Direct Monte Carlo Simulation of Time- Dependent Problems
|
|
- Job Singleton
- 6 years ago
- Views:
Transcription
1 the Technlgy Interface/Fall 007 Direct Mnte Carl Simulatin f Time- Depent Prblems by Matthew. N. O. Sadiku, Cajetan M. Akujubi, Sarhan M. Musa, and Sudarshan R. Nelatury Center f Excellence fr Cmmunicatin Systems Technlgy Research (CECSTR) Cllege f Engineering Prairie View A&M University, Prairie View, TX mnsadiku, cmakujubi, Schl f Engineering and Engineering Technlgy Pennsylvania State University Erie, PA srn3@psu.edu Abstract: Mnte Carl methd is well knwn fr slving static prblems such as Laplace s r Pissn s equatin. In this paper, we ext the applicability f the cnventinal Mnte Carl methd t slve time-depent (heat) prblems. We illustrate this with sme examples and present results in ne-dimensin (-D) and tw-dimensin (-D) that agree with the exact slutins. I. Intrductin Mnte Carl methds are nndeterministic mdeling appraches fr slving physical and engineering prblems. They have been applied successfully fr slving differential and integral equatins, fr finding eigenvalues, fr inverting matrices, and fr evaluating multiple integrals [-5]. Mnte Carl methds are well knwn fr slving static prblems such as Laplace s r Pissn s equatin. They are hardly applied in slving parablic and hyperblic partial differential equatins. The s-called Mnte Carl simulatin f Maxwell s equatin [6-9] gives the impressin that Mnte Carl methd is being applied t time-depent prblems. This is nt a direct r explicit slutin f Maxwell equatins like the finite-difference time-dmain (FDTD) scheme [0-]. In this paper, we ext the applicability f the cnventinal Mnte Carl methd t slve directly time-depent (heat) prblems. We deal with the case f rectangular slutin regins. We cmpare Mnte Carl slutins with the finite difference and exact slutins. Our results fr ne-dimensin (-D) and tw-dimensin (-D) prblems agree
2 the Technlgy Interface/Fall 007 with the exact slutins. The Mnte Carl treatment is s straightfrward that it can be presented t undergraduate students withut difficulties. II. Diffusin Equatin Cnsider the skin effect n a slid cylindrical cnductr. The current density distributin within a gd cnducting wire ( σ / ω >> ) beys the diffusin equatin J J () =μσ We may derive the diffusin equatin directly frm Maxwell s equatins. We recall that H = J+ J () d D where J = σ E is the cnductin current density and J d = is the displacement current t density. Fr σ / ω >>, J d is negligibly small cmpared with J. Hence Als, H J (3) H E =μ E= E E=μ H (4) Since E = 0, intrducing eq. (3), we btain Replacing E with J/σ, eq. (5) becmes which is the diffusin (r heat) equatin. J E (5) = μ J J =μσ We nw cnsider the Mnte Carl slutin f the diffusin (r heat) equatin in nedimensinal (-D) and tw-dimensinal (-D) frms in rectangular crdinate system.
3 the Technlgy Interface/Fall 007 III. One-Dimensinal Heat Equatin T be cncrete, cnsider the ne-dimensinal heat equatin: Uxx = U, 0< x <, t > 0 (6) Bundary cnditins: U(0, t) = 0 = U(, t), t > 0 (7a) Initial cnditin: U( x,0) = 00, 0< x < (7b) In eq. (6), U xx indicates secnd partial derivative with respect t x, while Ut indicates partial derivative with respect tt. The prblem mdels temperature distributin in a rd r eddy current in a cnducting medium [3]. In rder t slve this prblem using the Mnte Carl methd, we first need t btain the finite difference equivalent f the partial differential equatin in eq.(6). Using the central-space and backward-time scheme, we btain Ui ( +, n) ( Uin (, ) + Ui (, n) Uin (, ) Uin (, ) = (8) ( Δx) Δt where x = iδ x and t = nδ t. If we let ( Δx) α = (9) Δ t eq.(8) becmes Uin (, ) = PUi ( +, n) + PUi (, n) + PUin (, ) (0) x+ x t where P = Px =, + α x+ P t = α +α () Ntice that Px+ + Px + Pt =. Equatin (0) can be given a prbabilistic interpretatin. If a randm-walking particle is instantaneusly at the pint ( x, y ), it has prbabilities P x +, Px, and P t f mving frm ( x, t ) t ( x + Δ xt, ), ( x Δ xt, ), and ( x, t Δ t) respectively. The particle can nly mve tward the past, but never tward the future. A means f determining which way the particle shuld mve is t generate a randm number r, 0< r <, and instruct the particle t walk as fllws: ( x, t) ( x+δ x, t) if (0 < r < 0.5) ( x, t) ( xδ x, t) if (0.5 < r < 0.5) ( x, t) ( x, tδ t) if ( (0.5 < r < ) ) () where it is assumed that α =. Mst mdern sftware such as MATLAB have a randm number generatr t btain r. T calculate U at pint ( x, t ), we fllw the fllwing randm walk algrithm:. Begin a randm walk at ( x, t) = ( x, t ). 3
4 the Technlgy Interface/Fall 007. Generate a randm number 0< r <, and mve t the next pint using eq. (). 3(a). If the next pint is nt n the bundary, repeat step. 3(b). If the randm walk hits the bundary, terminate the randm walk. Recrd U b at the bundary and g t step and begin anther randm walk. 4. After N randm walks, determine N U( x, t ) = U ( K) (3) b N K = where N, the number f randm walks is assumed large. A typical randm walk is illustrated in Fig.. Example As a numerical example, cnsider the slutin f the prblem in eqs. (6) and (7). We selectα =, Δ x = 0., s that Δ t = and Px+ = Px =, Pt =. 4 We calculate U at x 0 = 0.4, t = 0.0, 0.0, 0.03, 0.04, 0.0. The MATLAB cde fr the prblem is shwn in Fig.. As evident in the prgram, N = 000. As shwn in Table, we cmpare the results with the finite different slutin and exact slutin [4]: 400 ( n π t) U( x, t) = sin( nπ x) e, n= K + (4) π n K = 0 Fr the exact slutin in eq. (4), the infinite series was truncated at K = 0. U(0, t ) = 0 U(, t ) = 0 x = 0 U( x,0) = 00 x = Fig. A typical randm walk in rectangular dmain. 4
5 the Technlgy Interface/Fall 007 Table Cmparing Mnte Carl (MCM) slutin with finite difference (FD) and exact slutin ( x = 0.4) t Exact MCM FD % This prgram slves ne-dimensinal diffusin (r heat) equatin % using Mnte Carl methd nrun = 000; delta = 0.; % deltat=*delta^; deltat = 0.005; A=.0; x=0.4; t=0.; i=x/delta; j=t/deltat; n=t/deltat; imax=a/delta; sum=0; fr k=:nrun i=i; n=n; while i<=imax & n<=n r=rand; %randm number between 0 and if (r >= 0.0 & r <= 0.5) i=i+; if (r >= 0.5 & r <= 0.5) i=i-; if (r >= 0.5 & r <=.0) n=n-; if (n < 0) break; % check if (i,n) is n the bundary if(i == 0.0) 5
6 the Technlgy Interface/Fall 007 sum=sum+ 0.0; break; if(i == imax) sum=sum+ 0.0; break; if(n == 0.0) sum=sum+ 00; break; % while u=sum/nrun Fig. MATLAB prgram fr Example. IV. Tw-Dimensinal Heat Equatin Suppse we are interested in the slutin f the tw-dimensinal heat equatin: Uxx + Uyy = Ut, 0< x <, 0 < y <, t > 0 (5) Bundary cnditins: U(0, y, t) = 0 = U(, y, t), 0 < y <, t > 0 (6a) Initial cnditin: U( x,0, t) = 0 = U( x,, t), 0< x <, t > 0 (6b) U( x, y,0) = 0xy, 0< x <, 0 < y < (6c) Using the central-space and backward-time scheme, we btain the finite difference equivalent as ( i+, j, n) U( i, j, n) + U( i, j, n) ( i, j+, n) U( i, j, n) + U( i, j, n) ( i, j, n) U( i, j, n) + = ( Δx) ( Δy) ( Δt) (7) Let Δ x=δ y =Δ and α = Δ (8) Δ t eq. (7) becmes Ui (, jn, ) = PUi x+ ( +, jn, ) + PUi x (, jn, ) + PUi y+ (, j+, n) + PUi y (, j, n) + PUi t (, jn, ) where Px+ = Px = Py+ = Py = (0a) 4 + α (9) 6
7 the Technlgy Interface/Fall 007 P = α t (0b) 4 +α Nte that Px+ + Px + Py+ + Py + Pt = s that a prbabilistic interpretatin can be given t eq. (9). A randm walking particle at pint ( x, yt, ) mves t ( x +Δ, yt, ), ( x Δ, yt, ), ( x, y+δ, t), ( x, yδ, t), ( x, yt, Δ t) with prbabilities, P x +, Px, P y +, Py, and Pt respectively. By generating a randm number 0< r <, we instruct the particle t mve as fllws: (, x yt,) ( x+δ, yt,) if (0 < r < 0.) (, x yt,) ( xδ, yt,) if (0. < r < 0.4) ( x, yt, ) ( xy, +Δ, t) if (0.4 < r < 0.6) () ( x, yt, ) ( xy, Δ, t) if (0.6 < r < 0.8) ( x, yt, ) ( xyt,, Δ t) if ( (0.8 < r < ) ) assuming thatα =. Therefre, we take the fllwing steps t calculate U at pint ( x, y, t ) :. Begin each randm walk at ( x, yt, ) = ( x, y, t).. Generate a randm number 0< r <, and mve the next pint accrding t eq. (). 3(a). If the next pint is nt n the bundary, repeat step. 3(b). If the randm walk hits the bundary, terminate the randm walk. Recrd at the bundary and g t step and begin anther randm walk. 3. After N randm walks, determine N U( x, y, t ) = U ( K) () b N K = U b The nly difference between -D and -D is that there are three kinds f displacements in -D while there are five displacements (fur spatial nes and ne tempral ne) in -D. Example As a numerical example, cnsider the slutin f the prblem in eqs. (5) and (6). We selectα =, Δ= 0., s that Δ t = 0.0 and we calculate U at x = 0.5, y = 0.5, t = 0.05, 0., 0.5, 0., 0.5, 0.3. In Mnte Carl simulatins, we used N = 000. As shwn in Table, we cmpare the results frm the Mnte Carl methd (MCM) with the finite difference (FD) slutin and exact slutin [5]: 40 cs( mπ)cs( nπ) mn U( x, y, t) = sin( m x) sin( n y) e t π ( λ ) π π, (3) m= n= mn where λ ( ) ( ) mn = mπ + nπ. In the exact slutin in eq. (3), the infinite series was truncated at m = 0 and n = 0. Due t the randmness f the Mnte Carl slutin, each MCM result in Tables and was btained by running the simulatin five times and taking the average. 7
8 the Technlgy Interface/Fall 007 Table Cmparing Mnte Carl slutin with finite difference and exact slutin t Exact MCM FD V. Cnclusin In this paper, we have demnstrated hw the cnventinal Mnte Carl methd (the fixed randm walk) can be applied t time-depent prblems such as the heat equatin in bth rectangular and cylindrical crdinates. Fr -D and -D cases, we ntice that the Mnte Carl slutins agree well with the finite difference slutin and the exact analytical slutins and it is easier t understand and prgram than the finite difference methd. The methd des nt require the need fr slving large matrices and is trivially easy t prgram s that even undergraduates can understand it. The randmness f the MCM results can be eliminated if we apply the Exdus methd, anther Mnte Carl technique [6, 7]. The idea can be exted t ther time-depent prblems such as Maxwell s equatins r the wave equatin. VI. References [] T. E. Price and D. P. Stry, Mnte Carl simulatin f numerical integratin, J. Statistical Cmputatin and Simulatin, vl. 3, 985, pp [] J. Padvan, Slutin f anistrpic heat cnductin prblems by Mnte Carl prcedures, Trans. ASME, August 974, pp [3] K. K. Sabelfeld, Mnte Carl Methds in Bundary Value Prblems. Berlin/New Yrk: Springer-Verlag, 99, p.. [4] N. Metrplis and S. Ulam, "The Mnte Carl Methd," Jur. Amer. Stat. Assc. vl. 44, n. 47, 949, pp [5] I. M. Sbl, A Primer fr the Mnte Carl Methd. Bca Ratn: CRC Press, 994. [6] J. Gu et al., Frequency depence f scattering by dense media f small particles based n Mnte Carl simulatin f Maxwell equatins, IEEE Trans. Gescience & Remte Sensing, vl. 40, n., Jan. 00, pp [7] L. Tsang, C. E. Mandt, and K. H. Ding, Mnte Carl simulatins f the extinctin rate f dense media with randmly distributed dielectric spheres based n slutin f Maxwell s equatins, Optical Letters, vl. 7, n. 5, 99, pp
9 the Technlgy Interface/Fall 007 [8] R. L. Wagner et al., Mnte Carl simulatin f electrmagnetic scattering frm twdimensinal randm rugh surfaces, IEEE Trans. Antennas and Prpagatin, vl. 45, n., Feb. 997, pp [9] K. W. Lam et al., On the analysis f statistical distributins f UWB signal scattering by randm rugh surfaces based n Mnte Carl simulatins f Maxwell equatins, IEEE Trans. Antennas and Prpagatin, vl. 5, n., Dec. 004, pp [0] K. S. Yee, Numerical slutin f initial bundary-value prblems invlving Maxwell s equatins in istrpic media, IEEE Trans. Antennas and Prpagatin, vl. 4, May. 966, pp [] A. Taflve and M. E. Brdwin, Numerical slutin f steady-state electrmagnetic scattering prblems using the time-depent Maxwell s equatins, IEEE Micrwave Thery & Techniques, vl. 3, n.8, Aug. 975, pp [] M. Okniewski, Vectr wave equatin D-FDTD methd fr guided wave equatin, IEEE Micrwave Guided Wave Letters, vl. 3, n. 9, Sept. 993, pp [3] D. Netter, J. Levenque, P. Massn and A. Rezzug, Mnte Carl methd fr transient eddy-current calculatins, IEEE Trans. Magnetics, vl.40, n.5, Sept. 004, pp [4] M. N. O. Sadiku, Numerical Techniques in Electrmagnetics. Bca Ratn, FL: CRC Press, nd editin, 00, pp [5] D. L. Pwers, Bundary Value Prblems. New Yrk: Academic Press, 97, pp [6] M. N. O. Sadiku. and D. Hunt, "Slutin f Dirichlet prblems by the Exdus Methd," IEEE Transactins f Micrwave Thery & Techniques, vl. 40, n., Jan. 99, pp [7] M. N. O. Sadiku, S. O. Ajse, and F. Zhiba, "Applying the Exdus Methd t slve Pissn's Equatin," IEEE Transactins f Micrwave Thery & Techniques, V. 4, N. 4, April 994, pp
Numerical Simulation of the Thermal Resposne Test Within the Comsol Multiphysics Environment
Presented at the COMSOL Cnference 2008 Hannver University f Parma Department f Industrial Engineering Numerical Simulatin f the Thermal Respsne Test Within the Cmsl Multiphysics Envirnment Authr : C. Crradi,
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationTechnology, Dhauj, Faridabad Technology, Dhauj, Faridabad
STABILITY OF THE NON-COLLINEAR LIBRATION POINT L 4 IN THE RESTRICTED THREE BODY PROBLEM WHEN BOTH THE PRIMARIES ARE UNIFORM CIRCULAR CYLINDERS WITH EQUAL MASS M. Javed Idrisi, M. Imran, and Z. A. Taqvi
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationinitially lcated away frm the data set never win the cmpetitin, resulting in a nnptimal nal cdebk, [2] [3] [4] and [5]. Khnen's Self Organizing Featur
Cdewrd Distributin fr Frequency Sensitive Cmpetitive Learning with One Dimensinal Input Data Aristides S. Galanpuls and Stanley C. Ahalt Department f Electrical Engineering The Ohi State University Abstract
More informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
More informationthe results to larger systems due to prop'erties of the projection algorithm. First, the number of hidden nodes must
M.E. Aggune, M.J. Dambrg, M.A. El-Sharkawi, R.J. Marks II and L.E. Atlas, "Dynamic and static security assessment f pwer systems using artificial neural netwrks", Prceedings f the NSF Wrkshp n Applicatins
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationSchedule. Time Varying electromagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 only) 6.3 Maxwell s equations
chedule Time Varying electrmagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 nly) 6.3 Maxwell s equatins Wave quatin (3 Week) 6.5 Time-Harmnic fields 7.1 Overview 7.2 Plane Waves in Lssless
More informationELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322
ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationQuantum Harmonic Oscillator, a computational approach
IOSR Jurnal f Applied Physics (IOSR-JAP) e-issn: 78-4861.Vlume 7, Issue 5 Ver. II (Sep. - Oct. 015), PP 33-38 www.isrjurnals Quantum Harmnic Oscillatr, a cmputatinal apprach Sarmistha Sahu, Maharani Lakshmi
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationThe Electromagnetic Form of the Dirac Electron Theory
0 The Electrmagnetic Frm f the Dirac Electrn Thery Aleander G. Kyriaks Saint-Petersburg State Institute f Technlgy, St. Petersburg, Russia* In the present paper it is shwn that the Dirac electrn thery
More informationPressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects
Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University 13518 Benha EGYPT Abstract:-The nnlinear differential
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationOn Boussinesq's problem
Internatinal Jurnal f Engineering Science 39 (2001) 317±322 www.elsevier.cm/lcate/ijengsci On Bussinesq's prblem A.P.S. Selvadurai * Department f Civil Engineering and Applied Mechanics, McGill University,
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 1-3 weeks 6-10 Majr Learning Indicatrs Identify situatins relevant
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationPhysical Layer: Outline
18-: Intrductin t Telecmmunicatin Netwrks Lectures : Physical Layer Peter Steenkiste Spring 01 www.cs.cmu.edu/~prs/nets-ece Physical Layer: Outline Digital Representatin f Infrmatin Characterizatin f Cmmunicatin
More informationECE 497 JS Lecture - 14 Projects: FDTD & LVDS
ECE 497 JS Lecture - 14 Prjects: FDTD & LVDS Spring 2004 Jse E. Schutt-Aine Electrical & Cmputer Engineering University f Illinis jse@emlab.uiuc.edu 1 ECE 497 JS - Prjects All prjects shuld be accmpanied
More informationAdvanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
6.5 Natural Cnvectin in Enclsures Enclsures are finite spaces bunded by walls and filled with fluid. Natural cnvectin in enclsures, als knwn as internal cnvectin, takes place in rms and buildings, furnaces,
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationYou need to be able to define the following terms and answer basic questions about them:
CS440/ECE448 Sectin Q Fall 2017 Midterm Review Yu need t be able t define the fllwing terms and answer basic questins abut them: Intr t AI, agents and envirnments Pssible definitins f AI, prs and cns f
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationRADIATED EM FIELDS FROM A ROTATING CURRENT- CARRYING CIRCULAR CYLINDER: 2-DIMENSIONAL NUMERICAL SIMULATION
Prgress In Electrmagnetics Research M, Vl. 18, 103 117, 2011 RADIATED EM FIELDS FROM A ROTATING CURRENT- CARRYING CIRCULAR CYLINDER: 2-DIMENSIONAL NUMERICAL SIMULATION M. H Department f Electrnic Engineering
More informationA NOTE ON THE EQUIVAImCE OF SOME TEST CRITERIA. v. P. Bhapkar. University of Horth Carolina. and
~ A NOTE ON THE EQUVAmCE OF SOME TEST CRTERA by v. P. Bhapkar University f Hrth Carlina University f Pna nstitute f Statistics Mime Series N. 421 February 1965 This research was supprted by the Mathematics
More informationHypothesis Tests for One Population Mean
Hypthesis Tests fr One Ppulatin Mean Chapter 9 Ala Abdelbaki Objective Objective: T estimate the value f ne ppulatin mean Inferential statistics using statistics in rder t estimate parameters We will be
More informationAP Statistics Notes Unit Two: The Normal Distributions
AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres).
More informationELECTROMAGNETIC CHARACTERISTICS OF CON- FORMAL DIPOLE ANTENNAS OVER A PEC SPHERE
Prgress In Electrmagnetics Research M, Vl. 26, 85 1, 212 ELECTROMAGNETIC CHARACTERISTICS OF CON- FORMAL DIPOLE ANTENNAS OVER A PEC SPHERE J. S. Meiguni *, M. Kamyab, and A. Hsseinbeig Faculty f Electrical
More informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More information5 th grade Common Core Standards
5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin
More informationSection 6-2: Simplex Method: Maximization with Problem Constraints of the Form ~
Sectin 6-2: Simplex Methd: Maximizatin with Prblem Cnstraints f the Frm ~ Nte: This methd was develped by Gerge B. Dantzig in 1947 while n assignment t the U.S. Department f the Air Frce. Definitin: Standard
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More information(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f
1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments
More informationThe influence of a semi-infinite atmosphere on solar oscillations
Jurnal f Physics: Cnference Series OPEN ACCESS The influence f a semi-infinite atmsphere n slar scillatins T cite this article: Ángel De Andrea Gnzález 014 J. Phys.: Cnf. Ser. 516 01015 View the article
More informationON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT
17 Kragujevac J. Sci. 8 (006) 17-4. ON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT Hazem Ali Attia
More informationThe Sputtering Problem James A Glackin, James V. Matheson
The Sputtering Prblem James A Glackin, James V. Mathesn I prpse t cnsider first the varius elements f the subject, next its varius parts r sectins, and finally the whle in its internal structure. In ther
More informationSPH3U1 Lesson 06 Kinematics
PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.
More informationPure adaptive search for finite global optimization*
Mathematical Prgramming 69 (1995) 443-448 Pure adaptive search fr finite glbal ptimizatin* Z.B. Zabinskya.*, G.R. Wd b, M.A. Steel c, W.P. Baritmpa c a Industrial Engineering Prgram, FU-20. University
More informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationMATCHING TECHNIQUES. Technical Track Session VI. Emanuela Galasso. The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Emanuela Galass The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Emanuela Galass fr the purpse f this wrkshp When can we use
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ Mdeling with differential equatins Mdeling strategy Fcus
More informationIntroduction: A Generalized approach for computing the trajectories associated with the Newtonian N Body Problem
A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering
More informationChapter 6. Dielectrics and Capacitance
Chapter 6. Dielectrics and Capacitance Hayt; //009; 6- Dielectrics are insulating materials with n free charges. All charges are bund at mlecules by Culmb frce. An applied electric field displaces charges
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationSIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST. Mark C. Otto Statistics Research Division, Bureau of the Census Washington, D.C , U.S.A.
SIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST Mark C. Ott Statistics Research Divisin, Bureau f the Census Washingtn, D.C. 20233, U.S.A. and Kenneth H. Pllck Department f Statistics, Nrth Carlina State
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More informationANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES
ANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES CONSISTING OF LONG PARALLEL CONDUCTORS B. de Halleux, O. Lesage, C. Mertes and A. Ptchelintsev Mechanical Engineering Department Cathlic University
More informationA Simple Set of Test Matrices for Eigenvalue Programs*
Simple Set f Test Matrices fr Eigenvalue Prgrams* By C. W. Gear** bstract. Sets f simple matrices f rder N are given, tgether with all f their eigenvalues and right eigenvectrs, and simple rules fr generating
More informationLesson Plan. Recode: They will do a graphic organizer to sequence the steps of scientific method.
Lessn Plan Reach: Ask the students if they ever ppped a bag f micrwave ppcrn and nticed hw many kernels were unppped at the bttm f the bag which made yu wnder if ther brands pp better than the ne yu are
More informationEDA Engineering Design & Analysis Ltd
EDA Engineering Design & Analysis Ltd THE FINITE ELEMENT METHOD A shrt tutrial giving an verview f the histry, thery and applicatin f the finite element methd. Intrductin Value f FEM Applicatins Elements
More informationINTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 3, 2010
Prbabilistic Analysis f Lateral Displacements f Shear in a 20 strey building Samir Benaissa 1, Belaid Mechab 2 1 Labratire de Mathematiques, Université de Sidi Bel Abbes BP 89, Sidi Bel Abbes 22000, Algerie.
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More informationGAUSS' LAW E. A. surface
Prf. Dr. I. M. A. Nasser GAUSS' LAW 08.11.017 GAUSS' LAW Intrductin: The electric field f a given charge distributin can in principle be calculated using Culmb's law. The examples discussed in electric
More informationRay tracing equations in transversely isotropic media Cosmin Macesanu and Faruq Akbar, Seimax Technologies, Inc.
Ray tracing equatins in transversely istrpic media Csmin Macesanu and Faruq Akbar, Seimax Technlgies, Inc. SUMMARY We discuss a simple, cmpact apprach t deriving ray tracing equatins in transversely istrpic
More informationCESAR Science Case The differential rotation of the Sun and its Chromosphere. Introduction. Material that is necessary during the laboratory
Teacher s guide CESAR Science Case The differential rtatin f the Sun and its Chrmsphere Material that is necessary during the labratry CESAR Astrnmical wrd list CESAR Bklet CESAR Frmula sheet CESAR Student
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationI. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is
Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More information( ) kt. Solution. From kinetic theory (visualized in Figure 1Q9-1), 1 2 rms = 2. = 1368 m/s
.9 Kinetic Mlecular Thery Calculate the effective (rms) speeds f the He and Ne atms in the He-Ne gas laser tube at rm temperature (300 K). Slutin T find the rt mean square velcity (v rms ) f He atms at
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationCHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS
CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS 1 Influential bservatins are bservatins whse presence in the data can have a distrting effect n the parameter estimates and pssibly the entire analysis,
More information4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
More informationA New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation
III-l III. A New Evaluatin Measure J. Jiner and L. Werner Abstract The prblems f evaluatin and the needed criteria f evaluatin measures in the SMART system f infrmatin retrieval are reviewed and discussed.
More informationYeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN
Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 321-3424 Rm: N-110 Fax : (901) 321-3402
More informationFigure 1a. A planar mechanism.
ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,
More informationGENESIS Structural Optimization for ANSYS Mechanical
P3 STRUCTURAL OPTIMIZATION (Vl. II) GENESIS Structural Optimizatin fr ANSYS Mechanical An Integrated Extensin that adds Structural Optimizatin t ANSYS Envirnment New Features and Enhancements Release 2017.03
More informationProgress In Electromagnetics Research M, Vol. 9, 9 20, 2009
Prgress In Electrmagnetics Research M, Vl. 9, 9 20, 2009 WIDE-ANGLE REFLECTION WAVE POLARIZERS USING INHOMOGENEOUS PLANAR LAYERS M. Khalaj-Amirhsseini and S. M. J. Razavi Cllege f Electrical Engineering
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationMATCHING TECHNIQUES Technical Track Session VI Céline Ferré The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Céline Ferré The Wrld Bank When can we use matching? What if the assignment t the treatment is nt dne randmly r based n an eligibility index, but n the basis
More informationIntroduction to Three-phase Circuits. Balanced 3-phase systems Unbalanced 3-phase systems
Intrductin t Three-hase Circuits Balanced 3-hase systems Unbalanced 3-hase systems 1 Intrductin t 3-hase systems Single-hase tw-wire system: Single surce cnnected t a lad using tw-wire system Single-hase
More informationChapter 4. Unsteady State Conduction
Chapter 4 Unsteady State Cnductin Chapter 5 Steady State Cnductin Chee 318 1 4-1 Intrductin ransient Cnductin Many heat transfer prblems are time dependent Changes in perating cnditins in a system cause
More informationLecture 17: Free Energy of Multi-phase Solutions at Equilibrium
Lecture 17: 11.07.05 Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical
More informationCOMP 551 Applied Machine Learning Lecture 11: Support Vector Machines
COMP 551 Applied Machine Learning Lecture 11: Supprt Vectr Machines Instructr: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted fr this curse
More informationEASTERN ARIZONA COLLEGE Introduction to Statistics
EASTERN ARIZONA COLLEGE Intrductin t Statistics Curse Design 2014-2015 Curse Infrmatin Divisin Scial Sciences Curse Number PSY 220 Title Intrductin t Statistics Credits 3 Develped by Adam Stinchcmbe Lecture/Lab
More informationMath Foundations 10 Work Plan
Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame
More informationTHE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS
THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS M.S. Chng Department f Mechanical Engineering The University f Melburne Victria 3010 AUSTRALIA min@unimelb.edu.au J.P. Mnty
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationThe standards are taught in the following sequence.
B L U E V A L L E Y D I S T R I C T C U R R I C U L U M MATHEMATICS Third Grade In grade 3, instructinal time shuld fcus n fur critical areas: (1) develping understanding f multiplicatin and divisin and
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More informationmaking triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=
Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More informationLecture 24: Flory-Huggins Theory
Lecture 24: 12.07.05 Flry-Huggins Thery Tday: LAST TIME...2 Lattice Mdels f Slutins...2 ENTROPY OF MIXING IN THE FLORY-HUGGINS MODEL...3 CONFIGURATIONS OF A SINGLE CHAIN...3 COUNTING CONFIGURATIONS FOR
More informationCOMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification
COMP 551 Applied Machine Learning Lecture 5: Generative mdels fr linear classificatin Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Jelle Pineau Class web page: www.cs.mcgill.ca/~hvanh2/cmp551
More informationPlan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations
STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationAn RWG Basis Function Based Near- to Far-Field Transformation for 2D Triangular-Grid Finite Difference Time Domain Method
An RWG Basis Functin Based Near- t Far-Field Transfrmatin fr D Triangular-Grid Finite Difference Time Dmain Methd Xuan Hui Wu, Student Member, IEEE, Ahmed A. Kishk, Fellw, IEEE, and Allen W. Glissn, Fellw,
More information