Advanced FDTD Algorithms
|
|
- Maria Davidson
- 6 years ago
- Views:
Transcription
1 EE 5303 Elecromagneic Analsis Using Finie Difference Time Domain Lecure #5 Advanced FDTD Algorihms Lecure 5 These noes ma conain coprighed maerial obained under fair use rules. Disribuion of hese maerials is sricl prohibied Slide 1 Lecure Ouline Alernaing Direcion Implici (ADI) Algorihm Pseudospecral Time Domain (PSTD) M4 Algorihm Inroducion Formulaion Performance Improvemen Lecure 5 Slide 1
2 Alernaing Direcion Implici Algorihm Lecure 5 Slide 3 Some Limiaions of Ordinar FDTD Recall he Couran Sabili Condiion c 1 min c0 3 x 0 Problem If he cell sie is much less han he wavelengh, hen a prohibiivel large number of ieraions will be required due o he exremel small ime sep ha ensures sabili. Low frequenc bioelecromagneics Simulaion of VLSI circuis We would like o exceed he Couran limi b more han 10. How? Lecure 5 Slide 4
3 New Sabili Condiion In an alernaing direcion implici (ADI) algorihm, we no longer have o consider he grid resoluion. We need onl look a he ccle ime of he highes frequenc. min min N N fmax 1 0 We can ge awa wih exremel fine grid resoluion wihou having o reduce for sabili! ADI FDTD is uncondiionall sable, bu his does no mean uncondiionall accurae. Lecure 5 Slide 5 Alernaing Direcion Implici Mehod Suppose we have he following PDE: x u u u u i, i, n1 n We have so far solved his using he Crank Nicolson scheme i i i i un1 un un1 un u x,,,, Insead, we can spli his ino wo ime seps, each of duraion /. u u u u nn1 : x i, i, i, i, n1 n n1 n u u u u n1n1: x i, i, i, i, n1 n1 n1 n1 Lecure 5 Slide 6 3
4 Zheng/Chen/Zhang ADI Algorihm Spaial Derivaives: Fields are saggered on an ordinar Yee grid. Time Derivaives: Fields are collocaed in ime. Original Finie Difference Equaion: E 1, 1, 1, 1, 1,, 1 1,, 1 1,, 1,, i k i k i k i k i k i k x E 1 x 1 H H H n 1 n 1 H n 1 n n n1 ADI Finie Difference Equaions (now wo seps): E E H H i1,, k i1,, k i1, 1, k i1, 1, k i1,, k1 i1,, k1 x n 1 x n 1 H H n1 n1 n n i1,, k i1,, k i1, 1, k i1, 1, k i1,, k1 Ex E n 1 x n 1 1 H H n 1 H n 1 H n1 i1,, k1 n1 Never calculaed No calculaed e Lecure 5 Slide 7 Complee Se of Spli Finie Difference Equaions Subieraion #1 Subieraion # Lecure 5 Slide 8 4
5 Derivaion of ADI Updae Equaions (1 of ) Subieraion #1 We subsiue Eq. (4.100) ino Eq. (4.99) o eliminae he H fields a he n+1/ ime seps. We sill reain Eq. (4.100). Lecure 5 Slide 9 Derivaion of ADI Updae Equaions ( of ) Subieraion # We subsiue Eq. (4.10) ino Eq. (4.101) o eliminae he H fields a he n+1 ime seps. We sill reain Eq. (4.10). Lecure 5 Slide 10 5
6 ADI Finie Difference Equaions Subieraion #1 Subieraion # Noe: H field updae equaions remain unchanged. Lecure 5 Slide 11 Soluion o ADI Finie Difference Equaions (1 of ) Subieraion #1 This equaion is wrien once for each occurrence of E x a a consan posiion. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion k. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion i. This se of equaions has he form of a ridiagonal marix and is easil solved. Lecure 5 Slide 1 6
7 Soluion o ADI Finie Difference Equaions ( of ) This equaion is wrien once for each occurrence of E x a a consan posiion k. This se of equaions has he form of a ridiagonal marix and is easil solved. Subieraion # This equaion is wrien once for each occurrence of E a a consan posiion i. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion. This se of equaions has he form of a ridiagonal marix and is easil solved. Lecure 5 Slide 13 Noes in ADI FDTD ADI FDTD is uncondiionall sable for all so he Couran sabili condiion no longer applies. ADI FDTD has accurac issues. Dispersion error increases seadil above he Couran sabili condiion. Increasing error wih increasing. ADI FDTD no well suied for elecricall large simulaions. Bes applied o elecricall small problems requiring ver fine grids. Lecure 5 Slide 14 7
8 Pseudospecral Time Domain Lecure 5 Slide 15 Purpose of PSTD Numerical dispersion is a serious problem ha is paricularl severe in elecricall large simulaions. I arises due o he numerical error arising from approximaing he spaial derivaives in Maxwell s equaions. Specral accurac is achieved when he fields are represened b rigonomeric funcions or Chebshev polnomials. This means numerical dispersion decreases exponeniall wih sampling densi. Lecure 5 Slide 16 8
9 Opions for Approximaing Spaial Derivaives Finie Difference Approximaion dfi fi 1 fi 1 dx x Requires a minimum of 10 o 0 poins per wavelengh. Fourier (Trigonomeric) Approximaion df FFT dx Nx 1 nfft f Requires a minimum of poins per wavelengh. Chebshev Approximaion Requires a minimum of poins per wavelengh. Lecure 5 Slide 17 Achieving Specral Accurac Single Domain PSTD Inernal medium mus be coninuousl inhomogeneous. coninuousl inhomogeneous piecewise inhomogeneous Mulidomain PSTD When he inernal medium is piecewise inhomogeneous, singledomain is applied o each subdomain and hen mached a he boundaries. Lecure 5 Slide 18 9
10 Noes Wraparound Effec When rigonomeric funcions are used, he grid becomes inherenl periodic. This can be miigaed b using a PML a he boundaries. Gibb s Phenomenon When he field has disconinuiies, like a a boundar of an obec, a significan overshoo and ringing is inroduced in he vicini of he boundar. Lecure 5 Slide 19 M4 Algorihm Daa and diagrams in his secion were borrowed from M. F. Hadi, M. Pike Ma, A Modified FDTD (,4) Scheme for Modeling Elecricall Large Srucures wih High Phase Accurac, IEEE Trans. on An. and Prop., Vol. 45, No., pp , Lecure 5 Slide 0 10
11 Wh M4? Problem excessive phase error ha accumulaes during an FDTD simulaion. Waves on a grid propagae differenl han phsical waves. Paricularl severe for large srucures. (,4) scheme means nd order differences in ime and 4 h order differences in space. (4,4) scheme means 4 h order differences in ime and 4 h order differences in space. These higher order schemes suffer from insabili and more complicaed boundar condiions. Lecure 5 Slide 1 Noaion L# 1 # L algorihm (S=sandard, M=modified) # 1 order of accurac in ime # order of accurac in space S Sandard FDTD wih nd order differences in ime and nd order differences in space. This is wha we learned his semeser. S4, S44 Improved formulaions, bu wih some problems. M4 Modified FDTD wih nd order differences in ime and 4 h order differences in space. Currenl sae of he ar. Lecure 5 Slide 11
12 S4 Updae Equaion (1 of ) Recall our S updae equaion for E. i, i, i, i1, i, i, 1 E H H H x H E x c 0 i, x We can wrie a similar equaion, bu wih 4 h order accurae finiedifferences. i1, i, i1, i, H 7 H 7 H H,, i i E E c 0 4 x i, i, 1 i, i, 1 i, Hx 7 Hx 7 H x H x 4 Lecure 5 Slide 3 Rearrange S4 Updae Equaion For simplici, le h = x = i, i, i1, i, i1, i, i, 1 H 7 H 7 H H E E c, 1,, 1, 0 4 h i i i i + Hx 7 Hx 7 H x H x The righ hand side can be rearranged as follows i, i, i, i, 1 i, i1, i, i, i, 1 i, i1, E E Hx Hx H H H x Hx H H c 0 4h 4h i, i, i, E E 9 i, i, 1 i, i1, 1 i, 1 i, i1, i, H x c 0 8h Hx H H Hx Hx H H 4h,, 1 i, i1, i, i, i, E E 9 i i 1 i, 1 i, i1, i, hh 3 x hh x h H h H hh x 3hH x 3 3 c hh hh 0 8h 89h Lecure 5 Slide 4 1
13 S4 Conains Closed Conour Line Inegrals We recognie ha he righ hand side of our finie difference equaion has wo expressions in he form of closed conour line inegrals. E 9 1 H d H d c0 8h 89h C1,, 1 i, i1,, 1, i1, i, hh x hh x h H h H hh x hh x hh hh i, i, i, E E 9 i i 1 i i c 0 8h 89h C Lecure 5 Slide 5 Maxwell s Equaions in Inegral Form Recall Maxwell s equaions in inegral form B E Ed Bds L S D H H d Dds L S These equaions le us calculae he line inegrals as surface inegrals. Lecure 5 Slide 6 13
14 Use Surface Inegral Insead of Line Inegral We calculae he line inegrals b insead calculaing he he surface inegrals over he area enclosed b each conour. H d Dds C1 S1 S1 E E ds S1 S1 E h E h E ds ds E H d ds C S E 3h E 9 h Lecure 5 Slide 7 Compile New Equaion We sar wih our S4 equaion derived wih line inegrals. E 9 1 H d H d c0 8h 89h C C1 We replace he line inegrals wih our new surface inegrals. E 9 E 1 E h 9 h c h 0 8 C 89h 1 C Now we simplif. E 9 E 1 E c FDTD C 1 C We see his is us a weighed sum of wo applicaions of Ampere s circui law. The coefficiens add up o uni (-1/8 + 9/8 = 1) so ha he inegri of Maxwell s equaions is preserved. Lecure 5 Slide 8 14
15 Spli The Ouer Loop Here, we spli he ouer loop ino wo disinc loops. Noe, half of he erms are included in he firs ouer loop and he remaining are included in he second ouer loop. E 9 E 1 E c0 8 8 FDTD C 1 C E 9 E 1 E 1 E c FDTD C 1 C C3 Lecure 5 Slide 9 Assign Arbirar Weighs We need more degrees of freedom in order o reduce numerical error. To do his, we assign arbirar weighs o he erms in our equaion. E E E E K K K c FDTD C 1 C C3 Noe ha in order o preserver he inegri of Maxwell s equaions, we require ha K 1 + K + K 3 = 1. To enforce his, our equaion is wrien as E E E E 1K K K K c FDTD C 1 C C3 Lecure 5 Slide 30 15
16 M4 Updae Equaion for E (1 of ) Saring wih E E E E 1K K K K c FDTD C 1 C C3 Each erm on he righ is calculaed as E 1,, 1 i, i 1, i i Hx Hx H H C h 1 E 1, 1, i 1, i, i i Hx Hx H H 3 C h E 1 i1, i1, i1, 1 i1, 1 i1, 1 i1, 1 i, 1 i, 1 Hx Hx H x Hx H H H H 6 C h 3 Lecure 5 Slide 31 M4 Updae Equaion for E ( of ) So he overall updae equaion is now i, i, i, E E 1,, 1 i, i 1, K1 K i i Hx Hx H H c 0 h K 1,, 1 i, 1 i, i i H x H x H H 3h H H H K 6h H H H H i1, i1, i1, 1 i1, 1 x x x H x i1, 1 i1, 1 i, 1 i, 1 Lecure 5 Slide 3 16
17 Opimum Values for K 1 and K NRES ki Global Phase Error 1 ki k i k d i 0 ki ki phsical wave number k numerical wave number i angle of wave hrough grid Lecure 5 Slide 33 Global Phase Error Vs. Frequenc Lecure 5 Slide 34 17
18 Global Phase Error Vs. NRES NRES Lecure 5 Slide 35 Memor Requiremens NRES min Lecure 5 Slide 36 18
Lecture Outline. Introduction Transmission Line Equations Transmission Line Wave Equations 8/10/2018. EE 4347 Applied Electromagnetics.
8/10/018 Course Insrucor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@uep.edu EE 4347 Applied Elecromagneics Topic 4a Transmission Line Equaions Transmission These Line noes
More informationReview of EM and Introduction to FDTD
1/13/016 5303 lecromagneic Analsis Using Finie Difference Time Domain Lecure #4 Review of M and Inroducion o FDTD Lecure 4These noes ma conain coprighed maerial obained under fair use rules. Disribuion
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationWindowing and Grid Techniques
EE 533 Elecromagneic Analsis Using Finie Difference Time Domain Lecure #12 Windowing and Grid Techniques Lecure 12 These noes ma conain coprighed maerial obained under fair use rules. Disribuion of hese
More informationFinite Difference Time Domain
11/9/016 5303 lecromagneic Analsis Using Finie Difference Time Domain Lecure #3 Finie Difference Time Domain Lecure 3 These noes ma conain coprighed maerial obained under fair use rules. Disribuion of
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationUnsteady Flow Problems
School of Mechanical Aerospace and Civil Engineering Unseady Flow Problems T. J. Craf George Begg Building, C41 TPFE MSc CFD-1 Reading: J. Ferziger, M. Peric, Compuaional Mehods for Fluid Dynamics H.K.
More informationln 2 1 ln y x c y C x
Lecure 14 Appendi B: Some sample problems from Boas Here are some soluions o he sample problems assigned for Chaper 8 8: 6 Soluion: We wan o find he soluion o he following firs order equaion using separaion
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationIntroduction to Physical Oceanography Homework 5 - Solutions
Laure Zanna //5 Inroducion o Phsical Oceanograph Homework 5 - Soluions. Inerial oscillaions wih boom fricion non-selecive scale: The governing equaions for his problem are This ssem can be wrien as where
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationThe fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation
Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationElementary Differential Equations and Boundary Value Problems
Elemenar Differenial Equaions and Boundar Value Problems Boce. & DiPrima 9 h Ediion Chaper 1: Inroducion 1006003 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationLecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits
Lecure 13 RC/RL Circuis, Time Dependen Op Amp Circuis RL Circuis The seps involved in solving simple circuis conaining dc sources, resisances, and one energy-sorage elemen (inducance or capaciance) are:
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationOutline of Topics. Analysis of ODE models with MATLAB. What will we learn from this lecture. Aim of analysis: Why such analysis matters?
of Topics wih MATLAB Shan He School for Compuaional Science Universi of Birmingham Module 6-3836: Compuaional Modelling wih MATLAB Wha will we learn from his lecure Aim of analsis: Aim of analsis. Some
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
More informationMA Study Guide #1
MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()
More informationLinear Dynamic Models
Linear Dnamic Models and Forecasing Reference aricle: Ineracions beween he muliplier analsis and he principle of acceleraion Ouline. The sae space ssem as an approach o working wih ssems of difference
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationEE 301 Lab 2 Convolution
EE 301 Lab 2 Convoluion 1 Inroducion In his lab we will gain some more experience wih he convoluion inegral and creae a scrip ha shows he graphical mehod of convoluion. 2 Wha you will learn This lab will
More informationExam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp
Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..
More informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
More information6.003 Homework #8 Solutions
6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
More informationThe equation to any straight line can be expressed in the form:
Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he
More informationMulti-scale 2D acoustic full waveform inversion with high frequency impulsive source
Muli-scale D acousic full waveform inversion wih high frequency impulsive source Vladimir N Zubov*, Universiy of Calgary, Calgary AB vzubov@ucalgaryca and Michael P Lamoureux, Universiy of Calgary, Calgary
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationEECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits
EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationCH.7. PLANE LINEAR ELASTICITY. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.7. PLANE LINEAR ELASTICITY Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Plane Linear Elasici Theor Plane Sress Simplifing Hpohesis Srain Field Consiuive Equaion Displacemen Field The Linear
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationTheory of! Partial Differential Equations-I!
hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More information( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationMorning Time: 1 hour 30 minutes Additional materials (enclosed):
ADVANCED GCE 78/0 MATHEMATICS (MEI) Differenial Equaions THURSDAY JANUARY 008 Morning Time: hour 30 minues Addiional maerials (enclosed): None Addiional maerials (required): Answer Bookle (8 pages) Graph
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationES.1803 Topic 22 Notes Jeremy Orloff
ES.83 Topic Noes Jeremy Orloff Fourier series inroducion: coninued. Goals. Be able o compue he Fourier coefficiens of even or odd periodic funcion using he simplified formulas.. Be able o wrie and graph
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationCh.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationFrom Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
More informationDifferential Geometry: Numerical Integration and Surface Flow
Differenial Geomery: Numerical Inegraion and Surface Flow [Implici Fairing of Irregular Meshes using Diffusion and Curaure Flow. Desbrun e al., 1999] Energy Minimizaion Recall: We hae been considering
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationA Shooting Method for A Node Generation Algorithm
A Shooing Mehod for A Node Generaion Algorihm Hiroaki Nishikawa W.M.Keck Foundaion Laboraory for Compuaional Fluid Dynamics Deparmen of Aerospace Engineering, Universiy of Michigan, Ann Arbor, Michigan
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationMA 366 Review - Test # 1
MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationKriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number
More informationDistance Between Two Ellipses in 3D
Disance Beween Two Ellipses in 3D David Eberly Magic Sofware 6006 Meadow Run Cour Chapel Hill, NC 27516 eberly@magic-sofware.com 1 Inroducion An ellipse in 3D is represened by a cener C, uni lengh axes
More information10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
More informationOrdinary differential equations. Phys 750 Lecture 7
Ordinary differenial equaions Phys 750 Lecure 7 Ordinary Differenial Equaions Mos physical laws are expressed as differenial equaions These come in hree flavours: iniial-value problems boundary-value problems
More informationMetals and Alternative Grid Schemes
EE 5303 Elecromagneic Analysis Using Finie Difference Time Domain Lecure #18 Meals and Alernaive Grid Schemes Lecure 18 These noes may conain copyrighed maerial obained under fair use rules. Disribuion
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationBU Macro BU Macro Fall 2008, Lecture 4
Dynamic Programming BU Macro 2008 Lecure 4 1 Ouline 1. Cerainy opimizaion problem used o illusrae: a. Resricions on exogenous variables b. Value funcion c. Policy funcion d. The Bellman equaion and an
More informationEECS 2602 Winter Laboratory 3 Fourier series, Fourier transform and Bode Plots in MATLAB
EECS 6 Winer 7 Laboraory 3 Fourier series, Fourier ransform and Bode Plos in MATLAB Inroducion: The objecives of his lab are o use MATLAB:. To plo periodic signals wih Fourier series represenaion. To obain
More informationEE650R: Reliability Physics of Nanoelectronic Devices Lecture 9:
EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he
More informationMost Probable Phase Portraits of Stochastic Differential Equations and Its Numerical Simulation
Mos Probable Phase Porrais of Sochasic Differenial Equaions and Is Numerical Simulaion Bing Yang, Zhu Zeng and Ling Wang 3 School of Mahemaics and Saisics, Huazhong Universiy of Science and Technology,
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationCHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More information