An optimized predictor–corrector scheme for fast 3D crack growth simulations

An optimized predictor–corrector scheme for the accelerated simulation of 3D fatigue crack growth is presented. Based on experimental evidence, it is assumed that the crack front shape ensures a constant energy release rate. Starting from a crack front satisfying this requirement a predictor step is performed. Usually, the new crack front does not fulfill the requirement of a constant energy release rate. Therefore, several corrector steps are needed. Within the new predictor–corrector scheme the history of crack growth is taken into account to reduce the number of corrector steps. The efficiency of the new scheme is shown on two numerical examples providing a speed up of a factor above three.     

Wide-field prime focus corrector for the Anglo-Australian telescope

A wide-field (2°) prime focus corrector has been designed and built for the Anglo-Australian Telescope. The corrector incorporates an atmospheric dispersion compensator for the spectral range 365-1014 nm. A four-element lens system is the basis of the corrector, and it provides a satisfactory state of correction over a full field of 2°. The design of the lens system is described. The choice of layout is related directly to the intrinsic properties of each component.     

用可编程ASIC实现的波前校正器过压保护网络

波前校正器是自适应光学系统中的关键部件之一，制作困难，且在加电压不当时容易损坏。本文阐述了波前校正器的过压损坏机理，提出了利用分压网络避免波前校正器过压损坏的方案，介绍了基于可编程ＡＳＩＣ的过压保护电路的实现。此电路经测试，取得了较满意的结果。     

Spherical primary telescope with aspheric correction at a small internal pupil

The aspheric plate at the center of curvature of a spherical primary is replaced by a small aspheric corrector at a minified pupil located inside a reimaging camera. The correctors are identical for each reimaging camera because the spherical aberration of the primary sphere is identical and symmetrical for all field positions. The magnitude of the field aberrations is evaluated over a range of primary focal ratios and minified pupil diameters. The major term is the increased field angle through the minified aspheric corrector. The field and chromatic aberrations in such a camera are compared with the equivalent full-aperture Schmidt corrector. Field-of-view partitioning enables each subfield to be designed for specific observational requirements, such as multiple-fiber spectrography or CCD imaging. Field partitioning is shown to be a powerful means for the replacement of the large aspheric corrector of a Schmidt telescope by a multiplicity of small reimaging subsystems. The cost to fill the typical wide field of a Schmidt telescope with reimaging modules is approximately 1% the cost of a Schmidt aspheric plate.     

Design of Achromatic Corrector Plates for Schmidt Cameras and Related Optical Systems

Abstract

An analytical method for the design of achromatic corrector plates for either classical Schmidt cameras or related types of systems is presented. The method may be applied to the design of achromatic doublet correcting plates allowing for the attainment of any order of asphericity for the aspheric surfaces of the corrector plate.     

Reflective liquid crystal wavefront corrector used with tilt incidence

To allow angular separation of the beam reflected off a liquid crystal wavefront corrector from the incident beam, it is convenient to introduce a small incident angle. This avoids using a beam splitter and the associated energy losses. The effect of the tilt incidence on the liquid crystal wavefront corrector was investigated in this paper. For a parallel aligned liquid crystal wavefront corrector, a simplified model was established and used to analyze the change of the phase modulation under the tilt incidence. The simulated results showed that the effect of the tilt incidence on the phase modulation can be ignored when the angle of tilt incidence is less than 6 degrees. The phase modulation related to the incident angle was measured and the changing trend was similar to the calculated results. The effect of the tilt incidence on the diffraction efficiency of the liquid crystal wavefront corrector was also discussed. The simulated results indicated that the reduction of the diffraction efficiency is less than 1% for incidence angles under 3 degrees. Last, a closed loop correction experiment was done with an incident angle of 1 degrees. After correction, the averaged peak to valley (PV) and root mean square (RMS) of the wavefront were down to 0.15 lambda and 0.03 lambda, respectively, and a resolvable image was acquired.     

Multi‐scale approach for simulating time‐delay biochemical reaction systems

This study presents a multi‐scale approach for simulating time‐delay biochemical reaction systems when there are wide ranges of molecular numbers. The authors construct a new efficient approach based on partitioning into slow and fast subsets in conjunction with predictor–corrector methods. This multi‐scale approach is shown to be much more efficient than existing methods such as the delay stochastic simulation algorithm and the modified next reaction method. Numerical testing on several important problems in systems biology confirms the accuracy and computational efficiency of this approach.Inspec keywords: biochemistry, delays, biological techniques, predictor‐corrector methodsOther keywords: multiscale approach, time‐delay biochemical reaction systems, predictor–corrector methods, delay stochastic simulation algorithm, modified next reaction method, numerical testing, systems biology, method accuracy, computational efficiency     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Phase-correction of turbulent distortions of an optical wave propagating under conditions of strong intensity fluctuations

Phase correction of a plane wave and a spatiolimited beam propagating through a turbulent layer of atmosphere were considered. The required adaptive corrector element size and the system bandwidth were found by numerical simulation. These requirements were determined to be the same as for a weak-intensity scintillation approximation. The size of the required segmented mirror element was found to be equal to Fried length r0, whereas the tolerable time lag was r0/V, where V is the wind velocity. However, the local slope sensors then became impractical, as did tip-tilt correction over the corrector subapertures.     

An improved numerical scheme for the sine‐Gordon equation in 2+1 dimensions

A rational approximant of order 4, which is applied to a three‐time‐level recurrence relation, is used to transform the initial/boundary‐value problem associated with the two‐dimensional sine‐Gordon (SG) equation arising in the Josephson junctions problem. The resulting non‐linear system, which is analyzed for stability, is solved using an appropriate predictor–corrector (P–C) scheme, in which an explicit scheme of order 2 is used as predictor. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. In this P–C scheme, a modification in the corrector has been proposed according to which the already evaluated corrected values are considered. The behavior of this P–C scheme is tested numerically on line and ring solitons known from the bibliography regarding the SG equation and conclusions for both undamped and damped problems are derived. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order predictor–corrector algorithms

New predictor–corrector algorithms are presented for the computation of solution paths of non‐linear partial differential equations. The predictors and the correctors are based on perturbation techniques and Padé approximants. This extends the Asymptotic Numerical Method (ANM), which is an efficient high‐order continuation technique without corrector. The efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier–Stokes equations. Many variants have been tested to establish an optimal algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

An Interior Method for Nonconvex Semidefinite Programs

In several applications, semidefinite programs arise in which the matrix depends nonlinearly on the unknown variables. We propose a new solution method for such semidefinite programs that also applies to other smooth nonconvex programs. The method is an extension of a primal predictor corrector interior method to nonconvex programs. The predictor steps are based on Dikin ellipsoids of a “convexified” domain. The corrector steps are based on quadratic subprograms that combine aspects of line search and trust region methods. Convergence results are given, and some preliminary numerical experiments suggest a high robustness of the proposed method.     

New predictor/corrector algorithms with improved energy balance for a recent formulation of dynamic vehicle/structure interaction

A new class of predictor/corrector algorithms is proposed to solve the complex system of differential equations that arises from a Galerkin spatial discretization of the equations of motion in a recent formulation of dynamic vehicle/structure interaction. The applicability of the concept of a building-block vehicle/structure interaction model developed in our previous work—where the vehicle nominal motion is not prescribed a priori, but is part of the unknown motion of the system—is demonstrated through the construction of a simple vehicle model. In the new algorithms, the presence of the accelerations of the vehicle component is eliminated in the predictor structural equations, making these equations different from the corrector structural equations. The special treatment of the predicted axial motion that provides an artificial damping to eliminate unstable oscillations in the numerical results as proposed in the old algorithms is avoided. Accurate results from numerical simulations using the new algorithms are obtained, and there are no unstable oscillations that were observed in some other predictor/corrector schemes. The system energy balance is also better preserved compared with the old algorithms.     

Nonlinear predictors and hybrid corrector for fast continuation power flow

Continuation power flow is a powerful tool to simulate power system steady-state stationary behaviours with respect to a given power injection variation scenario. Although continuation power flow methods have been implemented in several commercial packages, they may be still too slow for online applications. The authors aim to improve the continuation power flow methods, mainly their speed and, to a less extent, their reliability. Nonlinear predictors are developed based on the polynomial interpolations. The authors' numerical studies show that continuation power flow with the proposed nonlinear predictors can be much faster than that with traditional linear predictors such as tangent or secant predictors. Of the nonlinear predictors, second-order polynomial approximation-based and third-order-based nonlinear predictors show their superior performance in speed. Continuation power flow with second-order nonlinear predictors is generally slightly faster than that with third-order nonlinear predictors. In addition, a hybrid corrector is developed and incorporated into continuation power flow. It is numerically shown on several test systems ranging from 118-bus to 1648-bus that continuation power flow with the proposed hybrid corrector can be much faster than that with traditional correctors such as the Newton method and the fast decoupled method. Finally, an improved continuation power flow with the developed nonlinear predictor and hybrid corrector is presented and evaluated.     

All-spherical catadioptric telescope design for wide-field imaging

The current trend in building medium-size telescopes for wide-field imaging is to use a Ritchey-Chrétien (RC) design with a multilens corrector near the focus. Our goal is to find a cost-effective alternative design to the RC system for seeing-limited observations. We present an f/4.5 all-spherical catadioptric system with a 1.5° field of view. The system consists of a 0.8 m spherical primary and 0.4 m flat secondary mirror combined with a meniscus lens and followed by a three-lens field corrector. The optical performance is comparable to an equivalent f/4.5 RC system. We conclude that, for telescopes with apertures up to 2 m, the catadioptric design is a good alternative to the RC system.     

A GENERALIZED VARIABLE FORMULATION FOR GRADIENT DEPENDENT SOFTENING PLASTICITY

A mesh-independent finite element method for elastoplastic problems with softening is proposed. The regularization of the boundary value problem is achieved introducing in the yield function the second order gradient of the plastic multiplier. The backward-difference integrated finite-step problem enriched with the gradient term is given a variational formulation where the consitutive equations are treated in weak form as well as the other field equations. A predictor–corrector scheme is proposed for the solution of the non-linear algebraic problem resulting from the finite element discretization of the functional. The expression of the consistent tangent matrix is provided and the corrector phase is formulated as a Linear Complementarity Problem. The effectiveness of the proposed methodology is verified by one- and two-dimensional tests.     

Path following and critical points for contact problems

This work is concerned with the problem of tracing the equilibrium path in large displacement frictionless contact problems. Conditions for the detection of critical points along the equilibrium path are also given. By writing the problem as a system of non-linear B-differentiable functions, the non-differentiability due to the presence of the unilateral contact constraints is overcome. The path-following algorithm is given as a predictor-corrector method, where the corrector part is performed using Newton's method for B-differentiable functions. A new type of displacement constraints are introduced where the constraining displacement node may change during the corrector iterations. Furthermore it is shown that, in addition to the usual bifurcation and limit points, bifurcation is possible or the equilibrium path may have reached an end point even if the stiffness matrix is non-singular.     

Diffractive-refractive hybrid corrector for achro- and apochromatic corrections of optical systems

The possibility is shown of achro- and apochromatic correction of an optical system with any residual chromatism by completing the system with a diffractive-refractive hybrid corrector comprising one diffractive lens and one or two refractive lenses.     

A predictor–corrector-type technique for the approximate parameterization of intersection curves

We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.