Grid Generation around a Cylinder by Complex Potential Functions
|
|
- Gwendolyn Chase
- 6 years ago
- Views:
Transcription
1 Research Journal of Appled Scences, Engneerng and Technolog 4(): , 0 ISSN: Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around a Clnder b Comple Potental Functons Hassan Davar, Mahd Chekan and Al Asghar Davar Department of Mechancal Engneerng, Islamc Azad Unverst, Roudan Branch, Roudan, Iran Abstract: In ths stud, orthogonal and structured grd generaton around a clnder s descrbed b usng potental functons. In ths method, the orthogonalt of and n functons are used for grd generaton. Frst, coordnates of ponts are gven b usng the algebrac method on clnder boundares and then, accordng to known potental functons n terms of and values n the eternal flow around a clnder, coordnates of other network's ponts are calculated through solvng a sstem of two nonlnear equatons wth two unknowns. Iteraton methods are used for solvng ths sstem of equatons. The generated grd besdes orthogonal propert has small dstances on the surface of clnder and graduall as t goes farther awa from the clnder, the dstance betweeodes rses. Ths knd of grd can be useful n solvng the flow feld around a clnder. Ke words: Comple potental functon, grd generaton, orthogonal grd INTRODUCTION Computatonal flud dnamcs s a branch of Mechancal Engneerng n whch numercal method s used to solve equatons governng flud dnamcs and heat transfer. Equatons governng flud moton are usuall of partal dfferental tpe that partal dervatves should be appromated to solve these knds of equatons. Wth these appromatons, the equatons of partal dervatves are converted to fnte dfference epressons that dfferental equatons are converted n to algebrac equatons. Obtaned algebrac equatons are called fnte dfference equatons that should be solved n the specal grd and network. So wthn the related doman and on ts borders, some grd ponts are determned. Furthermore, the effect of grd's qualt on accurac and convergence of numercal methods has caused partcular mportance of computatonal grd generaton. Generated grds are classfed nto two groups of structured and unstructured. Structured grd s generated n a wa that grd ponts can be easl dentfed on a regular bass compared to grd lnes that are defned regularl. In other words, each pont of the grd can be defned wth a certan raton of and j. There are dfferent methods to generate grds. The smplest method s algebrac method generated b usng smple algebrac equatons wth respect to geometr of the approprate grd doman. Another method s Comple Varable method on two-dmensonal spaces developed b Churchll (948), Morett (979), Davs (979) and Ives (98). The thrd common method of generatng structured grds s the method of partal dfferental equatons. The dea of usng dfferental equatons s based on research works of Crowle (96) and Wnslow (966) and also based on dea of changng phscal domato a computatonal doman. For solvng dfferental equatons on some specfc spaces wth respect to phscs of the problem, orthogonal grds can have ver benefcal and reduce calculaton's sze. For eample, when we nvestgate flow around an arfol, we know that pressure gradent n the drecton perpendcular to the surface s zero;.e. pressure n the vertcal drecton on the surface wll not change. Accordng to orthogonal grd generaton b methods of dfferental equatons, ths method has been used etensvel. In ths stud b choosng a dfferent method we are lookng for generatng an orthogonal grd around a clnder n order to help us be able to analze the flow around t. In ths stud, orthogonal propert of R and n functons s used for grd generaton. In fact, we wll generate an orthogonal grd b help of potental functons around a clnder and ts relaton wth and values. It should be noted that ths method can be used etensvel for all domans that ther R and n functons are known. MATERIALS AND METHODS Phscal doman: It s assumed that we want to generate an orthogonal grd around a clnder b Comple Potental Functons method. We assume clnder's the radus as m and defne phscal space as show Fg.. As we know, ths method s based on ths assumpton that R and n functons are orthogonal. Therefore, t s enough just to know the functons propert based on and, then ou can easl generate an orthogonal grd: Snce shape has smmetr, frst t s possble to consder one fourth of t as t s show Fg.. Net, we can use mrrorng aganst the horzontal and orthogonal as of the grd n order to make the whole shape. Mesh Correspondng Author: Hassan Davar, Department of Mechancal Engneerng, Islamc Azad Unverst, Roudan Branch, Iran 53
2 Res. J. Appl. Sc. Eng. Technol., 4(): , 0 Start Fg. : The Phscal doman Insert N, N, R N = / N- = ( +) = r cos = r sn prnt, No Yes End N No -RN- -R = 0R = 0 X + = X +r - prnt, Fg. : One forth of the phscal doman should be generated n a wa that n areas close to the clnder s there wll be less space betweeodes and graduall as t goes farther awa from the clnder, the dstance betweeodes rses. Grd generaton method: Frst assume grd for the j =,.e. ABC lne. In ths area, the shape wll be dvded nto two parts: AB arc: We dvde ths arc nto N parts. Therefore, t gves: N ( ) r cos rsn () () (3) (4) BC lne: As t can be observed, the value on ths lne s equal wth zero and space betweeodes should be n a wa that as t goes farther awa from the clnder, the dstance betweeodes rses. Therefore, we should calculate the grd rato n ths stage. Node spaces on the AB arc wll be presented wth * and the frst node on the BC lne wll be consdered wth the same amount. Then graduall we ncrease the space betweeodes accordng to the rato of R n order to get the C pont. We wll show number of nodes on ths lne wth N. N R R BC (5) =+ =+ Fg. 3: Grd algorthm generated on j: R or grd raton for ever j =, = N+N! s known. Therefore, the onl unknow ths equaton s R. Ths equaton s a knd of frst-degree non-lnear equaton that can be solved b teraton method of Newton- Raphson, and R value can be calculated. Wth gven values of and t s possble to use and n values through relatons of comple potental functons for the flow around the clnder: A r sn A r A r cos A r (6) (7) Parameter A s a fed number. It s assumed as. Therefore, t wll be possble to calculate values of and on the j = lne. You can observe grd algorthm generated on j = lne n the Fg. 3. In the net step, we consder the AE lne. Space betweeodes and on the AB arc equals wth * that s a known value. Frst node on AE lne wll be assumed wth the same space and then space betweeodes wll be ncreases graduall wth R rato. We want to generate N3 nodes on ths lne. R rato n ths case can be calculated b followng relaton: N 3 R AE R (8) It s obvous that coordnates of the ponts located on AE lne segment equals wth zero. Therefore, and values can be calculated on ths lne too. So (, j) and (, j) for ever ( =, j = N3-) s known. 53
3 Res. J. Appl. Sc. Eng. Technol., 4(): , 0 To solve the sstem's equatons, we assume that (", $ ) s the desred answer of the sstem and ( 0, 0 ) s an appromaton of (, ). So, we can wrte: " = 0 +h 0 (7) Fg. 4: The fed R lne Now we assume the j = lne. Because and s known for (I =, j = ) pont, then t wll be possble to calculate R and n based on gven relaton. In Fg. 4 ou can see the fed lne: Now we move on the fed R lne or j =. It s obvous that: R(, ) = R(, ) (9) Now we move on the fed R lne R or move perpendcular to the drecton: n(, ) = n(, ) (0) Wth known values of and for ths pont, t wll be possble to calculate and values for these ponts through solvng two equatons and two unknown sstem (equatons). Method of solvng nonlnear two equatons and two unknown sstem s gve the net secton. We can contnue n the same wa: n(3, ) = n(3, ) () R(3, ) = R(, ) () Therefore, on j=constant lnes we wll have: n(, j) = n(, j-) (3) R(, j) = R(-, j) (4) Therefore, b the above-mentoned relaton, and n values can be easl used for varous ponts. Furthermore, and values can be calculated through solvng two equatons and two unknown sstem. Method of solvng two equatons and two unknown sstem: Our equaton sstem s as follows n whch wth known values of and, the value of and or nodes coordnates wll be calculated (A and B parameters are assumed as ): f (, ) 0 (5) $ = 0 +k 0 (8) we use Talor Epanson method to calculate h 0 and k 0 parameters n Eq. (3) and (4). In the performed epanson, the epressons above second order are gnored: ƒ(", $) = ƒ( 0 + h 0, 0 + k 0 ) (9) f ( 0, 0) f ( 0, 0) f ( 0, 0) k0 Snce (", $) s an appromaton for answers of the problem, then we wll have: 0 f ( 0, 0) f (, ) f ( 0, 0) k0 It wll be calculated n the same wa for g functon: g ( 0, 0) g ( 0, 0) k0 0 g ( 0, 0) (0) () B substtuton of ƒ and g values n Eq. (6) and (7), we wll have: k ( ) ( ) 0 0 k0 ( 0 0 ) ( 0 0 ) () (3) The ntal assumpton of 0 = 0 = s consdered. Therefore, Eq. (8) and (9) can be smplfed as follows: h 3 0 k0 k0 (4) (5) The above mentoned sstem s a sstem of lnear equatons based on h 0 and k 0 that wll be solved b Kramer Order: g (, ) 0 (6) h (6) 533
4 Res. J. Appl. Sc. Eng. Technol., 4(): , 0 k (7) B gettng h 0 and k 0 values, then 0 + h 0 value wll be a better appromaton of 0 for " and also 0 + k 0 wll be a better appromaton of 0 for $. Therefore, = 0 + h 0 (8) 0 = 0 + k 0 (9) Fg. 5: The generated grd for the one-fourth of the shape related to the N: 50; N: 50; N3: 50 case Now we repeat the same operaton for and and calculate and appromatons. Generall, f n and n to be calculated, then b solvng the followng sstem: n n hn k 0 n n n ( ) ( ) n n n hn kn n ( n) ( ) n (30) (3) Fg. 6:The generated grd for the one-half of the shape related to the N:50; N:50; N3:50 case The h n and k n values wll be calculated and we wll have: n+ = n + h n (3) n+ = n +k n (33) We repeat the teraton operaton to the pont that: n+! n < (34) Fg. 7: Magnfcaton of some parts of Fg. 6 n+! n < (35) RESULTS AND DISCUSSION In ths stud b usng comple potental functon method we are lookng for generatng an orthogonal grd around a clnder n order to help us be able to analze the flow around t. The generated mesh s defned n a wa that n areas close to the clnder s there wll be less space betweeodes and graduall as t goes farther awa from the clnder, the dstance betweeodes ncreases. Accordng to smmetr of the phscal space, frst grd s produced for one fourth of t and then s generalzed to the whole problem accordng to the smmetr. In Fg. 5, ou can see the generated grd for the one fourth and one-half of the shape related to the N = 50, N = 50, N3 = 50 case. To represent orthogonal grds more clearl, ou can observe magnfcaton of some parts of Fg. 6 n the Fg. 7. As ou can observe n these fgures, as we go Fg. 8: The generated grd for the whole shape related to the N = 50, N = 50, N3 = 50 case farther awa from the clnder surface, the dstance betweeodes ncreases because our calculaton's sze decreases. On the other hand, the generated mesh s orthogonal that can be used effcentl to solve the flow feld n the eternal flows. In the Fg. 8, ou can see generated grd for the whole shape related to the N = 50, N = 50, N3 = 50 case. 534
5 Res. J. Appl. Sc. Eng. Technol., 4(): , 0 CONCLUSION In ths stud, an orthogonal mesh s generated around a clnder b usng orthogonal propert of R and n functons. The generated grd besdes orthogonal propert has small dstances on the surface of clnder and graduall as t goes farther awa from the clnder, the dstance betweeodes ncreases. Ths knd of grd can be useful n solvng the flow feld around a clnder. Ths knd of mesh besdes ts orthogonal propert s generated n a wa that there are man lttle dstances on the clnder surface, and as we go farther awa from the clnder surface, the dstances betweecreases, because the closer dstances est betweeodes, the more accurate answers wll generate for solvng mesh equatons. Snce responses on the surface of the clnder and n the closer ponts are of more mportance, then fewer dstances are consdered betweeodes. Most mportant advantage of ths method s orthogonal mesh generaton wth a low amount of calculatons n comparson wth other methods of grd generaton lke partal dervatves method. However, dsadvantage of ths method s that n specal spaces we consder R and n functons based on r and or and, or n other words t s possble to calculate the Functon Comple Potental. RECOMMENDATIONS In ths stud onl the grd generated around a clnder s descrbed b usng potental functons method. It s suggested that ths ssue to be consdered n other spaces wth known functons lke wedge and obtaned results to be analzed. On the other hand, t s suggested to solve a flow equaton around a clnder wth generated mesh for nvestgaton of valdt of the method. Net, obtaned results wll be compared wth results of other methods. REFERENCES Churchll, R.V., 948. Introducton to Comple Varables. McGraw-Hll, New York. Crowle, W.P., 96. Internal Memorandum, Lawrence Radaton Laborator, Lvermore, Calforna. Davs, R.T., 979. Numercal Methods for Coordnate Generaton Based on Schwarz-Chrstoffel Transformatons. AIAA Paper , Wllamsburg, Vrgna. Ives, D.C., 98. Conformal Grd Generaton, Numercal Grd Generaton, Proceedngs of a Smposum on the Numercal Generaton of Curvlnear Coordnate Sstems and Ther Use n the Numercal Soluton of Partal Dfferental Equatons. Thompson, J.F., (Ed.), Elsever, New York, pp: Morett, G., 979. Conformal Mappngs for the Computaton of Stead Three-Dmensonal Supersonc Flows, Numercal j Laborotor Computer Methods n Flud Mechancs. Pourng, A.A. and V.I. Shah, (Eds.), ASME, New York, pp: 3-8. Wnslow, A., 966. Numercal Soluton of the quaslnear posson equaton. J. Comput. Phs., :
PART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationFinite Difference Method
7/0/07 Instructor r. Ramond Rump (9) 747 698 rcrump@utep.edu EE 337 Computatonal Electromagnetcs (CEM) Lecture #0 Fnte erence Method Lecture 0 These notes ma contan coprghted materal obtaned under ar use
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationMean Field / Variational Approximations
Mean Feld / Varatonal Appromatons resented by Jose Nuñez 0/24/05 Outlne Introducton Mean Feld Appromaton Structured Mean Feld Weghted Mean Feld Varatonal Methods Introducton roblem: We have dstrbuton but
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More information2.29 Numerical Fluid Mechanics
REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationPre-Calculus Summer Assignment
Pre-Calculus Summer Assgnment Dear Future Pre-Calculus Student, Congratulatons on our successful completon of Algebra! Below ou wll fnd the summer assgnment questons. It s assumed that these concepts,
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationME 501A Seminar in Engineering Analysis Page 1
umercal Solutons of oundary-value Problems n Os ovember 7, 7 umercal Solutons of oundary- Value Problems n Os Larry aretto Mechancal ngneerng 5 Semnar n ngneerng nalyss ovember 7, 7 Outlne Revew stff equaton
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationLecture 2 Solution of Nonlinear Equations ( Root Finding Problems )
Lecture Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Root Fndng
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More information36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to
ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More information: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:
764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationSolutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1
Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton
More informationMeasurement Indices of Positional Uncertainty for Plane Line Segments Based on the ε
Proceedngs of the 8th Internatonal Smposum on Spatal ccurac ssessment n Natural Resources and Envronmental Scences Shangha, P R Chna, June 5-7, 008, pp 9-5 Measurement Indces of Postonal Uncertant for
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationTHE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES
THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationNumerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method
Appled Mathematcs, 6, 7, 5-4 Publshed Onlne Jul 6 n ScRes. http://www.scrp.org/journal/am http://.do.org/.436/am.6.77 umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationAPPENDIX 2 FITTING A STRAIGHT LINE TO OBSERVATIONS
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationApplication to Plane (rigid) frame structure
Advanced Computatonal echancs 18 Chapter 4 Applcaton to Plane rgd frame structure 1. Dscusson on degrees of freedom In case of truss structures, t was enough that the element force equaton provdes onl
More informationarxiv: v1 [math.ho] 18 May 2008
Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc
More informationAERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY
LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY The Bot-Savart Law The velocty nduced by the sngular vortex lne wth the crculaton can be determned by means of the Bot- Savart formula
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION
Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More information9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw
More information5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I
5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationElshaboury SM et al.; Sch. J. Phys. Math. Stat., 2015; Vol-2; Issue-2B (Mar-May); pp
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 Scholars Journal of Phscs Mathematcs Statstcs Sch. J. Phs. Math. Stat. 5; (B):69-75 Scholars Academc Scentfc Publshers
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationMeasurement and Uncertainties
Phs L-L Introducton Measurement and Uncertantes An measurement s uncertan to some degree. No measurng nstrument s calbrated to nfnte precson, nor are an two measurements ever performed under eactl the
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationDEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica
demo8.nb 1 DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA Obectves: - defne matrces n Mathematca - format the output of matrces - appl lnear algebra to solve a real problem - Use Mathematca to perform
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationNumerical Simulation of Wave Propagation Using the Shallow Water Equations
umercal Smulaton of Wave Propagaton Usng the Shallow Water Equatons Junbo Par Harve udd College 6th Aprl 007 Abstract The shallow water equatons SWE were used to model water wave propagaton n one dmenson
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationPHYS 1443 Section 003 Lecture #17
PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationMA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials
MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationThe Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions
5-6 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9,
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationVEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82
VEKTORANAY Kursvecka 3 GAU THEOREM and TOKE THEOREM Kaptel 6 7 dor 51 82 TARGET PROBEM Do magnetc monopoles est? EECTRIC FIED MAGNETIC FIED N +? 1 TARGET PROBEM et s consder some EECTRIC CHARGE 2 - + +
More informationA PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS
HCMC Unversty of Pedagogy Thong Nguyen Huu et al. A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS Thong Nguyen Huu and Hao Tran Van Department of mathematcs-nformaton,
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
More information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationSolving Fractional Nonlinear Fredholm Integro-differential Equations via Hybrid of Rationalized Haar Functions
ISSN 746-7659 England UK Journal of Informaton and Computng Scence Vol. 9 No. 3 4 pp. 69-8 Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalzed Haar Functons Yadollah Ordokhan
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More information