An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

攀枝花机场北东角滑坡整治措施研究

攀枝花机场地形及地质条件复杂,在修建过程中实施了大量的挖填方工程,填方边坡形成的沉降变形和滑坡成为机场主要的工程病害。本文阐述了机场北东角滑坡的基本特征及成因,综合反分析演算和试验结果确定了滑动面的抗剪强度参数,并对滑坡稳定性进行了参数敏感性分析。在滑坡推力计算的基础上,对滑坡整治方案进行对比论证,从而确立了以抗滑桩、预应力锚索框架、削坡减载和地表防排水为主的综合整治措施。     

A new high-accuracy scheme for compressible turbulent flows

Shock-capturing and broad-bandwidth scale resolutions are two main challenges of compressible turbulent flow simulation. To meet the rigorous requests, a novel fifth-order hybrid scheme based on a uniform hybrid framework is designed. With the help of a continuous weight operator, the new scheme combines an upwind compact scheme for smooth regions and a compact-reconstruction weighted essentially non-oscillatory scheme for discontinuous regions. Numerical analyses and canonical numerical tests confirm that the new scheme has high accuracy, spectral-like resolution property and shock-capturing capability. Besides, the new scheme shows high computational efficiency compared to the related shock-capturing schemes and hybrid ones.     

高陡岩质料场边坡稳定性与支护设计研究

水电站料场高边坡具有高度大、坡度陡、卸荷速度快等特点,因多按临时边坡进行设计,故施工期变形破坏事例频发。基于这一现状,依托瀑布沟水电站两岩质料场边坡,通过两年多跟踪施工过程的支护设计工作,总结出一套操作性强的料场高边坡稳定性及支护设计方法。针对料场边坡存在的受软弱结构面控制的边坡整体稳定性、浅表层块体稳定性、碎裂岩体稳定性三种工程地质问题,在跟踪施工过程开展岩体结构调查的基础上,按照先整体后局部的稳定性评价思路,开展高边坡稳定性评价。施工期动态支护设计按照“保证整体稳定,控制局部变形,顾全潜在失稳区域”的理念,通过定性评价确定不稳定区域并优先设计提交施工;针对施工中最易出现的块体变形和碎裂岩体变形,建立了合理的支护设计原则和严格的施工规定;对稳定性差、施工风险高、支护造价大的潜在不稳定区域,应及时地调整开挖方案,减少工程造价。实践表明,这套方法保证了料场高边坡的快速施工安全,减少了工程投资。     

有限谱ENO格式及其应用

首先对王健平提出的有限谱法^[1-3]做了进一步的理论研究，发现了一些新的有限谱法的插值基函数组，并互将有限谱法应用在ENO格式中，构造了有限谱ENO计算格式，然后通过对一维Euler方程的几个经典的模型问题和二维湍流与弱激波相互作用问题的数值计算，并且与理论解或准精确解进行比较分析，从而表明了此格式对于激波和其他间断具有较高的分辨率，在激波附近基本上没有明显的数值振荡，而且对于流场中的细致结构也具有相当高的精度。     

Adaptive multi-resolution central-upwind schemes for systems of conservation laws

In this article, we develop an adaptive scheme for solving systems of hyperbolic conservation laws. In this scheme nonlinear shock and linear contact waves will be treated differently. The proposed scheme uses the Kurganov central-upwind scheme. Fourth-order non-oscillatory reconstruction is employed near shock only while the unlimited fifth-order reconstruction is used for smooth regions and linear contact waves. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multi-resolution technique. The advantages of the scheme are high resolution and computationally efficient since limiters are used only for shocks. Numerical experiments with one- and two-dimensional problems are presented which show the robustness of the proposed scheme.     

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

In this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd.     

A slope modification method for shallow water equations

A slope modification method is proposed for non-oscillatory schemes based on the Lax-Friedrich solver. The modified scheme is proved to be total-variation-diminishing (TVD) and second-order accurate. Application of the scheme to the shallow water equations produces sharp profiles for shocks and achieves high accuracy in the smooth regions of the solution.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

一种基于稳定性评价的岩质边坡坡体结构分类方法

对目前岩质边坡勘测研究中所使用的结构术语及概念进行了梳理和厘定,提出了基于边坡稳定性评价的坡体结构的概念; 结合我国近年来水电工程岩质边坡勘测实践,以边坡主控结构面和潜在变形失稳模式为核心,总结提出了坡体结构类型划分体系,即将岩质边坡划分为层状坡体结构、中陡裂隙(面)控制坡体结构、楔形坡体结构和均质坡体结构四个大类九个亚类。该坡体结构概念及其分类体系对于复杂岩质边坡稳定性的定性评价、计算方法选择、边界条件确定以及稳定控制方案的制定具有指导意义。     

水平层状岩体边坡动力响应中的结构面效应研究

根据应力波传播原理分析了水平层状岩体边坡中应力波传播特征,建立了应力波在该类边坡中传播的模型。利用离散元软件UDEC分析了不同频率垂向压缩应力波作用下边坡动力响应规律中的结构面效应。结果表明:边坡中的水平层面对坡顶的动力响应有明显影响。低频应力波作用下,水平层状岩体边坡坡顶的垂向峰值速度较均质坡体相同部位的峰值速度的增加值随坡高增加而增大。较高频率应力波作用时,边坡顶部靠近坡面的垂向峰值速度高于无结构面边坡相同部位的峰值速度,远离坡面时情况相反; 坡顶垂向峰值速度大小呈周期性变化,输入应力波频率越大该变化频次越高。研究结果将有助于进一步揭示各种不同岩体结构类型边坡在动力荷载作用下损伤机理及破坏模式。     

斜坡地质灾害的空间预测问题

结合典型实例阐明斜坡地质灾害的空间预测是灾害防治工作的前提条件与成败关键。继而讨论了“斜坡地质灾害空间预测”的内涵与工作程序,以及与斜坡稳定性评价预测相关的斜坡破坏型式,斜坡结构类型,斜坡破坏的环境条件和人类活动的影响等问题。在斜坡地质灾害空间预测的定义中,笔者将预测时段仅限于“工程年代”之内,且前提是保持现有斜坡结构和环境条件,以及可预料的人类活动影响;预测程序是先进行斜坡稳定性预测,然后再对稳定性差或较差的坡段进行灾害危险性(度)预测。关于斜坡破坏型式的划分,一是从它的影响范围和“量”与“质”的关系看都需重视其单次规模;二是要与斜坡的成因结构类型相结合,才能对现场地质调查与研究具有指导意义。为了展示斜坡的成因结构类型对稳定性的控制意义,本文提供了长江三峡工程库区和西部大开发中总结出来的碎屑岩和副变质岩斜坡的新类型系统,并对三类土质斜坡的典型结构特征和稳定性进行了简要比较。关于环境条件对斜坡稳定性影响的量化研究,笔者更看重多个非自体(非本坡段)天然模型观测和滑坡反算的成果。同时建议在全面准确的现场调研的基础上建立确定性地质模型并应用基础学科的相关原理进行单要素的量化分析和单元分析,以便进一步揭示斜坡变形破坏机理。     

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

Improvement of multistep WENO scheme and its extension to higher orders of accuracy

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non‐oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth‐order accurate multistep methods are devised, providing weighted essentially non‐oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non‐oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock‐capturing problems. Copyright © 2016 John Wiley & Sons, Ltd.     

三阶WENO格式精度分析与改进

高分辨率激波捕捉格式对含激波流场的数值模拟具有重要意义。通过理论推导分析了WENO-JS3格式和WENO-Z3格式的精度,发现两种格式在光滑流场区域（包含极值点处）具有相同的理论精度且均低于三阶设计精度,WENO-Z3格式由于增大了非光滑模板的非线性权重使其计算精度有所提高。在理论推导的基础上,提出了WENO-Z+3格式及其改进格式（WENO-Z+3P1和WENO-Z+3P32）,且改进格式在光滑流场区域能满足所设计的三阶精度要求。选用一维平面黎曼问题及双马赫反射等经典算例,验证了本文提出的WENO-Z+3格式及其改进格式相较其他格式具有耗散低和对流场结构分辨率高的特性。     

高山峡谷地区高陡边坡深部裂缝一般特征分析

中国西南部,尤其是水能资源较为集中的青藏高原周边地区是近期工程建设的热点地区。由于强烈的内外动力地质作用,这些地区形成了高山耸立、沟谷深切的地貌景观,高山峡谷区的高陡岩质斜坡变形与稳定将是工程建设所面临的主要工程地质问题之一。高陡边坡的变形与稳定对工程建设的可行性起到决定性作用,并影响着施工周期和工程造价。本文分析了深裂缝现象的一般特征,认为深裂缝是在特定地质环境下内外动力地质共同作用的产物,一般均具有"三高"、"继生"这一共性。在工程处理方面,深裂缝可按山体加固的方法进行工程处理,预应力锚固洞可较好处理深裂缝问题。     

松弛方法在计算凝聚炸药爆轰问题中的应用

利用松弛近似,将非线性的凝聚炸药爆轰控制方程转化为线性的松弛方程组,并采用五阶WENO格式和五阶线性多步显隐格式对线性松弛方程组进行空间方向和时间方向的离散,由此建立具有高精度和高分辨率性质的计算凝聚炸药爆轰的松弛方法。建立的松弛方法可以避免求解Riemann问题及计算非线性通量的Jacobi矩阵,同时无需分裂处理反应源项。通过对凝聚炸药的平面一维定常爆轰波结构及球面一维聚心、散心爆轰起爆和传播过程的数值模拟,验证了所建立的松弛方法能够很好地计算凝聚炸药爆轰问题。     

西南某大型水电站拱肩槽边坡开挖的变形响应研究

对基于结构网格的Euler方程及N-S方程求解器和基于非结构网格的Euler方程求解 器,采用结构模态分析方法和柔度矩阵方法,对无人机大展弦比机翼在Ma=0.6, α=2?, 飞行高度20km的巡航状态下的静气动弹性特性进行了数值模 拟. 验证了两种求解器对静气动弹性模拟的准确性. 同时,对模态分析方法和柔度 矩阵方法进行了对比研究,发现柔度矩阵方法更适用于静气动弹性数值模拟. 另外, 对应用物面法向偏转方法替代网格变形技术模拟静气动弹性进行了研究,计算表明 物面法向偏转方法可以大大提高静气动弹性计算效 率和克服机翼结构变形过大时动网格技术无法处理的不足.     

三峡库区边坡结构及失稳模式研究

随着三峡库区移民迁建工程的实施,移民城市与集镇建设和路桥工程施工形成了大量人工高边坡。据统计,移民迁建区已形成的高边坡多达3000余处,由于地质条件复杂,90%的边坡安全性达不到要求,对移民工程构成严重威胁。与我国其他地区不同,三峡移民区的边坡顶部、下部,甚至边坡中部大都要修建建筑物,因此,在治理边坡的同时,往往要考虑采用退层建筑方式利用边坡的问题,同时,三峡库区的城市功能以旅游开发为主,结合边坡整治来实施绿色工程和开发旅游资源非常重要。为了尽可能利用有限的土地资源,移民工程往往是依山就势,一般都采用前挡后挖的模式进行场平。本文系统研究了三峡库区边坡结构类型,划分为顺层边坡、平缓软硬岩层互层边坡、滑崩堆积体边坡、溶塌角砾岩边坡、层状碎裂岩体边坡等,并对典型边坡开挖前后变形破坏过程进行了研究。结合三峡移民迁建工程,介绍了基于防护的边坡利用技术,并进行了示范。     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.