of the propagator contains adjustments designed to eliminate artifacts that can develop at very low temperatures. In the low temperature regime, the approximation accommodates tunneling and zero-point motion with a small number of discretization points in the path integral. In the limit of high temperature, the LPR propagator approaches the form of the standard high temperature propagator. Both the single- and multi-dimensional formulations are discussed in this paper. The accuracy of the Monte Carlo path integrals is demonstrated in the calculation of the equilibrium average potential energies for a set of model systems with one degree of freedom, and for a system of ten coupled double-well oscillators. Also, for a one-dimensional quartic oscillator system, the LPR approximation results are compared with those of the approximations of Messina, Garrett and Schenter [J. Chem. Phys. 100, 6570 (1994)], Mak and Andersen [J. Chem. Phys. 92, 2953 (1990)], and Zhang, Levy and Freisner [Chem. Phys. Lett. 144, 236 (1988)]. It is anticipated that this approach to constructing propagators will be useful for multi-dimensional barrier-crossing problems.

Improved finite difference calculation of magnetic field effects; inclusion of analytic asymptotics

Abstract In the usual finite difference calculations of magnetic field effects the accuracy decreases and the number of required discretization points increases with decreasing magnetic interaction. By matching the numerical results with the asymptotic analytic solution we obtain a consistently high accuracy for a very small number of discretization points (of the order of 10) independent of the size of the magnetic interaction. The matching is incorporated into the standard algorithm by changing one of the boundary conditions which becomes spin dependent. A simple rule is given for the optimal position of the matching boundary condition. The method is tested both for free diffusion and for diffusion in a strong Coulombic field.

On some representations of solutions of dynamical problems in elasticity theory

The divergence and rotation of a translation vector field, which are solutions of wave equations, are used as auxiliary potentials for numerical solution of dynamical problems in elasticity theory. In a particular case, the problems of longitudinal translation are tested in two variants of numerical algorithms with their programming realization, some suggestions are given regarding the choice of the optimal algorithm for achieving highly precise calculations with a small number of discretization points.

A Simple Explicit form of Finite Differences to Compute the Effective Gain in Degenerate Two-wave Mixing; Comparison with the Shooting Method

Abstract A numerical study is presented of the effective gain in degenerate two-wave mixing, for a reflection geometry in a photorefractive crystal. Apart from the so-called ‘shooting method’ we employ a simple explicit method of ‘finite differences’ to obtain numerical solutions of the nonlinear coupled-wave equations. A comparison between the two indicates that the computed results from the finite differences method are still in reasonable agreement with those obtained from the shooting method. At the same time the finite differences method is more straightforward and usually found to be faster than the shooting method for a small number of discrete points. The dependence of the effective gain on various parameters is shown at the saturation as well as non-saturation (i.e. below saturation) stage of the refractive-index modulation. A significant difference in gain is found to exist between these two regimes of modulation. Numerical results are compared and presented as graphs.