近场动力学方法及其应用

近场动力学(peridynamics,PD)是一种新兴的基于非局部作用思想建立模型并通过求解空间积分方程描述物质力学行为的方法.它兼有分子动力学方法和无网格方法的优点,避免了基于连续性假设建模和求解空间微分方程的传统宏观方法在面临不连续问题时的奇异性,又突破了经典分子动力学方法在计算尺度上的局限,在宏/微观不连续力学问题分析中均表现出很高的求解精度和效率.首先概述了PD方法的理论基础、建模思路和计算体系;进而介绍了PD方法在不同尺度不连续力学问题中的应用,包括均匀与非均匀材料和结构的大变形、损伤、断裂、冲击、穿透和失稳问题,结晶相变动力学问题以及纳米材料和结构的破坏问题;最后讨论了PD方法在理论、计算和应用等方面值得进一步研究的问题.     

近场动力学框架下钢轨疲劳裂纹萌生预测的数值方法研究

经典连续介质力学在求解由裂纹引起的不连续问题时,会出现数学构架失效的情况.为克服这一难题,基于近场动力学理论,构建铁路钢轨疲劳裂纹萌生的数值预测方法,可实现钢轨疲劳裂纹萌生寿命与位置的预测.当未出现疲劳裂纹时,通过与经典连续介质力学模型的结果对比,验证近场动力学模型的正确性和适用性.分析了车轮全滑动、黏着-滑动和无摩擦三种状态对钢轨疲劳裂纹萌生的影响规律,结果表明:车轮由全滑动向无摩擦转变的过程中,裂纹萌生位置由钢轨表层转移到内部,裂纹萌生所需的荷载循环次数由0.45×10⁷次增至2.05×10⁷次,可见车轮滚滑状态会影响裂纹的萌生位置,并且较大的切向接触应力会显著降低钢轨的疲劳裂纹萌生寿命.     

近场动力学算子方法

提出一种基于非局部思想求解物理学问题的近场动力学算子方法 (peridynamic operator method, PDOM).运用PDOM可将任意阶局部微分及其乘积转化为相应的非局部积分形式,且无需额外地特殊处理间断点与奇异点等问题.近年来研究较多的两种非局部算子:近场动力学微分算子(peridynamic differential operator,PDDO)和非局部算子方法 (nonlocal operator method, NOM),均可视为PDOM的一种特例.以弹性力学问题为例,采用变分原理和拉格朗日方程,建立了适用于分析静/动态弹性力学问题的PDOM模型.理论分析表明,当分别限定相互作用域为与位置无关或位置相关的圆形域时,该PDOM弹性模型即可退化为近年来文献中常见的近场动力学(peridynamics, PD)模型或对偶域近场动力学(dual-horizon peridynamics, DH-PD)模型.通过3个典型实例:杆的拉伸与波动、亥姆霍兹方程和含孔板的拉伸,说明本方法的计算精度、收敛性与数值稳定性.PDOM方法适用于任意均匀或非均匀离散,且能有效避免零能模式以...     

钢筋混凝土结构破坏过程的近场动力学模拟

采用近场动力学(Peridynamics,PD)方法对钢筋混凝土结构破坏过程进行模拟,在"键"型近场动力学模型的基础上,考虑物质点对间的转动以突破泊松比的限制,采用能够描述混凝土材料的拉压异性和断裂特征的损伤模型,引入动态松弛、分级加载、平衡收敛准则和冲击接触等算法,建立了能准确描述钢筋混凝土结构破坏的近场动力学模型....     

含圆孔拉伸板的近场动力学数值模拟

首先推导了非普通状态为基础的近场动力学基本方程,引入损伤理论.在对非普通状态为基础的近场动力学基本方程进行离散化后,编制了非普通状态为基础的近场动力学数值分析程序.利用非普通状态为基础的近场动力学理论对圆孔拉伸板的断裂和非断裂问题进行了数值模拟.数值结果表明近场动力学理论不仅能模拟非断裂问题,也能模拟断裂问题,而且该方法不需要借助任何外部断裂准则就能很好地模拟裂纹的扩展和连接过程,相对于其它的数值模拟方法具有很大的优势,同时它也为材料断裂问题提供了一种新方法和新思路.     

基于近场动力学数值方法的冰-吊舱推进器接触判断研究

吊舱推进器在极地船舶中的应用, 可避免冰区航行中转向、调头困难等操纵问题, 是极地船舶广泛采用的推进形式. 在冰-吊舱推进器切削过程中, 吊舱推进器受到了极端冰载荷的作用, 对吊舱推进器结构强度和极地船舶的安全性带来严重的危害. 为了研究不同操纵状态的吊舱推进器与冰切削时冰载荷的变化规律, 首先, 详细介绍了近场动力学方法研究物体断裂问题的理论基础, 分析该方法模拟冰材料的可行性. 基于近场动力学方法和面元法耦合推导了适用于冰破碎问题模拟的材料破坏准则和冰载荷计算方法. 其次, 提出了不同操纵状态的吊舱推进器与冰的接触判断方法, 建立了冰-吊舱推进器切削状态的数值计算模型, 实现了冰-吊舱推进器切削动态变化过程的数值仿真. 最后, 分析了吊舱推进器在直航、斜航以及操舵状态与冰切削时冰块破碎、螺旋桨和桨叶冰载荷以及吊舱单元整体扭矩的变化情况. 计算结果表明: 本文提出的不同操纵状态的吊舱推进器与冰切削时的接触判断方法能够真实地模拟冰-吊舱推进器的切削过程, 并能获得该过程中冰块的破坏现象和冰载荷变化特性, 可为冰区海洋结构冰载荷数值预报技术的发展、冰区吊舱推进器结构的优化设计和运营提供指导.      

准静态变形破坏的近场动力学分析

基于近场动力学理论,提出新的更能反映非局部长程力特性的物质点间作用力函数,并通过在物质点运动方程中引入局部阻尼、构造分级加载算法和系统失衡判断准则,实现了采用统一的近场动力学模型和算法进行从准静态变形、裂纹萌生和扩展直至结构破坏全过程的连续模拟和准确定量计算。     

钢筋混凝土界面的近场动力学方法

近场动力学方法已被广泛用于钢筋混凝土的开裂破坏研究,传统近场动力学方法的控制方程与参数是基于同种均质材料的能量方程确定,在处理不同种材料之间的相互作用时,无法合理反映其界面的力学行为.针对这一问题,通过分析钢筋混凝土界面的黏结-滑移机理,提出了近场动力学界面区材料点的相互作用模型,发展了考虑钢筋混凝土界面黏结的键基近场动力学方法.基于键基近场动力学与连续介质力学的能量密度等效方法,提出了界面微弹性参数的确定方法;根据钢筋肋间混凝土的应力分布规律,获得界面材料点域半径与受限楔体半径的等效关系;利用界面黏结-滑移曲线峰值应力对应的滑移变形,给出了界面临界拉伸常数确定方法.通过与2组钢筋混凝土构件的拉拔试验对比,验证了发展的界面近场动力学方法,并开展了不同条件下钢筋混凝土构件的数值试验.结果表明,发展的近场动力方法能够合理反映钢筋直径、锚固长度、混凝土强度以及肋间距对钢筋混凝土界面黏结行为的影响,体现了所提方法的合理性与优越性.     

近场动力学与有限元方法耦合求解热传导问题

求解含裂纹等不连续问题一直是计算力学的重点研究课题之一,以偏微分方程为基础的连续介质力学方法处理不连续问题时面临很大的困难. 近场动力学方法是一种基于积分方程的非局部理论,在处理不连续问题时有很大的优越性. 本文提出了求解含裂纹热传导问题的一种新的近场动力学与有限元法的耦合方法. 结合近场动力学方法处理不连续问题的优势以及有限元方法计算效率高的优势,将求解区域划分为两个区域,近场动力学区域和有限元区域. 包含裂纹的区域采用近场动力学方法建模,其他区域采用有限元方法建模. 本文提出的耦合方案实施简单方便,近场动力学区域与有限元区域之间不需要设置重叠区域. 耦合方法通过近场动力学粒子与其域内所有粒子(包括近场动力学粒子和有限元节点)以非局部方式连接,有限元节点与其周围的所有粒子以有限元方式相互作用. 将有限元热传导矩阵和近场动力学粒子相互作用矩阵写入同一整体热传导矩阵中,并采用Guyan缩聚法进一步减小计算量. 分别采用连续介质力学方法和近场动力学方法对一维以及二维温度场算例进行模拟,结果表明,本文的耦合方法具有较高的计算精度和计算效率. 该耦合方案可以进一步拓展到热力耦合条件下含裂纹材料和结构的裂纹扩展问题.      

岩石材料动态破坏的近场动力学方法数值模拟

裂纹的聚集、扩展、分叉是岩石等脆性材料破坏失效的关键因素，本文在验证了近场动力学方法在研究岩石类材料裂纹动态扩展方面的有效性之后，采用近场动力学方法分别对冲击载荷作用下含有双裂纹岩石材料和单轴压缩作用下含单斜裂纹的岩石材料进行数值模拟．结果表明，对于双垂直裂纹，其裂纹扩展路径大致与预制裂纹成70°夹角；对于单裂纹，裂纹的扩展路径随裂纹倾角的变化而变化，最终导致构件的整体破坏．数值模拟结果表明近场动力学方法可以很好地模拟岩石等脆性材料的裂纹扩展直至破坏的过程，反映裂纹扩展的物理机理；其作为一种新的基于非局部理论的数值方法，在地下岩体工程方面及页岩气的开采方面会有很好的发展前景．     

近场动力学框架下的键基对应模型

传统键基近场动力学模型存在泊松比限制的问题,为了解决这一问题发展了态基近场动力学模型。其中非常规态的近场动力学模型通过定义非局部的变形梯度将近场力和传统应力关联起来,方便使用传统本构,但是态基近场动力学计算效率低于键基近场动力学。结合态基模型和键基模型的优势,提出键基对应模型,定义了基于键的变形梯度,参考连续介质力学中变形梯度的极分解过程,将键的变形分为转动部分和伸长部分。从而进一步定义了应变,通过物理方程求应力,进而计算键传递的近场力。键基对应模型解决了键基近场动力学的泊松比限制问题,也不需要进行近场动力学微观材料常数的计算。数值算例和理论推导证明了键变形梯度定义以及近场力计算方式的正确性。     

A three‐dimensional boundary layer scheme: stability and accuracy analyses

We present a numerical scheme for the calculation of incompressible three‐dimensional boundary layers (3DBL), designed to take advantage of the 3DBL model's overall hyperbolic nature, which is linked to the existence of wedge‐shaped dependence and influence zones. The proposed scheme, explicit along the wall and implicit in the normal direction, allows large time steps, thus enabling fast convergence. In order to keep this partly implicit character, the control volumes for the mass and momentum balances are not staggered along the wall. This results in a lack of numerical viscosity, making the scheme unstable. The implementation of a numerical diffusion, suited to the local zone of influence, restores the stability of the boundary layer scheme while preserving second‐order space accuracy. The purpose of this article is to present the analytical and numerical studies carried out to establish the scheme's accuracy and stability properties. Copyright © 2002 John Wiley & Sons, Ltd.     

FURTHER DISCUSSION OF NUMERICAL ERRORS IN CFD

The methods of estimating numerical errors given in an earlier paper are extended in directions that make them useful in actual CFD applications. In particular, the method of estimating convergence error (the error due to insufficient iteration) is extended to allow the possibility of complex eigenvalues; an ad hoc method that can be applied to any case is also given. For the discretization error, which arises from the numerical approximation of the differential equation(s), methods that can be used on non-uniform drids are presented; they can be extended to unstructured grids as well. The utility of these methods is demonstrated for linear problems as well as solutions of the Navier-Stokes equations. The examples show that the estimation of errors is neither difficult nor expensive.     

Shock wave propagation through a model one dimensional heterogeneous medium

We study the problem of impact-induced shock wave propagation through a model one-dimensional heterogeneous medium. This medium is made of a model material with spatially varying parameters such that it is heterogeneous to shock waves but homogeneous to elastic waves. Using the jump conditions and maximal dissipation criteria, we obtain the exact solution to the shock propagation problem. We use it to study how the nature of the heterogeneity changes material response, the structure of the shock front and the dissipation.     

Numerical Continuation Applied to Panel Flutter

The panel flutter formulation studied here includes a geometricnonlinearity. In a physical sense, the nonlinearity represents couplingbetween out-of-plane bending and in-plane stretching. To investigate theinitial condition response, the pseudo-arclength continuation method isapplied to this problem. The continuation method is an alternatenumerical method that complements direct numerical integration. Thelatter procedure is not necessarily computationally efficient when theparameter space under consideration is large. One strength ofcontinuation methods then is the ability to efficiently and accuratelydisplay the solution space, including certain bifurcations, for dynamicsystems. Results from the procedure are in substantial agreement withearlier studies, where applicable, where the panel undergoes cylindricalbending.     

Multi‐resolution analysis for high accuracy and efficiency of Euler computation

A multi‐resolution analysis (MRA) is proposed for efficient flow computation with preserving the high‐order numerical accuracy of a conventional solver. In the MRA process, the smoothness of a flow pattern is assessed by the difference between original flow property values, and the values approximated by high‐order interpolating polynomial in decomposition. Insignificant data in smooth region are discarded, and flux computation is performed only where crucial features of a solution exist. The reduction of expensive flow computation improves the overall computational efficiency. In order to maintain the high‐order accuracy, modified thresholding procedure restricts the additional error introduced by the thresholding below the order of accuracy of a conventional solver. The practical applicability of the MRA method is validated in various continuous and discontinuous flow problems. The MRA stably computes the Euler equations for continuous and discontinuous flow problems and maintains the accuracy of a conventional solver. Overall, it substantially enhances the computational efficiency of the conventional third‐order accurate solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Gradient-dependent plasticity model and dynamic strain localisation analysis of saturated and partially saturated porous media: one dimensional model

Dynamic strain localisation in saturated and partially saturated porous media is investigated with a one-dimensional model in this paper. The porous medium is treated as a multiphase continuum, with the pores filled by water and air, this last one at atmospheric pressure. A gradient-dependent plasticity model is introduced to describe the plastic behaviour of the solid skeleton. Material instability due to the softening behaviour of the solid skeleton and the well-posedness of the initial value problem are studied. The advantages of the enhanced model are that the governing equations remain hyperbolic even in the softening regime and convergent solutions with mesh refinements are obtained. Moreover, the influence of permeability in the seepage process for the development of the localised zones is discussed. We find that the permeability plays an important part in the compressive wave propagation, but not in the shear wave cases. For numerical implementation of the present method, a parametric variational principle is introduced by which the original problem is reduced to a standard linear complementary problem in mathematical programming. The results of a one dimensional example are given to illustrate the efficiency of the techniques presented here.     

Three dimensional tide‐induced circulation model on a triangular mesh

In this paper a triangular mesh, which is generated by an automatic triangular mesh generator, is used to calculate the tide‐induced residual current in the Bohai Sea, China. The generator can modify the distribution of the node according to the water depth, thus ensuring the uniform distribution of the Courant number. The three dimensional nonlinear tide equation is employed to get the Eularian residual current and Stokes' drift. Then the Lagrangian residual current is obtained by adding the above two values. It has been proved in other papers that the Lagrangian residual current is more proper to represent the circulation and a numerical example is given. The application of these methods to the Bohai Sea shows that the results are in accordance with those obtained before. Copyright © 2002 John Wiley & Sons, Ltd.     

伽辽金型无网格法的数值积分方法

无网格法直接通过节点信息构造形函数,不依赖于节点之间的有序单元连接,能够建立任意高阶连续的整体协调形函数.与传统的有限元法相比,无网格法对大变形问题、移动边界问题和高阶问题的求解有比较明显的优势.伽辽金型无网格法是目前应用最为广泛的一类无网格法.虽然无网格形函数本身不依赖于单元,但伽辽金型无网格法需要采取合适的方法进行弱形式的数值积分.由于无网格形函数一般不是多项式,具有非插值性且影响域与背景积分网格通常不重合,伽辽金型无网格法通常需要采用高阶的高斯积分进行数值积分,导致了计算效率低下,难于求解大型实际问题.因此,如何通过建立高效积分方法提高无网格法的计算效率成为无网格法研究领域的一个核心问题.论文总结了伽辽金型无网格法中若干常用的数值积分方法,并对伽辽金型无网格法的数值积分方法领域存在的一些问题进行了探讨.