CIVL 8/ D Boundary Value Problems - Triangular Elements (T6) 1/8
|
|
- Baldwin O’Connor’
- 5 years ago
- Views:
Transcription
1 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () A quadracall nrpolad rangular lmn dfnd b nod, hr a h vrc and hr a h mddl a ach d. h mddl nod, dpndng on locaon, ma dfn a ragh ln or a quadrac ln. SI-ODE RIAGULAR ELEMES () h lmn ma b vr hlpful n modlng om gomr hr h u of a quadrlaral lmn ma rul n an undrabl dformaon of h lmn and cau problm n h Jacoban mappng. SI-ODE RIAGULAR ELEMES () ranformaon and Shap Funcon - hr ar o approach o dvlop h nrpolaon or hap funcon for h quadrac rangular lmn. h fr approach bad on rprnng h gomr and h dpndn varabl a a funcon of h global coordna and. SI-ODE RIAGULAR ELEMES () ranformaon and Shap Funcon - hr ar o approach o dvlop h nrpolaon or hap funcon for h quadrac rangular lmn. h cond approach bgn h h parn lmn h h nrpolaon and hap funcon prd n rm of h local ara coordna. SI-ODE RIAGULAR ELEMES () For h fr approach, condr a ragh-dd rangular lmn hon blo. h varaon of h dpndn varabl u ovr h lmn ma b prd a:, u a b c d f SI-ODE RIAGULAR ELEMES () Fng h pron for u o h dfnon of h -nod rangl gvn abov rqur:, u a b c d f,,
2 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () h abov quaon rn for ach nodal valu of and rulng n quaon n h unknon a, b, c, d,, and f. Solvng h of quaon gv h follong nrpolaon. SI-ODE RIAGULAR ELEMES () Whr u =[u, u, u, u, u, u ] and h nrpolaon funcon ar: u, u u hr = - and = -. SI-ODE RIAGULAR ELEMES () Whr u =[u, u, u, u, u, u ] and h nrpolaon funcon ar: hr = - and = -. SI-ODE RIAGULAR ELEMES () h gomr of h rangular lmn ma alo b dcrbd ung h abov nrpolaon a: = An oparamrc lmn ma b formd b ung a valu of = hch u h nrpolaon funcon gvn abov. Hovr, a ubparamrc lmn ma alo b dfnd b ng =. In h ca, h nrpolaon funcon dfnd for a hrnod rangular ar ud. SI-ODE RIAGULAR ELEMES () h gomr of h rangular lmn ma alo b dcrbd ung h abov nrpolaon a: h nrpolaon or hap funcon n global coordna and ar mahmacall clum and rarl ud n FEM anal. An quvaln form of h hap funcon ma b drvd n rm of h local parnal lmn coordna. h funcon hav a rlavl mpl mahmacal form and ar mor ffcn n compung h lmnal marc. = SI-ODE RIAGULAR ELEMES () A dcrb abov, h cond approach o dvlopng a of nrpolaon or hap funcon for a -nod quadrac rangular lmn bgn h h parn lmn n local coordna. Condr h follong -nod rangl n local coordna and.,,
3 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () In h parn lmn h nrpolaon funcon ar gvn a:,,,, SI-ODE RIAGULAR ELEMES () h parn lmn nrpolaon funcon (, ) hav o bac hap. h bhavor of h funcon,, and mlar cp rfrnc a dffrn nod. h hap funcon hon blo:,,,, SI-ODE RIAGULAR ELEMES () h parn lmn nrpolaon funcon (, ) hav o bac hap. h cond p of hap funcon vald for funcon,, and. h funcon hon blo: SI-ODE RIAGULAR ELEMES () In ordr o undrand h naur of h ranformaon from h parn lmn o h global lmn, h chan rul ud o form h dffrnal rlaonhp: In mar noaon, h drvav ma b rn a: J SI-ODE RIAGULAR ELEMES () h drmna of h Jacoban mar J : J J J SI-ODE RIAGULAR ELEMES () Whn J pov vrhr n a rgon, h ranformaon ma b nvrd o drmn = (, ) and = (, ). h man ha for a gvn pon (, ) n h rangl hr a unqu corrpondng pon (, ) n h parn lmn. h J(, ) a maur of h panon or conracon of a dffrnal ara: d d J, d d h drmnan of h Jacoban mar, J, a of h nvrbl of h ranformaon = (, ) and = (, ).
4 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () Eampl Condr h quadrac rangular lmn gvn blo. (,) (,) ( aa, ) a8 a, 8 h rulng Jacoban : J a (,) (,) (,) h coordna ranformaon gvn a: a a8 SI-ODE RIAGULAR ELEMES () Eampl L condr vral ca hr h J ma bcom ngav. h mappng a nod and hav h valu + = n common. Subung h corrpondng and coordna for nod and no h pron for J gv: J, 8a hch clarl ngav a a /. SI-ODE RIAGULAR ELEMES () Eampl For valu of and l han /, h drmnan of h Jacoban ngav. (, ) (, ) (,) (.,.) (,) nror angl = 0 (,).. EVALUAIO OF MARICES - RIAGULAR ELEMES h lmnal marc for h Poon problm ar: k a A da d f h A fda hd o ha for a = / h nror angl a nod and qual o zro. EVALUAIO OF MARICES - RIAGULAR ELEMES h dvlopmn of boh h k and f rm dncal o ha prnd for ohr p of lmn: k JJ hr JJ = (J J + J J ) J. d d f fda d d J f A A EVALUAIO OF MARICES - RIAGULAR ELEMES h rm J and J ar h fr and cond ro of h nvr of h Jacoban mar. J J hr J h drmnan of J.
5 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 EVALUAIO OF MARICES - RIAGULAR ELEMES h valu of h mar ma b compud a: For a gnral quadrac rangular lmn, h mar JJ a funconal of and h h Jacoban J n h dnomnaor. h rulng pron for JJ vr dffcul o valua acl and h ngraon ar uuall don numrcall. EVALUAIO OF MARICES - RIAGULAR ELEMES hrfor, h rm k and f ma b ca n h follong form: G, dd hr G(,) a funcon of h varabl and. In prncpl, ma b pobl o valua h f rm, hovr, numrcal ngraon pcall mor praccal. For h k rm, h apparanc of h Jacoban J n h ngrand gnrall ndca h u of numrcal quadraur. EVALUAIO OF MARICES - RIAGULAR ELEMES hrfor, h gnral pron for k and f ar: k, JJ,, dd,, f J d d EVALUAIO OF MARICES - RIAGULAR ELEMES h gnral form for an -rm Gauan quadraur : G, dd G, hr, ar h rangular Gau pon and h gh. h dvlopmn of Gauan quadraur for a rangl dncal o ha prnd n prvou con. = = = EVALUAIO OF MARICES - RIAGULAR ELEMES h gnral form for an -rm Gauan quadraur : G, dd G, hr, ar h rangular Gau pon and h gh. hrfor k and f ma b valuad b: k JJ d d JJ f J d d f J f EVALUAIO OF MARICES - RIAGULAR ELEMES h follong abl conan h Gau pon and gh for =,, and. Accurac
6 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 EVALUAIO OF MARICES - RIAGULAR ELEMES h follong abl conan h Gau pon and gh for = and 7. Accurac EVALUAIO OF MARICES - RIAGULAR ELEMES Condr h ngral a and h : a d hr h ngraon along a boundar gmn of h lmn. Snc, h ngraon compud along a ngl d of h quadrac lmn, h nrpolaon funcon ar quadrac. k n h m l h l l h h h h h h l l ll hd l EVALUAIO OF MARICES - RIAGULAR ELEMES h varaon of and a funcon of along h boundar gvn a: h dffrnal arc lngh dl : I J l l I J l l dl d d dl d l dl d l l EVALUAIO OF MARICES - RIAGULAR ELEMES h ngral dfnng a and h ar: a l d I J l l h rulng lmnal ffn mar conrbu o h global m quaon f h lmn ha a d a par of h boundar. h hd ldh h rulng lmnal load vcor conrbu o h global m quaon f h lmn ha a d a par of h boundar. EVALUAIO OF MARICES - RIAGULAR ELEMES h global m quaon ar compod from h follong ummaon: K k a G G G h rulng m quaon ar, n mar form, gvn a: Ku F G G G F f h G G G SI-ODE RIAGULAR ELEMES () Eampl Condr h quadrac rangular lmn gvn blo. (, ). (, ) (,) (,) (.,.) (,) Calcula h valu of h ffn mar k for h abov lmn hn =. and =... k JJ d d JJ = (J J + J J ) J.
7 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () 7/8 SI-ODE RIAGULAR ELEMES () Eampl k JJ d d h valu of h mar ma b compud a: SI-ODE RIAGULAR ELEMES () Eampl k JJ d d J and J ar h fr and cond ro of h nvr of h Jacoban mar. J J J J J J J SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ k mmrc k mmrc.077 SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ k mmrc.9787 k mmrc.079
8 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () 8/8 SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : 7 k JJ d d JJ k mmrc.0889 SI-ODE RIAGULAR ELEMES () Eampl h rul for = ar no vr accura. Snc onl on Gau pon ud n h quadraur, h rul no unpcd. h valu for k ung Gauan quadraur h = and ndca ha h accurac of h valuaon ncra a h numbr of Gau pon ncra. = = = SI-ODE RIAGULAR ELEMES () PROBLEM # - Wr a compu ubroun calld QUAD ha calcula h componn of h k mar for a gnral quadrac rangular lmn ung Gauan quadraur. Whl ou ll no r a compl fn lmn program, h ubroun a crcal componn of h compuaonal procdur. A mpl drvr program ll b rqurd o dbug and our ubroun and pa gomrc nformaon. our program hould allo h ur o pcf h numbr of Gauan quadraur pon and h coordna nformaon. Chck our ork h h problm n our book on pag 9 and h ampl prnd n h no. End of -od rangular Elmn ()
CIVL 7/ D Boundary Value Problems - Quadrilateral Elements (Q8) 1/9
CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES (Q) h nx n our lmn dvlopmn logcl xnon of h qudrlrl lmn o qudrclly nrpold qudrlrl lmn dfnd by gh nod, four h vrc nd four
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationCIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid
More informationProblem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d.
A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav POBLM S SOLIONS Par I Analcal Quon Problm : Condr h followng aonar daa gnraon proc for a random varabl - N..d. wh < and N -. a Oban h populaon man varanc
More informationSummary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns
Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral
More informationLecture 4 : Backpropagation Algorithm. Prof. Seul Jung ( Intelligent Systems and Emotional Engineering Laboratory) Chungnam National University
Lcur 4 : Bacpropagaon Algorhm Pro. Sul Jung Inllgn Sm and moonal ngnrng Laboraor Chungnam Naonal Unvr Inroducon o Bacpropagaon algorhm 969 Mn and Papr aac. 980 Parr and Wrbo dcovrd bac propagaon algorhm.
More informationChapter 13 Laplace Transform Analysis
Chapr aplac Tranorm naly Chapr : Ouln aplac ranorm aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d < aplac Tranorm Thr condon Unlaral on-dd aplac ranorm: aplac ranorm
More information8-node quadrilateral element. Numerical integration
Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 -nod quadrlatral lmnt. Numrcal ntgraton h tchnqu usd for th formulaton of th lnar trangl can b formall tndd to construct quadrlatral lmnts as wll
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationThe Variance-Covariance Matrix
Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o
More informationStatistical Analysis of Environmental Data - Academic Year Prof. Fernando Sansò
Scl nly of nvronmnl D - cdmc r 8-9 Prof. Frnndo Snò XRISS - PR 5 bl of onn Inroducon... xrc (D mprcl covrnc m)...7 xrc (D mprcl covrnc m)... xrc 3 (D mprcl covrnc m)... xrc 4 (D mprcl covrnc m)...3 xrc
More informationt=0 t>0: + vr - i dvc Continuation
hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationAdvanced Queueing Theory. M/G/1 Queueing Systems
Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld
More informationTheoretical Seismology
Thorcal Ssmology Lcur 9 Sgnal Procssng Fourr analyss Fourr sudd a h Écol Normal n Pars, augh by Lagrang, who Fourr dscrbd as h frs among Europan mn of scnc, Laplac, who Fourr rad lss hghly, and by Mong.
More informationOUTLINE FOR Chapter 2-2. Basic Laws
0//8 OUTLINE FOR Chapr - AERODYNAMIC W-- Basc Laws Analss of an problm n fld mchancs ncssarl nclds samn of h basc laws gornng h fld moon. Th basc laws, whch applcabl o an fld, ar: Consraon of mass Conn
More informationConventional Hot-Wire Anemometer
Convnonal Ho-Wr Anmomr cro Ho Wr Avanag much mallr prob z mm o µm br paal roluon array o h nor hghr rquncy rpon lowr co prormanc/co abrcaon roc I µm lghly op p layr 8µm havly boron op ch op layr abrcaon
More informationValuation and Analysis of Basket Credit Linked Notes with Issuer Default Risk
Valuaon and Analy of Ba Crd Lnd o wh ur Dfaul R Po-Chng Wu * * Dparmn of Banng and Fnanc Kanan Unvry Addr: o. Kanan Rd. Luchu Shang aoyuan 33857 awan R.O.C. E-mal: pcwu@mal.nu.du.w l.: 886-3-34500 x. 67
More information10.5 Linear Viscoelasticity and the Laplace Transform
Scn.5.5 Lnar Vclacy and h Lalac ranfrm h Lalac ranfrm vry uful n cnrucng and analyng lnar vclac mdl..5. h Lalac ranfrm h frmula fr h Lalac ranfrm f h drvav f a funcn : L f f L f f f f f c..5. whr h ranfrm
More informationFrequency Response. Response of an LTI System to Eigenfunction
Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc
More informationChapter 7 Stead St y- ate Errors
Char 7 Say-Sa rror Inroucon Conrol ym analy an gn cfcaon a. rann ron b. Sably c. Say-a rror fnon of ay-a rror : u c a whr u : nu, c: ouu Val only for abl ym chck ym ably fr! nu for ay-a a nu analy U o
More informationinnovations shocks white noise
Innovaons Tm-srs modls ar consrucd as lnar funcons of fundamnal forcasng rrors, also calld nnovaons or shocks Ths basc buldng blocks sasf var σ Srall uncorrlad Ths rrors ar calld wh nos In gnral, f ou
More informationWave Superposition Principle
Physcs 36: Was Lcur 5 /7/8 Wa Suroson Prncl I s qu a common suaon for wo or mor was o arr a h sam on n sac or o xs oghr along h sam drcon. W wll consdr oday sral moran cass of h combnd ffcs of wo or mor
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationELEN E4830 Digital Image Processing
ELEN E48 Dgal Imag Procssng Mrm Eamnaon Sprng Soluon Problm Quanzaon and Human Encodng r k u P u P u r r 6 6 6 6 5 6 4 8 8 4 P r 6 6 P r 4 8 8 6 8 4 r 8 4 8 4 7 8 r 6 6 6 6 P r 8 4 8 P r 6 6 8 5 P r /
More information9. Simple Rules for Monetary Policy
9. Smpl Ruls for Monar Polc John B. Talor, Ma 0, 03 Woodford, AR 00 ovrvw papr Purpos s o consdr o wha xn hs prscrpon rsmbls h sor of polc ha conomc hor would rcommnd Bu frs, l s rvw how hs sor of polc
More informationSIMEON BALL AND AART BLOKHUIS
A BOUND FOR THE MAXIMUM WEIGHT OF A LINEAR CODE SIMEON BALL AND AART BLOKHUIS Absrac. I s shown ha h paramrs of a lnar cod ovr F q of lngh n, dmnson k, mnmum wgh d and maxmum wgh m sasfy a cran congrunc
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationEngineering Circuit Analysis 8th Edition Chapter Nine Exercise Solutions
Engnrng rcu naly 8h Eon hapr Nn Exrc Soluon. = KΩ, = µf, an uch ha h crcu rpon oramp. a For Sourc-fr paralll crcu: For oramp or b H 9V, V / hoo = H.7.8 ra / 5..7..9 9V 9..9..9 5.75,.5 5.75.5..9 . = nh,
More informationImproved Exponential Estimator for Population Variance Using Two Auxiliary Variables
Improvd Epoal Emaor for Populao Varac Ug Two Aular Varabl Rajh gh Dparm of ac,baara Hdu Uvr(U.P., Ida (rgha@ahoo.com Pakaj Chauha ad rmala awa chool of ac, DAVV, Idor (M.P., Ida Flor maradach Dparm of
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationElementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationSupplementary Figure 1. Experiment and simulation with finite qudit. anharmonicity. (a), Experimental data taken after a 60 ns three-tone pulse.
Supplmnar Fgur. Eprmn and smulaon wh fn qud anharmonc. a, Eprmnal daa akn afr a 6 ns hr-on puls. b, Smulaon usng h amlonan. Supplmnar Fgur. Phagoran dnamcs n h m doman. a, Eprmnal daa. Th hr-on puls s
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More information( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationThe Hyperelastic material is examined in this section.
4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):
More informationEuler-Maruyama Approximation for Mean-Reverting Regime Switching CEV Process
Inrnaonal Confrnc on Appld Mahmac, Smulaon and Modllng (AMSM 6 Eulr-Maruyama Appromaon for Man-vrng gm Swchng CE Proc ung u* and Dan Wu Dparmn of Mahmac, Chna Jlang Unvry, Hangzhou, Chna * Corrpondng auhor
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationDynamic Behaviour of Plates Subjected to a Flowing Fluid
SES RNSCIONS on FLID MECHNICS Y. Krboua and. Lak Dnamc Bavour o la Subjcd o a Flowng Flud Y. KERBO ND.. LKIS Mcancal Engnrng Dparmn Écol olcnqu o Monral C.. 679, Succural Cnr-vll, Monral, Qubc, HC 7, CND
More informationComparative Study of Finite Element and Haar Wavelet Correlation Method for the Numerical Solution of Parabolic Type Partial Differential Equations
ISS 746-7659, England, UK Journal of Informaon and Compung Scnc Vol., o. 3, 6, pp.88-7 Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal Soluon of Parabolc Typ Paral Dffrnal Equaons S.
More informationSeptember 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline
Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons
More informationChapter 9 Transient Response
har 9 Transn sons har 9: Ouln N F n F Frs-Ordr Transns Frs-Ordr rcus Frs ordr crcus: rcus conan onl on nducor or on caacor gornd b frs-ordr dffrnal quaons. Zro-nu rsons: h crcu has no ald sourc afr a cran
More informationImproved Exponential Estimator for Population Variance Using Two Auxiliary Variables
Rajh gh Dparm of ac,baara Hdu Uvr(U.P.), Ida Pakaj Chauha, rmala awa chool of ac, DAVV, Idor (M.P.), Ida Flor maradach Dparm of Mahmac, Uvr of w Mco, Gallup, UA Improvd Epoal Emaor for Populao Varac Ug
More informationCONTINUOUS TIME DYNAMIC PROGRAMMING
Eon. 511b Sprng 1993 C. Sms I. Th Opmaon Problm CONTINUOUS TIME DYNAMIC PROGRAMMING W onsdr h problm of maxmng subj o and EU(C, ) d (1) j ^ d = (C, ) d + σ (C, ) dw () h(c, ), (3) whr () and (3) hold for
More informationNeutron electric dipole moment on the lattice
ron lcrc dol on on h lac go Shnan Unv. of Tkba 3/6/006 ron lcrc dol on fro lac QCD Inrodcon arar Boh h ha of CKM arx and QCD vac ffc conrb o CP volaon P and T volaon arar. CP odd QCD 4 L arg d CKM f f
More informationConvergence of Quintic Spline Interpolation
Inrnaonal Journal o ompur Applcaons 97 8887 Volum 7 No., Aprl onvrgnc o Qunc Spln Inrpolaon Y.P. Dub Dparmn O Mamacs, L.N..T. Jabalpur 8 Anl Sukla Dparmn O Mamacs Gan Ganga ollg O Tcnog, Jabalpur 8 ASTRAT
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationA Review of Term Structure Estimation Methods
A Rvw of Trm Srucur Emaon Mhod Sanay Nawalkha, Ph.D Inbrg School of Managmn Glora M. Soo, Ph.D Unvry of Murca 67 Inroducon Th rm rucur of nr ra, or h TSIR, can b dfnd a h rlaonhp bwn h yld on an nvmn and
More informationFAULT TOLERANT SYSTEMS
FAULT TOLERANT SYSTEMS hp://www.cs.umass.du/c/orn/faultolransysms ar 4 Analyss Mhods Chapr HW Faul Tolranc ar.4.1 Duplx Sysms Boh procssors xcu h sam as If oupus ar n agrmn - rsul s assumd o b corrc If
More informationSOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
More informationSouthern Taiwan University
Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationMA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.
MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ
More informationEE243 Advanced Electromagnetic Theory Lec # 10: Poynting s Theorem, Time- Harmonic EM Fields
Appl M Fall 6 Nuruhr Lcur # r 9/6/6 4 Avanc lcromagnc Thory Lc # : Poynng s Thorm Tm- armonc M Fls Poynng s Thorm Consrvaon o nrgy an momnum Poynng s Thorm or Lnar sprsv Ma Poynng s Thorm or Tm-armonc
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationThe Fourier Transform
/9/ Th ourr Transform Jan Baptst Josph ourr 768-83 Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s.
More informationCHAPTER 7d. DIFFERENTIATION AND INTEGRATION
CHAPTER 7d. DIFFERENTIATION AND INTEGRATION A. J. Clark School o Engnrng Dpartmnt o Cvl and Envronmntal Engnrng by Dr. Ibrahm A. Assakka Sprng ENCE - Computaton Mthods n Cvl Engnrng II Dpartmnt o Cvl and
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More informationEquil. Properties of Reacting Gas Mixtures. So far have looked at Statistical Mechanics results for a single (pure) perfect gas
Shool of roa Engnrng Equl. Prort of Ratng Ga Mxtur So far hav lookd at Stattal Mhan rult for a ngl (ur) rft ga hown how to gt ga rort (,, h, v,,, ) from artton funton () For nonratng rft ga mxtur, gt mxtur
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
(Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons of Falurs ovr Tm Rlably Modlng Exampls of Dsrbuon Funcons Th xponnal
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
Safy and Rlably of Embddd Sysms (Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Safy and Rlably of Embddd Sysms Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationLecture 4: Parsing. Administrivia
Adminitrivia Lctur 4: Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationDerivation of Eigenvalue Matrix Equations
Drivation of Eignvalu Matrix Equations h scalar wav quations ar φ φ η + ( k + 0ξ η β ) φ 0 x y x pq ε r r whr for E mod E, 1, y pq φ φ x 1 1 ε r nr (4 36) for E mod H,, 1 x η η ξ ξ n [ N ] { } i i i 1
More informationErgodic Capacity of a SIMO System Over Nakagami-q Fading Channel
DUET Journal Vol., Issu, Jun Ergodc apac of a SIO Ssm Ovr Nakagam-q Fadng hannl d. Sohdul Islam * and ohammad akbul Islam Dp. of Elcrcal and Elcronc Engnrng, Islamc Unvrs of Tchnolog (IUT, Gazpur, Bangladsh
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationSome Transcendental Elements in Positive Characteristic
R ESERCH RTICE Scnca 6 ( : 39-48 So Trancndna En n Pov Characrc Vchan aohaoo Kanna Kongaorn and Pachara Ubor Darn of ahac Kaar Unvry Bango 9 Thaand Rcvd S 999 ccd 7 Nov 999 BSTRCT Fv rancndna n n funcon
More informationAn N-Component Series Repairable System with Repairman Doing Other Work and Priority in Repair
Mor ppl Novmbr 8 N-Compo r Rparabl m h Rparma Dog Ohr ork a ror Rpar Jag Yag E-mal: jag_ag7@6om Xau Mg a uo hg ollag arb Normal Uvr Yaq ua Taoao ag uppor b h Fouao or h aural o b prov o Cha 5 uppor b h
More informationH = d d q 1 d d q N d d p 1 d d p N exp
8333: Sacal Mechanc I roblem Se # 7 Soluon Fall 3 Canoncal Enemble Non-harmonc Ga: The Hamlonan for a ga of N non neracng parcle n a d dmenonal box ha he form H A p a The paron funcon gven by ZN T d d
More informationBethe-Salpeter Equation Green s Function and the Bethe-Salpeter Equation for Effective Interaction in the Ladder Approximation
Bh-Salp Equaon n s Funcon and h Bh-Salp Equaon fo Effcv Inacon n h Ladd Appoxmaon Csa A. Z. Vasconcllos Insuo d Físca-UFRS - upo: Físca d Hadons Sngl-Pacl Popagao. Dagam xpanson of popagao. W consd as
More informationIntroduction to Inertial Dynamics
nouon o nl Dn Rz S Jon Hokn Unv Lu no on uon of oon of ul-jon oo o onl W n? A on of o fo ng on ul n oon of. ou n El: A ll of l off goun. fo ng on ll fo of gv: f-g g9.8 /. f o ll, n : f g / f g 9.8.9 El:
More informationVertical Sound Waves
Vral Sond Wavs On an drv h formla for hs avs by onsdrn drly h vral omonn of momnm qaon hrmodynam qaon and h onny qaon from 5 and hn follon h rrbaon mhod and assmn h snsodal solons. Effvly h frs ro and
More informationTIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's) TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE
TIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE J. Lanoë*, M. M. Ny*, S. L Magur* * Laboraory of Elcroncs and Sysms for Tlcommuncaons (LEST, GET-ENST
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationIntroduction to Laplace Transforms October 25, 2017
Iroduco o Lplc Trform Ocobr 5, 7 Iroduco o Lplc Trform Lrr ro Mchcl Egrg 5 Smr Egrg l Ocobr 5, 7 Oul Rvw l cl Wh Lplc rform fo of Lplc rform Gg rform b gro Fdg rform d vr rform from bl d horm pplco o dffrl
More informationCIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7
CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More informationHopf Bifurcation Analysis for the Comprehensive National Strength Model with Time Delay Xiao-hong Wang 1, Yan-hui Zhai 2
opf Bfurcaon Analy for h Comprhnv Naonal Srnh ol wh m Dlay Xao-hon an Yan-hu Zha School of Scnc ann Polychnc Unvry ann Abrac h papr manly mof an furhr vlop h comprhnv naonal rnh mol By mofyn h bac comprhnv
More informationHomework: Introduction to Motion
Homwork: Inroducon o Moon Dsanc vs. Tm Graphs Nam Prod Drcons: Answr h foowng qusons n h spacs provdd. 1. Wha do you do o cra a horzona n on a dsancm graph? 2. How do you wak o cra a sragh n ha sops up?
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationSource code. where each α ij is a terminal or nonterminal symbol. We say that. α 1 α m 1 Bα m+1 α n α 1 α m 1 β 1 β p α m+1 α n
Adminitrivia Lctur : Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
More information3.4 Repeated Roots; Reduction of Order
3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r &
More informationSun and Geosphere, 2008; 3(1): ISSN
Sun Gosphr, 8; 3(): 5-56 ISSN 89-839 h Imporanc of Ha Conducon scos n Solar Corona Comparson of Magnohdrodnamc Equaons of On-Flud wo-flud Srucur n Currn Sh Um Dn Gor Asronom Spac Scncs Dparmn, Scnc Facul,
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationChapter 5. Introduction. Introduction. Introduction. Finite Element Modelling. Finite Element Modelling
Chaptr 5 wo-dimnsional problms using Constant Strain riangls (CS) Lctur Nots Dr Mohd Andi Univrsiti Malasia Prlis EN7 Finit Elmnt Analsis Introction wo-dimnsional init lmnt ormulation ollows th stps usd
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationLaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationFinite Element Analysis
Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis
More informationState Observer Design
Sa Obsrvr Dsgn A. Khak Sdgh Conrol Sysms Group Faculy of Elcrcal and Compur Engnrng K. N. Toos Unvrsy of Tchnology Fbruary 2009 1 Problm Formulaon A ky assumpon n gnvalu assgnmn and sablzng sysms usng
More informationPotential Games and the Inefficiency of Equilibrium
Optmzaton and Control o Ntork Potntal Gam and th Inny o Equlbrum Ljun Chn 3/8/216 Outln Potntal gam q Rv on tratg gam q Potntal gam atom and nonatom Inny o qulbrum q Th pr o anarhy and lh routng q Rour
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More information