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CONVERGENCE ANALYSIS FOR SPECTRAL APPROXIMATION TO A SCALAR TRANSPORT EQUATION WITH A RANDOM WAVE SPEED

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Journal of Computational Mathematics 2012年第6期30卷 643-656页

作者： taoZhou tao tang LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China CSQI-MATHICSE Ecole Polytechnique Federale de Lausanne Switzerland Department of Mathematics Hong Kong Baptist University Kowloon Hong Kong

This paper is concerned with the initial-boundary value problems of scalar transport equations with uncertain transport velocities. It was demonstrated in our earlier works that regularity of the exact solutions in the random spaces （or the parametric spaces） can be determined by the given data. In turn, these regularity results are crucial to convergence analysis for high order numerical methods. In this work, we will prove the spectral conver- gence of the stochastic Galerkin and collocation methods under some regularity results or assumptions. As our primary goal is to investigate the errors introduced by discretizations in the random space, the errors for solving the corresponding deterministic problems will be neglected.

关键词： Scalar transport equations Analytic regularity Stochastic Galerkin Stochasticcollocation Spectral convergence.

Characteristics and Formation Mechanism for Stainless Steel Fiber with Periodic Micro-fins

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Chinese Journal of Mechanical Engineering 2016年第3期29卷 564-570页

作者： tang tao WAN Zhenping LU Longsheng tang Yong School of Mechanical and Automotive Engineering South China University of Technology

Metal fibers have been widely used in many industrial applications due to their unique advantages. In certain applications, such as catalyst supports or orthopedic implants, a rough surface or tiny outshoots on the surface of metal fibers to increase surface area are needed. However, it has not been concerned about the surface morphologies of metal fiber in the current research of metal fiber manufacturing. In this paper, a special multi-tooth tool composed of a row of triangular tiny teeth is designed. The entire cutting layer of multi-tooth tool bifurcates into several thin cutting layers due to tiny teeth involved in cutting. As a result, several stainless steel fibers with periodic micro-fins are produced simultaneously. Morphology of periodic micro-fins is found to be diverse and can be classified into three categories： unilateral plane, unilateral tapering and bilateral. There are two forming mechanisms for the micro-fins. One is that periodic burrs remained on the free side of cutting layer of a tiny tooth create micro-fins of stainless steel fiber produced by the next neighboring tiny tooth; the other is that the connections between two fibers stuck together come to be micro-fins if the two fibers are finally detached. Influence of cutting conditions on formation of micro-fins is investigated. Experimental results show that cutting depth has no significant effect on micro-fin formation, high cutting speed is conducive to micro-fin formation, and feed should be between 0.12 mm/r and 0.2 mm/r to reliably obtain stainless steel fiber with micro-fins. This research presents a new pattern of stainless steel fiber characterized by periodic micro-fins formed on the edge of fiber and its manufacturing method.

关键词： stainless steel fiber periodic micro-fins burrs chip morphology

Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed

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Communications in Computational Physics 2010年第6期8卷 226-248页

作者： tao tang tao Zhou Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong Institute of Computational Mathematics Academy of Mathematics and System ScienceChinese Academy of SciencesBeijing 100190China

For a simple model of a scalar wave equation with a random wave speed,Gottlieb and Xiu[***.,3(2008),pp.505-518]employed the generalized polynomial chaos(gPC)method and demonstrated that when uncertainty causes the change of characteristic directions,the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative ***,a consistent method of imposing the boundary conditions is proposed and its convergence is established under the assumption that the expansion coefficients decay fast *** this work,we investigate stochastic collocation methods for the same type of scalar wave equation with random wave *** will be demonstrated that the rate of convergence depends on the regularity of the solutions;and the regularity is determined by the random wave speed and the initial and boundary *** examples are presented to support the analysis and also to show the sharpness of the assumptions on the relationship between the random wave speed and the initial and boundary *** accuracy enhancement technique is investigated following the multi-element collocation method proposed by Foo,Wan and Karniadakis[***.,227(2008),pp.9572-9595].

关键词： Hyperbolic equation stochastic collocation methods convergence analysis

DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2023年第2期41卷 325-344页

作者： Hong-lin Liao tao tang tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

This is one of our series works on discrete energy analysis of the variable-step BDF *** this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM ***.,58:2294-2314]and[***.,90:1207-1226]for our previous works on the BDF2 *** this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation ***-robust stability and convergence analysis in the L^(2) norm are then *** the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step *** also present numerical tests to support our theoretical results.

关键词： Diffusion equations Variable-step third-order BDF scheme Discrete gradient structure Discrete orthogonal convolution kernels Stability and convergence

IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE

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Journal of Computational Mathematics 2016年第5期34卷 451-461页

作者： tao tang Jiang Yang Department of Mathematics Southern University of Science and Technology Shenzhen Guangdong 518055 China Department of Applied Mathematics Columbia University New York NY 10027 USA

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

关键词： Allen-Cahn Equations Implicit-explicit scheme Maximum principle Nonlin-ear energy stability.

不同内固定系统治疗股骨转子间骨折的力学稳定性

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中国组织工程研究 2025年第9期29卷 1783-1788页

作者：陈曦汤涛陈铜兵李青张文常州市第一人民医院江苏省常州市213003 苏州大学附属第三医院江苏省常州市213003 苏州大学苏州医学院骨科研究所江苏省苏州市215006

背景:股骨转子间骨折的骨折类型和固定方式多样,各固定系统间的力学稳定性相差较大。使用有限元分析方法对各固定系统开展生物力学研究具有科学的临床意义。目的:通过有限元方法对比分析多种内固定应用于股骨A031-A2.1型转子间骨折的力... 详细信息

背景:股骨转子间骨折的骨折类型和固定方式多样,各固定系统间的力学稳定性相差较大。使用有限元分析方法对各固定系统开展生物力学研究具有科学的临床意义。目的:通过有限元方法对比分析多种内固定应用于股骨A031-A2.1型转子间骨折的力学稳定性。方法:在已验证有效性的股骨有限元模型基础上,对模型进行必要的切割,造模成股骨A031-A2.1型转子间骨折,模拟临床手术方法置入不同的内固定系统,分别建立股骨近端防旋髓内钉、动力髋螺钉、经皮加压钢板和股骨近端锁定钢板固定模型。约束4组模型股骨远端下所有节点,在股骨头上施加700,1400和2100N的压缩载荷,通过计算分析,观察各组模型的等效应力分布和压缩刚度,比较各组模型之间的力学稳定性。结果与结论:①通过计算分析,对比各组模型的变形量计算压缩刚度后,在各级载荷作用下,各组模型上压缩刚度呈现的趋势:生理组>股骨近端防旋髓内钉组>股骨近端锁定钢板组>经皮加压钢板组>动力髋螺钉组;完整的生理组模型的压缩刚度明显高于所有的手术组模型;②观察应力指标,因存在应力遮挡效应,致使各固定组的应力峰值均高于生理组,最大峰值均集中分布于各内固定上;其中股骨近端防旋髓内钉固定组应力峰值最小,动力髋螺钉固定组应力最高,应力分布趋势呈现:生理组<股骨近端防旋髓内钉固定组<经皮加压钢板固定组<股骨近端锁定钢板固定组<动力髋螺钉固定组;③分布于骨质模型的应力,因内固定置入位置不同呈现不同的分布结果;④提示对于股骨转子间骨折,各种内固定均能起到有效的固定,结合有限元分析结果得出股骨近端防旋髓内钉组是较好的一种内固定选择,呈现变形量小、应力峰值低,应力分布均匀的特性;经皮加压钢板组和股骨近端锁定钢板组的力学效果优良,固定效果接近股骨近端防旋髓内钉固定;动力髋螺钉固定效果欠佳,同比于其他内固定,其力学稳定性较差。

关键词：股骨转子间骨折内固定系统力学稳定性有限元分析等效应力压缩刚度

Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

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Frontiers of Mathematics in China 2012年第1期7卷 69-84页

作者： Xianjuan LI tao tang College of Mathematics and Computer Science Fuzhou University Fuzhou 350108 China Department of Mathematics Hong Kong Baptist University Hong Kong China

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel p（t, s） = （t - s）^-μ. In an earlier work of Y. Chen and T. tang [J. Comput. Appl. Math., 2009, 233：938 950], the error analysis for this approach is carried out for 0 〈 μ 〈 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and tang to the Abel-ype but also establish the error estimates under a more general regularity assumption on the exact solution.

关键词： Jacobi spectral collocation method Abel-Volterra integralequation convergence analysis

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

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Science China Mathematics 2024年第2期67卷 237-252页

作者： Hong-Lin Liao tao tang tao Zhou School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China SUSTech International Center for Mathematics Shenzhen 518055China NCMIS&LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution *** precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

关键词： discrete convolution kernels positive definiteness variable time-stepping orthogonal convolution kernels complementary convolution kernels

ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

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Journal of Computational Mathematics 2008年第6期26卷 825-837页

作者： tao tang Xiang XU Jin Cheng Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong China School of Mathematics Fudan University Shanghai 200433 China

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.

关键词： Legendre-spectral method Second kind Volterra integral equations Convergence analysis.