Numerical calculation of dispersion relation for linear internal waves

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算术平均牛顿法的英文

算术平均牛顿法的英文Arithmetic-Geometric Mean Newton's Method.The arithmetic-geometric mean (AGM) Newton's method is an iterative algorithm used in numerical analysis to approximate the solution of equations, particularly those involving transcendental functions. This method is avariant of the classical Newton's method, which uses the tangent line to the function at a given point to approximate the root of the function. The AGM Newton's method incorporates the arithmetic-geometric mean (AGM) iteration, which is itself a fast converging method for computing the square root of a number.Background on Newton's Method:Newton's method is based on the Taylor series expansion of a function. Given a function f(x) and its derivativef'(x), the method starts with an initial guess x0 and iteratively updates the approximation using the formula:x_{n+1} = x_n f(x_n) / f'(x_n)。

温度场计算中差分法的应用

温度场计算中差分法的应用2005年第39卷No8温度场计算中差分法的应用*19苗恩铭合肥工业大学摘要:在温度场计算中,对变步距划分网格的差分法进行了分析,通过使用等步距和变步距划分网格的方法进行对比计算,得出了变步距划分网格的方法.该法既减轻了计算工作量又可获得较好的计算精度,在生产实践中具有较高的实用价值.关键词:差分法,温度场,步距ApplianceofCalculusofDifferenceinTemperatureneIdMiaoEnmingAbstract:Thispaperanalysesunequalstepgnddingincalculatingoftemperaturefieldandco mparesisometryandunequal stepddiTlgtoeduceunequalstepgriddingmeans.Thisunequalstepgnd~ngmeansnotonlyde ducecalculateworkload,butal-SOobtainmoreprecisioncomputer.Thismethodhasgoodapplicationinpractice. Keywords:calculusofdifference,temperaturefield,step~iddmg1引言温度场计算中的诸多问题使用解析法是很难求出结果的,因此对于工程上许多重要的问题是使用各种数值解法…1给以解决.特别是计算机的发展使得此方向的应用得到了极大的发展.差分法,变分法_2j2,有限元等方法都是使用较多的数值解法_3j.差分法简单实用,应用面广,使用时间也较久.该法在实际应用时通常是将弹性体进行等间距划分网格,把基本方程和边界条件近似地改用代数方程来*国家自然科学基金资助项目(项目编号:50475069)合肥工业大学校基金资助项目(项目编号:040102F)收稿日期:2OO4年12月计算,这对于确定的边界条件及确定的几何尺寸比例关系的弹性体而言是较好的解决方法.若几何尺寸比例关系任意的弹性体(形体尺寸不确定),想解出其在一定的边界条件下的变形,该法是有较大难度的.因为将弹性体进行等间距划分网格,而几何尺寸比例关系任意且形体尺寸不确定,这使得网格划分困难,一般是采用细分网格的方法给以解决,这必然带来计算量增加的副作用.对此,提出变步划分网格的方法,以解决上述问题.2变步差分法原理对弹性体先用网格进行划分,如图1所示,在轴向间隔为h,在Y轴向间隔为r.设/:,y)为15GSUpdhyayaeto!I.SinteringofsubmicronWC-10wt%hard metalscontainingnickelandiron.MaterialsScienceandEngi—neering,1988,249～25616张立,孙宝琦.国外WC.Co硬质合金中钴的替代研究概况.湖北冶金,1995,(6):54—6217AlmondEA,RoebuckB.Identificationofoptimumbinder phasecompositionsforimprovedWChardmetals.Materials ScienceandEngineering(A),1987,105/106:237～24818黄家明,李维,黄劲松等.预弥散强化镍粉一硬质合金的新型粘结剂.稀有金属与硬质合金,1999,(137):1～319BWittmann,WSchubert,BLux.WCgraingrowthandgr~in growthinhibitioninnickelandironbinderhardmetals.Inter.nationalJournalofRefractoryMetalsandHardMaterials, 2002,20(1):51～6020熊继,沈保罗.混合稀土CeLa对WC.Ni硬质合金性能和显微结构的影响.工具技术,2003,37(6):18～2021贾佐诚.烧结态硬质合金的HIP处理.粉末冶金工业, 2000,10(5):22～2522SjomUhrenluseto!I.OnthecompositionofFNi—Co-WC—basedcementedcarbides.InternationalJournalofRefractory MetalsandHardMaterials,1997,15:139～14923方名尧.无磁硬质台金模具的开发与生产.江西冶金, 1993,13(2):43～4524孙景,鲁颖炜,郭小南等.无磁硬质合金研究进展.磁性材料及器件,2004,35(2):6～4525刘寿荣.WC.Ni硬质合金游离碳的无损测定原理.硬质合金,2001,18(1):33～3426杨宣增,卢伟民,张立.WC.Ni系大直径无磁硬质合金模具的研制.稀有金属与硬质合金,2OO0,120:1～5第一作者:郭智兴,在读硕士研究生,四川大学制造科学与工程学院材料成型与控制工程系,610065成都市弹性体某一连续函数,它可能是某应力分量或位移分量,温度等,该函数在邻近结点0处,将其展开为台劳级数如下:lXI125l1宫._}19136I10}jB^r14,=fo+().(X--X0)+(蓦).X--X0)+().X--X0)3+者().X--X0 (1)在结点3及结点1,分别等于0一h及0+h,代入式(1),得,3=,0一()0+譬(嘉等()0+丢(…(2)A=fo+()0+譬()0+壁6r~3x3,/.+丢(券)0+ (3)假设h是充分小的,因而可以不计它的三次幂及更高幂的各项,则式(2),(3)可简化为f3=fo)0+譬().(4)=fo川).+萼(~---f).(5)联立求解().及(翥).,得差分公式().=(6)(嘉(7)在结点4及结点2处,y分别等于0一r及0+ r,用上述方法可得().=(8)(雾吐(9)式(6),(7),(8),(9)是基本差分公式,可从中导出其它的差分公式黝:瓣==[(,6+f8)一(+fO]2hJ8J5J7(10)一一4,6一\'\工具技术Ox2---f)]o=盟±二.±厶!二,厶±二'一h2————————————一=去[6,0—4(+,3)+(,9+?)](11)---fy.)O=[盟+每一2Irr一^=[一4(,2+,4)+(flo+2)](12)盟矗±一.,7+fs一2f3,2+,4—2fo11-^一二一[4l,0—2(+,2+,3+,4)+(,5+,6+,7+,8)](13)式(6),(7),(8),(9)在此称为变步差分法基本公式,下面讨论如何在平面问题中使用此法.3平面稳定温度场的差分法解在无热源的平面稳定温度场中,有一长8m,宽6m的平面,边界温度如图2所示,现求内部结点T2, T5处的温度值.80533520J27乞.191232320128圜2半回稳疋温厦场等『日J跑划分9.9硌4×3 对此求解分两种,一种是通常差分法[],即是将弹性体进行等间距划分网格(见图2),分成4X3网格,差分法基本公式为().=(14)(嘉)().=(16)(雾(17)由热传导微分方程简化而得的调和方稗为2005年第39卷No8等2+等(18)a.au.化简得1++一4%=0(19)经计算得T1=36.2,:26.0,7"3=24.5,:29.9,:25.3,=22.2现提高其划分网格数,分成8X6网格,如图3所示,仍用差分法得T1=50.76,=26.79,7"3=38.33,7"4=29.64,=25.68,=24.47,=23.44,=25.18,7"9=29.12,TA=44.73,TB=36.90,Tc=30.77,=25.10,=25.18,=26.28,Tc=32.59,TH=30.10,TI=27.72,Tj:23.99,TK=22.51, TL=20.96,=38.27,TN=33.76,To:29.76,Tp=26.81, TQ=25.01,TR=24.13,rs=23.84,TT=26.99,Tu=26.35, =25.32,Tw=24.19,Tx=22.78,=20.94,Tz=l8.47 553625l8l92331403427,l9JKLl6uvxYz122232323222Ol6l2—8.图3平面稳定温度场等间距划分网格8×64平面稳定温度场的变步差分法解上述问题另一种解法就是变步差分法,具体解法如下:由热传导微分方程简化而得的调和方程2+蓦a.a在使用该法之前,根据需要在温度场上织成网格,为便于比较,在本例中使用两种分法,分别为3×2,3X8,如图4,5所示.将式(7),(9)代人式(18),得+堡:0,'0,一v8053352036l8234027l923232O12图4平面稳定温度场划分3×2网格80533520J271922232323222Ol612—8—21图5平面稳定温度场划分3×8网格化简,得r2(T1+)+hz(+)一2(r+hz)=0(20)对于图4,由式(19)得rz(27+53)+hz(+18)一2r2(r+h)=0r2(19+35)+hz(23+)一2rs(r+h)=0其中h=4m,r=2m,则上两式求解得=25.05,=24.62对于图5,由式(19)得r(B+53)+h2(Tc+55)一2TA(r2+h)=0r2(A+)+h2(TH+36)一2TB(r+hz)=0r2(+)+h(+23)一2r5(r+h)=0其中h=lm,r=2m,则上式联立求解得=44.64,TB=36.67,Tc=30.51,To=24.78,Tr=24.80, =25.89,Tc:23.69,TH=30.08,TI:27.58,Tj=23.78, =22.27,TL:20.70,=26.47,:25.505两种解法比较按差分法理论[,划分网格数越多,计算的精度也越高.现以差分法8X6网格为计算标准,对其它算法进行比较可以看出,精度排序依次是变步差分法8X3网格,差分法4X3网格,变步差分法2X3网格(见表1).表1差分法和变步差分法计算结果对比\解法追度(℃)\差分法差分法变步差分法变步差分法8X6网格4×3网格2×3网格8X3网格26.7926.O25.O526.4725.6825.324.6225.5O由此可见,变步差分法和差分法的计算精度完全和划分网格数有关,网格划分越多,精度越高,二者算法原理相同,故误差因素也相同.差分法划分方法呆板,变步差分法同其比较就显得较为灵活,且当变步差分法的步距看作相同时,即式(6),(7),(8),(9)中r=h,此时就是差分法基本方程.实质上,现行的差分法是变步差分法的一种特例.参考文献1铁摩辛柯,古地尔.弹性理论.北京:高等教育出版社,1990:628～660∞:g∞北:号∞玻璃和45钢的切削力特性与刀具磨损的研究*工具技术宋欠芽刘亚俊汤勇陈平杨马生华南理工大学摘要:对比研究了切削玻璃和45钢时切削力与刀具磨损之间的关系,描述了玻璃和45钢在切削过程中塑性变形和脆性断裂的不同,优化了玻璃等脆性材料的切削加工参数.其研究结果对加工玻璃,陶瓷等难加工的脆性材料具有一定的实际意义.关键词:玻璃,45钢,切削力,刀具磨损StudyonCuttingForcePerfornmnceandToolWearinCuttingGlassand45SteelS0ngQianyaLiuY ajunTangY ongetalAbstract:Therelationshipwithcuttingforceandtoolwearincuttingglassand45steelisstudie d.Duringthecuttingpro—cess,thedifferenceofplasticdeformationandbrittlefractureisdescribed.Thebrittlematerial' Scuttingparameters,suchasglassareoptimized.Theresearchresultisveryusefultodirectthecuttingof45steelandthebrittlema terialsthatareverydifficulttomachine,suchasglassandcerainics.Keywords:glass,45steel,cuttingforce,toolwear1引言不.由于玻璃等脆性材料本身的物理化学性能与45钢等常见的塑性材料有较大的不同,所以使得其加工方法和效率也受到了一定的限制.目前国内外普遍使用以金刚石工具为主的锯切,磨削,研磨等加工工艺,但由于加工过程复杂,影响因素多,使得目前对上述材料的加工,在国内外均存在加工效率低,质量难以保证,环境污染大,材料浪费严重等突出问题.本文着重对比研究了玻璃和45钢的不同切削性能,分析了在特定的切削参数下,切削力,刀具磨损和表面粗糙度之间的对应关系,为找到脆性材料的高效,合理的加工方法提供有益的参考.2试验条件(1)试验所用的设备:CM6140车床;测力装置:Kisfler车削测力仪;KisflerS019A电荷放大器和研华PCI1701采集卡等,切削力测量采集系统如图l所*广东省自然科学基金博士启动基金资助项目(项目编号:04300123) 收稿日期:2OO4年l1月———1车削测力仪广.1电荷放大器r]数据采集卡r_1订鼻矾图1切削力测量采集系统图(2)工件材料:钠钙玻璃和45钢.(3)刀具:采用YG6硬质合金刀片.(4)切削参数:钠钙玻璃:切削速度=14.6m/min,进给量厂=0.08mm/r,切削深度口.=0.1mm;45钢:切削速度=95.8m/min,进给量厂=0.08mm/r,切削深度口.=0.1mm.(5)采样参数:采样频率为5kHz,采样点数为8192.(6)测量与观察:用TR200粗糙度仪测量工件表面粗糙度;LEO1530VP场发射型扫描电镜(SEM)观察刀具表面磨损形貌.3试验结果与分析3.1切削力的对比关系首先从切削力的角度来分析和比较玻璃和452祝同江,谈天民.工程数学——变分法.北京理工大学出版社,1994:25—1203盛宏玉,高荣誉.非均匀变温时两端固支叠层闭口厚柱壳的热应力分析.工程力学,2000,17(4):1171234王建梅,黄庆学,侯建亮等.热连轧机工作辊轴承座的热应力研究.轴承,2002(1):145刘萍.数值计算方法.北京:人民邮电出版社,2002:150～2506李景.有限元法.北京:北京邮电大学出版社,1999:51—787[美]JE艾金.有限元法的应用与实现.北京:科学出版社,1992:2662948林瑞泰编着.热传导理论与方法.天津:天津大学出版社. 1992:124—178作者:苗恩铭,教授,合肥工业大学仪器仪表学院, 230009合肥市。

不规则波下船舶运动响应特征数值分析

㊀㊀文章编号：１００５⁃９８６５（２０２１）０３⁃００３１⁃１１不规则波下船舶运动响应特征数值分析杜一豪１，纪㊀翀１，姜胜超１，２，顾丹丹３，米晓林３（１．大连理工大学船舶工程学院，辽宁大连㊀１１６０２４；２．大连理工大学海岸和近海工程国家重点实验室，辽宁大连㊀１１６０２４；３．电磁散射重点实验室，上海㊀２００４３８）摘㊀要：针对不规则波浪作用下Ｗｉｇｌｅｙ型船的运动响应问题进行了系统的研究，采用统计学方法深入探讨了船舶不规则运动幅值和响应周期的分布规律，并通过傅里叶变换对船舶运动响应进行了频谱特征分析㊂结果表明，船舶横摇方向与升沉和纵摇方向随机运动的响应特征有显著差异㊂在升沉与纵摇方向，波浪谱峰频率远离自振频率，前十分之一大振幅运动对应周期离散性较小，基本稳定在波浪谱峰周期附近，但小振幅运动周期分布离散性较大，频谱分析指出船舶升沉与纵摇运动响应频谱在波浪谱峰频率附近出现明显峰值㊂而在横摇方向，波浪谱峰频率与自振频率相耦合，不同振幅的横摇运动响应周期均稳定在自振周期附近，且周期离散性较小，频谱分析也表明横摇运动响应频谱主要集中于船舶运动自振频率附近㊂关键词：不规则波；运动响应；时域模拟；ＨｙｄｒｏＳｔａｒ；频谱分析中图分类号：Ｕ６６１．３２㊀㊀㊀文献标志码：Ａ㊀㊀㊀ＤＯＩ：１０．１６４８３／ｊ．ｉｓｓｎ．１００５⁃９８６５．２０２１．０３．００４收稿日期：２０２０⁃０５⁃１６基金项目：国家自然科学基本资助项目（５１９０９０２７，６１９０１２６９）；中央高校基本科研业务费资助（ＤＵＴ２１ＬＫ１２）作者简介：杜一豪（１９９５⁃），男，山东临沂人，硕士研究生，主要从事波浪对海上建筑物作用方面的研究㊂Ｅ⁃ｍａｉｌ：ｄｕｙｉｈａｏ＠ｍａｉｌ．ｄｌｕｔ．ｅｄｕ．ｃｎ通信作者：姜胜超（１９８４⁃），男，辽宁沈阳人，副教授，主要从事波浪对海上建筑物作用方面的研究㊂Ｅ⁃ｍａｉｌ：ｊｉａｎｇｓｈｅｎｇｃｈａｏ＠ｆｏｘｍａｉｌ．ｃｏｍＮｕｍｅｒｉｃａｌａｎａｌｙｓｉｓｏｆｓｈｉｐｍｏｔｉｏｎｒｅｓｐｏｎｓｅｃｈａｒａｃｔｅｒｉｓｔｉｃｓｉｎｉｒｒｅｇｕｌａｒｗａｖｅｓＤＵＹｉｈａｏ１，ＪＩＣｈｏｎｇ１，ＪＩＡＮＧＳｈｅｎｇｃｈａｏ１，２，ＧＵＤａｎｄａｎ３，ＭＩＸｉａｏｌｉｎ３（１．ＳｃｈｏｏｌｏｆＮａｖａｌＡｒｃｈｉｔｅｃｔｕｒｅ，ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｄａｌｉａｎ１１６０２４，Ｃｈｉｎａ；２．ＳｔａｔｅＫｅｙＬａｂｏｒａｔｏｒｙｏｆＣｏａｓｔａｌａｎｄＯｆｆｓｈｏｒｅＥｎｇｉｎｅｅｒｉｎｇ，ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｄａｌｉａｎ１１６０２４，Ｃｈｉｎａ；３．ＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｎＥｌｅｃｔｒｏｍａｇｎｅｔｉｃＳｃａｔｔｅｒｉｎｇＬａｂｏｒａｔｏｒｙ，Ｓｈａｎｇｈａｉ２００４３８，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＴｈｅｍｏｔｉｏｎｒｅｓｐｏｎｓｅｓｏｆＷｉｇｌｅｙｓｈｉｐｉｎｉｒｒｅｇｕｌａｒｗａｖｅｓａｒｅｉｎｖｅｓｔｉｇａｔｅｄｓｙｓｔｅｍａｔｉｃａｌｌｙ．Ｔｈｅｄｉｓｔｒｉｂｕｔｉｏｎｏｆｔｈｅａｍｐｌｉｔｕｄｅａｎｄｒｅｓｐｏｎｓｅｐｅｒｉｏｄｏｆｒａｎｄｏｍｍｏｔｉｏｎｉｎｉｒｒｅｇｕｌａｒｗａｖｅｓｉｓｄｉｓｃｕｓｓｅｄｂｙｓｔａｔｉｓｔｉｃａｌｍｅｔｈｏｄ，ｔｈｅｓｐｅｃｔｒａｏｆｓｈｉｐｍｏｔｉｏｎａｒｅａｌｓｏｏｂｔａｉｎｅｄｂｙｓｐｅｃｔｒａｌｃａｌｃｕｌａｔｉｏｎ．Ｔｈｅｎｕｍｅｒｉｃａｌｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｔｈｅｒｅａｒｅｓｉｇｎｉｆｉｃａｎｔｄｉｆｆｅｒｅｎｃｅｓｉｎｔｈｅｒｅｓｐｏｎｓｅｃｈａｒａｃｔｅｒｉｓｔｉｃｓｂｅｔｗｅｅｎｔｈｅｒａｎｄｏｍｍｏｔｉｏｎｉｎｔｈｅｒｏｌｌｉｎｇｄｉｒｅｃｔｉｏｎ，ｈｅａｖｉｎｇｄｉｒｅｃｔｉｏｎａｎｄｐｉｔｃｈｉｎｇｄｉｒｅｃｔｉｏｎ．Ｆｏｒｈｅａｖｉｎｇａｎｄｐｉｔｃｈｉｎｇｍｏｔｉｏｎ，ｂｅｃａｕｓｅｔｈｅｐｅａｋｆｒｅｑｕｅｎｃｙｏｆｗａｖｅｓｐｅｃｔｒｕｍｉｓｆａｒａｗａｙｆｒｏｍｔｈｅｎａｔｕｒａｌｆｒｅｑｕｅｎｃｙ，ｔｈｅｄｉｓｐｅｒｓｉｏｎｏｆｔｈｅｐｅｒｉｏｄｃｏｒｒｅｓｐｏｎｄｉｎｇｔｏｌａｒｇｅ⁃ａｍｐｌｉｔｕｄｅｍｏｔｉｏｎｉｓｓｍａｌｌ，ｗｈｉｃｈｉｓｂａｓｉｃａｌｌｙｓｔａｂｌｅｎｅａｒｔｈｅｐｅａｋｐｅｒｉｏｄｏｆｔｈｅｗａｖｅｓｐｅｃｔｒｕｍ，ｗｈｉｌｅｔｈｅｄｉｓｐｅｒｓｉｏｎｏｆｔｈｅｐｅｒｉｏｄｄｉｓｔｒｉｂｕｔｉｏｎｃｏｒｒｅｓｐｏｎｄｉｎｇｔｏｓｍａｌｌ⁃ａｍｐｌｉｔｕｄｅｍｏｔｉｏｎｉｓｌａｒｇｅ．ＡｎｄＦｏｕｒｉｅｒａｎａｌｙｓｉｓｉｎｄｉｃａｔｅｓｔｈａｔｔｈｅｒｅｓｐｏｎｓｅｆｒｅｑｕｅｎｃｙｏｆｈｅａｖｉｎｇａｎｄｐｉｔｃｈｉｎｇｍｏｔｉｏｎｈａｓｏｂｖｉｏｕｓｐｅａｋｖａｌｕｅｓｎｅａｒｔｈｅｐｅａｋｆｒｅｑｕｅｎｃｙｏｆｗａｖｅｓｐｅｃｔｒｕｍ．Ｆｏｒｒｏｌｌｉｎｇｍｏｔｉｏｎ，ｓｉｎｃｅｔｈｅｐｅａｋｆｒｅｑｕｅｎｃｙｏｆｗａｖｅｓｐｅｃｔｒｕｍｉｓｃｏｕｐｌｅｄｗｉｔｈｔｈｅｎａｔｕｒａｌｆｒｅｑｕｅｎｃｙ，ｔｈｅｒｅｓｐｏｎｓｅｐｅｒｉｏｄｏｆｒｏｌｌｉｎｇｍｏｔｉｏｎｗｉｔｈｄｉｆｆｅｒｅｎｔａｍｐｌｉｔｕｄｅｓｉｓｓｔａｂｌｅｎｅａｒｔｈｅｎａｔｕｒａｌｖｉｂｒａｔｉｏｎｐｅｒｉｏｄａｎｄｔｈｅｐｅｒｉｏｄｄｉｓｐｅｒｓｉｏｎｉｓｓｍａｌｌ．Ｔｈｅｓｐｅｃｔｒｕｍａｎａｌｙｓｉｓａｌｓｏｓｈｏｗｓｔｈａｔｔｈｅｒｅｓｐｏｎｓｅｓｐｅｃｔｒｕｍｏｆｒｏｌｌｉｎｇｍｏｔｉｏｎｉｓｍａｉｎｌｙｃｏｎｃｅｎｔｒａｔｅｄｎｅａｒｔｈｅｎａｔｕｒａｌｆｒｅｑｕｅｎｃｙｏｆｓｈｉｐｍｏｔｉｏｎ．Ｋｅｙｗｏｒｄｓ：ｉｒｒｅｇｕｌａｒｗａｖｅｓ；ｍｏｔｉｏｎｒｅｓｐｏｎｓｅｓ；ｔｉｍｅｄｏｍａｉｎｓｉｍｕｌａｔｉｏｎ；ＨｙｄｒｏＳｔａｒ；ｓｐｅｃｔｒｕｍａｎａｌｙｓｉｓ波浪是船舶在海上航行时经受的主要外部载荷，直接影响船舶与海洋结构物在海上的航行状态和工作性能㊂船舶在波浪中的运动响应一直是船舶水动力学和试验研究的热点问题㊂长时间以来，人们采用规则第３９卷第３期２０２１年５月海洋工程ＴＨＥＯＣＥＡＮＥＮＧＩＮＥＥＲＩＮＧＶｏｌ．３９Ｎｏ．３Ｍａｙ２０２１波对船舶耐波性问题进行研究，包括船舶水动力性能㊁波浪载荷和船舶运动响应等问题，并取得了比较完备的研究方法与大量的成果［１⁃４］㊂但是，船舶耐波性问题是动力学问题，将真实海况随机波浪简化为规则波的方法实际上是将具有一定频率分布的波浪谱简化为狄拉克函数谱，忽略了大量动力学影响㊂因此，使用不规则波理论来模拟，才能更为准确地描述出船舶在真实海况的运动响应情况㊂目前，大多数研究采用长峰不规则波或短峰不规则波来模拟真实海况㊂Ｈｕａ等［５］基于线性理论，采用长峰不规则波研究了船舶在不规则波中的强非线性横摇性能㊂Ｃａｒｒｉｃａ等［６⁃７］使用ＣＦＤ方法研究了黏性流中船舶在长峰不规则波作用下的运动响应㊂Ｌｕｑｕｅｔｈ等［８］将黏性流理论中ＳＷＥＮＳＥ方法推广到不规则波６自由度舰船的仿真中㊂马洁等［９］采用ＩＴＴＣ单参数标准海浪谱模拟了不规则海浪作用下船舶线性横摇㊁纵摇运动，讨论了不同船型和不同航态下船舶运动规律㊂吴乘胜等［１０⁃１１］基于黏性数值波浪水池技术对不规则波进行了模拟并研究了船舶顶浪航行的运动响应问题㊂郑文涛等［１２］开展了船舶在随机波浪作用下运动的模型试验，着重探讨了短峰波的方向扩散性对船舶运动的影响㊂陈京普等［１３⁃１４］建立了船舶在长峰和短峰不规则波中时域数值模拟方法，并研究了１０万ｔ油轮在长峰和短峰不规则波中的运动响应问题㊂ＳＨＥＮ等［１５］利用ＵＲＡＮＳ⁃ＶＯＦ方法对长峰波中船舶运动响应进行数值预测，通过白噪声谱生成入射波进而对船舶运动进行预报㊂石博文等［１６］在利用ＣＦＤ方法生成具有较高精度的不规则波的基础上，对ＤＴＭＢ５５１２船模的顶浪纵摇和垂荡运动进行了数值模拟㊂相比于规则波，不规则波由于所含的频率范围较大，一旦不规则波列中包含有较多船舶响应敏感的频率成分，则可导致船舶激烈的运动，船舶运动响应规律会更加复杂㊂但是目前对不规则波数值模拟的研究，多数只给出船舶运动历时曲线，而对不规则波作用下船舶运动统计规律缺乏深入的研究㊂实际上，海上波浪变化多样，不仅需要模拟高精度的波浪环境，船舶运动响应的统计学规律对预测实际船舶的耐波性也至关重要㊂基于线性势流理论，以Ｗｉｇｌｅｙ型船为研究对象，采用Ｊｏｎｓｗａｐ谱的长峰不规则波，借助水动力软件ＨｙｄｒｏＳｔａｒ对随机波浪中船舶升沉㊁横摇和纵摇运动进行数值模拟，对运动响应的统计学规律进行了深入研究㊂以５级海况条件为典型计算工况，深入剖析不规则波条件下船舶运动振幅和响应周期等响应规律，并比较了不同波浪谱峰频率对运动响应的影响㊂１㊀控制方程和数值方法基于势流理论，在流体无黏且运动无旋假设下，对于不可压缩流体，流场存在速度势Φ（ｘ，ｔ），且满足拉普拉斯方程：Ñ２Φ＝０（１）对于简谐波与结构物作用问题，结构的运动响应也应是同频率下的简谐运动㊂从速度势Φ（ｘ，ｔ）中分离出时间因子ｅ－ｉωｔ，即：Φ（ｘ，ｙ，ｚ，ｔ）＝Ｒｅ［φ（ｘ，ｙ，ｚ）ｅ－ｉωｔ］（２）式中：φ为空间复速度势，φ仍满足拉普拉斯方程㊂对于运动的物体，可以将φ分解为入射势φｉ㊁绕射势φｄ和辐射势φｒ，即：φ＝φｉ＋φｄ＋φｒ（３）辐射势φｒ按物体运动的６个分量分解为：φｒ＝ð６ｊ＝１－ｉωξｊφｊ，㊀ｊ＝１，２，，６（４）采用边界元方法进行求解，记φ７＝φｄ，对绕射势和辐射势可建立如下边界积分方程：１２φｊ（ｘ０）－∬Ｓｂφｊ（ｘ）∂Ｇ（ｘ，ｘ０）∂ｎｄＳ＝－∬ＳｂＧ（ｘ，ｘ０）ｎｊｄＳ，ｊ＝１，２，３，４，５，６∬ＳｂＧ（ｘ，ｘ０）∂φ０（ｘ）∂ｎｄＳ，ｊ＝７ìîíïïïïïï（５）２３海㊀㊀洋㊀㊀工㊀㊀程第３９卷根据式（５）可求得绕射势和辐射势，进而由伯努利方程获得流体压强㊂通过对瞬时湿物面Ｓ进行流体压强积分得到波浪作用力，进一步将其展开到物体表面平均位置Ｓｍ，使用线性化伯努利方程得到物体在静水中的湿表面ＳＢ上的波浪作用力为：ｆ＝∬ＳＢｐｎｄＳ＝－ρ∬ＳＢｇｚ－ｉω（φｉ＋φｄ＋φｒ）[]ｎｄＳ－ρｇ∬ＳＢξ＋αˑ（ｘ－ｘｃ）[]㊃ｎ３ｎｄＳ＋Ｏ（ε２）（６）式中：ξ为平动位移；α为转角㊂刚性浮体的运动响应通过刚体运动方程确定：－ω２（Ｍ＋ａ）－ｉω（Ｂ＋ｂ）＋（Ｋ＋Ｃ）[]ξ{}＝ｆｅｘ{}（７）采用脉动响应函数方法，利用频域下激振力㊁附加质量和辐射阻尼，通过傅里叶变换求得时域下结构物的运动响应㊂物体在时域内的方程为：ð６ｊ＝１Ｍｋｊ＋ｍｋｊ()ξ㊃㊃ｊ（ｔ）＋ʏｔ０ξ㊃ｊ（τ）Ｋｋｊ（ｔ－τ）ｄτ＋Ｂｋｊξ㊃ｊ（ｔ）＋Ｃｋｊξｊ（ｔ）[]＝Ｆｋ（ｔ）（８）式中：Ｍｋｊ和Ｃｋｊ为物体广义质量和恢复力系数，Ｂｋｊξ㊃ｊ（ｔ）为系统的黏性因素等产生的阻尼，Ｆｋ（ｔ）为波浪激振力㊂２㊀数值模拟设置及其验证采用上述数值模型对细长体Ｗｉｇｌｅｙ船模型在波浪作用下运动响应展开数值研究，其中，Ｗｉｇｌｅｙ型船模型定义为：２ｙｘ，ｚ()Ｂ＝１－ｚＤæèçöø÷２éëêêùûúú１－２ｘＬｐｐæèçöø÷２éëêêùûúú１＋０．２２ｘＬｐｐæèçöø÷２éëêêùûúú（９）式中：Ｌｐｐ，Ｂ，Ｄ分别表示静水时的水线长㊁船宽和吃水㊂计算所用Ｗｉｇｌｅｙ船模型尺寸为Ｌｐｐ＝１００ｍ，Ｂ＝１０ｍ，Ｄ＝６．２５ｍ，对应其他力学参数及自振频率如表１所示㊂表１㊀船模质量属性Ｔａｂ．１㊀Ｑｕａｌｉｔｙａｔｔｒｉｂｕｔｅｓ参㊀数Ｗｉｇｌｅｙ型船排水体积Δ／ｍ３２７７０．０５重心高度ｈ／ｍ５．６７横摇惯性矩ｋｘｘ／ｍ３．５０纵摇惯性半径ｋｙｙ／ｍ２５．００参㊀数Ｗｉｇｌｅｙ型船升沉自振频率ω３／（ｒａｄ㊃ｓ－１）１．３１横摇自振频率ω４／（ｒａｄ㊃ｓ－１）１．０６纵摇自振频率ω５／（ｒａｄ㊃ｓ－１）１．２４㊀㊀首先对网格收敛性进行了验证㊂分别使用３组不同尺寸的网格进行数值模拟，网格数量分别为５０４０个（Ｍｅｓｈ１）㊁９１２０个（Ｍｅｓｈ２）和１９０４０个（Ｍｅｓｈ３）㊂通过频域求解得到Ｗｉｇｌｅｙ船的附加质量㊁辐射阻尼和激振力，如图１㊁图２㊁图３所示㊂从图中可以看出，３组网格所得数据相互吻合较好，证明了方法所采用网格的正确性㊂综合考虑计算精度和计算成本，数值模拟均采用９１２０网格方案（Ｍｅｓｈ２）进行，计算网格如图４所示㊂图１㊀附加质量Ｆｉｇ．１㊀Ａｄｄｅｄｍａｓｓ３３第３期杜一豪，等：不规则波下船舶运动响应特征数值分析图２㊀辐射阻尼Ｆｉｇ．２㊀Ｒａｄｉａｔｉｏｎｄａｍｐｉｎｇ图３㊀波浪激振力Ｆｉｇ．３㊀Ｗａｖｅｅｘｃｉｔａｔｉｏｎｆｏｒｃｅ图４㊀计算网格Ｆｉｇ．４㊀Ｃｏｍｐｕｔａｔｉｏｎａｌｇｒｉｄｓ为了验证该数值模拟的可靠性，将数值模拟计算的Ｗｉｇｌｅｙ船水动力系数与文献［１７］中的试验值进行对比，文献中的船模主尺度如表２所示，根据重力相似准则，无因次化后的对比结果如图５㊁图６所示，可以看到，数值模拟结果与试验值在大部分频率范围内能较好地吻合，说明该数值模拟有较高的可靠性㊂进一步计算求得Ｗｉｇｌｅｙ船模的升沉㊁横摇㊁纵摇ＲＡＯ曲线，如图７所示，从图中可以看到升沉和横摇曲线峰值频率分别为１．３０ｒａｄ／ｓ㊁１．０５ｒａｄ／ｓ，纵摇曲线第二个峰值频率为１．２５ｒａｄ／ｓ，与计算所得的自振频率相互符合㊂表２㊀文献［１７］中船模主尺度Ｔａｂ．２㊀Ｐｒｉｎｃｉｐａｌｄｉｍｅｎｓｉｏｎｓ参数船长Ｌ／ｍ船宽Ｂ／ｍ吃水Ｄ／ｍ排水体积Δ／ｍ３纵摇惯性半径ｋｙｙ／ｍ数值３．００．３０．１８７５０．０７８０．７５图５㊀附加质量Ｆｉｇ．５㊀Ａｄｄｅｄｍａｓｓ㊀㊀４３海㊀㊀洋㊀㊀工㊀㊀程第３９卷图６㊀辐射阻尼Ｆｉｇ．６㊀Ｒａｄｉａｔｉｏｎｄａｍｐｉｎｇ图７㊀船舶运动ＲＡＯ曲线Ｆｉｇ．７㊀ＭｏｔｉｏｎＲＡＯｓｏｆｓｈｉｐ３㊀不规则波条件下船舶运动响应分析３．１㊀海况条件与船舶运动响应基本特征为研究不规则波浪作用下船舶运动响应特征，采用Ｊｏｎｓｗａｐ谱为入射波浪谱，选取５级海况作为典型海图８㊀５级海况条件下对应的波浪谱Ｆｉｇ．８㊀Ｗａｖｅｓｐｅｃｔｒｕｍｏｆｆｉｖｅ⁃ｌｅｖｅｌｓｅａｓｔａｔｅ况对船舶运动响应情况进行数值研究，其中，５级海况对应的有效波高和谱峰周期分别为Ｈｓ＝３．０３ｍ㊁Ｔｐ＝６．２０ｓ，实际Ｊｏｎｓｗａｐ谱如图８所示㊂在时域模拟中，模拟时长为３６００ｓ，时间步长取０．１ｓ㊂Ｗｉｇｌｅｙ船升沉㊁横摇与纵摇３个方向的典型运动历时曲线如图９所示㊂从图中可以看出，船模运动振幅呈现出不规则波运动的状态，采用上跨零点法统计三者分别为５２２㊁６００㊁５３４个完整运动响应组成，说明取得的运动响应结果满足不规则波平稳性与各态历经性的统计要求㊂进一步对１２００２４００ｓ㊁２４００３６００ｓ两个时间段的运动响应特征进行统计分析，如表３所示，可以看出，两者统计数据接近，说明１２００３６００ｓ时间段内，船体运动已经进入稳定状态，统计结果不随时间的变化改变㊂上述结果说明文中计算结果可以作为典型运动响应工况进行统计分析㊂以下分析选１５００３０００ｓ时间历程作为样本进行统计㊂表３㊀船舶运动响应统计数据Ｔａｂ．３㊀Ｒｅｓｕｌｔｓｏｆｓｈｉｐｍｏｔｉｏｎｒｅｓｐｏｎｓｅｓ运动时间段ＡａｖｅＡ１／３ＴａｖｅＴ１／３升沉１２００２４００ｓ０．１８ｍ０．１１ｍ７．２９ｓ６．８１ｓ２４００３６００ｓ０．１７ｍ０．１１ｍ７．３８ｓ６．８２ｓ横摇１２００２４００ｓ３６．３７ʎ２２．１４ʎ５．５９ｓ５．９４ｓ２４００３６００ｓ３６．３５ʎ２３．４２ʎ５．９４ｓ５．９７ｓ纵摇１２００２４００ｓ１．２２ʎ０．７５ʎ７．２１ｓ６．６３ｓ２４００３６００ｓ１．１６ʎ０．７４ʎ７．２３ｓ６．７４ｓ５３第３期杜一豪，等：不规则波下船舶运动响应特征数值分析图９㊀船舶运动响应历程线Ｆｉｇ．９㊀Ｍｏｔｉｏｎｒｅｓｐｏｎｓｅｓｏｆｓｈｉｐ３．２　船舶运动历时曲线概率密度分析对样本内船舶运动历时曲线的概率密度进行统计，历时曲线概率密度如图１０１２所示，从图中可以看出，船舶升沉㊁横摇㊁纵摇３个方向上的运动时间历程概率密度分布基本符合高斯分布，高斯分布位置函数为ζ（ｔ）＝０，即船舶平衡位置㊂进一步对比历时曲线概率密度，可以看出，不同角度波浪作用时，将不同程度的影响历时曲线概率密度的形状㊂对于升沉运动，对比图１０（ａ）㊁（ｂ）和（ｃ）可以发现，历时曲线概率密度的形状发生显著变化，图１０（ａ）和（ｂ）中相关系数Ｒ２分别为０．９８１７９和０．９７２６１，此时Ｒ２＜０．９９，说明此时概率密度分布与高斯分布吻合程度稍差；而图１０（ｃ）中Ｒ２＝０．９９６５５＞０．９９，说明此时概率密度分布与高斯分布吻合程度较高㊂而对于横摇运动和纵摇运动，从图１１和１２可以看出，历时曲线概率密度的形状变化较小，图１１和１２中相关系数Ｒ２＞０．９９，说明横摇和纵摇运动时间历程的概率密度分布与高斯分布高斯积分吻合程度较高㊂此外，不同角度波浪作用会影响不同运动曲线点出现的概率值，并且影响高斯分布的宽度，实际反映了船舶运动最大振幅㊂以升沉运动为例，从图１０可以发现，不同角度波浪作用时船舶运动振幅不同，横浪时升沉运动幅度最大㊂图１０㊀升沉运动历时曲线概率密度Ｆｉｇ．１０㊀Ｐｒｏｂａｂｉｌｉｔｙｄｅｎｓｉｔｙｏｆｈｅａｖｅｍｏｔｉｏｎｒｅｓｐｏｎｓｅ６３海㊀㊀洋㊀㊀工㊀㊀程第３９卷图１１㊀纵摇运动历时曲线概率密度Ｆｉｇ．１１㊀Ｐｒｏｂａｂｉｌｉｔｙｄｅｎｓｉｔｙｏｆｐｉｔｃｈｍｏｔｉｏｎｒｅｓｐｏｎｓｅ图１２㊀横摇运动历时曲线概率密度Ｆｉｇ．１２㊀Ｐｒｏｂａｂｉｌｉｔｙｄｅｎｓｉｔｙｏｆｒｏｌｌｍｏｔｉｏｎｒｅｓｐｏｎｓｅ３．３　船舶运动响应振幅和周期分布为了对船舶运动响应振幅进行定量描述，采用上跨零点法对样本内船舶运动振幅与对应周期进行统计，将振幅按照从大到小顺序排列，并给出其对应周期如图１３所示㊂图１３㊀船舶运动幅度分布直方图Ｆｉｇ．１３㊀Ｄｉｓｔｒｉｂｕｔｉｏｎｏｆｍｏｔｉｏｎａｍｐｌｉｔｕｄｅｓａｎｄｐｅｒｉｏｄｓ７３第３期杜一豪，等：不规则波下船舶运动响应特征数值分析首先从图１３（ａ）和（ｂ）中可以看出，不同振幅所对应的周期不同，具有随机的特征，但结合表５和表６的振幅和周期的统计特征可以看出，对于升沉㊁横摇和纵摇运动，均有Ａ１／１０ʈＡ４％和Ａ１／３（Ａｓ）ʈＡ１３％的统计关系，符合随机过程的统计特征㊂从图１３（ａ）（ｄ）可以看到，船舶升沉和纵摇运动周期离散性较大，但较大的运动振幅对应周期基本相同㊂表４给出了船舶运动周期标准差的统计数据，记前十分之一大振幅运动对应周期的标准差为σ１，后十分之一小振幅运动对应周期的标准差为σ２㊂以５级海况波浪０ʎ作用为例，升沉运动和纵摇运动对应的σ１分别为０．３７ｓ和０．３３ｓ，数值较小，说明大振幅响应周期基本相同㊂结合表５和图１３（ａ）（ｄ）可以看到，升沉运动前十分之一大振幅响应周期稳定在６．８３ｓ左右，纵摇运动前十分之一大振幅响应周期稳定在６．７５ｓ左右㊂而σ２分别为１．２８ｓ和２．２２ｓ，数值较大，说明小振幅对应周期差别较大㊂而对于横摇运动，从图１３（ｅ）和（ｆ）可以看出，横摇运动周期离散性较小㊂从表４可以看到５级海况波浪９０ʎ作用时，σ１＝０．２９ｓ，σ２＝０．３１ｓ，数值较小，并且σ１ʈσ２，说明船舶横摇运动响应周期基本相同，进一步从表５和图１３（ｅ）㊁（ｆ）得知，横摇运动响应周期稳定在６．１２ｓ左右㊂表４㊀船舶运动周期标准差统计表Ｔａｂ．４㊀Ｓｔａｎｄａｒｄｄｅｖｉａｔｉｏｎｏｆｐｅｒｉｏｄｓ运动浪向／（ʎ）σ１／ｓσ２／ｓ升沉运动００．３７１．２８４５０．３９２．２６９００．３９１．４６纵摇运动００．３３２．２２４５０．２５１．５１横摇运动９００．２９０．３１４５０．１４０．２３㊀㊀以升沉运动为例，对比不同波浪入射角度下的升沉运动振幅和响应周期，从表５和６可以看到，不同角度波浪入射对应的振幅不同，波浪入射角度从０ʎ到９０ʎ，升沉运动振幅Ａ１／１０和Ａ１／３显著增大，但对应周期则变化较小，基本稳定在５．７６．８ｓ范围内㊂横摇与纵摇运动振幅和周期变化与升沉运动相似㊂表５㊀运动振幅统计表Ｔａｂ．５㊀Ａｍｐｌｉｔｕｄｅｓｏｆｍｏｔｉｏｎ运动浪向／（ʎ）Ａ１／１０Ａ１／３（Ａｓ）ＡａｖｅＡ４％Ａ１３％升沉运动／ｍ００．２１０．１７０．１１０．２１０．１８４５０．４６０．３８０．２３０．４８０．３９９０２．２１１．７８１．１３２．２６１．８８纵摇运动／（ʎ）０１．４４１．１７０．７６１．４９１．１８４５３．１２２．５０１．５７３．２４２．６３横摇运动／（ʎ）９０４６．０９３６．８０２３．０５４７．７９３７．２３４５１８．７４１４．９０９．３８１８．５４１５．５１㊀㊀８３海㊀㊀洋㊀㊀工㊀㊀程第３９卷表６㊀纵摇运动振幅周期统计表Ｔａｂ．６㊀Ａｍｐｌｉｔｕｄｅｓａｎｄｐｅｒｉｏｄｓｏｆｐｉｔｃｈｍｏｔｉｏｎ运动浪向／（ʎ）Ｔ１／１０Ｔ１／３ＴａｖｅＴ４％Ｔ１３％升沉运动／ｍ０６．８３６．８１６．８３６．３８６．９４４５６．７８６．９１６．６１６．４２７．０６９０５．６８５．７８５．５９６．００６．２６纵摇运动／（ʎ）０６．７５６．８２６．７３７．０４７．２２４５６．３４６．３３６．２８６．５１６．１４横摇运动／（ʎ）９０６．１１６．１２６．１３５．９７５．８０４５６．４２６．４４６．４２６．３６６．９３３．４㊀不同谱峰频率下的船舶运动响应频谱分析为考虑不同不规则波浪参数对船舶运动响应特征的影响，基于５级海况，选取有效波高Ｈｓ＝３．０３ｍ，谱峰频率ωｐ＝０．５０ｒａｄ／ｓ㊁１．００ｒａｄ／ｓ㊁１．５０ｒａｄ／ｓ条件下船舶运动进行数值模拟，波浪谱如图１４所示㊂采用快速傅里叶变换（ＦＦＴ）算法分别对３种不同谱峰频率作用下船舶运动历时曲线进行分析，获得运动响应频谱曲线，并与波浪谱曲线进行对比，如图１５所示㊂从图中可以看出，波浪谱峰频率可以对船舶运动响应频率产生显著影响㊂对于升沉与纵摇运动的情况，ωｐ＝０．５０ｒａｄ／ｓ条件下波浪谱与船舶运动响应谱几乎重合㊂当ωｐ＝１．００ｒａｄ／ｓ与１．５０ｒａｄ／ｓ时，船舶运动响应频率均呈现双峰的特征，其中高频峰值出现在船舶自振频率附近，说明升沉和纵摇运动在波浪谱峰频率和自振频率范围内运动㊂对于横摇运动的情况，在横摇自振频率，即ω４＝１．０６ｒａｄ／ｓ附近，船舶运动频率均出现较大峰值，包括谱峰频率为ωｐ＝０．５０ｒａｄ／ｓ与１．５０ｒａｄ／ｓ时的情况㊂进一步分析可以看出，当ωｐ＝０．５０ｒａｄ／ｓ时，谱峰频率低于船舶自振频率，船舶运动响应谱在０．５０ｒａｄ／ｓ附近没有明显的峰值，运动响应谱仍为单峰特征，说明低频波浪能量对横摇运动贡献较小，横摇运动依然主要按照自振频率运动；当ωｐ＝１．５０ｒａｄ／ｓ时，谱峰频率高于船舶自振频率，船舶运动响应谱在波浪谱峰频率附近存在明显的峰值，运动响应谱呈现双峰的特征㊂图１４㊀不同谱峰频率对应的波浪谱Ｆｉｇ．１４㊀Ｗａｖｅｓｐｅｃｔｒｕｍｓｏｆｄｉｆｆｅｒｅｎｔｐｅａｋｆｒｅｑｕｅｎｃｉｅｓ９３第３期杜一豪，等：不规则波下船舶运动响应特征数值分析图１５㊀运动响应频域分析结果Ｆｉｇ．１５㊀Ａｎａｌｙｓｉｓｒｅｓｕｌｔｓｏｆｍｏｔｉｏｎｒｅｓｐｏｎｓｅｉｎｆｒｅｑｕｅｎｃｙｄｏｍａｉｎ４㊀结㊀语采用数值方法，对不规则波浪作用下Ｗｉｇｌｅｙ型船升沉㊁横摇和纵摇运动响应进行了模拟研究，采用统计学方法对时域运动响应特征进行了统计分析，通过分析比较运动时历的概率密度函数㊁运动振幅和响应周期的分布变化情况以及频谱变换的结果，得到了船舶运动的统计规律㊂数值结果表明，在不规则波作用下，船舶升沉㊁横摇㊁纵摇３个方向上运动时间历程的概率密度函数均基本符合高斯分布㊂船舶横摇方向与升沉及纵摇方向随机运动的响应特征有显著差异㊂在升沉与纵摇方向，波浪谱峰频率远离自振频率，前十分之一大振幅运动对应周期离散性较小，基本稳定在波浪谱峰周期附近，但小振幅运动周期分布离散性较大㊂而在横摇方向，波浪谱峰频率与自振频率相耦合，不同振幅的横摇运动响应周期均稳定在自振周期附近，且周期离散性较小㊂傅里叶分析表明，船舶运动频谱与入射波浪谱有显著差异，主要体现在船舶运动自振频率附近出现较大的频谱峰值㊂通过改变谱峰频率，比较船舶运动响应频谱发现，谱峰频率为０．５０ｒａｄ／ｓ时，升沉和纵摇运动主要依照谱峰频率运动，响应谱能量集中在０．５ｒａｄ／ｓ附近；谱峰频率为１．５０ｒａｄ／ｓ时，升沉和纵摇运动频率主要分布在１．２５１．５０ｒａｄ／ｓ范围内㊂而谱峰频率在０．５０ｒａｄ／ｓ和１．５０ｒａｄ／ｓ时，横摇运动频率则均包含自振频率和谱峰频率㊂总之，不规则波作用下船舶横摇运动响应与升沉和纵摇方向具有不同的统计特征㊂文中的研究方法可适用于船舶在不规则波中的６自由度运动，为船舶时域运动预报提供了重要参考依据㊂参考文献：［１］㊀ＦＡＬＴＩＮＳＥＮＯＭ，ＭＩＣＨＥＬＳＥＮＦＣ．Ｍｏｔｉｏｎｓｏｆｌａｒｇｅｓｔｒｕｃｔｕｒｅｓｉｎｗａｖｅｓａｔｚｅｒｏｆｒｏｕｄｅｎｕｍｂｅｒ［Ｊ］．ＩｎｔｅｒｎａｔｉｏｎａｌＳｙｍｐｏｓｉｕｍｏｎｔｈｅＤｙｎａｍｉｃｓｏｆＭａｒｉｎｅＶｅｈｉｃｌｅｓａｎｄＳｔｒｕｃｔｕｒｅｓｉｎＷａｖｅｓ，１９７５，９０：３⁃１８．［２］㊀ＷＩＬＳＯＮＲ，ＬＥＩＪ，ＪＲＳＫ，ｅｔａｌ．Ｓｉｍｕｌａｔｉｏｎｏｆｌａｒｇｅａｍｐｌｉｔｕｄｅｓｈｉｐｍｏｔｉｏｎｓｆｏｒｐｒｅｄｉｃｔｉｏｎｏｆｆｌｕｉｄ⁃ｓｔｒｕｃｔｕｒｅｉｎｔｅｒａｃｔｉｏｎ［Ｃ］ʊＰｒｏｃｅｅｄｉｎｇｓｏｆ２７ｔｈＯＮＲＳｙｍｏｓｉｕｍｏｎＮａｖａｌＨｙｄｒｏｄｙｎａｍｉｃｓ．２００８：２１６⁃２２７．［３］㊀李积德．船舶耐波性［Ｍ］．哈尔滨：哈尔滨工程大学出版社，２００７．（ＬＩＪｉｄｅ．Ｓｈｉｐｓｅａｋｅｅｐｉｎｇ［Ｍ］．Ｈａｒｂｉｎ：ＨａｒｂｉｎＥｎｇｉｎｅｅｒｉｎｇＵｎｉｖｅｒｓｉｔｙＰｒｅｓｓ，２００７．（ｉｎＣｈｉｎｅｓｅ））［４］㊀郭浩，王建华，万德成．不同波长下ＫＣＳ船运动响应与波浪增阻数值研究［Ｊ］．海洋工程，２０２０，３８（６）：１１⁃２３．（ＧＵＯＨａｏ，ＷＡＮＧＪｉａｎｈｕａ，ＷＡＮＤｅｃｈｅｎｇ．ＮｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｎｏｆｗａｖｅｍｏｔｉｏｎｓａｎｄａｄｄｅｄｒｅｓｉｓｔａｎｃｅｏｆＫＣＳｆｏｒｄｉｆｆｅｒｅｎｔｗａｖｅｌｅｎｇｔｈｓ［Ｊ］．ＴｈｅＯｃｅａｎＥｎｇｉｎｅｅｒｉｎｇ，２０２０，３８（１６）：１１⁃２３．（ｉｎＣｈｉｎｅｓｅ））０４海㊀㊀洋㊀㊀工㊀㊀程第３９卷［５］㊀ＨＵＡＪ．ＦａｓｔｓｉｍｕｌａｔｉｏｎｏｆｎｏｎｌｉｎｅａｒＧＭ⁃ｖａｒｉａｔｉｏｎｏｆａｓｈｉｐｉｎｉｒｒｅｇｕｌａｒｗａｖｅｓ［Ｊ］．ＪｏｕｒｎａｌｏｆＳｈｉｐＭｅｃｈａｎｉｃｓ，２０００，３（４）：２５⁃３４．［６］㊀ＣＡＲＲＩＣＡＰＭ，ＷＩＬＳＯＮＲＶ，ＮＯＡＣＫＲ，ｅｔａｌ．Ａｄｙｎａｍｉｃｏｖｅｒｓｅｔｓｉｎｇｌｅ⁃ｐｈａｓｅＬｅｖｅｌ⁃ｓｅｔａｐｐｒｏａｃｈｆｏｒｖｉｓｃｏｕｓｓｈｉｐｆｌｏｗｓａｎｄｌａｒｇｅａｍｐｌｉｔｕｄｅｍｏｔｉｏｎｓａｎｄｍａｎｅｕｖｅｒｉｎｇ［Ｃ］ʊＰｒｏｃｅｅｄｉｎｇｓｏｆ２６ｔｈＳｙｍｐｏｓｉｕｍｏｎＮａｖａｌＨｙｄｒｏｄｙｎａｍｉｃｓ．２００６：１７⁃２２．［７］㊀ＣＡＲＲＩＣＡＰＭ，ＰＡＬＫＫＪ，ＨＯＳＳＥＩＮＩＨＳ，ｅｔａｌ．ＵＲＡＮＳａｎａｌｙｓｉｓｏｆａｂｒｏａｃｈｉｎｇｅｖｅｎｔｉｎｉｒｒｅｇｕｌａｒｑｕａｒｔｅｒｉｎｇｓｅａｓ［Ｊ］．ＪｏｕｒｎａｌｏｆＭａｒｉｎｓＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙ，２００８，１３（４）：３９５⁃４０７．［８］㊀ＬＵＱＵＥＴＲ，ＤＵＣＲＯＺＥＴＧ，ＧＥＮＴＡＺＬ，ｅｔａｌ．ＡｐｐｌｉｃａｔｉｏｎｏｆｔｈｅＳＷＥＮＳＥｍｅｔｈｏｄｔｏｓｅａｋｅｅｐｉｎｇｓｉｍｕｌａｔｉｏｎｓｉｎｉｒｒｅｇｕｌａｒＷａｖｅｓ［Ｃ］ʊＰｒｏｃｅｅｄｉｎｇｓｏｆ９ｔｈＩｎｔｅｒｎａｔｉｏｎａｌＣｏｎｆｅｒｅｎｃｅｉｎＮｕｍｅｒｉｃａｌＳｈｉｐＨｙｄｒｏｄｙｎａｍｉｃｓ．２００７．［９］㊀马洁，韩蕴韬，李国斌．不同航太下船舶运动规律仿真研究［Ｊ］．船舶科学技术，２００６，２８（１）：３２⁃３６．（ＭＡＪｉｅ，ＨＡＮＹｕｎｔａｏ，ＬＩＧｕｏｂｉｎ．Ｔｈｅｓｉｍｕｌａｔｉｏｎｓｔｕｄｉｅｓｏｎｓｈｉｐｓｍｏｖｅｍｅｎｔｐａｔｔｅｒｎｕｎｄｅｒｔｈｅｖａｒｉｏｕｓｓａｉｌｉｎｇｓｉｔｕａｔｉｏｎｓ［Ｊ］．ＳｈｉｐＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙ，２００６，８（１）：３２⁃３６．（ｉｎＣｈｉｎｅｓｅ））［１０］吴乘胜，朱德祥，顾民，等．数值波浪水池及顶浪中船舶水动力计算［Ｊ］．船舶力学，２００８，１２（２）：１６８⁃１７９．（ＷＵＣｈｅｎｇｓｈｅｎｇ，ＺＨＵＤｅｘｉａｎｇ，ＧＵＭｉｎ，ｅｔａｌ．Ｃｏｍｐｕｔａｔｉｏｎｏｆｈｙｄｒｏｄｙｎａｍｉｃｆｏｒｃｅｓｆｏｒａｓｈｉｐｉｎｒｅｇｕｌａｒｈｅａｄｉｎｇｗａｖｅｓｂｙａｖｉｓｃｏｕｓｎｕｍｅｒｉｃａｌｗａｖｅｔａｎｋ［Ｊ］．ＪｏｕｒｎａｌｏｆＳｈｉｐＭｅｃｈａｎｉｃｓ，２００８，１２（２）：１６８⁃１７９．（ｉｎＣｈｉｎｅｓｅ））［１１］吴乘胜，朱德祥，顾民，等．数值波浪水池中船舶顶浪运动模拟研究［Ｊ］．船舶力学，２００８，１２（５）：６９２⁃６９６．（ＷＵＣｈｅｎｇｓｈｅｎｇ，ＺＨＵＤｅｘｉａｎｇ，ＧＵＭｉｎ，ｅｔａｌ．Ｎ⁃ＳＣＦＤｓｉｍｕｌａｔｉｏｎｏｆｗａｖｅ⁃ｉｎｄｕｃｅｄｓｈｉｐｍｏｔｉｏｎｓｉｎｒｅｇｕｌａｒｈｅａｄｗａｖｅｓ［Ｊ］．ＪｏｕｒｎａｌｏｆＳｈｉｐＭｅｃｈａｎｉｃｓ，２００８，１２（５）：６９２⁃６９６．（ｉｎＣｈｉｎｅｓｅ））［１２］郑文涛，匡晓锋，缪泉明，等．船舶在长峰波和短峰波中运动响应的模型试验研究［Ｃ］／／第九届全国水动力学学术会议暨第二十二届全国水动力学研讨会论文集．２００９：３５９⁃３６４．（ＺＨＥＮＧＷｅｎｔａｏ，ＫＵＡＮＧＸｉａｏｆｅｎｇ，ＭＩＡＯＱｕａｎｍｉｎｇ，ｅｔａｌ．Ｍｏｔｉｏｎｓｔｅｓｔｓｔｕｄｙｏｆｓｈｉｐｍｏｔｉｏｎｓｉｎｌｏｎｇ⁃ｃｒｅｓｔｅｄａｎｄｓｈｏｒｔ⁃ｃｒｅｓｔｅｄｉｒｒｅｇｕｌａｒｗａｖｅｓ［Ｃ］ʊＰｒｏｃｅｅｄｉｎｇｓｏｆｔｈｅ９ｔｈＮａｔｉｏｎａｌＣｏｎｇｒｅｓｓｏｎＨｙｄｒｏｄｙｎａｍｉｃｓａｎｄ２２ｎｄＮａｔｉｏｎａｌＣｏｎｆｅｒｅｎｃｅｏｎＨｙｄｒｏｄｙｎａｍｉｃｓ．２００９：３５９⁃３６４．（ｉｎＣｈｉｎｅｓｅ））［１３］ＣＨＥＮＪ，ＺＨＵＤ．Ｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｓｏｆｗａｖｅ⁃ｉｎｄｕｃｅｄｓｈｉｐｍｏｔｉｏｎｓｉｎｔｉｍｅｄｏｍａｉｎｂｙａＲａｎｋｉｎｅｐａｎｅｌｍｅｔｈｏｄ［Ｊ］．ＪｏｕｒｎａｌｏｆＨｙｄｒｏｄｙｎａｍｉｃｓ，２０１０，２２（３）：３７３⁃３８０．［１４］陈京普，魏锦芳，朱德祥．船舶在长峰和短峰不规则波中运动的三维时域数值模拟［Ｊ］．水动力学研究与进展，Ａ辑，２０１１，２６（５）：５８９⁃５９６．（ＣＨＥＮＪｉｎｇｐｕ，ＷＥＩＪｉｎｆａｎｇ，ＺＨＵＤｅｘｉａｎｇ．Ｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｓｏｆｓｈｉｐｍｏｔｉｏｎｓｉｎｌｏｎｇ⁃ｃｒｅｓｔｅｄａｎｄｓｈｏｒｔ⁃ｃｒｅｓｔｅｄｉｒｒｅｇｕｌａｒｗａｖｅｓｂｙａ３Ｄｔｉｍｅｄｏｍａｉｎｍｅｔｈｏｄ［Ｊ］．ＪｏｕｒｎａｌｏｆＨｙｄｒｏｄｙｎａｍｉｃｓ，２０１１，２６（５）：５８９⁃５９６．（ｉｎＣｈｉｎｅｓｅ））［１５］ＳＨＥＮＺ，ＹＥＨ，ＷＡＮＤ．ＵＲＡＮＳｓｉｍｕｌａｔｉｏｎｓｏｆｓｈｉｐｍｏｔｉｏｎｒｅｓｐｏｎｓｅｓｉｎｌｏｎｇ⁃ｃｒｅｓｔｉｒｒｅｇｕｌａｒｗａｖｅｓ［Ｊ］．ＪｏｕｒｎａｌｏｆＨｙｄｒｏｄｙｎａｍｉｃｓ，２０１４，２６（３）：４３６⁃４４６．［１６］石博文，刘正江，吴明．船模不规则波中顶浪运动数值模拟研究［Ｊ］．船舶力学，２０１４（８）：９０６⁃９１５．（ＳＨＩＢｏｗｅｎ，ＬＩＵＺｈｅｎｇｊｉａｎｇ，ＷＵＭｉｎｇ．Ｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｏｆｓｈｉｐｍｏｔｉｏｎｓｉｎｉｒｒｅｇｕｌａｒｈｅａｄｗａｖｅｓ［Ｊ］．ＪｏｕｒｎａｌｏｆＳｈｉｐＭｅｃｈａｎｉｃｓ，２０１４（８）：９０６⁃９１５．（ｉｎＣｈｉｎｅｓｅ））［１７］ＪＯＵＲＮＥＥＪ．ＥｘｐｅｒｉｍｅｎｔｓａｎｄｃａｌｃｕｌａｔｉｏｎｓｏｎｆｏｕｒＷｉｇｌｅｙｈｕｌｌｆｏｒｍｓｉｎｈｅａｄｗａｖｅｓ［Ｒ］．Ｄｅｌｆｔ：ＤｅｌｆｔＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙＲｅｐｏｒｔ，１９９２．１４第３期杜一豪，等：不规则波下船舶运动响应特征数值分析。

CMG-STARS功能介绍

CMG STARS 功能介绍STARS/Builder/Results的功能及模拟选项Functionality and modeling options of STARS/Builder/Results网格、几何形状及结构模型复杂型、加密和粗化Grids, geometry and structural model complexity, upgrading/upscaling径向网格角点网格连接到静态模型包Link to static model package灵活的局部网格加密（LGR）定义混和LGR多级LGR * Is this the same as nesting LGRs directly inside of each other?网格中的断层显示表示Fault explicit representation in grid (YES)倾斜断层铲状断层非相邻网格连接Non-neighboring connections局部加密LGR存在情况下的粗化Upscaling in presence of LGR * In vertical direction (YES) 处理非常复杂网格结构的能力 Ability to handle very complex grid structures追踪前缘移动的自动动态网格调整Automatic dynamic grid adjustment to track front movement 动态网格技术油藏性质处理灵活的岩石类型定义与岩石类型有关的性质Properties corresponding to rock types静态及动态岩石性质的相关性Correlations between static and dynamic rock properties基本的线性压实Basic linear compaction基本的非线性压实Basic non-linear compaction压实相关渗透率Compaction dependent permeability先进的压实特征（滞后）Advanced compaction (hysteresis)耦合地质力学及流体模拟Coupled geomechanics and fluid flow modeling压实驱替Compaction drive下沉／岩石坍塌Subsidence/rock collapse动态裂缝模型 Dynamic fracture modeling多点流量计算Multi-point flux calculation解析水层灵活的水层参数调整双孔和双渗模型沿井筒深度定义User definable shape factor基质－裂缝连接选项Matrix-fracture connection options基质－裂缝传质Matrix-fracture transfer区别于裂缝加密的基质加密 Refinement of matrix separate from fracture refinement温度定义 Temperature specification热性质及岩石地质性质的相关性Correlation between thermal properties and geological rock properties注蒸汽模式Steam injection pattern盖层及岩石热损失Cap and base rock heat losses化学反应组分及反应方案 Chemical reaction components and reaction schemes反应动力学Reaction kinetics非平衡反应动力学 Non-equilibrium reaction kinetics组分的吸附和扩散Adsorption and dispersion of components渗透率及扩散张量Permeability and dispersion tensors速度相关扩散张量 Velocity dependent dispersion tensors 在原来的分子扩散的基础上加入了速度相关扩散饱和度性质函数Saturation property functions表及解析定义Table and analytical definitions每一相相对渗透率表的输入Input of relative permeability tables for each phaseStone 模型Stone models线性等渗透率线 Linear isoperms参数相关比例缩放Parameter dependent scaling相对渗透率比例缩放 Relative permeability scaling滞后作用 Hysteresis流体性质模型 Fluids property modeling黑油／挥发油Black oil/volatile oilPVT 跟踪PVT tracking负示踪剂 Passive tracers STARS multi components and IFE正示踪剂Active tracers示踪剂注入模式Tracer injection pattern数值弥散控制Numerical dispersion controls TVD enhancement流线计算及可视化 Streamline calculation and visualization聚合物流变学 Polymer rheologies聚合物堵塞注入模式Polymer slug injection pattern泡沫模拟 Foam modelling状态方程选项 EOS model options多个状态方程模型 Multiple EOS models多个状态方程区 Multiple EOS regionsJ函数 J - Function状态方程输入选项 EOS data input options烃相密度修正 Hydrocarbon phase density correction烃相粘度 Hydrocarbon phase viscosity水相密度模型 Aqueous phase density model水相粘度模型 Aqueous phase viscosity model水相溶解度模型 Aqueous phase solubility model界面张力模型 Interfacial tension models出砂模型 Sand production model非碳水组份Non-HC components稠油模型 Heavy oil model冷采过程 Cold production processes蒸汽能力 Steam capability燃烧功能 Combustion capability定义多个固相能力 Ability to define multiple solids 固相反应的精度得到提高模块化PVT组分 Modular PVT component非达西流动模型 Non Darcy flow model单相非达西 Single phase Non-Darcy多气组份CBM能力 Multiple gaseous component CBM capabilities沥青沉淀及堵塞模型 Asphaltene precipitation model and plugging model蒸汽添加剂 Steam Additives凝胶 Gel化学平衡 Chemical equilibrium低温氧化 Low Temperature Oxidation尖趾注空气 THAI裂缝油藏中的气相扩散 Gas phase diffusion in fractured reservoirs泡沫油 Foamy oil、微生物 Microbial注空气 Air Injection三元复合驱 ASP电加热 Electrical蜡 Wax界面张力影响 Interfacial tensor effects微粒＆乳状液 Fines & Emulsions矿物分解及沉淀 Mineral dissolution and precipitation二氧化碳吸收 CO2 Sequestration热值选项 Heating value option9点离散 9 point discretization蒸汽圈闭 Stream trap流体性质－润湿性 Fluid properties - wettability井模型／管理 Well modeling/managementBSW and GOR monitors井限制选项 Well constraint options沿井筒深度定义 Along hole depth definition重新完井的历史数据输入 Historical recompletion data input井筒相关性、流动曲线、流动摩阻Wellbore correlations, flow curves, flow friction举升曲线 Lift curves气举优化 Lift gas optimization溶剂循环选项 Solvent recycling option水循环选项 Water recycling option井组 Well groups分段井模型 Segmented well model -SAM 计算井筒摩阻和热损失多级井, 分枝井, 水平井等等Horizontal, snake, hook, and multi-lateral wells 可以模拟多级井和分枝井修井作业以及智能油井作业Workover operations and intelligent well operations油管和套管的分别模拟 Separate modelling of tubing and annulus循环井Circulating wells从井底到井口能量, 相态計算Full energy and phase calculation to top hole position紊流表皮Turbulence Skin离散化井筒Discretized wellbore生産期分数On-time fractions窜流和回流Cross-flow and Back flow蒸气温度保證計算, 蒸汽圈闭Steam Trap数据数入、输出、管理、连接以及界面 Data input, output, management, links, and interfaces关键字及输入命令语言 Keywords and input command language输入文件包括，输入及运行文件路径寻找Input file including, input or run file path searching单位选项及用户定义单位和量Unit options and user-defined units and quantities语法检查 Syntax check检查及测量诊断信息 Checks and measures, diagnostic messages网格加密的量化数据输入Scalable data input for grid refinement重新启动功能Restart mechanism输出及运行文件的用户定义目录 User defined directory for output and run files输出到ASCII文件Export to files输出到Excel文件Export to Excel容易调整报告和绘图（图标、比例尺）Easy to adjust reports and plots (legends, scales, labels)用户定义的习惯输出方式User-defined custom output用户定义的数据结构User-defined data structures容易与新的计算模块连接，例如PVT模块 Easy to link to new calculations modules, e.g. PVT modeling交互式建模及模拟Interactive model building and full simulation运行时间分维（自动）Run time dimensioning (Automatic) 组分数、反应数、岩石流体数的维数限制取消了交互式网格定义和更新 Interactive grid defining/updating交互式岩石性质／分区定义／指定／更新Interactive rock properties/regiondefining/assigning/updating交互式饱和度函数／流体界面定义／更新 Interactive saturation function/fluid contacts defining/updating交互式断层／水体定义／更新 Interactive fault/aquifer properties defining/updating全交互式模拟 Fully interactive simulation运行时间监测结果Run time monitor results动态重启Dynamic Restarts数据分析、解释及可视化Data analysis, interpretation, and visualization多窗口可视化环境用于空间数据的快速分析Visual environment with multiple linked widows for quick analysis of spatial data井特征与时间关系图 Compact representation of well performance plots versus time敏感性分析选项Sensitivity analysis option复杂非均质性的解释（例如通道的几何形状）Interpretation of complex heterogeneities (e.g. channel geometry)多井模拟Multi-well modeling三维可视化到QC输入／输出 3D visualization to QC inputs/outputs三维可视化来分析流动模式3D visualization to analyze flow patterns from the simulations 泡状图来显示产量分布Bubble plots to show production distribution空间分布的历史拟合指标的可视化 Visualization of spatially located history match performance3D和曲线分析图直接链接能力 Direct link between 3D and line plotting analysis capabilities三维立体可视化3D stereoscopic visualization三维可视化的时间动画Time lapse Animation in 3D Stereo灵活的剪切面Flexible cutting plane3D物性探测显示功能 3D probe with additional property list capabilities快捷的栅栏图Easy Fence diagrams不同性质的多窗口／视图Multiple Windows/Views for various propertiesAVI电影生成功能AVI Movie generation capabilities透明Transparency交互旋转及缩放Interactive Rotate and Zoom用户定义参数选择User defined preferences用户定义的函数用于额外的性质分析User defined functions for additional properties analysis讲稿风格图及打印 Presentation style plots and printing等值线绘图能力Contour plotting capabilities井眼轨迹Well Trajectories流动向量Flow Vectors快捷键功能Shortcut hot keys计算性能及多平台Computation performance and multi-platform全隐式及自适应隐式解法Fully implicit and adaptive implicit formulation全隐式井Fully implicit wells稳定性和可靠性Stability and reliabilityWindows 98, NT, 2000, XPIBM, Sun, SG Simulators only32/64位版本32/64 bit versions 三十二位版本性能提高,并推出64位平台并行版本 Parallel运行统计以及性能报告 Run statistics and performance reporting支持及文献Support and documentation在线和搜索（标题、索引）帮助On-line and searchable (topic, index) help操作的浮动窗口描述 Floating window description of operations流程向导以及内容敏感性专家分析Workflow wizards and context sensitive expert guidance练习 Tutorial用户指南User’s guide参考手册 Reference manual用户可以与模拟器开发者沟通User access to simulator development staff。

A Numerical Method for Solution of Ordinary Differential Equations of Fractional Order

β α a It (a It y (t)) α+β β +α β α = a It y (t) = a It y (t) = a It (a It y (t)) .
(6)
In the general case the Riemann-Liouville fractional derivatives do not commute
β α+β β α α (a Dt (a Dt y ))(t) = (a Dt )(t) = (a Dt (a Dt y ))(t) .
(7)
Extending our considerations, the integer operator commutes with the fractional operator
β α−β α (a Dt (a It y ))(t) = (a Dt )(t) ,
(9)
but they do not commute in the opposite way. We turn our attention to the composition rule in two following ways. First of all we found in literature [10,11] the fact, that authors neglect the general property of fractional derivatives given by eqn. (7). Regarding to solution of fractional diﬀerential equations we will apply above properties in the next sections.
α y )(t) = (a It

A Review of Water Hammer Theory and Practice

Mohamed S.Ghidaouiemail:ghidaoui@ust.hkMing Zhaoemail:cezhm@ust.hk Department of Civil Engineering,The Hong Kong University of Science and Technology,Hong Kong,ChinaDuncan A.McInnis Surface Water Group,Komex International Ltd., 450016th Avenue,Suite100,N.W.Calgary,Alberta T3B0M6,CanadaDavid H.Axworthy 163N.Marengo Avenue,#316,Pasadena,CA91101email:bm300@ A Review of Water Hammer Theory and PracticeHydraulic transients in closed conduits have been a subject of both theoretical study and intense practical interest for more than one hundred years.While straightforward in terms of the one-dimensional nature of pipe networks,the full description of transientﬂuidﬂows pose interesting problems inﬂuid dynamics.For example,the response of the turbulence structure and strength to transient waves in pipes and the loss ofﬂow axisymmetry in pipes due to hydrodynamic instabilities are currently not understood.Yet,such under-standing is important for modeling energy dissipation and water quality in transient pipe ﬂows.This paper presents an overview of both historic developments and present day research and practice in theﬁeld of hydraulic transients.In particular,the paper dis-cusses mass and momentum equations for one-dimensional Flows,wavespeed,numerical solutions for one-dimensional problems,wall shear stress models;two-dimensional mass and momentum equations,turbulence models,numerical solutions for two-dimensional problems,boundary conditions,transient analysis software,and future practical and re-search needs in water hammer.The presentation emphasizes the assumptions and restric-tions involved in various governing equations so as to illuminate the range of applicabil-ity as well as the limitations of these equations.Understanding the limitations of current models is essential for(i)interpreting their results,(ii)judging the reliability of the data obtained from them,(iii)minimizing misuse of water-hammer models in both research and practice,and(iv)delineating the contribution of physical processes from the contribution of numerical artifacts to the results of waterhammer models.There are134refrences cited in this review article.͓DOI:10.1115/1.1828050͔1IntroductionThus the growth of knowledge of the physical aspect of reality cannot be regarded as a cumulative process.The basic Gestalt of this knowledge changes from time to time...During the cumulative periods scientists behave as if reality is exactly as they know it except for missing details and improvements in accuracy.They speak of the laws of nature,for example,which are simply models that explain their experience of reality at a certain ter generations of scientists typically discover that these conceptions of reality embodied certain implicit as-sumptions and hypotheses that later on turned out to be incor-rect.Vanderburg,͓1͔Unsteadyﬂuidﬂows have been studied since manﬁrst bent water to his will.The ancient Chinese,the Mayan Indians of Cen-tral America,the Mesopotamian civilizations bordering the Nile, Tigris,and Euphrates river systems,and many other societies throughout history have developed extensive systems for convey-ing water,primarily for purposes of irrigation,but also for domes-tic water supplies.The ancients understood and appliedﬂuidﬂow principles within the context of‘‘traditional,’’culture-based tech-nologies.With the arrival of the scientiﬁc age and the mathemati-cal developments embodied in Newton’s Principia,our under-standing ofﬂuidﬂow took a quantum leap in terms of its theoretical abstraction.That leap has propelled the entire develop-ment of hydraulic engineering right through to the mid-twentieth century.The advent of high-speed digital computers constituted another discrete transformation in the study and application of ﬂuids engineering principles.Today,in hydraulics and other areas, engineersﬁnd that their mandate has taken on greater breadth and depth as technology rapidly enters an unprecedented stage of knowledge and information accumulation.As cited in The Structure of Scientiﬁc Revolutions,Thomas Kuhn͓2͔calls such periods of radical and rapid change in our view of physical reality a‘‘revolutionary,noncumulative transi-tion period’’and,while he was referring to scientiﬁc views of reality,his remarks apply equally to our technological ability to deal with a revised or more complex view of the physical uni-verse.It is in this condition that theﬁeld of closed conduit tran-sientﬂow,and even more generally,the hydraulic analysis,de-sign,and operation of pipeline systems,currentlyﬁnds itself.The computer age is still dawning,bringing with it a massive development and application of new knowledge and technology. Formerly accepted design methodologies,criteria,and standards are being challenged and,in some instances,outdated and revised. Computer aided analysis and design is one of the principal mecha-nisms bringing about these changes.Computer analysis,computer modeling,and computer simula-tion are somewhat interchangeable terms,all describing tech-niques intended to improve our understanding of physical phe-nomena and our ability to predict and control these phenomena. By combining physical laws,mathematical abstraction,numerical procedures,logical constructs,and electronic data processing, these methods now permit the solution of problems of enormous complexity and scope.This paper attempts to provide the reader with a general his-tory and introduction to waterhammer phenomena,a general com-pendium of key developments and literature references as well as an updated view of the current state of the art,both with respect to theoretical advances of the last decade and modeling practice.2Mass and Momentum Equations forOne-Dimensional Water Hammer FlowsBefore delving into an account of mathematical developments related to waterhammer,it is instructive to brieﬂy note the societal context that inspired the initial interest in waterhammer phenom-ena.In the late nineteenth century,Europe was on the cusp of the industrial revolution with growing urban populations and indus-tries requiring electrical power for the new machines of produc-tion.As the fossil fuel era had not begun in earnest,hydroelectric generation was still the principal supply of this important energy source.Although hydroelectric generation accounts for a much smaller proportion of energy production today,the problems asso-Transmitted by Associate Editor HJS Fernando.ciated with controlling theﬂow of water through penstocks and turbines remains an important application of transient analysis. Hydrogeneration companies contributed heavily to the develop-ment ofﬂuids and turbomachinery laboratories that studied, among other things,the phenomenon of waterhammer and its con-trol.Some of Allievi’s early experiments were undertaken as a direct result of incidents and failures caused by overpressure due to rapid valve closure in northern Italian power plants.Frictionless approaches to transient phenomena were appropriate in these early developments because͑i͒transients were most inﬂuenced by the rapid closure and opening of valves,which generated the majority of the energy loss in these systems,and͑ii͒the pipes involved tended to have large diameters and theﬂow velocities tended to be small.By the early1900s,fuel oils were overtaking hydrogeneration as the principal energy source to meet society’s burgeoning de-mand for power.However,the fascination with,and need to un-derstand,transient phenomena has continued unabated to this day. Greater availability of energy led to rapid industrialization and urban development.Hydraulic transients are critical design factors in a large number ofﬂuid systems from automotive fuel injection to water supply,transmission,and distribution systems.Today, long pipelines transportingﬂuids over great distances have be-come commonplace,and the almost universal development of sprawling systems of small pipe diameter,high-velocity water dis-tribution systems has increased the importance of wall friction and energy losses,leading to the inclusion of friction in the governing equations.Mechanically sophisticatedﬂuid control devices,in-cluding many types of pumps and valves,coupled with increas-ingly sophisticated electronic sensors and controls,provide the potential for complex system behavior.In addition,the recent knowledge that negative pressure phases of transients can result in contamination of potable water systems,mean that the need to understand and deal effectively with transient phenomena are more acute than ever.2.1Historical Development:A Brief Summary.The prob-lem of water hammer wasﬁrst studied by Menabrea͓3͔͑although Michaud is generally accorded that distinction͒.Michaud͓4͔ex-amined the use of air chambers and safety valves for controlling water hammer.Near the turn of the nineteenth century,researchers like Weston͓5͔,Carpenter͓6͔and Frizell͓7͔attempted to develop expressions relating pressure and velocity changes in a pipe.Fri-zell͓7͔was successful in this endeavor and he also discussed the effects of branch lines,and reﬂected and successive waves on turbine speed regulation.Similar work by his contemporaries Joukowsky͓8͔and Allievi͓9,10͔,however,attracted greater at-tention.Joukowsky͓8͔produced the best known equation in tran-sientﬂow theory,so well known that it is often called the‘‘fun-damental equation of water hammer.’’He also studied wave reﬂections from an open branch,the use of air chambers and surge tanks,and spring type safety valves.Joukowsky’s fundamental equation of water hammer is as fol-lows:⌬PϭϮ␳a⌬V or⌬HϭϮa⌬Vg(1)where aϭacoustic͑waterhammer͒wavespeed,Pϭ␳g(HϪZ)ϭpiezometric pressure,Zϭelevation of the pipe centerline from a given datum,Hϭpiezometric head,␳ϭﬂuid density,Vϭ͐A udA ϭcross-sectional average velocity,uϭlocal longitudinal velocity, Aϭcross-sectional area of the pipe,and gϭgravitational accelera-tion.The positive sign in Eq.͑1͒is applicable for a water-hammer wave moving downstream while the negative sign is applicable for a water-hammer wave moving upstream.Readers familiar with the gas dynamics literature will note that⌬PϭϮ␳a⌬V is obtain-able from the momentum jump condition under the special case where theﬂow velocity is negligible in comparison to the wavespeed.The jump conditions are a statement of the conserva-tion laws across a jump͑shock͓͒11͔.These conditions are ob-tained either by directly applying the conservation laws for a con-trol volume across the jump or by using the weak formulation of the conservation laws in differential form at the jump.Allievi͓9,10͔developed a general theory of water hammer fromﬁrst principles and showed that the convective term in the momentum equation was negligible.He introduced two important dimensionless parameters that are widely used to characterize pipelines and valve behavior.Allievi͓9,10͔also produced charts for pressure rise at a valve due to uniform valve closure.Further reﬁnements to the governing equations of water hammer appeared in Jaeger͓12,13͔,Wood͓14͔,Rich͓15,16͔,Parmakian͓17͔, Streeter and Lai͓18͔,and Streeter and Wylie͓19͔.Their combined efforts have resulted in the following classical mass and momen-tum equations for one-dimensional͑1D͒water-hammerﬂowsa2gץVץxϩץHץtϭ0(2)ץVץtϩgץHץxϩ4␳D␶wϭ0(3) in which␶wϭshear stress at the pipe wall,Dϭpipe diameter,x ϭthe spatial coordinate along the pipeline,and tϭtemporal coor-dinate.Although Eqs.͑2͒and͑3͒were fully established by the 1960s,these equations have since been analyzed,discussed,red-erived and elucidated in numerous classical texts͑e.g.,͓20–23͔͒. Equations͑2͒and͑3͒constitute the fundamental equations for1D water hammer problems and contain all the physics necessary to model wave propagation in complex pipe systems.2.2Discussion of the1D Water Hammer Mass and Mo-mentum Equations.In this section,the fundamental equations for1D water hammer are derived.Special attention is given to the assumptions and restrictions involved in various governing equa-tions so as to illuminate the range of applicability as well as the limitations of these equations.Rapidﬂow disturbances,planned or accidental,induce spatial and temporal changes in the velocity͑ﬂow rate͒and pressure͑pi-ezometric head͒ﬁelds in pipe systems.Such transientﬂows are essentially unidirectional͑i.e.,axial͒since the axialﬂuxes of mass,momentum,and energy are far greater than their radial counterparts.The research of Mitra and Rouleau͓23͔for the lami-nar water hammer case and of Vardy and Hwang͓25͔for turbulent water-hammer supports the validity of the unidirectional approach when studying water-hammer problems in pipe systems.With the unidirectional assumption,the1D classical water ham-mer equations governing the axial and temporal variations of the cross-sectional average of theﬁeld variables in transient pipe ﬂows are derived by applying the principles of mass and momen-tum to a control volume.Note that only the key steps of the derivation are given here.A more detailed derivation can be found in Chaudhry͓20͔,Wylie et al.͓23͔,and Ghidaoui͓26͔.Using the Reynolds transport theorem,the mass conservation ͑‘‘continuity equation’’͒for a control volume is as follows͑e.g.,͓20–23͔͒ץץt͵cv␳d᭙ϩ͵cs␳͑v"n͒dAϭ0(4)where cvϭcontrol volume,csϭcontrol surface,nϭunit outward normal vector to control surface,vϭvelocity vector.Referring to Fig.1,Eq.͑4͒yieldsץץt͵x xϩ␦x␳Adxϩ͵cs␳͑v"n͒dAϭ0(5)The local form of Eq.͑5͒,obtained by taking the limit as the length of the control volume shrinks to zero͑i.e.,␦x tends to zero͒,isץ͑␳A ͒ץt ϩץ͑␳AV ͒ץxϭ0(6)Equation ͑6͒provides the conservative form of the area-averagedmass balance equation for 1D unsteady and compressible ﬂuids in a ﬂexible pipe.The ﬁrst and second terms on the left-hand side of Eq.͑6͒represent the local change of mass with time due to the combined effects of ﬂuid compressibility and pipe elasticity and the instantaneous mass ﬂux,respectively.Equation ͑6͒can be re-written as follows:1␳D ␳Dt ϩ1A DA Dt ϩץVץxϭ0or1␳A D ␳A Dt ϩץVץxϭ0(7)where D /Dt ϭץ/ץt ϩV ץ/ץx ϭsubstantial ͑material ͒derivative inone spatial dimension.Realizing that the density and pipe area vary with pressure and using the chain rule reduces Eq.͑7͒to the following:1␳d ␳dP DP Dt ϩ1A dA dP DP Dt ϩץVץxϭ0or1␳a 2DP Dt ϩץVץxϭ0(8)where a Ϫ2ϭd ␳/dP ϩ(␳/A )dA /dP .The historical development and formulation of the acoustic wave speed in terms of ﬂuid and pipe properties and the assumptions involved in the formulation are discussed in Sec.3.The momentum equation for a control volume is ͑e.g.,͓20–23͔͒:͚F ext ϭץץt͵cv␳v ᭙ϩ͵cs␳v ͑v "n ͒dA (9)Applying Eq.͑9͒to the control volume of Fig.2;consideringgravitational,wall shear and pressure gradient forces as externally applied;and taking the limit as ␦x tends to zero gives the follow-ing local form of the axial momentum equation:ץ␳AV ץt ϩץ␤␳AV 2ץx ϭϪA ץPץxϪ␲D ␶w Ϫ␥A sin ␣(10)where ␥ϭ␳g ϭunit gravity force,␣ϭangle between the pipe and the horizontal direction,␤ϭ͐A u 2dA /V 2ϭmomentum correction coefﬁing the product rule of differentiation,invoking Eq.͑7͒,and dividing through by ␳A gives the following nonconser-vative form of the momentum equation:ץV ץt ϩV ץV ץx ϩ1␳A ץ͑␤Ϫ1͒␳AV 2ץx ϩ1␳ץP ץx ϩg sin ␣ϩ␶w ␲D ␳A ϭ0(11)Equations ͑8͒and ͑11͒govern unidirectional unsteady ﬂow of a compressible ﬂuid in a ﬂexible tube.Alternative derivations of Eqs.͑8͒and ͑11͒could have been performed by applying the unidirectional and axisymmetric assumptions to the compressible Navier-Stokes equations and integrating the resulting expression with respect to pipe cross-sectional area while allowing for this area to change with pressure.In practice,the order of magnitude of water hammer wave speed ranges from 100to 1400m/s and the ﬂow velocity is of order 1to 10m/s.Therefore,the Mach number,M ϭU 1/a ,in water-hammer applications is often in the range 10Ϫ2–10Ϫ3,where U 1ϭlongitudinal velocity scale.The fact that M Ӷ1in wa-ter hammer was recognized and used by Allievi ͓9,10͔to further simplify Eqs.͑8͒and ͑11͒.The small Mach number approxima-tion to Eqs.͑8͒and ͑11͒can be illustrated by performing an order of magnitude analysis of the various terms in these equations.To this end,let ␳0aU 1ϭwater hammer pressure scale,␳0ϭdensity of the ﬂuid at the undisturbed state,and T ϭ␨L /a ϭtime scale,where L ϭpipe length,X ϭaT ϭ␨L ϭlongitudinal length scale,␨ϭa positive real parameter,␳f U 12/8ϭwall shear scale,and f ϭDarcy-Weisbach friction factor T d ϭradial diffusion time scale.The parameter ␨allows one to investigate the relative magnitude of the various terms in Eqs.͑8͒and ͑11͒under different time scales.For example,if the order of magnitude of the various terms in the mass momentum over a full wave cycle ͑i.e.,T ϭ4L /a )is desired,␨is set to 4.Applying the above scaling to Eqs.͑8͒and ͑11͒gives␳0␳DP *Dt ϩץV *ץx *ϭ0or␳0␳ͩץP *ץt *ϩM V *ץP *ץx *ͪϩץV *ץx *ϭ0(12)ץV *ץt *ϩM V *ץV *ץx *ϩM 1␳A ץ͑␤Ϫ1͒␳AV *2ץx *ϩ␳0␳ץP *ץx *ϩg ␨LUa sin ␣ϩ␨L D M f 2␶w *ϭ0(13)where the superscript *is used to denote dimensionless quantities.Since M Ӷ1in water hammer applications,Eqs.͑12͒and ͑13͒become␳0␳ץP *ץt *ϩץV *ץx *ϭ0(14)ץV *ץt *ϩ␳0␳ץP *ץx *ϩg ␨L Ua sin ␣ϩ␨L D M f2ϩ␨ͩT d L /aͪ␶w *ϭ0.(15)Rewriting Eqs.͑14͒and ͑15͒in dimensional form gives1␳a 2ץP ץt ϩץVץxϭ0(16)ץV ץt ϩ1␳ץP ץx ϩg sin ␣ϩ␶w ␲D ␳Aϭ0(17)Using the Piezometric head deﬁnition ͑i.e.,P /g ␳0ϭH ϪZ ),Eqs.͑16͒and ͑17͒becomeFig.1Control volume diagram used for continuity equationderivationFig.2Control volume diagram used for momentum equationderivationg␳0␳a2ץHץtϩץVץxϭ0(18)ץVץtϩg ␳0␳ץHץxϩ␶w␲D␳Aϭ0(19)The change in density in unsteady compressibleﬂows is of the order of the Mach number͓11,27,28͔.Therefore,in water hammer problems,where MӶ1,␳Ϸ␳0,Eqs.͑18͒and͑19͒becomeg a2ץHץtϩץVץxϭ0(20)ץVץtϩg ץHץxϩ␶w␲D␳Aϭ0(21)which are identical to the classical1D water hammer equations given by Eqs.͑2͒and͑3͒.Thus,the classical water hammer equa-tions are valid for unidirectional and axisymmetricﬂow of a com-pressibleﬂuid in aﬂexible pipe͑tube͒,where the Mach number is very small.According to Eq.͑15͒,the importance of wall shear,␶w,de-pends on the magnitude of the dimensionless parameter⌫ϭ␨L M f/2Dϩ␨T d/(L/a).Therefore,the wall shear is important when the parameter⌫is order1or larger.This often occurs in applications where the simulation time far exceeds theﬁrst wave cycle͑i.e.,large␨͒,the pipe is very long,the friction factor is signiﬁcant,or the pipe diameter is very small.In addition,wall shear is important when the time scale of radial diffusion is larger than the wave travel time since the transient-induced large radial gradient of the velocity does not have sufﬁcient time to relax.It is noted that T d becomes smaller as the Reynolds number increases. The practical applications in which the wall shear is important and the various␶w models that are in existence in the literature are discussed in Sec.4.If⌫is signiﬁcantly smaller than1,friction becomes negligible and␶w can be safely set to zero.For example,for the case L ϭ10,000m,Dϭ0.2m,fϭ0.01,and Mϭ0.001,and T d/(L/a)ϭ0.01the condition⌫Ӷ1is valid when␨Ӷ4.That is,for the case considered,wall friction is irrelevant as long as the simulation time is signiﬁcantly smaller than4L/a.In general,the condition ⌫Ӷ1is satisﬁed during the early stages of the transient͑i.e.,␨is small͒provided that the relaxation͑diffusion͒time scale is smaller than the wave travel time L/a.In fact,it is well known that waterhammer models provide results that are in reasonable agree-ment with experimental data during theﬁrst wave cycle irrespec-tive of the wall shear stress formula being used͑e.g.,͓29–32͔͒. When⌫Ӷ1,the classical waterhammer model,given by Eqs.͑20͒and͑21͒,becomesg a2ץHץtϩץVץxϭ0(22)ץVץtϩg ץHץxϭ0(23)which is identical to the model thatﬁrst appeared in Allievi͓9,10͔.The Joukowsky relation can be recovered from Eqs.͑22͒and ͑23͒.Consider a water hammer moving upstream in a pipe of length L.Let xϭLϪat deﬁne the position of a water hammer front at time t and consider the interval͓LϪatϪ⑀,LϪatϩ⑀͔, where⑀ϭdistance from the water hammer front.Integrating Eqs.͑22͒and͑23͒from xϭLϪatϪ⑀to xϭLϪatϩ⑀,invoking Leib-nitz’s rule,and taking the limit as⑀approaches zero gives⌬HϭϪa⌬Vg(24)Similarly,the relation for a water hammer wave moving down-stream is⌬Hϭϩa⌬V/g.3Water Hammer…Acoustic…Wave SpeedThe water hammer wave speed is͑e.g.,͓8,20,23,33,34͔͒,1a2ϭd␳dPϩ␳AdAdP(25)Theﬁrst term on the right-hand side of Eq.͑25͒represents the effect ofﬂuid compressibility on the wave speed and the second term represents the effect of pipeﬂexibility on the wave speed.In fact,the wave speed in a compressibleﬂuid within a rigid pipe is obtained by setting dA/dPϭ0in Eq.͑25͒,which leads to a2ϭdP/d␳.On the other hand,the wave speed in an incompressible ﬂuid within aﬂexible pipe is obtained by setting d␳/dPϭ0in ͑25͒,which leads to a2ϭAdP/␳dA.Korteweg͓33͔related the right-hand side of Eq.͑25͒to the material properties of theﬂuid and to the material and geometrical properties of the pipe.In particular,Korteweg͓33͔introduced the ﬂuid properties through the state equation dP/d␳ϭK f/␳,which was already well established in the literature,where K fϭbulk modulus of elasticity of theﬂuid.He used the elastic theory of continuum mechanics to evaluate dA/dP in terms of the pipe radius,thickness e,and Young’s modulus of elasticity E.In his derivation,he͑i͒ignored the axial͑longitudinal͒stresses in the pipe͑i.e.,neglected Poisson’s effect͒and͑ii͒ignored the inertia of the pipe.These assumptions are valid forﬂuid transmission lines that are anchored but with expansion joints throughout.With as-sumptions͑i͒and͑ii͒,a quasi-equilibrium relation between the pressure force per unit length of pipe DdP and the circumferential ͑hoop͒stress force per unit pipe length2ed␴␪is achieved,where ␴␪ϭhoop stress.That is,DdPϭ2ed␴␪or dpϭ2ed␴␪/ing the elastic stress-strain relation,dAϭ␲d␰D2/2,where d␰ϭd␴␪/Eϭradial͑lateral͒strain.As a result,AdP/␳dAϭeE/D␳and1a2ϭ␳K fϩ␳EeDor a2ϭK f␳1ϩK f DeE(26)The above Korteweg formula for wave speed can be extended to problems where the axial stress cannot be neglected.This is accomplished through the inclusion of Poisson’s effect in the stress-strain relations.In particular,the total strain becomes d␰ϭd␴␪/EϪ␯p d␴x/E,where␯pϭPoisson’s ratio and␴xϭaxial stress.The resulting wave speed formula is͑e.g.,͓17,23͔͒a2ϭK f␳1ϩcK f DeE(27)where cϭ1Ϫ␯p/2for a pipe anchored at its upstream end only, cϭ1Ϫ␯p2for a pipe anchored throughout from axial movement, and cϭ1for a pipe anchored with expansion joints throughout, which is the case considered by Korteweg͑i.e.,␴xϭ0).Multiphase and multicomponent water hammerﬂows are com-mon in practice.During a water hammer event,the pressure can cycle between large positive values and negative values,the mag-nitudes of which are constrained at vapor pressure.Vapor cavities can form when the pressure drops to vapor pressure.In addition, gas cavities form when the pressure drops below the saturation pressure of dissolved gases.Transientﬂows in pressurized or sur-charged pipes carrying sediment are examples of multicomponent water hammerﬂows.Unsteadyﬂows in pressurized or surcharged sewers are typical examples of multiphase and multicomponent transientﬂows in closed conduits.Clearly,the bulk modulus and density of the mixture and,thus,the wave speed are inﬂuenced by the presence of phases and components.The wave speed for mul-tiphase and multicomponent water hammerﬂows can be obtainedby substituting an effective bulk modulus of elasticity K e and an effective density␳e in place of K f and␳in Eq.͑27͒.The effective quantities,K e and␳e,are obtained by the weighted average of the bulk modulus and density of each component,where the partial volumes are the weights͑see,͓23͔͒.While the resulting math-ematical expression is simple,the explicit evaluation of the wave speed of the mixture is hampered by the fact that the partial vol-umes are difﬁcult to estimate in practice.Equation͑27͒includes Poisson’s effect but neglects the motion and inertia of the pipe.This is acceptable for rigidly anchored pipe systems such as buried pipes or pipes with high density and stiff-ness,to name only a few.Examples include major transmission pipelines like water distribution systems,natural gas lines,and pressurized and surcharged sewerage force mains.However,the motion and inertia of pipes can become important when pipes are inadequately restrained͑e.g.,unsupported,free-hanging pipes͒or when the density and stiffness of the pipe is small.Some ex-amples in which a pipe’s motion and inertia may be signiﬁcant include fuel injection systems in aircraft,cooling-water systems, unrestrained pipes with numerous elbows,and blood vessels.For these systems,a fully coupledﬂuid-structure interaction model needs to be considered.Such models are not discussed in this paper.The reader is instead directed to the recent excellent review of the subject by Tijsseling͓35͔.4Wall Shear Stress ModelsIt was shown earlier in this paper that the wall shear stress term is important when the parameter⌫is large.It follows that the modeling of wall friction is essential for practical applications that warrant transient simulation well beyond theﬁrst wave cycle͑i.e., large␨͒.Examples include͑i͒the design and analysis of pipeline systems,͑ii͒the design and analysis of transient control devices,͑iii͒the modeling of transient-induced water quality problems,͑iv͒the design of safe and reliableﬁeld data programs for diag-nostic and parameter identiﬁcation purposes,͑v͒the application of transient models to invertﬁeld data for calibration and leakage detection,͑vi͒the modeling of column separation and vaporous cavitation and͑vii͒systems in which L/aӶT d.Careful modeling of wall shear is also important for long pipes and for pipes with high friction.4.1Quasi-Steady Wall Shear Models.In conventional transient analysis,it is assumed that phenomenological expres-sions relating wall shear to cross-sectionally averaged velocity in steady-stateﬂows remain valid under unsteady conditions.That is, wall shear expressions,such as the Darcy-Weisbach and Hazen-Williams formulas,are assumed to hold at every instant during a transient.For example,the form of the Darcy-Weisbach equation used in water hammer models is͑Streeter and Wylie͓36͔͒␶w͑t͒ϭ␶wsϭ␳f͑t͉͒V͑t͉͒V͑t͒8(28)where␶ws(t)ϭquasi-steady wall shear as a function of t.The use of steady-state wall shear relations in unsteady prob-lems is satisfactory for very slow transients—so slow,in fact,that they do not properly belong to the water hammer regime.To help clarify the problems with this approach for fast transients,con-sider the case of a transient induced by an instantaneous and full closure of a valve at the downstream end of a pipe.As the wave travels upstream,theﬂow rate and the cross-sectionally averaged velocity behind the wave front are zero.Typical transient velocity proﬁles are given in Fig.3.Therefore,using Eq.͑28͒,the wall shear is zero.This is incorrect.The wave passage creates aﬂow reversal near the pipe wall.The combination ofﬂow reversal with the no-slip condition at the pipe wall results in large wall shear stresses.Indeed,discrepancies between numerical results and ex-perimental andﬁeld data are found whenever a steady-state based shear stress equation is used to model wall shear in water hammer problems͑e.g.,͓25,30,32,37,38͔͒.Let␶wu(t)be the discrepancy between the instantaneous wall shear stress␶w(t)and the quasi-steady contribution of wall shear stress␶ws(t).Mathematically␶w͑t͒ϭ␶ws͑t͒ϩ␶wu͑t͒(29)␶wu(t)is zero for steadyﬂow,small for slow transients,and sig-niﬁcant for fast transients.The unsteady friction component at-tempts to represent the transient-induced changes in the velocity proﬁle,which often involveﬂow reversal and large gradients near the pipe wall.A summary of the various models for estimating ␶wu(t)in water hammer problems is given below.4.2Empirical-Based Corrections to Quasi-Steady Wall Shear Models.Daily et al.͓39͔conducted laboratory experi-ments and found␶wu(t)to be positive for acceleratingﬂows and negative for deceleratingﬂows.They argued that during accelera-tion the central portion of the stream moved somewhat so that the velocity proﬁle steepened,giving higher shear.For constant-diameter conduit,the relation given by Daily et al.͓39͔can be rewritten asK uϭK sϩ2c2LV2ץVץt(30) where K uϭunsteadyﬂow coefﬁcient of boundary resistance and momentumﬂux of absolute local velocity and K sϭf L/Dϭsteady state resistance coefﬁcient.Daily et al.͓39͔noted that the longi-tudinal velocity and turbulence nonuniformities are negligible and K uϷKϭF/␳AV2/2ϭunsteadyﬂow coefﬁcient of boundary resis-tance,where Fϭ2␲DL␶wϭwall resistance force.Therefore,Eq.͑30͒becomes Fig.3Velocity proﬁles for steady-state and af-ter wave passages。

多相流模型经验谈

多相流模型经验谈多相流的介绍：Currentlytherearetwoapproachesforthenumericalcalculationofmultiphaseflows:theEuler-La grangeapproachandtheEuler-Eulerapproach.TheEuler-LagrangeApproach:TheLagrangiandiscretephasemodelinFLUENTfollowstheEuler-Lagr angeapproach,thisapproachisinappropriateforthemodelingofliquid-liquidmixtures,fluidizedbeds,oranyapplicationwhMIteration,theparticlesourcetermsarerecalculated.LengthScale:controlstheintegrationtimestepsizeusedtointegratetheequationsofmotionfort heparticle.Asmaller valuefortheLengthScaleincreasestheaccuracyofthetrajectoryandheat/masstransfercalculat ionsforthediscretephase.LengthScalefactor:AlargervaluefortheStepLengthFactordecreasesthediscretephaseintegrat iontimestep.颗粒积分方法：numerics叶中trackingscheme选项1）implicitusesanimplicitEulerintegrationofEquation23.2-1whichisunconditionallystablefor allparticlerelaxationtimes.2）trapezoidalusesasemi-implicittrapezoidalintegration.（梯形积分）3）analyticusesananalyticalintegrationofEquation23.2-1wheretheforcesareheldconstantdurin gtheintegration.4）runge-kuttafacilitatesa5thorderRungeKuttaschemederivedbyCashandKarp[47]. Youcaneitherchooseasingletrackingscheme,orswitchbetweenhigherorderandlowerordertracki ngschemesusingan12FluidFlowTimeSteptoinjecttheparticles,orwhetheryoupreferaParticleTimeStepSizeindepend entofthefluidflowtimestep.Withthelatteroption,youcanusetheDiscretePhaseModelincombinationwithchangesin thetimestepforthecontinuousequations,asitisdonewhenusingadaptiveflowtimestepping.随机轨道模型的参数：numberoftries:AninputofzerotellsFLUENTtocomputetheparticletrajectorybasedonthemeancon tinuousphasevelocityfield(Equation23.2-1),ignoringtheeffectsofturbulenceontheparticletrajectories.Aninput of1orgreatertellsFLUENTtoincludeturbulentvelocityfluctuationsintheparticleforcebalanceasinEquation23.2 -20.Ifyouwantthecharacteristiclifetimeoftheeddytoberandom(Equation23.2-32),enabletheRando mEddyLifetimeoption.YouwillgenerallynotneedtochangetheTimeScaleConstant(CLinEquation23.2-23)fromitsdefaul tvalueof0.15,unlessyouareusingtheReynoldsStressturbulencemodel(RSM),inwhichcaseavalueof0.3isrecomm ended.液滴颗粒碰撞与破碎碰撞：破碎：有两种模型，TAB模型适合低韦伯数射流雾化以及低速射流进入标态空气中的情况。

The dispersion equation for a wave number of the electromagnetic wave in a random medium of

Introduction
Under certain conditions the electrochemical anodization of metals leads to the formation of the oxide layer with parallel cylindrical pores on the surface of metal. The radius of the cylinders is much less than the thickness of the layer. These oxide films are used as corrosion-protective, strengthening, decorative coatings and attract much attention in microelectronics, in particular, for the creation of non-linear and active thin-film elements, and also for substrates of large hybrid microcircuits. Porous oxides are perspective for multilevel systems of interconnections. In dependence on the conditions of anodization and kind of metal it is possible to obtain films with different degree of ordering in pore placement of pores. The placement of pores is maximally chaotic when the pores are placed randomly and could be overlapped. The overlapping pores form the cavities of complex configurations. The placement of pores is maximally ordered, when the points of the cylinder axes intersection with the plane perpendicular to the axes lay in the nodes of some lattice, for example a hexagonal one. An investigation of the optical properties of oxide films allows to find a degree of pores ordering. If the wavelength is much larger than the radius of pores, the optical properties of the porous layer can be described by the effective refractive index. This index is to be found from the dispersion equations. The dispersion equation for the effective refractive index of the system of parallel nonoverlapping cylindrical fibres in media was deduced in the work [1]. The equation allows to describe the optical properties of the layer with the regular pores placement. In present work we obtain the dispersion equation for a medium with chaotic distribution of pores (the pores are located randomly and can be overlapped). We also analyse an influence of regularity on the effective refractive index.

AP_calculus参考书

Score
AP Score 5 4 3 2 1 Qualification Extremely well qualified Well qualified Qualified Possibly qualified No recommendation
Topic Outline for Calculus BC
1
by taking sufficiently close values of the domain.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). * Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. II. Derivatives Concept of the derivative • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. • Instantaneous rate of change as the limit of average rate of change. • Approximate rate of change from graphs and tables of values. Derivative as a function • Corresponding characteristics of graphs of ƒ and ƒ∙. • Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’. • The Mean Value Theorem and its geometric interpretation. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Second derivatives • Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’∙. • Relationship between the concavity of ƒ and the sign of ƒ’∙. • Points of inflection as places where concavity changes. Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration. • Optimization, both absolute (global) and relative (local) extrema. • Modeling rates of change, including related rates problems. • Use of implicit differentiation to find the derivative of an inverse function. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. + Numerical solution of differential equations using Euler’s method. + L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series. Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential, logarithmic,

金融博士书目

金融博士书目经济学、金融学博士书目（A：数学分析微分方程矩阵代数）微观金融学包括金融市场及金融机构研究、投资学金融工程学金融经济学、公司金融财务管理等方面，宏观金融学包括货币经济学货币银行学、国际金融学等方面，实证和数量方法包括数理金融学、金融计量经济学等方面，以下书目侧重数学基础、经济理论和数理金融学部分。

◎函数与分析《什么是数学》，牛津丛书●集合论Paul R. Halmos，Naive Set Theory 朴素集合论（美）哈莫斯（好书，深入浅出但过简洁）集合论(英文版)Thomas Jech（有深度）Moschovakis，Notes on Set Theory集合论基础（英文版）——图灵原版数学·统计学系列（美）恩德滕●数学分析○微积分Tom M. Apostol, Calculus vol Ⅰ&Ⅱ（数学家写的经典高等微积分教材/参考书，写法严谨，40年未再版，致力于更深刻的理解，去除微积分和数学分析间隔，衔接分析学、微分方程、线性代数、微分几何和概率论等的学习，学实分析的前奏，线性代数应用最好的多元微积分书，练习很棒，对初学者会难读难懂，但具有其他教材无法具备的优点。

Stewart 的书范围相同，也较简单。

）Carol and Robert Ash，The Calculus Tutoring Book（不错的微积分辅导教材）R. Courant, F. John, Introduction to Calculus and Analysis vol Ⅰ&Ⅱ（适合工科，物理和应用多）Morris Kline，Calculus, an intuitive approachRon LarsonCalculus (With Analytic Geometry（微积分入门教材，难得的清晰简化，与Stewart同为流行教材）《高等微积分》Lynn H.Loomis / Shlomo StermbergMorris Kline，Calculus: An Intuitive and Physical Approach （解释清晰的辅导教材）Richard Silverman，Modern Calculus with Analytic GeometryMichael，Spivak，Calculus（有趣味，适合数学系，读完它或者Stewart的就可以读Rudin 的Principles of Mathematical Analysis 或者Marsden的Elementary Classical Analysis，然后读Royden的Real Analysis学勒贝格积分和测度论或者Rudin的Functional Analysis 学习巴拿赫和希尔伯特空间上的算子和谱理论）James Stewart，Calculus（流行教材，适合理科及数学系，可以用Larson书补充，但解释比它略好，如果觉得难就用Larson的吧）Earl W. Swokowski，Cengage Advantage Books: Calculus: The Classic Edition（适合工科）Silvanus P. Thompson，Calculus Made Easy（适合微积分初学者，易读易懂）○实分析（数学本科实变分析水平）（比较静态分析）Understanding Analysis，Stephen Abbott，（实分析入门好书，虽然不面面俱到但清晰简明，Rudin, Bartle, Browder等人毕竟不擅于写入门书，多维讲得少）T. M. Apostol, Mathematical AnalysisProblems in Real Analysis 实分析习题集(美)阿里普兰斯，（美）伯金肖《数学分析》方企勤，北大胡适耕，实变函数《分析学》Elliott H. Lieb / Michael LossH. L. Royden, Real AnalysisW. Rudin, Principles of Mathematical AnalysisElias M.Stein，Rami Shakarchi, Real Analysis：MeasureTheory，Integration and Hilbert Spaces，实分析(英文版) 《数学分析八讲》辛钦《数学分析新讲》张筑生，北大社周民强，实变函数论，北大周民强《数学分析》上海科技社○测度论（与实变分析有重叠）概率与测度论（英文版）（美）阿什（Ash.R.B.），(美)多朗-戴德（Doleans-Dade,C.A.）?Halmos，Measure Theory，测度论(英文版)（德）霍尔姆斯○傅里叶分析（实变分析和小波分析各有一半）小波分析导论（美）崔锦泰H. Davis, Fourier Series and Orthogonal FunctionsFolland，Real Analysis：Modern Techniques and Their ApplicationsFolland，Fourier Analysis and its Applications，数学物理方程：傅里叶分析及其应用（英文版）——时代教育.国外高校优秀教材精选（美）傅兰德傅里叶分析（英文版）——时代教育·国外高校优秀教材精选（美）格拉法科斯B. B. Hubbard, The World According to Wavelets: The Story of a Mathematical Technique in the MakingKatanelson，An Introduction to Harmonic AnalysisR. T. Seeley, An Introduction to Fourier Series and IntegralsStein，Shakarchi，Fourier Analysis：An Introduction○复分析（数学本科复变函数水平）L. V. Ahlfors, Complex Analysis ，复分析——华章数学译丛，（美）阿尔福斯（Ahlfors,L.V.）Brown，Churchill，Complex Variables and Applications Convey, Functions of One Complex Variable Ⅰ&Ⅱ《简明复分析》龚升, 北大社Greene，Krantz，Function Theory of One Complex VariableMarsden，Hoffman，Basic Complex AnalysisPalka，An Introduction to Complex Function TheoryW. Rudin, Real and Complex Analysis 《实分析与复分析》鲁丁（公认标准教材，最好有测度论基础）Siegels，Complex VariablesStein，Shakarchi，Complex Analysis 《复变函数》庄坼泰●泛函分析（资产组合的价值）○基础泛函分析（实变函数、算子理论和小波分析）实变函数与泛函分析基础，程其衰，高教社Friedman，Foundations of Modern Analysis《实变与泛函》胡适耕《泛函分析引论及其应用》克里兹格泛函分析习题集（印）克里希南Problems and methods in analysis，Krysicki夏道行，泛函分析第二教程，高教社夏道行，实变函数与泛函分析《数学分析习题集》谢惠民，高教社泛函分析·第6版（英文版） K.Yosida《泛函分析讲义》张恭庆，北大社○高级泛函分析（算子理论）J.B.Conway, A Course in Functional Analysis，泛函分析教程(英文版)Lax，Functional AnalysisRudin，Functional Analysis，泛函分析（英文版）[美]鲁丁（分布和傅立叶变换经典，要有拓扑基础）Zimmer，Essential Results of Functional Analysis○小波分析Daubeches，Ten Lectures on WaveletsFrazier，An Introduction to Wavelets Throughout Linear Algebra Hernandez，《时间序列的小波方法》PercivalPinsky，Introduction to Fourier Analysis and WaveletsWeiss，A First Course on WaveletsWojtaszczyk，An Mathematical Introduction to Wavelets Analysis●微分方程（期权定价、动态分析）○常微分方程和偏微分方程（微分方程稳定性，最优消费组合）V. I. Arnold, Ordinary Differential Equations，常微分方程（英文版）（现代化，较难）W. F. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems《数学物理方程》陈恕行，复旦E. A. Coddington, Theory of ordinary differential equationsA. A. Dezin, Partial differential equationsL. C. Evans, Partial Differential Equations丁同仁《常微分方程教程》高教《常微分方程习题集》菲利波夫，上海科技社G. B. Folland, Introduction to Partial Differential EquationsFritz John, Partial Differential Equations《常微分方程》李勇The Laplace Transform: Theory and Applications，Joel L. Schiff（适合自学）G. Simmons, Differntial Equations With Applications and Historecal Notes索托梅约尔《微分方程定义的曲线》《常微分方程》王高雄，中山大学社《微分方程与边界值问题》Zill○偏微分方程的有限差分方法（期权定价）福西斯，偏微分方程的有限差分方法Kwok，Mathematical Models of Financial Derivatives（有限差分方法美式期权定价）?Wilmott，Dewynne，Howison，The Mathematics of Financial Derivatives （有限差分方法美式期权定价）○统计模拟方法、蒙特卡洛方法Monte Carlo method in finance （美式期权定价）D. Dacunha-Castelle, M. Duflo，Probabilités et Statistiques IIFisherman，Monte Carlo Glasserman，Monte Carlo Mathods in Financial Engineering （金融蒙特卡洛方法的经典书，汇集了各类金融产品）Peter Jaeckel，Monte Carlo Methods in Finance（金融数学好，没Glasserman的好）?D. P. Heyman and M. J. Sobel, editors，Stochastic Models, volume 2 of Handbooks in O. R. and M. S., pages 331-434. Elsevier Science Publishers B.V. (North Holland) Jouini，Option Pricing，Interest Rates and Risk ManagementD. Lamberton, B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance （连续时间）N. Newton，Variance reduction methods for diffusion process :H. Niederreiter，Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Appl. Math. SIAMW.H. Press and al.，Numerical recepies.B.D. Ripley. Stochastic SimulationL.C.G. Rogers et D. Talay, editors，Numerical Methods in Finance. Publicationsof the Newton Institute.D.V. Stroock, S.R.S. Varadhan，Multidimensional diffusion processesD. Talay，Simulation and numerical analysis of stochastic differential systems, a review. In P. Krée and W. Wedig, editors,Probabilistic Methods in Applied Physics, volume 451 of Lecture Notes in Physics, chapter 3, pages 54-96.P.Wilmott and al.，Option Pricing (Mathematical models and computation). Benninga，Czaczkes，Financial Modeling ○数值方法、数值实现方法Numerical Linear Algebra and Its Applications，科学社K. E. Atkinson, An Introduction to Numerical AnalysisR. Burden, J. Faires, Numerical Methods《逼近论教程》CheneyP. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge Texts in Applied MathematicsA. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics 《数值逼近》蒋尔雄《数值分析》李庆杨，清华《数值计算方法》林成森J. Stoer, R. Bulirsch, An Introduction to Numerical AnalysisJ. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations L. Trefethen, D. Bau, Numerical Linear Algebra《数值线性代数》徐树芳，北大其他（不必）《数学建模》Giordano《离散数学及其应用》Rosen《组合数学教程》Van Lint◎几何学和拓扑学（凸集、凹集）●拓扑学○点集拓扑学Munkres，Topology：A First Course《拓扑学》James R.MunkresSpivak，Calculus on Manifolds◎代数学（深于数学系高等代数）（静态均衡分析）○线性代数、矩阵论（资产组合的价值）M. Artin，AlgebraAxler, Linear Algebra Done RightCurtis，Linear Algeria：An Introductory ApproachW. Fleming, Functions of Several VariablesFriedberg, Linear Algebra Hoffman & Kunz, Linear AlgebraP.R. Halmos，Finite-Dimensional Vector Spaces（经典教材，数学专业的线性代数，注意它讲抽象代数结构而不是矩阵计算，难读）J. Hubbard, B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified ApproachN. Jacobson，Basic Algebra Ⅰ&ⅡJain《线性代数》Lang，Undergraduate AlgeriaPeter D. Lax，Linear Algebra and Its Applications（适合数学系）G. Strang, Linear Algebra and its Applications（适合理工科，线性代数最清晰教材，应用讲得很多，他的网上讲座很重要）●经济最优化Dixit，Optimization in Economic Theory●一般均衡Debreu，Theory of Value●分离定理Hildenbrand，Kirman，Equilibrium Analysis（均衡问题一般处理）Magill，Quinzii，Theory of Incomplete Markets（非完备市场的均衡）Mas-Dollel，Whinston，Microeconomic Theory（均衡问题一般处理）Stokey，Lucas，Recursive Methods in Economic Dynamics （一般宏观均衡）经济学、金融学博士书目（B：概率论、数理统计、随机）◎概率统计●概率论（金融产品收益估计、不确定条件下的决策、期权定价）○基础概率理论（数学系概率论水平）《概率论》（三册）复旦Davidson，Stochastic Limit TheoryDurrett，The Essential of Probability，概率论第3版（英文版）W. Feller，An Introduction to Probability Theory and its Applications概率论及其应用（第3版）——图灵数学·统计学丛书《概率论基础》李贤平，高教G. R. Grimmett, D. R. Stirzaker, Probability and Random ProcessesRoss，S. A first couse in probability，中国统计影印版；概率论基础教程（第7版）——图灵数学·统计学丛书（例子多）《概率论》汪仁官，北大王寿仁，概率论基础和随机过程，科学社《概率论》杨振明，南开，科学社○基于测度论的概率论测度论与概率论基础，程式宏，北大D. L. Cohn, Measure TheoryDudley，Real Analysis and ProbabilityDurrett，Probability：Theory and ExamplesJacod，Protter，Probability Essentials Resnick，A Probability PathShirayev，Probability严加安，测度论讲义，科学社钟开莱，A Course in Probability Theory○随机过程微积分Introduction of diffusion processes (期权定价)K. L. Chung, Elementary Probability Theory with Stochastic ProcessesCox，Miller，The Theory of StochasticR. Durrett, Stochastic calculus黄志远，随机分析入门黄志远《随机分析学基础》科学社姜礼尚，期权定价的数学模型和方法，高教社《随机过程导论》KaoKarlin，Taylor，A First Course in Stochastic Prosses（适合硕士生）Karlin，Taylor，A Second Course in Stochastic Prosses（适合硕士生）随机过程，劳斯，中国统计J. R. Norris，Markov Chains（需要一定基础）Bernt Oksendal, Stochastic differential equations（绝佳随机微分方程入门书，专注于布朗运动，比Karatsas和Shreve的书简短好读，最好有概率论基础，看完该书能看懂金融学术文献，金融部分没有Shreve的好）Protter，Stochastic Integration and Differential Equations （文笔优美）D. Revuz, M. Yor, Continuous martingales and Brownian motion（连续鞅）Ross，Introduction to probability model（适合入门）Steel，Stochastic Calculus and Financial Application（与Oksendal的水平相当，侧重金融，叙述有趣味而削弱了学术性，随机微分、鞅）《随机过程通论》王梓坤，北师大○概率论、随机微积分应用（连续时间金融）Arnold，Stochastic Differential Equations《概率论及其在投资、保险、工程中的应用》BeanDamien Lamberton,Bernard Lapeyre. Introduction to stochastic calculus applied t o finance.David Freedman.Browian motion and diffusion.Dykin E. B. Markov Processes.Gihman I.I., Skorohod A. V.The theory of Stochastic processes 基赫曼，随机过程论，科学Lipster R. ,Shiryaev A.N. Statistics of random processes.Malliaris，Brock，Stochastic Methods in Economics and FinanceMerton，Continuous-time FinanceSalih N. Neftci，Introduction to the Mathematics of Financial DerivativesSteven E. Shreve ，Stochastic Calculus for Finance I: The Binomial Asset Pric ing Model；II: Continuous-Time Models（最佳的随机微积分金融（定价理论）入门书，易读的金融工程书，没有测度论基础最初几章会难些，离散时间模型，比Naftci的清晰，S hreve的网上教程也很优秀）Sheryayev A. N. Ottimal stopping rules.Wilmott p., J.Dewynne,S. Howison. Option Pricing: Mathematical Models and Compu tations.Stokey，Lucas，Recursive Methods in Economic Dynamics Wentzell A. D. A Course in the Theory of Stochastic Processes.Ziemba，Vickson，Stochastic Optimization Models in Finance○概率论、随机微积分应用（高级）Nielsen，Pricing and Hedging of Derivative SecuritiesRoss，《数理金融初步》An Introduction to Mathematical Finance：Options and othe r TopicsShimko，Finance in Continuous Time：A Primer○概率论、鞅论P. Billingsley，Probability and MeasureK. L. Chung & R. J. Williams，Introduction to Stochastic IntegrationDoob，Stochastic Processes严加安，随机分析选讲，科学○概率论、鞅论Stochastic processes and derivative products （高级）J. Cox et M. Rubinstein : Options MarketIoannis Karatzas and Steven E. Shreve，Brownian Motion and Stochastic Calculu s（难读的重要的高级随机过程教材，若没有相当数学功底，还是先读Oksendal的吧，结合Rogers & Williams的书读会好些，期权定价，鞅）M. Musiela - M. Rutkowski : (1998) Martingales Methods in Financial Modelling ?Rogers & Williams，Diffusions, Markov Processes, and Martingales: Volume 1, F oundations；Volume 2, Ito Calculus （深入浅出，要会实复分析、马尔可夫链、拉普拉斯转换，特别要读第1卷）David Williams，Probability with Martingales（易读，测度论的鞅论方法入门书，概率论高级教材）○鞅论、随机过程应用Duffie，Rahi，Financial Market Innovation and Security Design：An Introduction，Journal of Economic Theory Kallianpur，Karandikar，Introduction to Option Pricing TheoryDothan，Prices in Financial Markets （离散时间模型）Hunt，Kennedy，Financial Derivatives in Theory and Practice何声武，汪家冈，严加安，半鞅与随机分析，科学社Ingersoll，Theory of Financial Decision MakingElliott Kopp，Mathematics of Financial Markets（连续时间）Marek Musiela，Rutkowski，Martingale Methods in Financial Modeling（资产定价的鞅论方法最佳入门书，读完Hull书后的首选，先读Rogers & Williams、Karatzas and Sh reve以及Bjork打好基础）○弱收敛与随机过程收敛Billingsley，Convergence of Probability MeasureDavidson，Stochastic Limit TheoremEthier，Kurtz，Markov Process：Characterization and Convergence Hall，Marting ale Limit TheoremsJocod，Shereve，Limited Theorems for Stochastic Process Van der Vart，Weller，Weak Convergence and Empirical Process◎运筹学●最优化、博弈论、数学规划○随机控制、最优控制（资产组合构建）Borkar，Optimal control of diffusion processesBensoussan，Lions，Controle Impulsionnel et Inequations Variationnelles Chiang，Elements of Dynamic Optimization Dixit，Pindyck，Investment under UncertaintyFleming，Rishel，Deterministic and Stochastic Optimal ControlHarrison，Brownian Motion and Stochastic Flow SystemsKamien，Schwartz，Dynamic OptimizationKrylov，Controlled diffusion processes○控制论（最优化问题）●数理统计（资产组合决策、风险管理）○基础数理统计（非基于测度论）R. L. Berger, Cassell, Statistical InferenceBickel，Dokosum，Mathematical Stasistics：Basic Ideas andSelected TopicsBirrens，Introdution to the Mathematical and Statistical Foundation of Econom etrics数理统计学讲义，陈家鼎，高教Gallant，An Introduction to Econometric TheoryR. Larsen, M. Mars, An Introduction to Mathematical Statistics《概率论及数理统计》李贤平，复旦社Papoulis，Probability，random vaiables，and stochastic processStone，《概率统计》《概率论及数理统计》中山大学统计系，高教社○基于测度论的数理统计(计量理论研究)Berger，Statistical Decision Theory and Bayesian Analysis陈希儒，高等数理统计Shao Jun，Mathematical StatisticsLehmann，Casella，Theory of Piont EstimationLehmann，Romano，Testing Statistical Hypotheses《数理统计与数据分析》Rice○渐近统计Van der Vart，Asymptotic Statistics○现代统计理论、参数估计方法、非参数统计方法参数计量经济学、半参数计量经济学、自助法计量经济学、经验似然经济学、金融学博士书目（C：计量经济学、数理金融）统计学基础部分1、《统计学》《探索性数据分析》 David Freedman等，中国统计（统计思想讲得好）2、Mind on statistics 机械工业（只需高中数学水平）3、Mathematical Statistics and Data Analysis 机械工业（这本书理念很好，讲了很多新东西）4、Business Statistics a decision making approach 中国统计（实用）5、Understanding Statistics in the behavioral science 中国统计回归部分1、《应用线性回归》中国统计（蓝皮书系列，有一定的深度，非常精彩）2、Regression Analysis by example，（吸引人，推导少）3、《Logistics回归模型——方法与应用》王济川郭志刚高教（不多的国内经典统计教材）多元1、《应用多元分析》王学民上海财大（国内很好的多元统计教材）2、Analyzing Multivariate Data，Lattin等机械工业（直观，对数学要求不高）3、Applied Multivariate Statistical Analysis，Johnson & Wichem，中国统计（评价很高）《应用回归分析和其他多元方法》Kleinbaum《多元数据分析》Lattin时间序列1、《商务和经济预测中的时间序列模型》弗朗西斯著（侧重应用，经典）2、Forecasting and Time Series an applied approach，Bowerman & Connell（主讲Box-Jenkins(ARIMA)方法，附上了SAS和Minitab程序）3、《时间序列分析：预测与控制》 Box，Jenkins 中国统计《预测与时间序列》Bowerman抽样1、《抽样技术》科克伦著（该领域权威，经典的书。

南开大学光学工程专业英语重点词汇汇总

光学专业英语部分refraction [rɪˈfrækʃn]n.衍射reflection [rɪˈflekʃn]n.反射monolayer['mɒnəleɪə]n.单层adj.单层的ellipsoid[ɪ'lɪpsɒɪd]n.椭圆体anisotropic[,ænaɪsə(ʊ)'trɒpɪk]adj.非均质的opaque[ə(ʊ)'peɪk]adj.不透明的；不传热的；迟钝的asymmetric[,æsɪ'metrɪk]adj.不对称的；非对称的intrinsic[ɪn'trɪnsɪk]adj.本质的，固有的homogeneous[,hɒmə(ʊ)'dʒiːnɪəs;-'dʒen-] adj.均匀的；齐次的；同种的；同类的，同质的incidentlight入射光permittivity[,pɜːmɪ'tɪvɪtɪ]n.电容率symmetric[sɪ'metrɪk]adj.对称的；匀称的emergentlight出射光；应急灯.ultrafast[,ʌltrə'fɑ:st,-'fæst]adj.超快的；超速的uniaxial[,juːnɪ'æksɪəl]adj.单轴的paraxial[pə'ræksɪəl]adj.旁轴的；近轴的periodicity[,pɪərɪə'dɪsɪtɪ]n.[数]周期性；频率；定期性soliton['sɔlitɔn]n.孤子，光孤子；孤立子；孤波discrete[dɪ'skriːt]adj.离散的，不连续的convolution[,kɒnvə'luːʃ(ə)n]n.卷积；回旋；盘旋；卷绕spontaneously:[spɒn'teɪnɪəslɪ] adv.自发地；自然地；不由自主地instantaneously:[,instən'teinjəsli]adv.即刻；突如其来地dielectricconstant[ˌdaiiˈlektrikˈkɔnstənt]介电常数，电容率chromatic[krə'mætɪk]adj.彩色的；色品的；易染色的aperture['æpətʃə;-tj(ʊ)ə]n.孔，穴；（照相机，望远镜等的）光圈，孔径；缝隙birefringence[,baɪrɪ'frɪndʒəns]n.[光]双折射radiant['reɪdɪənt]adj.辐射的；容光焕发的；光芒四射的; photomultiplier[,fəʊtəʊ'mʌltɪplaɪə]n.[电子]光电倍增管prism['prɪz(ə)m]n.棱镜；[晶体][数]棱柱theorem['θɪərəm]n.[数]定理；原理convex['kɒnveks]n.凸面体；凸状concave['kɒnkeɪv]n.凹面spin[spɪn]n.旋转；crystal['krɪst(ə)l]n.结晶，晶体；biconical[bai'kɔnik,bai'kɔnikəl] adj.双锥形的illumination[ɪ,ljuːmɪ'neɪʃən] n.照明；[光]照度；approximate[ə'prɒksɪmət] adj.[数]近似的；大概的clockwise['klɒkwaɪz]adj.顺时针方向的exponent[ɪk'spəʊnənt；ek-] n.[数]指数；even['iːv(ə)n]adj.[数]偶数的；平坦的；相等的eigenmoden.固有模式；eigenvalue['aɪgən,væljuː]n.[数]特征值cavity['kævɪtɪ]n.腔；洞，凹处groove[gruːv]n.[建]凹槽，槽；最佳状态；惯例；reciprocal[rɪ'sɪprək(ə)l]adj.互惠的；相互的；倒数的，彼此相反的essential[ɪ'senʃ(ə)l]adj.基本的；必要的；本质的；精华的isotropic[,aɪsə'trɑpɪk]adj,各向同性的；等方性的phonon['fəʊnɒn]n.[声]声子cone[kəʊn]n.圆锥体，圆锥形counter['kaʊntə]n.柜台；对立面；计数器；cutoff['kʌt,ɔːf]n.切掉；中断；捷径adj.截止的；中断的cladding['klædɪŋ]n.包层；interference[ɪntə'fɪər(ə)ns]n.干扰，冲突；干涉borderline['bɔːdəlaɪn]n.边界线，边界；界线quartz[kwɔːts]n.石英droplet['drɒplɪt]n.小滴，微滴precision[prɪ'sɪʒ(ə)n]n.精度，[数]精密度；精确inherently[ɪnˈhɪərəntlɪ]adv.内在地；固有地；holographic[,hɒlə'ɡræfɪk]adj.全息的；magnitude['mægnɪtjuːd]n.大小；量级；reciprocal[rɪ'sɪprək(ə)l]adj.互惠的；相互的；倒数的，彼此相反的stimulated['stimjə,letid]v.刺激（stimulate的过去式和过去分词）cylindrical[sɪ'lɪndrɪkəl]adj.圆柱形的；圆柱体的coordinates[kəu'ɔ:dineits]n.[数]坐标；external[ɪk'stɜːn(ə)l;ek-]n.外部；外观；scalar['skeɪlə]n.[数]标量；discretization[dɪs'kriːtaɪ'zeɪʃən]n.[数]离散化synthesize['sɪnθəsaɪz]vt.合成；综合isotropy[aɪ'sɑtrəpi]n.[物]各向同性；[物]无向性；[矿业]均质性pixel['pɪks(ə)l;-sel]n.（显示器或电视机图象的）像素（passive['pæsɪv]adj.被动的spiral['spaɪr(ə)l]n.螺旋；旋涡；equivalent[ɪ'kwɪv(ə)l(ə)nt]adj.等价的，相等的；同意义的; transverse[trænz'vɜːs;trɑːnz-;-ns-]adj.横向的；横断的；贯轴的;dielectric[,daɪɪ'lektrɪk]adj.非传导性的；诱电性的;n.电介质；绝缘体integral[ˈɪntɪɡrəl]adj.积分的；完整的criteria[kraɪ'tɪərɪə]n.标准，条件（criterion的复数）Dispersion:分散|光的色散spectroscopy[spek'trɒskəpɪ]n.[光]光谱学photovoltaic[,fəʊtəʊvɒl'teɪɪk]adj.[电子]光电伏打的，光电的polar['pəʊlə]adj.极地的；两极的；正好相反的transmittance[trænz'mɪt(ə)ns;trɑːnz-;-ns-] n.[光]透射比；透明度dichroic[daɪ'krəʊɪk]adj.二色性的；两向色性的confocal[kɒn'fəʊk(ə)l]adj.[数]共焦的；同焦点的rotation[rə(ʊ)'teɪʃ(ə)n]n.旋转；循环，轮流photoacoustic[,fəutəuə'ku:stik]adj.光声的exponential[,ekspə'nenʃ(ə)l]adj.指数的;fermion['fɜːmɪɒn]n.费密子（费密系统的粒子）semiconductor[,semɪkən'dʌktə]n.[电子][物]半导体calibration[kælɪ'breɪʃ(ə)n]n.校准；刻度；标度photodetector['fəʊtəʊdɪ,tektə]n.[电子]光电探测器interferometer[,ɪntəfə'rɒmɪtə]n.[光]干涉仪；干涉计static['stætɪk]adj.静态的；静电的；静力的;inverse相反的，反向的，逆的amplified['æmplifai]adj.放大的；扩充的horizontal[hɒrɪ'zɒnt(ə)l]n.水平线，水平面；水平位置longitudinal[,lɒn(d)ʒɪ'tjuːdɪn(ə)l;,lɒŋgɪ-] adj.长度的，纵向的；propagate['prɒpəgeɪt]vt.传播；传送；wavefront['weivfrʌnt]n.波前；波阵面scattering['skætərɪŋ]n.散射；分散telecommunication[,telɪkəmjuːnɪ'keɪʃ(ə)n] n.电讯；[通信]远程通信quantum['kwɒntəm]n.量子论mid-infrared中红外eigenvector['aɪgən,vektə]n.[数]特征向量；本征矢量numerical[njuː'merɪk(ə)l]adj.数值的；数字的ultraviolet[ʌltrə'vaɪələt]adj.紫外的；紫外线的harmonic[hɑː'mɒnɪk]n.[物]谐波。

Induced forests in regular graphs with large girth

An induced forest in a graph G is an acyclic induced subgraph of G. The problem of ﬁnding a large induced forest in a graph G has been a widely studied topic in graph theory, especially in its form known as the decycling set problem or the feedback vertex set problem. A decycling set of a graph is a subset of its vertices whose deletion yields an acyclic graph. From this deﬁnition, we deduce that a set S ⊆ V is such that G[S ] is an induced forest of G = (V, E ) if and only if V \ S is a decycling set of G. Therefore, ﬁnding a lower bound for τ (G), the maximum number of vertices in an induced forest of G, amounts to ﬁnding an upper bound for φ(G), the minimum cardinality of a decycling set of G. Historically, the problem of obtaining an acyclic subgraph of a graph G by removing vertices was already considered by Kirchhoﬀ in his work on spanning trees [11]. Erd¨ os et. al. also worked on this problem stated in terms of maximum induced trees in a graph [8]. However, ﬁnding a decycling set of a given size in a graph is inherently diﬃcult.

法国数学家拉格朗日著作《解析函数论》英文名

法国数学家拉格朗日著作《解析函数论》英文名全文共3篇示例，供读者参考篇1Title: French Mathematician Lagrange's Work "Analytical Function Theory"Introduction:Lagrange's "Analytical Function Theory" is a seminal work by the French mathematician Joseph-Louis Lagrange, also known as the Lagrange interpolation or Lagrange polynomial. In this work, Lagrange presents a detailed analysis of functions and their properties, laying the foundation for modern function theory. The book delves into topics such as series, limits, derivatives, and integrals of functions, providing a comprehensive study of mathematical functions.Chapter 1: Historical BackgroundJoseph-Louis Lagrange was born in Turin, Italy, in 1736 and later moved to Paris, where he made significant contributions to mathematics, mechanics, and astronomy. Lagrange's work in function theory was influenced by earlier mathematicians such as Euler, d'Alembert, and Legendre. His innovative approach toanalyzing functions set him apart as a pioneering figure in the field of mathematics.Chapter 2: Analytical Function TheoryIn "Analytical Function Theory," Lagrange explores the properties of functions through the use of calculus and algebraic techniques. He introduces the concept of series as a way to represent functions as infinite sums of terms, allowing for a more precise analysis of their behavior. Lagrange also discusses the importance of limits in determining the behavior of functions at particular points, laying the groundwork for modern calculus.Chapter 3: Applications and ImpactLagrange's work on function theory has had a lasting impact on mathematics, with his ideas forming the basis for modern theories in analysis and calculus. The concept of the Lagrange interpolation polynomial, named in his honor, is still widely used in numerical analysis and approximation techniques. His work has inspired generations of mathematicians to further explore the depths of function theory and its applications in various fields.Conclusion:In conclusion, Joseph-Louis Lagrange's work "Analytical Function Theory" stands as a testament to his brilliance and innovative contributions to the field of mathematics. Through his meticulous analysis of functions and their properties, Lagrange paved the way for future developments in function theory and calculus. His work remains a cornerstone of modern mathematics, continuing to inspire mathematicians worldwide to push the boundaries of knowledge in this dynamic field.篇2Title: French Mathematician Lagrange's Work "Analytic Functions Theory"IntroductionLagrange's work on Analytic Functions Theory is a significant contribution to the field of mathematics. This book, also known as "Théorie des Fonctions Analytiques" in French, was published in the year 1797 by the renowned French mathematician Joseph Louis Lagrange. In this seminal work, Lagrange established the foundation for the study of analytic functions and laid the groundwork for the development of complex analysis.Background of LagrangeJoseph Louis Lagrange, born in Turin, Italy in 1736, was a prominent mathematician who made groundbreaking contributions to various fields of mathematics, such as number theory, calculus, and celestial mechanics. He is widely regarded as one of the greatest mathematicians of all time and his work continues to influence modern mathematics.Content of the BookIn "Analytic Functions Theory", Lagrange delves into the study of functions that can be represented by a power series expansion. He introduces key concepts and theorems related to complex analysis, such as Cauchy's integral theorem, the residue theorem, and the fundamental theorem of algebra. Lagrange's work on the properties and behavior of analytic functions revolutionized the field of mathematics and paved the way for further developments in the study of complex numbers.Significance of the WorkLagrange's book is considered a seminal work in the field of mathematics and remains a cornerstone of complex analysis. His contributions to analytic functions theory have had a lasting impact on the field of mathematics and continue to influence contemporary research in areas such as number theory, physics,and engineering. The book has been widely studied and referenced by mathematicians and scientists around the world.ConclusionIn conclusion, Lagrange's work on Analytic Functions Theory stands as a testament to his brilliance and innovation in the field of mathematics. His insights and discoveries continue to shape our understanding of complex analysis and pave the way for new advancements in the field. The book remains a timeless classic in the realm of mathematics and serves as a source of inspiration for generations of mathematicians to come.篇3Title: The Analytic Function Theory by French Mathematician LagrangeIntroduction:Joseph-Louis Lagrange, a renowned mathematician from France, made significant contributions to the field of mathematics during the 18th century. One of his most influential works is the book "Analytic Function Theory," where he laid down the foundations for the study of complex functions. In this article, we will delve into the contents of this seminal work and discuss its impact on the development of mathematics.Overview of the Book:Lagrange's "Analytic Function Theory" is a comprehensive treatise on the analysis of complex functions, which play a crucial role in a variety of mathematical disciplines including calculus, differential equations, and number theory. The book is divided into several sections, each covering different aspects of the theory of analytic functions. Lagrange begins by introducing the basic concepts of complex numbers and functions, before delving into more advanced topics such as power series, contour integration, and the Cauchy-Riemann equations.Key Concepts and Theorems:One of the key contributions of Lagrange in this work is the development of the Cauchy Integral Formula, which provides a powerful method for calculating complex integrals over closed curves. This formula has important applications in the study of harmonic functions and the theory of residues. Lagrange also proved several important theorems in the book, including the Maximum Modulus Principle and the Riemann Mapping Theorem, which have been instrumental in the development of complex analysis.Impact on Mathematics:Lagrange's "Analytic Function Theory" is considered a seminal work in the field of complex analysis and has had a lasting impact on the development of mathematics. The insights and techniques introduced by Lagrange in this book have been instrumental in solving many mathematical problems in diverse areas such as physics, engineering, and computer science. The book continues to be studied and referenced by mathematicians and researchers around the world, highlighting the enduring legacy of Lagrange's contributions to the field.Conclusion:In conclusion, Joseph-Louis Lagrange's "Analytic Function Theory" stands as a cornerstone in the field of complex analysis, providing a solid foundation for the study of analytic functions and their applications. The insights and theorems introduced by Lagrange in this book have had a profound impact on the development of mathematics, shaping the way we understand and solve complex mathematical problems. As we continue to push the boundaries of mathematical research, Lagrange's work remains as relevant and influential as ever.。

Numerical solution of SDE

d n
f (t) dWt = lim
c
∆t→0
f (ti−1 )∆Wi
i=1
where ∆Wi = Wti − Wti−1 , a step of Brownian motion across the interval. Note a major diﬀerence: while the ti in the Riemann integral may be chosen at any point in the interval (ti−1 , ti ), the corresponding point for the Ito integral is required to be the left endpoint of that interval. Because f and Wt are random variables, so is the Ito integral I = d f (t) dWt . The diﬀerential dI is a notational convenience; thus c
t t
X (t) = X (0) +
0
a(s, y ) ds +
0
b(s, y ) dWs ,
where the meaning of the last integral, called an Ito integral, will be deﬁned next. Let c = t0 < t1 < . . . < tn−1 < tn = d be a grid of points on the interval [c, d]. The Riemann integral is deﬁned as a limit

双层搅拌气液反应器的CFD模拟

双层搅拌气液反应器的CFD模拟王嘉骏*，冯连芳，顾雪萍，杨富军（浙江大学化学工程与生物工程系聚合反应工程国家重点实验室，浙江杭州 310027）摘要：气液分散过程在工业中广泛应用，目前对多层叶轮在气液分散过程中应用的数值模拟研究很少。

首先应用流体力学软件FLUENT对搅拌槽内双层搅拌桨（提升式斜叶涡轮PTU和圆盘式涡轮DT）的单相流场进行了模拟，并与激光多普勒测速仪LDV测试结果进行了比较。

进一步对气液两相流进行三维CFD非稳态模拟，研究了带气体分布器的两层搅拌桨组合在不同转速和气速条件下的速度场和气含率分布等。

实验结果表明，底部提升式斜叶涡轮PTU和圆盘式涡轮DT可形成较强的气体再循环流，从而提高气含率。

模拟过程采用双流体模型和标准k－ε湍流模型，并采用滑移网格方法来处理搅拌器和挡板的相对移动。

最后分别采用两种曳力系数模型，考察气液间曳力对气液动力学的影响。

关键词：搅拌反应器气体分散 CFD 混合双流体模型引言由于搅拌反应器在工业中的广泛应用，近三十年来，对搅拌槽内气一液两相分散特性已得到了广泛研究。

目前已发表的文献中对两层桨的气－液分散特性的研究，多数仍是采用传统的六叶圆盘涡轮桨进行的，由于该桨型具有明显的分区特征，对于全槽的轴向混合极为不利[1]。

而对实际工业过程所需的由不同桨型构成的多层组合桨的气液分散特性研究较少。

另外，目前的大量研究仍局限于实验研究多层桨搅拌釜的混合时间、搅拌功率以及传热系数和气液分散等宏观现象，而对其数值模拟的微观研究还比较少。

采用计算流体力学方法CFD研究搅拌反应器内的流动场已经取得了很大进展，国内外对搅拌器的数值模拟方面已经做了不少的研究工作，但对搅拌反应器内多相体系的CFD模拟研究还很缺乏[2,3]。

本文应用LDV对多层搅拌桨的单相流场进行了测量，然后采用CFD方法对单相体系和气液两相体系进行模拟，与实验结果进行了比较，并分析了不同气液曳力模型的模拟效果。

Numerical Methods for Differentiations and Integrations

Lecture 2: Numerical Methods for Differentiations and IntegrationsAs we have discussed in Lecture 1 that numerical simulation is a set of carefully planed numerical schemes to solve initial value problems numerically. Let us consider the following three types of differential equations.dy (t )dt=f (t ) (2.1) dy (t )dt=f (y ,t )(2.2) ∂y (x ,t )∂t =f (t ,y ,∂y ∂x ,∂2y∂x2,...,ydx ,...∫)(2.3)The numerical methods for time integration of these equations will be discussed in the next section (Section 4). Before we conduct the time integration, we need to determine thedifferentiations∂y ∂x ,∂2y∂x 2,... and integrations ydx ,...∫ on the left hand side of the equation(2.3) at each grid point.In this section, we are going to discuss the following four types of numerical methods, which are commonly used in spatial differentiations and integrations.1. Finite Differences (based on Taylor’s expansion)2. FFT (Fast Fourier Transform)3. Cubic Spline2.1. Finite DifferencesFor convenience, we shall use the following notation in the rest of this lecture notes.f ijk n =f (x =i Δx ,y =j Δy ,z =k Δz ,t =n Δt )=f (x i ,y j ,z k ,t n )Given a tabulate functioni :12...N x i :x 1x 2...x N f i :f 1f 2...f NUsing finite difference method, we can obtain derivatives of a tabulated function f . Table 2.1 lists examples of the first-order and second-order finite-difference expressions of[d f /dx ]x =x i , [d 2f /dx 2]x =x i , and [d 3f /dx 3]x =x i . Table 2.2 lists examples of the finite∫,difference expressions of y=f dxTable 2.1. The numerical differentiations based on finite difference methodExercise 2.1.(a) Show that the Central Difference shown in Table 2.1 is a second-order scheme(b) Show that the Forward Difference and the backward difference shown in Table 2.1 arefirst-order schemes., and(c) Determine the forth order central difference expressions of [d f/dx]x=x i, based on the Taylor expansions of the function f.[d2f/dx2]x=x iExercise e the first-order, the second-order, and the forth-order finite-differences expressions to determine the d f/dx, and d2f/dx2, an analytical function f with a fixed Δx. Determine the numerical errors in your results. Compare the numerical errors obtained from different finite differences expressions.Figure 2.1. A summary diagram of the central difference, forward difference, and the backward difference schemes.Table 2.2. The spatial integrations based on finite difference methodExercise 2.3.Use the first order, the second order, and the forth order integration expressions listed in Table 2.2 to determine y (x =π/4) with dy (x )/dx =cos(x ) and boundary conditiony (x =0)=0. Determine the numerical errors in your results. Compare the numericalerrors obtained from different integration expressions.2.2. FFT (Fast Fourier Transform)A function can be expanded by a complete set of sine and cosine functions. In the Fast Fourier Transform, the sine and cosine tables are calculated in advance to save the CPU time of the simulation.For a periodic function f , one can use FFT to determine its differentiations and integrations, i.e.,d fdx=FFT −1{ik [FFT (f )]} fdx ∫=FFT −1{1ik [FFT (f )]} for k >0.Exercise 2.4.Use an FFT subroutine to determine the first derivatives of a periodic analytical functionf . Determine the numerical errors in your results.Exercise 2.5.Use an FFT subroutine to determine the first derivatives of a non-periodic analytical function f . Determine the numerical errors in your results.2.3. Cubic SplineA tabulate function can be fitted by a set of piece-wise continuous functions, in which the first and the second derivatives of the fitting functions are continuous at each grid point. One need to solve a tri-diagonal matrix to determine the piece-wise continuous cubic spline functions. The inversion of the tri-diagonal matrix depends only on the position of grid points. Thus, for simulations with fixed grid points, one can evaluate the inversion of the tri-diagonal matrix in advance to save the CPU time of the simulation.For a non-periodic function f, it is good to use the cubic spline method to determine its differentiations and integrations at each grid point.Results of differentiations obtained from the cubic spline show the same order of accuracy as the results obtained from the forth order finite differences scheme.Exercise 2.6.Use a Cubic Spline subroutine to determine the first derivatives of an analytical functionf. Determine the numerical errors in your results.ReferencesHildebrand, F. B., Advanced Calculus for Applications, 2nd edition,Prentice-Hall, Inc., Englewood, Cliffs, New Jersey, 1976.Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (in C or in FORTRAN and Pascal), Cambridg e University Press, Cambridge, 1988.System/360 Scientific Subroutine Package Version III, Programmer’s Manual,5th edition, IBM, New York, 1970.。

数值分析中常用数学词汇英中文对照翻译.

数值分析中常用数学词汇英中文对照abbreviation 简写符号;简写absolute error 绝对误差absolute value 绝对值accelerate 加速accumulation 累积accuracy 准确度act on 施于action 作用; 作用力add 加addition 加法addition formula 加法公式addition law 加法定律additive property 可加性adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式;代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algorithm 算法系统; 规则系统alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设analysis 分析;解析angle 角anti-clockwise direction 逆时针方向;返时针方向anti-derivative 反导数; 反微商anti-logarithm 逆对数;反对数anti-symmetric 反对称approach 接近;趋近approximate value 近似值approximation 近似;略计;逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积argument (1论证; (2辐角argument of a function 函数的自变量arithmetic 算术array 数组; 数组ascending order 递升序ascending powers of X X 的升幂assumption 假定;假设asymmetrical 非对称asymptote 渐近augmented matrix 增广矩阵average 平均;平均数;平均值axiom 公理back substitution 回代base (1底;(2基;基数basis 基belong to 属于bias 偏差;偏倚billion 十亿binary number 二进数binary operation 二元运算binary system 二进制binomial 二项式bisection method 分半法;分半方法boundary condition 边界条件boundary line 界(线;边界bounded 有界的bounded above 有上界的;上有界的bounded below 有下界的;下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度calculation 计算calculator 计算器;计算器calculus (1 微积分学; (2 演算cancel 消法;相消Cartesian coordinates 笛卡儿坐标category 类型;范畴centre 中心;心chain rule 链式法则chance 机会change of base 基的变换change of variable 换元;变量的换characteristic equation 特征(征方程characteristic function 特征(征函数characteristic root 特征(征根chart 图;图表check digit 检验数位checking 验算circle 圆classification 分类clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coefficient 系数cofactor 余因子; 余因式coincide 迭合;重合collection of terms 并项collinear 共线collinear planes 共线面column (1列;纵行;(2 柱column matrix 列矩阵column vector 列向量combination 组合common denominator 同分母;公分母common difference 公差common divisor 公约数;公约common factor 公因子;公因子common multiple 公位数;公倍comparable 可比较的complement 余;补余completing the square 配方complex number 复数complex number plane 复数平面complex root 复数根component 分量composite function 复合函数; 合成函数computation 计算computer 计算机;电子计算器concept 概念conclusion 结论condition 条件conditional 条件句;条件式conjugate 共轭constant 常数constant of integration 积分常数constraint 约束;约束条件continuity 连续性continuous function 连续函数contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数convex 凸convexity 凸性coordinate 坐标corollary 系定理; 系; 推论correspondence 对应counter clockwise direction 逆时针方向;返时针方向counter example 反例counting 数数;计数criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cube 正方体;立方;立方体cubic 三次方;立方;三次(的 cubic equation 三次方程cubic roots of unity 单位的立方根cumulative 累积的curve 曲线decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制definite integral 定积分definition 定义degree (1 度; (2 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of ODE 常微分方程次数delete 删除; 删去denary number 十进数denominator 分母dependence (1相关; (2应变derivable 可导derivative 导数determinant 行列式diagonal 对角线diagonal matrix 对角矩阵difference 差difference equation 差分方程differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiation 微分法digit 数字dimension 量; 量网; 维(数direction 方向; 方位discontinuity 不连续性discontinuous 间断(的;连续(的; 不连续(的discontinuous point 不连续点discrete 分立; 离散distance 距离diverge 发散divergence 发散(性divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divide 除dividend (1被除数;divisible 可整除division 除法division algorithm 除法算式divisor 除数;除式;因子dot 点dot product 点积echelon form 梯阵式echelon matrix 梯矩阵eigenvalue 本征值eigenvector 本征向量element 元素elementary row operation 基本行运算elimination 消法elimination method 消去法;消元法empty set 空集equivalent 等价(的error 误差error estimate 误差估计error term 误差项estimate 估计;估计量evaluate 计值exact 真确exact solution 准确解;精确解;真确解exact value 法确解;精确解;真确解example 例expand 展开experiment 实验;试验experimental 试验的exponent 指数exponential function 指数函数express…in terms of… 以………表达extreme point 极值点extreme value 极值extremum 极值factor 因子;因式;商factor method 因式分解法factorial 阶乘factorization 因子分解;因式分解fallacy 谬误FALSE 假(的falsehood 假值finite 有限finite sequence 有限序列first derivative 一阶导数first order differential equation 一阶微分方程fixed point 不动点fixed point iteration method 不动点迭代法for all X 对所有X for each /every X 对每一Xform 形式;型format 格式;规格formula(formulae 公式fraction 分数;分式function 函数fundamental theorem of calculus 微积分基本定理Gaussian elimination 高斯消去法general form 一般式;通式general solution 通解;一般解general term 通项given 给定;已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值gradient (1斜率;倾斜率;(2梯度graph 图像;图形;图表graphical method 图解法graphical representation 图示;以图样表达graphical solution 图解growth 增长higher order derivative 高阶导数horizontal 水平的;水平hypothesis 假设identity 等(式identity matrix 恒等矩阵if and only if/iff 当且仅当;若且仅若if…, then 若….则;如果…..则illustration 例证;说明image 像点;像imaginary number 虚数implicit function 隐函数imply 蕴涵;蕴含improper integral 广义积分; 非正常积分increase 递增;增加indefinite integral 不定积分independence 独立;自变inequality 不等式;不等inequality sign 不等号infinite 无限;无穷infinite sequence 无限序列;无穷序列infinite series 无限级数;无穷级数infinitesimal 无限小;无穷小infinity 无限(大;无穷(大initial approximation 初始近似值initial condition 原始条件;初值条件initial value 初值;始值initial-value problem 初值问题inner product 内积input 输入integer 整数integral 积分integrate 积;积分;......的积分integration 积分法integration by parts 分部积分法integration by substitution 代换积分法;换元积分法interchange 互换intermediate value theorem 介值定理interpolating polynomial 插值多项式interpolation 插值interval 区间intuition 直观invalid 失效;无效invariance 不变性invariant (1不变的;(2不变量;不变式inverse 反的;逆的inverse function 反函数;逆函数inverse matrix 逆矩阵inverse problem 逆算问题invertible 可逆的invertible matrix 可逆矩阵iterate (1迭代值; (2迭代iteration 迭代iterative method 迭代法known 己知Lagrange interpolating polynomial 拉格朗日插值多项代leading coefficient 首项系数leading diagonal 主对角线lemma 引理limit 极限limit of sequence 序列的极限line of best-fit 最佳拟合line segment 线段linear 线性;一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程;一次方程linear equation in two unknowns 二元一次方程;二元线性方程linearly dependent 线性相关的linearly independent 线性无关的local maximum 局部极大(值local minimum 局部极小(值logic 逻辑long division method 长除法loop 回路lower bound 下界lower triangular matrix 下三角形矩阵Maclaurin expansion 麦克劳林展开式magnitude 量;数量;长度;大小mantissa 尾数matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差mean value theorem 中值定理method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法;换元法method of successive substitution 逐次代换法; 逐次调替法minimize 极小minus 减modulus of a complex number 复数的模monomial 单项式multiple 倍数multiple root 多重根multiplication 乘法multiplicity 重数multiplier 乘数;乘式multiply 乘mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根;n次方根n the root of unity 单位的n次根natural logarithm 自然对数necessary and sufficient condition 充要条件necessary condition 必要条件negative 负neighborhood 邻域Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's method 牛顿方法non-linear 非线性non-linear equation 非线性方程non-negative 非负的non-singular (1满秩的; (2非奇异的non-singular matrix 满秩矩阵non-trivial 非平凡的non-zero 非零norm 模方; 范数normal (1垂直的;正交的;法线的(2正态的(3正常的;正规的normalize 正规化normalized form 标准型notation 记法;记号null 零; 空null set 空集null vector 零向量number 数numerator 分子numerical method 计算方法;数值法objective function 目标函数octant 卦限odd function 奇函数one-to-one 一个对一个one-one correspondence 一一对应operation 运算order of a matrix 矩阵的阶ordinary differential equation 常微分方程origin 原点orthogonal 正交orthogonality 正交性 outcome 结果 output 输出 parameter 参数；参变量parametric equation 参数方程 partition 分割; 划分 periodic function 周期函数permutation 排列 perpendicular 垂线；垂直(于 phase 相; 位相 pivot 支点 plot 绘图plus 加 point 点 polynomial 多项式 polynomial equation 多项式方程 positive 正 post-multiply 后乘; 自右乘 premultiply 前乘; 自左乘 prime 素 product 乘积；积 proper integral 正常积分 property 性质 quadratic convergence 二阶收敛性 quadratic formula 二次公式 quadratic function 二次函数 quadratic inequality 二次不等式 quadrature 求积法 quadrilateral 四边形 quotient 商；商式 quotient rule 商法则 R.H.S 右 rank 秩 rate of convergence 收敛率 ratio 比 ; 比率 rational function 有理函数 real number 实数 real part 实部 real root 实根 reciprocal 倒数 rectangle 长方形；矩形 recurrence formula 递推公式 recurrent 循环的 recurring decimal 循环小数 reduce 简化 region 区域 region of convergency 收敛区域 regular 正；规则 relative error 相对误差 remainder term 余项root 根 rotation 旋转 rounded number 舍数 rounding(off 舍入；四舍五入 row 行；棋行 row vector 行向量; 行矢量 rule 规则；法(则 satisfy 满足；适合 scalar 纯量; 无向量, 标量 scalar matrix 纯量矩阵 scale 比例尺；标度；图尺 scientific notation 科学记数法 secant (1正割; (2割线 secant method 正割法 second derivative 二阶导数 second order ordinary differential equation 二阶常微分方程 sentence 句；语句 sequence 序列series 级数 set 集 shaded portion 有阴影部分 shape 形状 shear 位移 side 边；侧 sign 符号；记号 signed number 有符号数 significant figure 有效数字 signum 正负号函数similar 相似 simplify 简化 Simpson's integral 森逊积分 Simpson's rule 森逊法则singular 奇的 singular matrix 奇异矩阵; 不可逆矩阵 span 生成 square (1平方；(2正方形 square bracket 方括号square matrix 方(矩阵 stability 稳度 stationary 平稳 stationary point 平稳点; 逗留点; 驻点 straight line 直线 subset 子集 substitute 代入 substitution 代入; 代入法subtract 减 subtraction 减法 successive approximation 逐次逼近法 successive derivative 逐次导数 successive differentiation 逐次微分法 sufficiency 充份性 sufficient and necessary condition 充要条件 sufficient condition 充份条件 sufficiently close 充份接近suffix 下标 sum 和 summation 求和法; 总和 symbol 符号; 记号 symmetry 对称; 对称性Taylor’s expansion 泰勒展开式 term 项 transpose 移项；转置 transpose of matrix 倒置矩阵；转置矩阵 trapezium 梯形 trapezoidal integral 梯形积分 trapezoidal rule 梯形法则 trial 试；试验 triangle 三角形 triangular matrix 三角矩阵 trigonometric equation 三角方程 trigonometric function 三角函数 triple 三倍 trivial solution 平凡解truncation error 截断误差 undefined 未下定义(的 undetermined coefficient 待定系数unequal 不等 unique solution 唯一解 uniqueness 唯一性 unit 单位 unit area 单位面积unit circle 单位圆 unknown 未知数；未知量 upper bound 上界 upper limit 上限 upper triangular matrix 上三角形矩阵 validity 真确性; 有效性 variable 变项；变量；元；变元；变数 vector 向量; 矢量 vector function 向量函数; 矢量函数 vector product 矢量积; 矢量积 vector space 向量空间 verify 证明；验证 weight (1重量；(2权 weighted average, weighted mean 加权平均数 without loss of generality 不失一般性 x-axis x 轴x-coordinate x 坐标 x-intercept x 轴截距 y-axis y 轴 y-coordinate y 坐标 y-intercept y轴截距 zero 零 zero factor 零因子 zero matrix 零矩阵 zero vector 零向量 zeros of a function 函数零值。

An algorithm for the numerical solution of differential equations of fractional order

$An algorithm for the numerical solution of differential equations of fractional order$

Electronic Transactions on Numerical Analysis.Volume 5, pp. 1-6, March 1997.Copyright ©1997, Kent State University.ISSN 1068-9613.ETNAKent State University etna@AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIALEQUATIONS OF FRACTIONAL ORDERKAI DIETHELMAbstract.Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws,diffusion processes,etc.A small number of algorithms for the numerical solution of these equations has been suggested,but mainly without any error estimates.In this paper,we propose an implicit algorithm for the approximate solution of an important class of these equations.The algorithm is based on a quadrature formula approach.Error estimates and numerical examples are given.Key words.Fractional derivative,Riemann-Liouville derivative,differential equation,numerical solution,quadrature formula,implicit method.AMS subject classiﬁcations.26A33,65L70,65L05.1.Introduction and Main Results.1.1.The differential equation.In this paper,we discuss a numerical method for the solution of the fractional differential equation 01(1.1)0(1.2)where 01,is a given function on the interval 01,0,and is the unknownfunction.Here,denotes the Riemann-Liouville fractional derivative of order of the function ,deﬁned by [6]:1Received November 28,1996.Accepted for publication February 5,municated by R.S.V arga.Institut f¨u r Mathematik,Universit¨a t Hildesheim,Marienburger Platz 22,D-31141Hildesheim,Germany (diethelm@informatik.uni-hildesheim.de ).12K.DiethelmOur algorithm is based on the observation[3]that we may interchange differentiation and integration in(1.3)to obtain11(1.4)where now the integral must be interpreted as a Hadamardﬁnite-part integral.Then,for a given,we introduce an equispaced grid on the interval where the solution of eq.(1.1)is sought.Discretizing with this grid and applying(1.4),we obtain for12111Now,for every,we replace the integral by aﬁrst-degree compound quadrature formula with the equispaced nodes0121,:11with remainder term11as proposed in[2].Since the quadrature formula uses both end points of the integration interval as nodes,we obtain an implicit scheme.Explicit expressions for the weights are given in Lemma2.1below.Ignoring the quadrature error,we may solve the resulting equation for the values which will be our approximations for(12).We obtain the following formulas:1Γ11Fractional differential equations3C OROLLARY1.2.If the functions involved are sufﬁciently smooth,we have the following global error estimate for the approximation method described above:max012The proof will be given in 2.The scheme has been tested on some numerical examples. The results are reported in 3.It is easily seen that the algorithm may be generalized to handle equations of the formwith non-constant.It may also be combined with an explicit scheme to form a predictor-corrector method for the more general nonlinear equationHowever,since the equation stated in(1.1)seems to be the most important case as far as applications are concerned,we shall not go into details about these two generalizations here.2.Proofs.2.1.Preliminaries.Before we come to the proofs of the main results,we state some auxiliary lemmas.L EMMA2.1.For the weights of the quadrature formula,1,we have11for0,211111for121, 1111for.Proof.This follows after a simple calculation from the deﬁnition of the quadrature formula.The following result is taken directly from[2,Theorem2.3and the remark following its proof].L EMMA2.2.Let01.(i)There exists a constant0such that,for every201,112(ii)If is convex,then11Upper and lower bounds for can also be found in[2,Theorem2.3].L EMMA2.3.For01,let the sequence be given by11and111123where is as in Lemma2.1.Then,1sin4K.DiethelmRemark1.This lemma also holds in the limit cases0and1,for then the recurrence relation reduces to1and11,respectively,which immediately implies1or.Remark2.A short calculation yields that1sin14for every01.Proof.The inequality1is an easy consequence of the fact that0for1 (cf.Lemma2.1).We prove the upper bound for by induction.SincesinsinUsing this result and the fact that0for1,we obtain11111111sin11sin010sinΓ1Γwhere0,and is the quadrature error.Thus,Fractional differential equations5 By construction,01111Γwhich,combined with(1.5),:1Γ01We now majorize this relation andﬁnd,using Lemmas2.1and2.2(i),that1012112Because of the initial condition,00,and therefore112.Let us now deﬁne a new sequence by11and1111,23.Then,1212This is obvious for1,and for23,it follows by a simple induction.Now,an application of Lemma2.3yields ourﬁnal result,viz.sin6K.DiethelmT ABLE3.1Results for1and222Γ3.075EOC EOC EOC 00208700530700062000077300231200019900028200099100006300010200042100002005025Error at1Error at1Error at1 5101291091502013811615840143120163REFERENCES[1]L.B LANK,Numerical treatment of differential equations of fractional order,Numerical Analysis Report287,Manchester Centre for Computational Mathematics,1996.[2]K.D IETHELM,Generalized compound quadrature formulae forﬁnite-part integrals,IMA J.Numer.Anal.,(toappear).[3] D.E LLIOTT,An asymptotic analysis of two algorithms for certain Hadamardﬁnite-part integrals,IMA J.Numer.Anal.,13(1993),pp.445–462.[4]L.G AUL,P.K LEIN,AND S.K EMPFLE,Damping description involving fractional operators,Mech.SystemsSignal Processing,5(1991),pp.81–88.[5]R.G ORENFLO AND R.R UTMAN,On ultraslow and intermediate processes,in Transform Methods and SpecialFunctions,P.Rusev,I.Dimovski,and V.Kiryakova,eds.,Soﬁa,1995,pp.61–81.[6]K.B.O LDHAM AND J.S PANIER,The Fractional Calculus,vol.111of Mathematics in Science and Engineering,Academic Press,New York,London,1974.。

数值逼近与微分方程的数值解中的英文单词和短语

estimation of error
收敛性
convergence
稳定性
stability
第三章
Chapter3 SplineInterpolation
龙格现象
Rungephenomenon
伯恩斯坦定理
Bernstein theorem
分段低次插值
piecewise lower-order interpolation
数值微分
numerical differential
数值积分
numerical integral
代数精度
algebraic accuracy
减消法
subtractive cancellation
外推法
extrapolation
理查森外推算法
Richardson extrapolation algorithm
orthogonal polynomial
加权函数
weighted function
勒让德多项式
Legendre polynomial
切比雪夫多项式
Chebyshevpolynomial
埃尔米特多项式
Hermitepolynomial
拉盖尔多项式
Laguerrepolynomial
第五章
Chapter5NumericalDifferentials and Numerical Integrals
样条函数
spline function
自然样条
natural spline
周期样条
periodic spline
B-样条
B-spline
第四章
Chapter4 BestApproximation