Application of cubic spline in soil erosion modeling from Narmada Watersheds,India

Soil erosion by water is ubiquitous, exhibits spatio-temporal variability, and is fundamental to determining sediment yield which is key to proper watershed management. In this study, we propose a relationship between the curve number and sediment yield index (SYI) using cubic splines. Using field data from four watersheds, the relation between observed and computed SYI is found to have a coefficient of determination (R²) value from 0.63 to 0.88 suggesting that such a relation can be used to determine SYI from the available CN value. It is found that cubic splines perform satisfactorily with Nash-Sutcliff efficiency ranging from 60.18 to 64.01%, absolute prediction error from 1.35 to 5.56%, integral square error from 1.21 to 5.82%, coefficient of correlation from 79.32 to 93.78%, and degree of agreement from 0.87 to 0.99%.     

Modelling soil erosion from a watershed using cubic splines

Erosion in a watershed exhibits spatial and temporal variability, and its determination is fundamental to determining sediment yield which is a key to proper watershed management. In this study, we propose a relationship between the curve number (SCS 1956) and Sediment Yield Index (SYI) using cubic splines. The method is illustrated with a case study of one watershed of Narmada Basin located in Mandla district of Madhya Pradesh, India. Cubic splines are found to perform satisfactorily with Nash efficiency of 63.64%, absolute prediction error of 2.64%, integral square error of 1.22%, coefficient of correlation of 93.78% and degree of agreement of 0.99%. The relation between observed and computed SYI values is correlated with a coefficient of determination (R ²) of 0.87. Such a relationship can be used to determine SYI from the available CN value, which may be quite useful in field applications.     

An interpolation method taking into account inequality constraints: I. Methodology

Several kinds of data can provide information about a variable measured on a one- or two-dimensional space; at some points, the value is known to be equal to a certain number. At other points, the only information may be that the variable is greater or smaller than a given value. The theory of splines provides interpolating functions that can take into account both equality and inequality data. These interpolating functions are presented. The parallel between splines and kriging is reviewed, using the formalism of dual kriging. Coefficients of dual kriging can be obtained directly by minimizing a quadratic form. By adding some inequality constraints to this minimization, an interpolating function may be calculated which takes into account inequality data and is more general than a spline. The method is illustrated by some simple one-dimensional examples.Work performed at Sohio Petroleum Company     

砂土中边坡附近单桩水平抗力的模型试验研究

针对平地p-y曲线对临近边坡桩基的不适用性,开展了砂土中边坡存在对单桩水平抗力影响的室内物理模型试验。依据边坡坡角及边坡到单桩距离的不同,共进行了11次3组试验。采用分段3次多项式对实测桩身应变分布进行拟合,以推算桩身变形和桩侧抗力,有效克服了数据拟合点的奇异。对比无边坡时的试验p-y曲线、美国API规范曲线和双曲线模型,确认了平地p-y曲线的计算表达式。其次获得了边坡条件下的单桩荷载-位移曲线,揭示了边坡坡角和边坡到桩的距离对水平受荷单桩桩身弯矩分布的影响规律。根据试验结果采用折减系数修正浅层桩侧极限抗力,并引入边坡对桩侧初始刚度的削弱表达式,得到边坡条件下水平受荷单桩浅层区域双曲线p-y曲线表达式。最后通过与离心试验结果对比,验证了给出的p-y曲线考虑边坡影响时的合理性。     

试件形状对冻结粉土抗压强度影响的试验研究

在不同温度和应变率条件下,分别对圆柱体和正方体冻结粉土进行了单轴抗压强度试验,分析比较了二者的应力-应变关系、抗压强度及破坏应变.结果发现:单轴压缩下正方体与圆柱体冻结粉土的力学性质存在差异.正方体冻结粉土的应力-应变曲线有明显的弹性屈服点,屈服点后曲线呈线性硬化,而圆柱体冻结粉土的应力-应变曲线无明显弹性屈服,全应力-应变曲线呈抛物线型.相同温度和应变率条件下,圆柱体试样的应变软化程度比正方体试样大.相同试验条件下正方体冻结粉土的抗压强度及破坏应变大于圆柱体的抗压强度和破坏应变,二者抗压强度及破坏应变的差异程度与温度和应变率无关.研究结果可为冻结壁的设计提供重要依据和参考.     

Evaluation of Curve Fitting Techniques in Deriving p–y Curves for Laterally Loaded Piles

The p–y method is one of the most popular methods for the analysis and design of laterally loaded piles. The mathematical relationship it provides between the bending moment, which can be easily measured at strain gauges along the pile, and the soil resistance and lateral pile displacement, facilitates the construction of p–y curves. Numerical techniques are required to fit smooth continuous curves to the discrete bending moment data in order to improve the accuracy of subsequent differentiation and integration operations. Due to the lack of guidance on the optimum positioning of strain gauges and the reliability and accuracy of curve fitting methods, a unifying study, inclusive of small (0.61 m) and large (3.8 and 7.5 m) diameter piles in clay, was carried out using 18 strain gauge layouts and cubic spline, cubic to quintic B-spline and 3rd to 10th degree global polynomial techniques. Bending moment data was obtained using 3D finite element analysis. Through a comprehensive evaluation, the cubic and cubic B-spline methods were found to be consistently accurate in deriving p–y curves for both the small and large diameter piles.     

Adaptive isogeometric finite element analysis of steady‐state groundwater flow

Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B‐splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B‐splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B‐splines should be locally refined. The error estimates are calculated based on recovery of the L₂‐projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two‐dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd.     

Preparing stream flow drought severity–duration–frequency curves using threshold level method

Drought is a natural phenomenon which occurs in different climate regimes. In the present study, hydrological drought of the Roud Zard basin has been investigated based on run theory. Daily runoff data of Mashin hydrometery station during 1970 to 2012 was assessed using 70 % of mean daily runoff as threshold level. Results showed that the maximum drought duration of 309 days occurred in 1998 and 1999 and max drought deficit of 117.217 million cubic meters per second in 1983 with 275 days duration. Time series of the annual maxima values of duration and volume deficit showed similar trend of increase and decreasing. Burr statistical distribution, as the most suitable one fitted to the drought duration data, forecasted 339 days duration for drought event with 20 years return period and generalized extreme value forecasted 37.9 million cubic meters of deficit volume for this return period. Severity-duration-frequency (SDF) curves were prepared, classifying drought durations to four intervals and fitting statistical distribution to each. Resulted SDF curves showed that, in each period, increase of duration resulted in increased value of the volume deficit with a non-linear trend, though predicted drought volume with 20 years return period was 2.1 million cubic meters for 1 to 10 days duration, 6.9 for 11 to 30 days, 34.5 for 31 to 120 days, and 79.1 for more than 120 days duration drought event. Drought deficit volume increasing rate was also different in each class of duration interval. Drought SDF curves derived in this study can be used to quantify water deficit for natural stream and reservoir. SDFs could also be extended to allow for drought regional frequency analysis to be used in ungauged sites.     

A CUBIC STRAIN QUADRILATERAL FINITE ELEMENT

In this paper, a fourth-order displacement (cubic strain) Serendipity quadrilateral element is presented. The shape functions and integration scheme are introduced, followed by a series of convergence tests, indicating that the element does not have any spurious zero-energy modes, and passes the patch test and single-element tests. Modifications are then made to enable the element's use in effective stress analysis problems. Examples are finally solved using the element and a comparison is made between some computed and closed-form solutions. It is also shown that the cubic strain quadrilateral may be used in conjunction with or as a substitute for the cubic strain triangle when predicting collapse loads for undrained plane or axisymmetric problems in the fully plastic range.     

High-Order Spatial Simulation Using Legendre-Like Orthogonal Splines

High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The high-order simulation approach does not require any transformation of initial data and makes no assumptions about any probability distribution function, while it introduces complex spatial relations to the simulated realizations via high-order spatial statistics. This paper presents a new extension where a conditional probability density function (cpdf) is approximated using Legendre-like orthogonal splines. The coefficients of spline approximation are estimated using high-order spatial statistics inferred from the available sample data, additionally complemented by a training image. The advantages of using orthogonal splines with respect to the previously used Legendre polynomials include their ability to better approximate a multidimensional probability density function, reproduce the high-order spatial statistics, and provide a generalization of high-order simulations using Legendre polynomials. The performance of the new method is first tested with a completely known image and compared to both the high-order simulation approach using Legendre polynomials and the conventional sequential Gaussian simulation method. Then, an application in a gold deposit demonstrates the advantages of the proposed method in terms of the reproduction of histograms, variograms, and high-order spatial statistics, including connectivity measures. The C++ course code of the high-order simulation implementation presented herein, along with an example demonstrating its utilization, are provided online as supplementary material.     

两种贯通潜孔锤反循环钻头应用效果及理论分析

对于矿区钻孔作业存在的粉尘污染严重、岩矿心采取困难等问题,常规金刚石钻进取心难以满足现场施工要求。采用贯通式潜孔锤反循环连续取心技术替代常规金刚石取心工艺,在保证了较高钻进效率的同时,可有效解决干旱缺水矿区存在的取心及其引起的污染问题,保障施工人员的身体健康。为此,笔者所在课题组研发了两种新型潜孔锤反循环钻头并进行了野外试验,同时,运用LSDYNA数值模拟对设计的两种反循环钻头进行了强度分析。试验结果表明,两种钻头均具有较好的碎岩钻进效果,但花键内喷孔式反循环钻头易发生断裂;数值模拟分析结果显示,钻头的最大应力远高于材料疲劳极限是造成钻头破坏的根本原因,数值模拟结果与试验结果取得了较好的一致性。由此证实了花键内喷孔式反循环钻头结构存在缺陷,设计时应尽量避免在内喷孔周围设置环槽,以保证钻头体强度。     

Estimation of the Mesozoic geological time scale

Geological time scales are constructed by combining the stratigraphic record with radiometric dates and their standard deviations. The stratigraphic record encompasses litho-, bio-, chrono-, and magnetostratigraphy. The statistical methods employed should embody concepts and data available for the systems considered. Recently, in order to estimate the ages of 31 Mesozoic stage boundaries, use was made of a database with chronostratigraphic classifications for 340 dates, biostratigraphic data including ammonite subzones, and information on seafloor spreading. This paper is primarily concerned with the propagation of errors through the successive steps of the data analysis. The following stepwise approach was taken for combining the different types of data: (1) maximum likelihood estimation with windows set around prior stage boundary estimates, (2) averaging of estimates with variable precision including magnetochronologic data, and (3) calibration by means of cubic smoothing splines assuming equal duration of ammonite subzones. The end product is a time-scale in which the stage boundary ages are accompanied by approximate 95 per cent confidence intervals.     

基于AutoCAD平台土样颗分数据处理样条函数方法

针对土的颗粒大小分析试验中人工计算及绘制颗粒大小分配曲线,并从曲线上查取有关值计算土样不均匀系数Cu和曲率系数Cc的问题,介绍了基于AutoCAD平台采用VB编程语言,应用三次样条函数数学原理进行二次开发,模拟土工颗分曲线,并自动查询、计算出Cu/Cc值的原理和方法。     

VTI介质中的地层弹性参数反演

在VTI介质中，用非双曲线可以很好地近似P－P波中长排列情况下的旅行时；而对P－SV波来说，应用非双曲线却不能很好地近似P－SV波的施行时。对P－P波、P－SV波的旅行时进行三项泰勒展开时，它们的系数却包含了地震波的各向异性信息及其垂直速度信息，这为地震波的弹性参数反演提供了基础。实验分析表明，在中等排列长度下可以精确地得到了P－P波，P－SV波旅行时的三项泰勒展式系数，从而可以精确地进行地层弹性参数反演。     

青金石中的结构有序化和调制及其与光性异常间的关系

青金石结构有无序和有序之别。无序青金石为光性均质体,但它的有序化及由此引起的结构调制,则有可能使青金石产生光性异常。使结构仍保持立方对称的有序化并不改变它光性上的均质性,而其他形式的有序化以及结构调制,如果它们是多方向的 且在空间取向上具有立方对称的特征,则晶体仍是光性均质体,否则便转变为光性非均质体 。此外,还提出了根据选区电子衍射花样对结构调制方向进行三维判识的可能性;报道了在所研究青金石中发现的沿［110］方向n=4的有公度调制的有序结构,以及所观察到的其他有序结构和无公度调制结构。     

Computer-aided geometric design of geologic surfaces and bodies

Bivariate quadratic simplicial B-splines are employed to obtain aC ¹-smooth surface from scattered positional or directional geological data given over a two-dimensional domain. Vertices are generated according to the areal distribution of data sites, and polylines are defined along real geological features. The vertices and the polylines provide a constrained Delaunay triangulation of the domain. Note that the vertices do not generally coincide with the data sites. Six linearly independent simplex B-splines are associated with each triangle. Their defining knots and finite supports are automatically deduced from the vertices. Specific knot configurations result in discontinuities of the surface or its directional derivatives. Coefficients of a simplex spline representation are visualized as geometric points controlling the shape of the surface. This approach calls for geologic modeling and interaction of the geologist up front to define vertices and polylines, and to move control points initially given by an algorithm. Thus, simplex splines associated with irregular triangles seem to be well-suited to approximate and allow further geometrically modeling of geologic surfaces, including discontinuities, from scattered data. Applications to mathematical test as well as to real geological data are given as examples.     

压剪作用下裂隙扩展过程渗流与应力耦合模型研究

裂隙的剪胀特性及扩展演化规律对岩体的渗流特性具有重要影响。为揭示裂隙剪胀及扩展演化对岩体渗流的影响,基于残余强度提出了一种能较好描述岩石硬化-软化特性的全剪切本构关系;结合剪切变形与裂隙开度的关系,利用最小势能原理和立方定律,建立了压剪作用下考虑裂隙剪胀特性的渗流应力耦合模型;假定压剪作用下裂隙发生Ⅰ型扩展,提出了伴有翼型裂纹的渗流模型,该模型不仅考虑了岩石的剪胀特性,更反映了裂隙扩展过程渗流的演化规律。对不同裂隙粗糙度的剪切应力-位移曲线进行分析,全剪切本构模型表现出较高的拟合精度。在剪切应力-位移关系基础上,通过剪切渗流试验数据对压剪作用下渗流模型进行验证,结果表明,该模型能较好地描述岩体剪胀阶段渗透性演化规律。利用等效裂隙简化裂隙网络,并通过试验数据进行验证,证实了裂隙扩展过程渗流-应力耦合模型的准确性与适用性。     

弯曲岩溶管道溶质运移的尺度效应研究

岩溶管道溶质运移的尺度效应研究对穿透曲线的正确解译非常重要,但目前针对单一弯曲管道中溶质运移尺度效应的研究仍比较缺乏。文章将岩溶管道和溶潭分别概化为透明软管和水箱,基于前期建立的水箱-管道系统(简称“管道系统”),在水箱下游设置不同长度的弯曲管道,通过示踪试验研究管道运移尺度对穿透曲线的影响,并采用暂时存储模型模拟试验曲线。结果表明：(1)随着水箱下游管道长度的增加,峰值质量浓度逐渐缓慢降低,穿透曲线上升段斜率无明显变化,穿透曲线拖尾逐渐缩短,表明运移管道长度增加对溶质运移的影响大于下游管道弯曲;(2)穿透曲线偏度系数、后段溶质羽穿透时间和溶质羽穿透时间与管道系统长度呈良好的负线性相关关系(R²>0.96);(3)当对称和不对称水箱管道系统长度分别增加至154.5 m和164.3 m时,偏度系数接近0,穿透曲线分布接近对称;(4)弥散系数、存储区截面积和交换系数与管道系统长度呈良好的负线性相关关系,当对称和不对称水箱管道系统长度分别增加至159.9 m和178.1 m时,存储区截面积接近0,水箱导致的溶质运移滞后效应基本消失。研究结果对野外岩溶管道穿透曲线的...     

On the soil water characteristic curves of poorly graded granular materials in aqueous polymer solutions

Tempe cell with a self-developed horizontal hanging column attachment was used to measure the soil water characteristic curves (SWCCs) of granular materials initially saturated with either water or polymer solutions. For SWCCs of six poorly graded granular materials with d ₅₀ ranging from 0.04 to 0.7 mm in water, it was found that (1) as grain size decreases, air entry value increases, and matric suction (ψ) for the funicular and pendular regimes increases and that (2) steep desaturation curves over narrow ψ range in the funicular regime were observed. Air entry values obtained from the fitting parameter 1/α in van Genuchten SWCC equation fall in the boundaries calculated from the pore throat sizes in both simple cubic and face-centered cubic packings. A toroidal meniscus water model, which incorporates the measured surface tension and contact angle values between aqueous solutions and solid surfaces, was proposed for the SWCC in pendular regime and was compared to well-received numerical methods. This toroid model successfully depicts SWCC of poorly graded granular materials in water. However, SWCCs predicted by this toroid model underestimate the degree of saturation in the pendular regime for Ottawa 20–30 sands in polymer solutions. Herschel–Bulkley fluid, which is a type of non-Newtonian fluids, is postulated to increase the ψ needed to drain the polymer solution due to the nonzero shear stress intercept. In addition, it is also postulated by scanning electron microscopy and optical confocal imaging results that the rough surfaces of Ottawa 20–30 sand, which have many micron-sized “kinks”, together with the possible chemical attractions, help retain the polymer solutions on the solid surfaces, or water film.     

TBM滚刀贯入过程中泥岩破坏特征试验研究

为研究无侧限和有侧限条件下软岩贯入破坏特征,结合声发射检测技术和电镜扫描试验进行泥岩立方体试样常截面滚刀贯入试验,分析试验过程中荷载-贯入度曲线和声发射参数特征,并对岩石渣块断口进行细观电镜扫描,研究了破坏时细观和宏观裂纹形成过程。试验结果表明,（1）无侧限和有侧限贯入试验中荷载-贯入度曲线在跃进荷载处均出现明显峰值,跃进荷载峰值以后无侧限试样完全破坏,有侧限试样尚未完全破坏;（2）当宏观裂纹产生时无侧限试样声发射参数出现明显峰值,主要源于该宏观裂纹在试样内部的扩展。有侧限试样破坏过程中出现多次声发射参数峰值,对应多条宏观裂纹形成和扩展;（3）常截面滚刀贯入过程中试样内部出现微滑动,剪切破坏现象比较明显,可以认为滚刀贯入过程中泥岩破坏模式以剪切破坏为主。