Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous Media
|
|
|
- Melanie Miles
- 5 years ago
- Views:
Transcription
1 Mixed Hybid Finite Element nd Itetive Metods fo Flo in Poous Medi E. Moue, C. Le Potie, P. Mugis,, L.V. Benet. Commissit à l'enegie Atomique, C.E. de Sly, Gif su Yvette Cedex, Fne Emil: : emmnuel.moue@e.f
2 Summy Cst3M Code Rid s eqution & MHFE Fomultion Itetive esolution Illusttions Ai Wte migtion + Tempetue Conlusion
3 Cst3M (CASTEM2000 ttp: //-st3m.e.f Finite Element Code (CEA PDE : Stutul menis, Fluid Menis, Temis, Objet oiented ode : Objet2 = Opeto Objet (options «Toll Box» (500 opetos 2 lnguges : use s (Gibine & develope s (Esope Pe & Post poessos FE, MHFE & FV
4 Cst3M (CASTEM2000 Poous medi : Dy (2d & 3d, Tnspot equtions (dv. disp. diff., (See eent ppe in omputtionl geosienes 2004, Gilles Bend Miel et l, bout te COUPLEX test se Bsi bik fo multipse flo : F A + BF +.J J = C F A, B,C ons tn ts = S(x, t Itetive Solutions Fo multipse flo F, G,
5 Rid s Eqution, Itetive Resolution Mixed Hybid Finite Element fomultion fo te flo in unstuted poous medi», C.Le Potie et l., CMWR XII, 998 C( =.U U = K( θ( + z MHFE : ed pessue, U : Dy veloity, C : Cpilly pity, K : pemebility, _ te ontent FE Mes (QUA4 K Ω Ω U.dΩ= C( Ω vdω= MHFE Mes EFMH (QUAF.dΩ Ω Ω.UvdΩ.ndΓ Element ente, ed Fe ente Veloity, ed
6 Rid s Eqution, Itetive Resolution Impliit time disetiztion & Pid lgoitm Time step : n Itetive step : i H = + z nd C( U,i,i+ H,i = K( + Δt,i n H =.U,i+ H,i+
7 Rid s Eqution, Itetive Resolution Time step sttegy : Δt = U Fe.n Fe sufe( Ω N Fes N volume( Ω k F k Pmete (X= C & K omogeneiztion : Diffeent types of mens : funtions of X vlues on te fes, X=F(X(Hfe, XA (itmeti, XG (geometi, XH (moni o enteed vlue X = X(Hente Element ente; X = X(Hente MHFE Mes MHFE (QUAF Fe ente; X =F(X(Hfe
8 Infilttion in Heteogeneous Soil Infilttion Infilttion nd Rege of n quife in eteogeneous soil 30mn Wte ontent Snd Cly lens Not t Sle dy Oveflo
9 Stution Rinfll on Slope (Cf DYNAS Rinfll on slope (Runoff, Infilttion, Rege, Oveflo, Lolized Rinfll Unifom Rinfll T0 Aquife T Runoff
10 Rid s Eqution, Itetive Resolution. Impotne of fist time step in te onvegene poess. If ell seleted, onvegene is ieved in 5-0 itetions. 2. Pmete omogeneiztion : Depending on te sitution : Aitmeti men o enteed vlue. Aitmeti seems to led to bette peision?!. 3. No poblem it te onvetive (gvity tem. 4. Heteogeneous medi my led to toug situtions : flo fom n impevious medium (lo yduli diffusivity, D=K/C to pevious medium (ig D 5. No poblem it te unstuted - stuted tnsition (see 4.
11 Ai Wte Migtion ( ( ( + θ ω = θ = θ α = ρ µ = = + ρ + θ + θ ρ = + θ α α α α α P P ( P, ( g P. k U 0 U H U. t H ( ( 0.U t : te, : i, H : Heny s onstnt, _ : density, _ : poosity
12 Ai Wte Migtion ( ( ( ( ( ( + θ ρ + θ + ρ θ ρ = ρ θ + ω θ + + θ = z ( K H. ( K H ( K. t C( H ( t H ( z ( K. t C( : te, : i, H : Heny s onstnt, _ : density, _ : poosity Reitten in tems of pilly ed & i ed pessues
13 Ai Wte Migtion It looks like te mtix system A A, 2, =.(U + A 2,2, + U,2 =.(U 2, + U 2,2 And solved te sme y s fo Rid s eqution A A,i,,i 2,2,i+,i Δt + Δt n n +.U +.U,i+,,i+ 2,2 =.U =.U,i,2,i+ 2, A,i 2,,i+ Δt n
14 Toum nd Vulin expeiment Exp. nd num. Anlysis of to-pse infilttion in ptilly stuted soil, TIPM ( 986 «Vetil infilttion in sndy olumn it no ltel i flo nd it i flo» Ai flo No Ai flo. Te te infilttion is dstilly sloed don by i
15 Ai mss onsevtion poblem THM nd Geo. Beviou of ly bie in diotive ste epositoies, Volket et l, CCE Repot EUR 6744 en 996 «Vetil infilttion in sndy olumn it no i flo nd soluble i» Ai mss onsevtion Ai Pessue
16 EVEGAS Euopen Pojet Cnnot find te efeene! «Podution of Hydogen Bubble in stuted poous medi» H2 H2 Does you ode see te Ct s es?
17 + Tempetue : Temo Hydulis Pysis Numeis A A A A A A, 2, 3, C C C I+ N+, I C, t I+ N+, I A 2, 2 N+, I 3, 3 t T t I+ + A + A + A,2 2,2 3,2 T +.(U A A + A + A 2,3 3,3, + U T T,2 + U +.(U +.(U,3 2, 3, + U + U + U, +. U =.( U + U + U A,4 2,2 3,2 N+ I+ N N N N+, I,, 2, 3, 4, 2 = 0 + U + U T t 2,3 3,3 + U + U I N, I N N N N N, I C N +. U2, 2 =.( U2, + U2, + U2, 3 + U2, 4 A 2, A, t I I N+, I+ N N N N+, I C +, +. U3, 3 =.( U3, + U3, 2 + U3, 4 A 3, A 3, 2 t t I 2,4 3,4 = 0 = , I 2 3 N I A T I t
18 Pollok D.W., WRR ( «Simultion of Fluid Flo & Enegy Tnspot Poesses ssoited it HLW Disposl in Unstuted Alluvium» Pollok s esults Stution y. Stution 0-00 y.
19 Conlusion Itetive metods ok quite ell. Not time onsuming, s omped to «globl» metods (benmking it THM odes. Sometimes gndm s tiks (oie of good vibles must be intodued My beome vey toug it some medi su s unstuted flo in ftued medi, geotemy, If you do it, ve fun! A ppe is in peption!
Properties and Formulas
Popeties nd Fomuls Cpte 1 Ode of Opetions 1. Pefom ny opetion(s) inside gouping symols. 2. Simplify powes. 3. Multiply nd divide in ode fom left to igt. 4. Add nd sutt in ode fom left to igt. Identity
Illustrating the space-time coordinates of the events associated with the apparent and the actual position of a light source
Illustting the spe-time oointes of the events ssoite with the ppent n the tul position of light soue Benh Rothenstein ), Stefn Popesu ) n Geoge J. Spi 3) ) Politehni Univesity of Timiso, Physis Deptment,
CHAPTER 7 Applications of Integration
CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves.......... Setion 7. Volume: The Disk Method................. Setion 7. Volume: The Shell Method................ Setion 7. A Length
Radial geodesics in Schwarzschild spacetime
Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using
Photographing a time interval
Potogaping a time inteval Benad Rotenstein and Ioan Damian Politennia Univesity of imisoaa Depatment of Pysis imisoaa Romania benad_otenstein@yaoo.om ijdamian@yaoo.om Abstat A metod of measuing time intevals
(conservation of momentum)
Dynamis of Binay Collisions Assumptions fo elasti ollisions: a) Eletially neutal moleules fo whih the foe between moleules depends only on the distane between thei entes. b) No intehange between tanslational
On the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
Equations from the Millennium Theory of Inertia and Gravity. Copyright 2004 Joseph A. Rybczyk
Equtions fo the illenniu heoy of Ineti nd vity Copyight 004 Joseph A. Rybzyk ollowing is oplete list of ll of the equtions used o deived in the illenniu heoy of Ineti nd vity. o ese of efeene the equtions
1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1
RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the
Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer
July, 008 PROGRESS IN PHYSICS Volume Detetion of te ltivisti Coetions to te Gvittionl Potentil using Sgn Intefeomete Ionnis Iklis Hns nd Miel Hney y Deptment of Pysis nd Astonomy, Yok Univesity, 4A Petie
2.25 Advanced Fluid Mechanics
MIT Depatment of Mechanical Engineeing 2.25 Advanced Fluid Mechanics Poblem 4.27 This poblem is fom Advanced Fluid Mechanics Poblems by A.H. Shapio and A.A. Sonin u(,t) pg Gas Liquid, density Conside a
1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physial insight in the sound geneation mehanism an be gained by onsideing simple analytial solutions to the wave equation One example is to onside aousti adiation
Influence of Hydraulic Hysteresis on Effective Stress in Unsaturated Clay
Intentionl Jounl of Envionmentl nd Eth Sciences 1:1 21 Influence of Hydulic Hysteesis on Effective Stess in Unstuted Cly Anuchit Uchipicht Abstct A compehensive pogm of lbotoy testing on compcted kolin
AP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.
AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function
Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
Fluids & Bernoulli s Equation. Group Problems 9
Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going
A Relativistic BCS Theory of S-Wave Superconductivity
A Relativisti BCS Theoy of S-Wave Supeondutivity Damien BERTRAND Jan GOVAERTS John MENDY UC-FYNU / Supeondutivity Goup ouvain-la-neuve - June 4 Outline Ginzbug-andau fundamentals Oiginal theoy Covaiant
Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system
436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique
OBSTACLE DETECTION USING RING BEAM SYSTEM
OBSTACLE DETECTION USING RING BEAM SYSTEM M. Hiaki, K. Takamasu and S. Ozono Depatment of Peision Engineeing, The Univesity of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan Abstat: In this pape, we popose
Mark Scheme (Results) January 2008
Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question
In the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information
m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque
( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x
SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.
On the Eötvös effect
On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This tile ppeed in jounl pulished y Elsevie. The tthed opy is funished to the utho fo intenl non-ommeil eseh nd edution use, inluding fo instution t the uthos institution nd shing with ollegues. Othe uses,
Mass Transfer (Stoffaustausch)
Mass Tansfe (Stoffaustaush) Examination 3. August 3 Name: Legi-N.: Edition Diffusion by E. L. Cussle: none nd 3 d Test Duation: minutes The following mateials ae not pemitted at you table and have to be
Lecture 10. Solution of Nonlinear Equations - II
Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution
Algebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016
Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw
ME306 Dynamics, Spring HW1 Solution Key. AB, where θ is the angle between the vectors A and B, the proof
ME6 Dnms, Spng HW Slutn Ke - Pve, gemetll.e. usng wngs sethes n nltll.e. usng equtns n nequltes, tht V then V. Nte: qunttes n l tpee e vets n n egul tpee e sls. Slutn: Let, Then V V V We wnt t pve tht:
Supplementary Information. Origin of Chains of Au-PbS Nano-Dumbbells in. Space
Supplementy Infomtion Oigin of Chins of Au-PbS Nno-Dumbbells in Spe Chndn Mondl, Ali Hossin Khn, Bidis Ds, Somobt Ahy* & Sujit Sengupt*, Cente fo Advned Mteils, Indin Assoition fo the Cultivtion of Siene,
ELECTRON ENERGY DISTRIBUTIONS AND NON-COLLISIONAL HEATING IN MAGNETICALLY ENHANCED INDUCTIVELY COUPLED PLASMAS*
ELECTRON ENERGY DISTRIUTIONS AND NON-COLLISIONAL HEATING IN MAGNETICALLY ENHANCED INDUCTIVELY COUPLED PLASMAS* Ronald L. Kinde and Mak J. Kushne Depatment of Electical and Compute Engineeing Ubana, IL
10.3 The Quadratic Formula
. Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti
π,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
Exercise 4: Adimensional form and Rankine vortex. Example 1: adimensional form of governing equations
Fluid Mechanics, SG4, HT9 Septembe, 9 Execise 4: Adimensional fom and Rankine votex Example : adimensional fom of govening equations Calculating the two-dimensional flow aound a cylinde (adius a, located
CHAPTER (6) Biot-Savart law Ampere s Circuital Law Magnetic Field Density Magnetic Flux
CAPTE 6 Biot-Svt w Ampee s Ciuit w Mgneti Fied Densit Mgneti Fu Soues of mgneti fied: - Pemnent mgnet - Fow of uent in ondutos -Time ving of eeti fied induing mgneti fied Cuent onfigutions: - Fiment uent
MATH Homework #1 Solution - by Ben Ong
MATH 46 - Homewok #1 Solution - by Ben Ong Deivation of the Eule Equations We pesent a fist pinciples deivation of the Eule Equations fo two-dimensional fluid flow in thee-dimensional cylndical cooodinates
Chapter Introduction to Partial Differential Equations
hpte 10.01 Intodtion to Ptil Diffeentil Eqtions Afte eding this hpte o shold be ble to: 1. identif the diffeene between odin nd ptil diffeentil eqtions.. identif diffeent tpes of ptil diffeentil eqtions.
An o5en- confusing point:
An o5en- confusing point: Recall this example fom last lectue: E due to a unifom spheical suface chage, density = σ. Let s calculate the pessue on the suface. Due to the epulsive foces, thee is an outwad
Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
Foundations of Chemical Kinetics. Lecture 9: Generalizing collision theory
Foundations of Chemical Kinetics Lectue 9: Genealizing collision theoy Mac R. Roussel Depatment of Chemisty and Biochemisty Spheical pola coodinates z θ φ y x Angles and solid angles θ a A Ω θ=a/ unit:
Friedmannien equations
..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions
Reference. Reference. Properties of Equality. Properties of Segment and Angle Congruence. Other Properties. Triangle Inequalities
Refeene opeties opeties of qulity ddition opety of qulity If =, ten + = +. Multiplition opety of qulity If =, ten =, 0. Reflexive opety of qulity = Tnsitive opety of qulity If = nd =, ten =. Suttion opety
Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 11. Chapter 3, CHEM 793, 2011 Fall, L. Ma
Cpte 3 Bsic Cystopy nd Eecton Diffction fom Cysts Lectue Pof. Sectmn: Nobe impossibe witout micoscope Isei ecipient of 0 cemisty Nobe Pize sys oundbein discoey of 'qusicysts' woud e been deyed fo yes witout
The Divergence Theorem
13.8 The ivegence Theoem Back in 13.5 we ewote Geen s Theoem in vecto fom as C F n ds= div F x, y da ( ) whee C is the positively-oiented bounday cuve of the plane egion (in the xy-plane). Notice this
This immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
Received 2 August 2014; revised 2 September 2014; accepted 10 September 2014
Ameicn Jounl of Computtionl Mthemtics, 4, 4, 357-365 Published Online eptembe 4 in cires. http://www.scip.og/jounl/jcm http://dx.doi.og/.436/jcm.4.443 Effect of Vible Viscosity, Dufou, oet nd Theml Conductivity
EECE 260 Electrical Circuits Prof. Mark Fowler
EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution
Forging Analysis - 2. ver. 1. Prof. Ramesh Singh, Notes by Dr. Singh/ Dr. Colton
Foging Analysis - ve. 1 Pof. ames Sing, Notes by D. Sing/ D. Colton 1 Slab analysis fictionless wit fiction ectangula Cylindical Oveview Stain adening and ate effects Flas edundant wo Pof. ames Sing, Notes
Lecture 13. Rotational motion Moment of inertia
Lectue 13 Rotational motion Moment of inetia EXAM 2 Tuesday Mach 6, 2018 8:15 PM 9:45 PM Today s Topics: Rotational Motion and Angula Displacement Angula Velocity and Acceleation Rotational Kinematics
8.022 (E&M) Lecture 13. What we learned about magnetism so far
8.0 (E&M) Letue 13 Topis: B s ole in Mawell s equations Veto potential Biot-Savat law and its appliations What we leaned about magnetism so fa Magneti Field B Epeiments: uents in s geneate foes on hages
mslumped-parameter (zero-dimensions!) groundwater model of Bangladesh
mslumed-pmete (zeo-dimensions!) goundwte model of Bngldesh You gol in this oblem set is to develo bucket model fo the hydology of ou study site in Bngldesh. Using this model, you will investigte how the
Chapter 21: Electric Charge and Electric Field
Chpte 1: Electic Chge nd Electic Field Electic Chge Ancient Gees ~ 600 BC Sttic electicit: electic chge vi fiction (see lso fig 1.1) (Attempted) pith bll demonsttion: inds of popeties objects with sme
ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:
LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6
A NOTE ON THE POCHHAMMER FREQUENCY EQUATION
A note on the Pohhmme feqeny eqtion SCIENCE AND TECHNOLOGY - Reseh Jonl - Volme 6 - Univesity of Mitis Rédit Mitis. A NOTE ON THE POCHHAMMER FREQUENCY EQUATION by F.R. GOLAM HOSSEN Deptment of Mthemtis
Answers to test yourself questions
Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E
Chapter 3 Basic Crystallography and Electron Diffraction from Crystals. Lecture 9. CHEM 793, 2008 Fall
Cpte 3 Bsic Cystopy nd Eecton Diffction fom Cysts Lectue 9 Top of tin foi Cyst pne () Bottom of tin foi B Lw d sinθ n Equtions connectin te Cyst metes (,, ) nd d-spcin wit bem pmetes () ( ) ne B Lw d (nm)
General momentum equation
PY4A4 Senio Sophiste Physics of the Intestella and Integalactic Medium Lectue 11: Collapsing Clouds D Gaham M. Hape School of Physics, TCD Geneal momentum equation Du u P Dt uu t 1 B 4 B 1 B 8 Lagangian
PROCESSING QUALITY OF NEW POTATO CULTIVARS FOLLOWING PROLONGED STORAGE
PRESSIG QUALITY F EW PTAT ULTIVARS FLLWIG PRLGED STRAGE Glenn E. Vogt J.R. Simplot o., aldwell, ID and Ei P. Eldedge and linton. Shok Malheu Expeiment Station egon State Univesity ntaio, R Intodution The
763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012
763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012 1. Continuous Random Walk Conside a continuous one-dimensional andom walk. Let w(s i ds i be the pobability that the length of the i th displacement
A L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
Real Asymmetric Matrix Eigenvalue Analysis
Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Rel Asymmetri Mtrix Eigenvlue Anlysis Heewoo Lee omputtionl Menis Lbortory Deprtment of Menil Engineering n Applie
Numerical Modeling in Biomedical Systems
Numeial Modeling in Biomedial Systems BME 15:35 Letue 7 9/6/17 Nonlinea Systems Dunn Chapte 5 Nonlinea equations Root finding Baketing methods Open methods Gaphial Bisetion False Position Newton s method
ITI Introduction to Computing II
ITI 1121. Intoduction to Computing II Mcel Tucotte School of Electicl Engineeing nd Compute Science Abstct dt type: Stck Stck-bsed lgoithms Vesion of Febuy 2, 2013 Abstct These lectue notes e ment to be
Analysis of Variance for Multiple Factors
Multiple Fto ANOVA Notes Pge wo Fto Anlsis Anlsis of Vine fo Multiple Ftos Conside two ftos (tetments) A nd B with A done t levels nd B done t levels. Within given tetment omintion of A nd B levels, leled
STATICS. CENTROIDS OF MASSES, AREAS, LENGTHS, AND VOLUMES The following formulas are for discrete masses, areas, lengths, and volumes: r c
STTS FORE foe is veto qutit. t is defied we its () mgitude, () oit of litio, d () dietio e kow. Te veto fom of foe is F F i F j RESULTNT (TWO DMENSONS) Te esultt, F, of foes wit omoets F,i d F,i s te mgitude
APPENDIX D COMPRESSIBILITY FACTOR EQUATIONS D.1 THE REDLICH KWONG EQUATION
AENDIX D COMRESSIBILIY FACOR EQUAIONS D.1 HE REDLICH KWONG EQUAION he Redlih-Kwong equation is atually an equation of state. It was fomulated by Otto Redlih and Joseph N. S. Kwong in 1949 [Chemial Review
Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f
C H A P T E R I G E N E S I S A N D GROWTH OF G U IL D S C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f i n a v a r i e t y o f f o r m s - s o c i a l, r e l i g i
A 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions
EN10: Continuum Mechanics Homewok 7: Fluid Mechanics Solutions School of Engineeing Bown Univesity 1. An ideal fluid with mass density ρ flows with velocity v 0 though a cylindical tube with cosssectional
Physics 217 Practice Final Exam: Solutions
Physis 17 Ptie Finl Em: Solutions Fll This ws the Physis 17 finl em in Fll 199 Twenty-thee students took the em The vege soe ws 11 out of 15 (731%), nd the stndd devition 9 The high nd low soes wee 145
Problem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by
Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,
Pulse Neutron Neutron (PNN) tool logging for porosity Some theoretical aspects
Pulse Neuton Neuton (PNN) tool logging fo poosity Some theoetical aspects Intoduction Pehaps the most citicism of Pulse Neuton Neuon (PNN) logging methods has been chage that PNN is to sensitive to the
Lecture 23. Representation of the Dirac delta function in other coordinate systems
Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.
Influence of the Magnetic Field in the Solar Interior on the Differential Rotation
Influene of the gneti Fiel in the Sol Inteio on the Diffeentil ottion Lin-Sen Li * Deptment of Physis Nothest Noml Univesity Chnghun Chin * Coesponing utho: Lin-Sen Li Deptment of Physis Nothest Noml Univesity
Supplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in
Supplementay Figue 1. Cicula paallel lamellae gain size as a function of annealing time at 50 C. Eo bas epesent the σ uncetainty in the measued adii based on image pixilation and analysis uncetainty contibutions
NXO a Spatially High Order Finite Volume Numerical Method for Compressible Flows
NXO Spty Hgh Ode Fnte Voume Nume Method fo Compebe Fow Jen-Me e Gouez One CFD Deptment Fouth HO CFD Wohop Heon June 4th B of the NXO heme Eue o Nve-Stoe fo pefet g w of tte dof pe e nd pe equton Voume
3-7 FLUIDS IN RIGID-BODY MOTION
3-7 FLUIDS IN IGID-BODY MOTION S-1 3-7 FLUIDS IN IGID-BODY MOTION We ae almost eady to bein studyin fluids in motion (statin in Chapte 4), but fist thee is one cateoy of fluid motion that can be studied
Continuous Charge Distributions
Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi
AY 7A - Fall 2010 Section Worksheet 2 - Solutions Energy and Kepler s Law
AY 7A - Fall 00 Section Woksheet - Solutions Enegy and Keple s Law. Escape Velocity (a) A planet is obiting aound a sta. What is the total obital enegy of the planet? (i.e. Total Enegy = Potential Enegy
GEOG 402 SURFACE ENERGY FLUXES
GEOG 402 SURFACE ENERGY FLUXES Some Point of Confuion Net Rdition Incident t the Sufce R n = K ( 1α + ( 4 εl εσ K L Veg Lye αk εl 4 εσ Soil Some Point of Confuion Veticl empetue Pofile Some Point of Confuion
Physics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
10. Groundwater in geotechnical problems
. Goundwate in geotechnical poblems Goundwate plays a key ole in many geotechnical poblems. We will look at; - land subsidence - dewateing open pits - Consolidation of sediments Remembe the stoage of wate
Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations
Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces
Mathematical Reflections, Issue 5, INEQUALITIES ON RATIOS OF RADII OF TANGENT CIRCLES. Y.N. Aliyev
themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev stt Some inequlities involving tios of dii of intenll tngent iles whih inteset the given line in fied points e studied
Water Tunnel Experiment MAE 171A/175A. Objective:
Wate Tunnel Expeiment MAE 7A/75A Objective: Measuement of te Dag Coefficient of a Cylinde Measuement Tecniques Pessue Distibution on Cylinde Dag fom Momentum Loss Measued in Wake it lase Dopple Velocimety
PHYS 172: Modern Mechanics. Fall Lecture 6 Fundamental Forces, Reciprocity Read
PHYS 172: Moden Mechanics Fall 2009 Lectue 6 Fundamental Foces, Recipocity Read 3.9 3.14 Exam 1 - Tuesday Septembe 15, 8:00-10:00 PM Elliott Hall of Music 1. The exam will be witten as a standad 1 hou
General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.
Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i
Fracture Mechanics: Linear Elastic Fracture Mechanics 2/2
Task 6 - Safety Review and Licensing On the Job Taining on Stess Analysis Factue Mechanics: Linea Elastic Factue Mechanics / Davide Mazzini Cio Santus Pisa (taly) June 15 July 14, 015 Table of content
SOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz
STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (
MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
Current, Resistance and
Cuent, Resistance and Electomotive Foce Chapte 25 Octobe 2, 2012 Octobe 2, 2012 Physics 208 1 Leaning Goals The meaning of electic cuent, and how chages move in a conducto. What is meant by esistivity
JURONG JUNIOR COLLEGE Physics Department Tutorial: Electric Fields (solutions)
JJ 5 H Physics (646) Electic Fields_tutsoln JURONG JUNIOR COLLEGE Physics Depatment Tutoial: Electic Fields (solutions) No Solution LO Electic field stength at a point in an electic field is defined as
CHAPTER 25 ELECTRIC POTENTIAL
CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When
Two dimensional polar coordinate system in airy stress functions
I J C T A, 9(9), 6, pp. 433-44 Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in
1D2G - Numerical solution of the neutron diffusion equation
DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3
DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl
WELL POSEDNESS OF PROBLEMS I
Finite Element Method 85 WELL POSEDNESS OF PROBLEMS I Consider the following generic problem Lu = f, where L : X Y, u X, f Y and X, Y are two Banach spaces We say that the above problem is well-posed (according