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Atomistic fracture in bcc iron revealed by active learning of Gaussian approximation potential

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npj Computational Materials 2023年第1期9卷 135-145页

作者： Lei Zhang Gábor Csányi Erik van der Giessen Francesco Maresca Engineering and Technology Institute Groningen Faculty of Science and EngineeringUniversity of GroningenNijenborgh 49747 AG GroningenThe Netherlands Engineering Laboratory University of CambridgeCambridge CB21PZUK Zernike Institute for Advanced Materials University of GroningenNijenborgh 49747 AG GroningenThe Netherlands

The prediction of atomistic fracture mechanisms in body-centred cubic(bcc)iron is essential for understanding its semi-brittle nature.Existing atomistic simulations of the crack-tip under mode-I loading based on empirical interatomic potentials yield contradicting predictions and artificial mechanisms.To enable fracture prediction with quantum accuracy,we develop a Gaussian approximation potential(GAP)using an active learning strategy by extending a density functional theory(DFT)database of ferromagnetic bcc iron.We apply the active learning algorithm and obtain a Fe GAP model with a converged model uncertainty over a broad range of stress intensity factors(SIFs)and for four crack systems.The learning efficiency of the approach is analysed,and the predicted critical SIFs are compared with Griffith and Rice theories.The simulations reveal that cleavage along the original crack plane is the atomistic fracture mechanism for{100}and{110}crack planes at T=0 K,thus settling a long-standing issue.Our work also highlights the need for a multiscale approach to predicting fracture and intrinsic ductility,whereby finite temperature,finite loading rate effects and pre-existing defects(e.g.,nanovoids,dislocations)should be taken explicitly into account.

关键词： approximation potential apply

approximation approach to the SRM based on root decomposition in the simulation of spatially varying ground motions

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Earthquake Engineering and Engineering Vibration 2013年第3期12卷 363-372页

作者： Wu Yongxin Gao Yufeng Li Dayong Xu Changjie Ali H Mahfouz Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai University Institute of Engineering Safety and Disaster Prevention Hohai University Geotechnical Research Institute Hohai University Department of Civil Engineering Shandong University of Science and Technology Research Center of Coastal and Urban Geotechnical Engineering Zhejiang University College of Engineering Jazan University

The spectral representation method （SRM） is widely used to simulate spatially varying ground motions. This study focuses on the approximation approach to the SRM based on root decomposition, which can improve the efficiency of the simulation. The accuracy of the approximation approach may be affected by three factors： matrix for decomposition, distribution of frequency interpolation nodes and elements for interpolation. The influence of these factors on the accuracy of this approach is examined and the following conclusions are drawn. The SRM based on the root decomposition of the lagged coherency matrix exhibits greater accuracy than the SRM based on the root decomposition of the cross spectral matrix. The equal energy distribution of frequency interpolation nodes proposed in this study is more effective than the counter pith with an equal spacing. Elements for interpolation do not have much of an effect on the accuracy, so interpolation of the elements of the decomposed matrix is recommended because it is less complicated from a computational efficiency perspective.

关键词： ground motions simulation spectral representation method spatially varying approximation root decomposition

approximation algorithm for multiprocessor parallel job scheduling

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Journal of Central South University of Technology 2002年第4期9卷 267-272页

作者：陈松乔黄金贵陈建二 College of Information Engineering Central South University

P k |fix| C max problem is a new scheduling problem based on the multiprocessor parallel job, and it is proved to be NP hard problem when k ≥3. This paper focuses on the case of k =3. Some new observations and new techniques for P 3 |fix| C max problem are offered. The concept of semi normal schedulings is introduced, and a very simple linear time algorithm Semi normal Algorithm for constructing semi normal schedulings is developed. With the method of the classical Graham List Scheduling, a thorough analysis of the optimal scheduling on a special instance is provided, which shows that the algorithm is an approximation algorithm of ratio of 9/8 for any instance of P 3|fix| C max problem, and improves the previous best ratio of 7/6 by M.X.Goemans.

关键词： multiprocessor parallel job scheduling approximation algorithm NP-hard problem

approximation OF SOLUTION OF LINEAR DIFFERENTIAL EQUATION WITH ALMOST PERIOD FUNCTION COEFFICIENTS

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Acta Mathematica Scientia 2001年第4期21卷 503-508页

作者：蔡海涛黄伯云 P/M Key State Laboratory Department of Applied Mathematics Central South University of Technology Changsha 410083 China Department of Applied Mathematics Central South University of Technology Changsha 410083 China

This paper is devoted to the study of approximation of the solution for the differential equation whose coefficients are almost period functions. To this end the authors establish the estimation of the solution of general linear differential equation for infinite interval case. For finite interval case, this equation was investigated by G. Tamarkin([1]) applying the Picard method of successive approximation.

关键词： approximation almost period function Picard method of successive approximation

approximation by semigroup of spherical operators

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Frontiers of Mathematics in China 2014年第2期9卷 387-416页

作者： Yuguang WANG FeUong CAO Department of Mathematics China Jiliang University Hangzhou 310018 China

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the （n ＋ 1）- dimensional Euclidean space for n ≥2. We prove that such operators form a strongly continuous contraction semigroup of class （l0） and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ＋Vt^γ and the rth Boolean of the generalized spherical Weierstrass operator ＋Wt^k for integer r ≥ 1 and reals γ, k∈ （0, 1] have errors ｜｜＋r Vt^γ- f｜｜X ω^rγ（f, t^1/γ）X and ｜｜＋rWt^kf - f｜｜X ω^2rk（f, t^1/（2k））X for all f ∈ X and 0 ≤t ≤2π, where X is the Banach space of all continuous functions or all L^p integrable functions, 1 ≤p ≤＋∞, on S^n with norm ｜｜·｜｜X, and ω^s（f,t）X is the modulus of smoothness of degree s 〉 0 for f ∈X. Moreover, ＋r^Vt^γ and ＋rWt^k have the same saturation class if γ= 2k.

关键词： Sphere semigroup approximation modulus of smoothness multiplier

approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

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Science China Mathematics 2007年第11期50卷 1583-1596页

作者： Da-chuan XU~(1+) Shu-zhong ZHANG~2 1 Department of Applied Mathematics,Beijing University of Technology,Beijing 100022,China 2 Department of Systems Engineering and Engineering Management,The Chinese University of Hong Kong,Shatin,Hong Kong,China Department of Applied Mathematics Beijing University of Technology Beijing China Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin Hong Kong China

In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the ratio is dependent on the data of the problem with α being a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δd if every weight is strictly positive,where δd ＞ 0 is a constant depending on the problem dimension and data.

关键词： quadratic maximization max-cut problem semidefinite programming relaxation approximation algorithm performance ratio

approximation Method for Flare Slamming Analysis of Large Container Ships in Parametric Rolling Conditions

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Journal of Marine Science and Application 2018年第3期17卷 406-413页

作者： Yuan Lin Ning Ma Xiechong Gu State Key Laboratory of Ocean Engineering Shanghai Jiao Tong UniversityShanghai 200240China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration Shanghai Jiao Tong UniversityShanghai 200240 China

Since the research of flare slamming prediction is seldom when parametric rolling happens, we present an efficient approximation method for flare slamming analysis of large container ships in parametric rolling conditions. We adopt a 6-DOF weakly nonlinear time domain model to predict the ship motions of parametric rolling conditions. Unlike previous flare slamming analysis, our proposed method takes roll motion into account to calculate the impact angle and relative vertical velocity between ship sections on the bow flare and wave surface. We use the Wagner model to analyze the slamming impact forces and the slamming occurrence probability. Through numerical simulations, we investigate the maximum flare slamming pressures of a container ship for different speeds and wave conditions. To further clarify the mechanism of flare slamming phenomena in parametric rolling conditions, we also conduct real-time simulations to determine the relationship between slamming pressure and 3-DOF motions, namely roll, pitch, and heave.

关键词： Flare slamming approximation method Parametric rolling Asymmetric effects Container ship

approximation algorithms for indefinite complex quadratic maximization problems

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Science China Mathematics 2010年第10期53卷 2697-2708页

作者： HUANG Yongwei1,& ZHANG Shuzhong2 1Department of Electronic and Computer Engineering,The Hong Kong University of Science and Technology,Kowloon,Hong Kong,China 2Department of Systems Engineering and Engineering Management,The Chinese University of Hong Kong,Shatin,Hong Kong,China Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Kowloon Hong Kong China Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin Hong Kong China

In this paper,we consider the following indefinite complex quadratic maximization problem: maximize zHQz,subject to zk ∈ C and zkm = 1,k = 1,...,n,where Q is a Hermitian matrix with trQ = 0,z ∈ Cn is the decision vector,and m 3.An (1/log n) approximation algorithm is presented for such problem.Furthermore,we consider the above problem where the objective matrix Q is in bilinear form,in which case a 0.7118 cos mπ 2approximation algorithm can be constructed.In the context of quadratic optimization,various extensions and connections of the model are discussed.

关键词： indefinite Hermitian matrix randomized algorithms approximation ratio semidefinite programming relaxation

approximation of partially smooth functions

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Science China Mathematics 2008年第4期51卷 553-561页

作者： John Erik FORN/ESS Erlend Fornaess WOLD Mathematics Department The University of Michigan Mathematisches Institut Universit■t Bern

In this paper we discuss approximation of partially smooth functions by smooth functions. This problem arises naturally in the study of laminated currents.

关键词： approximation currents test functions