Doyub Kim Fluid Engine Development.pdf
( Doyub Kim Fluid Engine Development.pdf )Doyub Kim Fluid Engine Development.pdfTaylor francisTaylor Francis Grouphttp://taylorandfrancis.comT& F Cat #K25519K25519C000— page l-10/21/2016-17:12LUID ENGINEDEVELOPMENTDOYUB KIM(CRC CRC PressTaylor & fiBoca raton london new yorkCRC Press is an imprint of theTaylor francis Group, an informa businessAN AK PETERS BOOKT& F Cat #K25519—K25519C000- page l-10/21/2016—17:12CRC PressTaylor Francis Group6000 Broken Sound parkway nw, suite 300Boca Raton, FL 33487-274@2017 by Taylor Francis Group, LLCCRC Press is an imprint of Taylor Francis Group, an Informa businessNo clain lo original U.S. Government worksPrinted on acid-free paperVersion date: 20161019International Standard book Number -13: 978-1-4987-1992-6(Pack- Book and ebook)This book contains information obtained from authentic and highly regarded sources. Reasonable effortshave been made to publish reliable data and information, but the author and publisher cannot assumeresponsibility for the validily of all materials or the consequences of their use. The authors and publisherhave attempted to trace the copyright holders of all material reproduced in this publication and apologize toc ht holders if permission to publish in this form has not been obtained If any cocht m aterial hnot been acknowledged please write and let us know so we may rectify in any future reprintExcept as permitted under U.S. Copyright L aw, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,including photocopying, microfilming, and recording, or in any information storage or retrieval system,without written permission from tblishTorpermissiontophotocopyorusematerialelectronicallyfromthisworkpleaseaccesswww.copyrightcom(http://www.copyright.com/)orcontacttheCopyrightClearanceceNter,inC.(ccc),222RosewoodDrive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses andregistration for a variety of users. For organizations that have been granted a photocopy license by the CCC,a separate system of payment has been arranged.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are usedonly for identification and explanation without intent to infringeLibrary of Congress Cataloging-in-Publication DataNames: Kim, Doyub, author.Title: Fluid engine development/Doyub KimDescription: Boca Raton: Taylor Francis, a CRC title, part of the TaylorFrancis imprint, a member of the taylor Francis group the academicdivision of T&F Informa, plc, [2017] Includes bibliographical referencesnd indIdentifiers: LCCN 2016028981 ISBN 9781498719926(hardback: acid-free paperSubjects: LCSH: Hydrodynamata processing. Fluids--Computer simulaticClassilication: LCC QA911 K485 2017 DDC 532.00285, 66--dc23Lcrecordavailableathttps://icCn.loc.sov/2016028981Visit the Taylor& Francis Webhttp://www.taylorandfrancis.comand the Crc Press Web site athttp!/www.crcpT&FCat#K25519K25519000— page iv-10/21/2016—17:12To my wifeT& F Cat #K25519K25519C000— page v-10/21/2016-1712Taylor francisTaylor Francis Grouphttp://taylorandfrancis.comT& F Cat #K25519K25519C000— page l-10/21/2016-17:12ContentsPresaList of FigureslI1 Basics1.1 Hello. Fluid SinIdulator1.1.1 Defining State1.1.2 Computing Motion1. 1.3 Boundary handling11.4 Visualization11134621.1.5 Final Rcsult1.2 How to read This book121.2.1 Getting the Codes131.2.2 Rcading codcs131. 2.2. 1 Languages131.2.2.2 Source code st131.2.2.3 Naming Conventions141.2.2.4 Constants151.2.2.5 Array151.2.3 Reading math Expressions01.2.3.1 Scalar. Vector and Matrix201.3 Math1.3. 1 Coordinate System211.3.2 Vector211.3.2.1 Basic Operations241.3.2.2 Dot and cross product261.3.2.3 More Operations273.3 Matri1.3.3.1 Basic Matrix Operations1.3.3.2 Sparse Matrix361.3.4 System of Linear Equations371.3.4.1 Direct Methods1.3.4.2 Indirect Methods381. 3.5 Field421.3.5.1 Partial derivative451.3.5.2 GradientT&FCat#K25519—K25519C000— page vll-10/21/2016-17:12Contents1.3.5.3 Divergence481.3.54C511.3.5.5 Laplacian531.3.6 Interpolation561.3.6.1 Nearest point581.3.6.2 Linear Interpolation591.3.6.3 Cat, mull-Rom Spline Interpolation61eometry631. 4.1 Surface671.4.3 Implicit Surface to Explicit Surface1.4.4 Explicit Surface to Implicit Surface1.5 Animation番普721.6 Physics-Based Animation1.6.1 Getting Started771.6. 2 Physics Animation with Example781.6.2. 1 Choosing a Model781. 6. 2.2 Simulation State791.6.2.3 Force and Motion801.6.2. 4 Time Integration871.6. 2.5 Constraints and collisions891.7 Fluid Animation911.7.1 Gravity1.7.2P1.7.3 Viscosity1.7.4 Density Constraint2 Particle-Based Simulation992.1 Seeing the World Like Seurat992 Data Structi2.2.1 Particle System Data1002.2.2 Particle System Example2.2.3 Neighbor1082.2.3.1 Searching Nearby Particlesl092.2.3.2 Caching Neighbors2.3 SOothed Particles116asics2.3.1.1 Kernel1172.3.1.2 Data Model119lerdO1202.3.1.4 Density1212.3.1.5 Differential Operators2.3.2 Dynamics1302.3.2. 1 Solver Over view1312.3.2.2 Pressure gradient Force133T&FCa#K25519—K25519C000— page vil-10/21/2016—17:12Contents2.3.2.3 Viscosity1372.3.2.4 Gravity and Drag Forces1372.3.3 Results and limitations1382.4 Incompressible SPii with Larger Time-Step12.4Predict and correct2.4.2 Implementation2.4.3 Results1472.5 Collision Handling1492.5.1 Defining colliders1502.6 Discussion and further reading1553 Grid-Based simulation1573.1 Pixelating the World1573. 2 Data Structures1573.2.1 Types of grid1593.2.2 Grid System D1663.3 Differential Operators1693.3.1 Finite difference1703.3.2 Gradient1723.3. 3 Divergence1733.3.4 Curl1763.3.5 Lap1783. 4 Fluid simulation1803.4.1 Collision Handling1823.4.1.1 Collider to Signed-Distance Field1823.4.1.2 Boundary Conditions1833.4.2 Advection1873.4.2.1 Semi-Lagrangian Method1883.4.2.2Ing Backg Accurac1923.4.2.3 Improving Interpolation Accuracy1943.4.2.4 Boundary handlingg1963.43G1983.4.4. 1 Solving Diffusion with Forward Euler1993.4.4.2 Stability of diffuusIonSol2013.4.4.3 Solving DiffusiOn with Backward Euler2033.4.4.4 Boundary Handling2083.4.5 Pressure and Incompressibility2113.4.5. 1 Building matrix2143.5 Smoke simulation2203.5.1 Buoyancy Force2223.5.2 Advection and Diffusioll3.6 Fluid with Surface2233.6.1 Defining surfacc on grids233.6.1.1 Tracking the Surface under the Flow263.6.1.2 Reinitializing Signed-Distance Field226T&FCat#K25519—K25519C000— page ix-10/21/2016—17:12
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北航矩阵论学习笔记
北京航空航天大学矩阵理论学习笔记，总结版，学霸总结，可以放心下载使用北京航空航天大学张京蕊工程系统工程系月录§0补充公式§1 Jordan(约当)标准形(简介)§2线性变换与矩阵.24§3欧式空间与QR分解.48§4常用矩阵分解●鲁D●●·,,,,,74§5范数与级数.81§6广义逆A..97§7直积拉直及应用105矩阵理论A笔记北京航空航天大学张京蕊工程系统工程系§0补充公式令A=(a)mxn∈C",风x)=4o+a1x0定义f(4)=a0+a1A+…+amAm,其中I=l若g(x)=bo+b1x+…+bkx,(x)g(x)=g(x)(x),则f(4)“g(A)=g(A)f(A)分块公式A10令A,A1,A2为方阵00 A(2)f(A),fx)为多项式令A=,A1,4为方阵AO(2)f(4)相似关系:A∽B,(PAP=B)则:(1)(P1AP)=P!AP,(k=0,1,2,(2)f(PAP)=PfA)P,f(x)为多项式许尔公式( schur):每个复方阼,A-(a)nxm都相似丁上三角形。共113页矩阵理论A笔记第1页北京航空航天大学张京蕊工程系统工程系即:P-1AP=其中41,,的次序可以任意指定Pf:用归纳法n=1时成立可以设为(n=1阶方阵成立对于n阶方阵A=(an)2×n设特征值为A,…,n取为对应的特征向量,记为a1≠0,A1=1ax1把a1扩展为可逆方阵Q=(a1,02,xn)22e又:g(a,a,…,.)=(Qa,Qba2,,Qan)其中Qe1,aQ0Q4=QA(a1a2,…an)2-I(Aa,,AAQ=(Qa,、+)…,(*)其中A1为(n-1所阶0人:0 A为由假设,对于A1必有(n-1)阶P,可推出PAPEg知n阶方阵A,适合A=0,则A+|=1共113页矩阵理论A笔记第2页北京航空航天大学张京蕊工程系统工程系Pf:A=0→任意特征值A=0→>=0即全体特征值为00,,00由需要P1AP=→PAP+7=1pAP+PP|=P(4+1)P=14+1→A+1=-1注(1)若AB(相似),则AB有相同特征值A,可引入记号:谱集(4)={2,2,…,λ}(全体特征值,含重复)A∽B→o()=o(B)(2)A∽B→1-A=1-B-(2-4元一2)…(-n),特征多项式PAP=B=A-A=p(1-A)P=A-B引理:若A0A2,则M-A|-|M1-4|-1-A1|2-A2→ar(4)=o(A)∪a(42k+1,Ak-2,…n1f(x2)设B,f(x)为多项式,则f(B)=o f(,)引理:若n阶方阵A的谱集(4)=1,42,…},则)的全体特社值为)2,…,),x)为多项式Pf:由许尔定理,A∽B→f(4)∽f(B)f(x)的全体特征值为(A1)(42),,()},fx)为多项式例如:4为A的特征值→x为4的特征值。(x)=x)共113页矩阵理论A笔记第3页北京航空航天大学张京蕊工程系统工程系引理:令B,f(x)=x-B|=(x-41)(x-12)….(x-n)则fB)=(B-1D(B-21)…(B-A1D=0Pf:当n=2时,B=0x2f(x)=(x-1)(x-2)000→f(B)-(B-41)(B-21)(2-元)0(00∴得证★ Cayley公式:设n阶方阵A的特征多项式为f(x)=|x-A|=a+a1x+…,+x则f4)=anl+a14+…,+4=0Pf:由许尔PAP=B=→P(4)P=fp3P)=f(B)=0(引理)定义若多项式x)使(4)=0,则称(x)为A的个零化式结论方阵A的特征多项式)=1x1-4为A的一个零化式g特征多项式fx)=x2可知:f(A)=A2+1=+I=00-1Hx)=|xI-A|=(x-)(x+i,(i=√-1,t2=-1)f(A)=(A-i)(4+i1=0也可取P=则PPAP=,对角形共113页矩阵理论A笔记第4页北京航空航天大学张京蕊工程系统工程系g:知A则A"=0Onxn由 Cayley特征多项式:f(x)=x"→f(4)=4"=0Ex 1. A=求P使得PP为对角阵,并验证 Cayley定理2.A=cd/,求fx)=x1-4验证f4)-0补充知识( schur公式、 Cayley公式)应用由A"=-(a0I+a1A+1A·AanA+a142+…+a.,A把①代入②→Am1=(-)+(+)4+…+(+)41可知:任何和(m≥n)都可写成,4,,A的线性组合任何多项式g(A),可写成lA,…,4的组合。Fg:若A|≠0,fx)=xI-A|=a0+a1x+…+x",ao=|-A|≠则A可用A的多项式表示∵a1A+a242+…+an21A-+A"--a072A(a1+a24+…+an-142+A)Aa1+…+an1A"2+A-1零化式定义:若g(x)=b+b1x+…+bnx,使得g(4)=bn+b14+…+bn4m=0,称g(x)为方阵A的零化式注:方阵A的零化式有无穷多个∴取特征多项式x)则4)=0任取式M(x),f(A(4)=0→f(x)(x)也是零化式极小式定义:在方阵A的零化式集合中,去次数最小的且首项系数为1的零化式m(x),称它为A的极小式共113页矩阵理论A笔记第5页北京航空航天大学张京蕊工程系统工程系注:极小式唯一性质:①极小式m(x)必为特征多项式fx)=|xI-A的因式。②特征多项式fx)=|x1-A的每个单因子(x-4)也是极小式的因子)f(x)=|x1-4=(x-x)(x-2)则极小式m(x)=(x-x)(x-2)y…(x-,),且1≤l1≤m1,1≤l2≤m2,…,1≤l≤n,41,A2…,n互不相同210EgA=020,B=020,求极小式mA(),m()解:(1)|xI-A|=(x-2)(x-1)极小式为:(x-2)(x-1)或(x-2)(x-1)计算:(4-2/)4-1)=000010k≠000000∴极小式为m4(x)=(x-2)2(x-1)(2)|-B|-=(x-2)2(x-1)00000计算:(B-2)B-1)=000010=000-1八000∴极小式为m(x)=(x-2)(x-1)Eg求下列极小式m(x)4604-60(1)A=-3-50,(2)B=2-303-6100210(3)C,(4)D=000010002000解:(1)特征多项式|x7-A|-(x-1)(x+2)极小式为:(x-1)(x+2)或(x-1)(x+2)共113页矩阵理论A笔记第6页北京航空航天大学张京蕊工程系统工程系验证:(4-D(A+2D=0∴极小式为m(x)-(x-1)(x+2)(3)解法如下引理:A1,A2的极小式为m1(x),m2(x)A10的极小式m(x)等丁m1(x),m2(x)的最小公倍式0A2(此引力可推广到A1,42,43)0100极小式为(x-1)2,0010极小式为(x-1)0取最小公倍式(x-1)2为C的极小式。460(5)F-/40,A1=020|,A00 A0123-6101O引理;设D=,则D的极小式m(x)O验证:先证D的性质(右推公式)设A-(an)xn=(a1,2,…,n)则有AD=(0,01,a2,,.m1)AD2=(0,0,∞1,,x12)AD=(0,….0.,a1,,axn)单位向量技巧:∵AI=A(en,e2…,en)=(el,leAen)=A=(a1, a2,. a,)∴Ae1=01,Ae2=(2,.,A→AD=A(0,e1,e2,…,en-1)=(0,a1,a2…,an-)同理AD2=(AD)D=(0,.01,.12)可知:D-1-(D)Dy2-(0.,0,,e1)≠0D"=(D)D1=0,而特征多项式(x)=|x1-D|=x,极小式为某个x共113页矩阵理论A笔记第7页
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