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Analysis and algorithms for the computation of the excited states of helium

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第8期235卷 2041-2062页

作者： Ding, Zhonghai Chen, Goong Univ Nevada Dept Math Sci Las Vegas NV 89154 USA Texas A&M Univ Dept Math College Stn TX 77843 USA Texas A&M Univ Inst Quantum Studies College Stn TX 77843 USA

In this paper, we study a dimensionally scaled helium atom model for excited states of helium. The mathematical analysis of the corresponding effective energy potential is presented. Two simple numerical algorithms are developed for the computation of the excited states of helium. Comparison between our numerical results and those in the existing literature is given to indicate the accuracy and efficiency of the proposed algorithms. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Helium atom Schrodinger equation Excited states Dimensional scaling numerical algorithm

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Simulation of damage evolution in ductile metals undergoing dynamic loading conditions

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INTERNATIONAL JOURNAL OF PLASTICITY 2011年第10期27卷 1598-1617页

作者： Bruenig, Michael Gerke, Steffen Univ Bundeswehr Munchen Inst Mech & Stat D-85577 Neubiberg Germany

A set of constitutive equations for large rate-dependent elastic-plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed. Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor-elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Damage and fracture Stress triaxiality dependence Dynamic loading numerical algorithm numerical simulations

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A robust implementation of a sequential quadratic programming algorithm with successive error restoration

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OPTIMIZATION LETTERS 2011年第2期5卷 283-296页

作者： Schittkowski, K. Univ Bayreuth Dept Comp Sci D-95440 Bayreuth Germany

We consider sequential quadratic programming methods for solving constrained nonlinear programming problems. It is generally believed that these methods are sensitive to the accuracy by which partial derivatives are provided. One reason is that differences of gradients of the Lagrangian function are used for updating a quasi-Newton matrix, e.g., by the BFGS formula. The purpose of this paper is to show by numerical experimentation that the method can be stabilized substantially. The algorithm applies non-monotone line search and internal and external restarts in case of errors due to inaccurate derivatives while computing the search direction. Even in case of large random errors leading to partial derivatives with at most one correct digit, termination subject to an accuracy of 10(-7) can be achieved in 90% of 306 problems of a standard test suite. On the other hand, the original version with monotone line search and without restarts solves only 30% of these problems under the same test environment. In addition, we show how initial and periodic scaled restarts improve the efficiency in situations with slow convergence.

关键词： SQP Sequential quadratic programming Nonlinear programming Line search Restart numerical algorithm Noisy functions

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Extended interval Newton method based on the precise quotient set

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COMPUTING 2011年第4期92卷 297-315页

作者： Chen, Chin-Yun Natl Chiayi Univ Dept Appl Math Chiayi 600 Taiwan

The interval Newton method can be used for computing an enclosure of a single simple zero of a smooth function in an interval domain. It can practically be extended to allow computing enclosures of all zeros in a given interval. This paper deals with the extended interval Newton method. An essential operation of the method is division by an interval that contains zero (extended interval division). This operation has been studied by many researchers in recent decades, but inconsistency in the research has occurred again and again. This paper adopts the definition of extended interval division redefined in recent documents (Kulisch in Arithmetic operations for floating-point intervals, 2009;Pryce in P1788: IEEE standard for interval arithmetic version 02.2, 2010). The result of the division is called the precise quotient set. Earlier definitions differ in the overestimation of the quotient set in particular cases, causing inefficiency in Newton's method and even leading to redundant enclosures of a zero. The paper reviews and compares some extended interval quotient sets defined during the last few decades. As a central theorem, we present the fundamental properties of the extended interval Newton method based on the precise quotient set. On this basis, we develop an algorithm and a convenient program package for the extended interval Newton method. Statements on its convergence are also given. We then demonstrate the performance of the algorithm through nine carefully selected very sensitive numerical examples and show that it can compute correct enclosures of all zeros of the functions with high efficiency, particularly in cases where earlier methods are less effective.

关键词： Interval computation Extended interval division Newton method numerical algorithm

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APPROXIMATION METHODS FOR HYBRID DIFFUSION SYSTEMS WITH STATE-DEPENDENT SWITCHING PROCESSES: numerical algorithmS AND EXISTENCE AND UNIQUENESS OF SOLUTIONS

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2010年第6期41卷 2335-2352页

作者： Yin, G. Mao, Xuerong Yuan, Chenggui Cao, Dingzhou Wayne State Univ Dept Math Detroit MI 48202 USA Univ Strathclyde Dept Stat & Modelling Sci Glasgow G1 1XH Lanark Scotland Univ Wales Swansea Dept Math Swansea SA2 8PP W Glam Wales

Focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state-dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. In contrast to existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing-stepsize algorithms together with their weak convergence are given. numerical experiments are also provided for demonstration.

关键词： switching diffusion numerical algorithm convergence existence and uniqueness of strong solution

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HIGH-ORDER BISECTION METHOD FOR COMPUTING INVARIANT MANIFOLDS OF TWO-DIMENSIONAL MAPS

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INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2011年第7期21卷 2017-2042页

作者： Goodman, Roy H. Wrobel, Jacek K. New Jersey Inst Technol Dept Math Sci Newark NJ 07102 USA

We describe an efficient and accurate numerical method for computing smooth approximations to invariant manifolds of planar maps, based on geometric modeling ideas from Computer Aided Geometric Design (CAGD). The unstable manifold of a hyperbolic fixed point is modeled by a piecewise Bezier interpolant (a Catmull-Rom spline) and properties of such curves are used to define a rule for adaptively adding points to ensure that the approximation resolves the manifold to within a specified tolerance. numerical tests on a variety of example mappings demonstrate that the new method produces a manifold of a given accuracy with far fewer calls to the map, compared with previous methods. A brief introduction to the relevant ideas from CAGD is provided.

关键词： Invariant manifold computer-aided geometric design numerical algorithm

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Implementation of an algorithm for general material behavior of steel taking interaction between plasticity and transformation-induced plasticity into account

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INTERNATIONAL JOURNAL FOR numerical METHODS IN ENGINEERING 2011年第12期87卷 1183-1206页

作者： Wolff, Michael Boehm, Michael Mahnken, Rolf Suhr, Bettina Univ Bremen Zentrum Technomath D-2800 Bremen 33 Germany Univ Paderborn Chair Engn Mech Paderborn Germany

In this article a semi-implicit algorithm (predictor-corrector approach) for incorporating the interaction between plasticity and transformation-induced plasticity (TRIP) in steel is developed. Contrary to the usual elasto-plasticity, the underlying model of material behavior of steel is far more complex. The interaction between plasticity and TRIP requires extensions of algorithms developed in Doghri (Int. J. Numer. Meth. Engng 1993;36: 3915-3932) and in Mahnken (Commun. Numer. Meth. Engng 1999;15: 745-754). A particular feature of the algorithm is that the inner iteration can be reduced to a single scalar equation. numerical examples illustrate the algorithm's capabilities. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： numerical algorithm steel non-linear isotropic and kinematic hardening interaction between plasticity and TRIP

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Acceleration of DEM algorithm for quasistatic processes

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INTERNATIONAL JOURNAL FOR numerical METHODS IN ENGINEERING 2011年第7期86卷 816-828页

作者： Padbidri, Jagan M. Mesarovic, Sinisa Dj. Washington State Univ Sch Mech & Mat Engn Pullman WA 99164 USA

The discrete element method (particle dynamics) is an invaluable tool for studying the complex behavior of granular matter. Its main shortcoming is its computational intensity, arising from the vast difference between the integration time scale and the observation time scale (similar to molecular dynamics). This problem is particularly acute for macroscopically quasistatic deformation processes. We first provide the proper definition of macroscopically quasistatic processes, on the basis of dimensional analysis, which reveals that the quasistatic nature of a process is size-dependent. This result sets bounds for application of commonly used method for computational acceleration, based on superficially increased mass of particles. Next, the dimensional analysis of the governing equations motivates the separation of time scales for the numerical integration of rotations and translations. We take advantage of the existence of fast and slow variables (rotations and translations) to develop a two-timescales algorithm based on the concept of inertial manifolds suggested by Gear and Kevrekidis. The algorithm is tested on a 2D problem with axial strain imposed by rigid plates and pressure on lateral boundaries. The benchmarking against the accurate short-time step results confirms the accuracy of the new algorithm for the optimal arrangement of short- and long-time steps. The algorithm provides moderate computational acceleration. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： particle dynamics size dependence numerical algorithm dimensional analysis inertial manifold

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A novel fast numerical algorithm for cascaded Raman fiber laser using the analytic approximate solution

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OPTICS COMMUNICATIONS 2007年第1期271卷 257-262页

作者： Huang Kaidi Zhou Xiaojun Qin Zujun Wu Haocheng Zhou Zili Univ Elect Sci & Technol China Sch Optoelect Informat Chengdu 610054 Peoples R China Guilin Inst Opt Commun Guilin 541004 Peoples R China

In this paper, we proposed a novel numerical algorithm for cascaded Raman fiber laser (CRFL) using the approximate analytic results as the initial values for shooting method, which effectively reduces the calculating time from several hours to a few minutes. With the algorithm, we obtained a numerical solution of the bilateral-pumping Ge-doped fifth-order CRFL which can avoid the so-called "end face damage" phenomenon efficiently. At the same time, we also simulated the unilateral-pumping one as a comparison, which showed both the characteristics of them are similar with each other. (c) 2006 Elsevier B.V. All rights reserved.

关键词： cascaded fiber Raman laser numerical algorithm shooting bilateral pump

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Simulation and Analysis of Coupled Vibration of Monorail Train-bridge-road Vehicle System 5

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5th International Symposium on Environmental Vibration (ISEV2011)

作者： Chen, Shangyou Liu, Gao Xia, He Lu, Ping CCCC Highway Consultants CO Ltd Beijing 100088 Peoples R China Beijing Jiaotong Univ Sch Civil Engn Beijing 100044 Peoples R China CCCC Water Transportat Consultants CO Ltd Beijing 100007 Peoples R China

ISBN: (纸本)9787030323736

A space dynamic analysis model of monorail train-bridge-vehicle coupled vibration system is established,in which the monorail train and road vehicles are idealized as the combination of a number of rigid bodies connected by a series of springs and dampers. The degree-of-freedom(DOF) of a car in monorail train is 15,and that of a road vehicle is 13 for 2-axle car or 17 for 3-axle car. Bridges are considered as an assemblage of beam elements with 6 DOFs at each node and link elements with 3 DOFs at each node. The contact between the bridge and the moving vehicle and train is assumed to be a point contact. Based on the displacement corresponding relationship of bridge and vehicle,the equations of motion of the coupled monorail train-bridge-road vehicle system are introduced. A numerical algorithm is then proposed for solving the nonlinear dynamic equations of the coupled system. A corresponding computer code is developed to get the dynamic response of the full bridge and trains and vehicles. With an actual bridge as an example, the dynamic responses of the bridge and riding comfort indexes of the trains and the vehicles are evaluated. The results show that the bridge is safety,and the proposed analysis model and the solution method are feasible.

关键词： monorail train train-bridge-vehicle system coupled vibration numerical algorithm dynamic responses

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