Numerical methods for PDEs FEM - abstract formulation, the Galerkin method
|
|
- Valerie Shields
- 5 years ago
- Views:
Transcription
1 Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Numerica methods for PDEs FEM - abstract formuation, the Gaerkin method Dr. Noemi Friedman
2 Contents of the course Fundamentas of functiona anaysis Abstract formuation FEM Appication to conrete formuations Convergence, reguarity Variationa crimes Impementation Mixed formuations (e.g. Stokes) Error indicators/estimation Adaptivity Gaerkin method Dr. Noemi Friedman PDE2 tutoria Seite 2
3 Abstract formuation, exampes Introduction From strong form to weak form Weak form of BVPs with inhomogenous BCs Best approximation by orthogona projection Orthogona projection Gaerkin method, minimized error for symmetric BVPs Bound of error of Gaerkin method (Céa s theorem) Gaerkin method Dr. Noemi Friedman PDE2 tutoria Seite 3
4 Introduction to Gaerkin method Matab code from Gockenbach book: Equations foowed throughout the semester [Chapter1]: Lapace/Poisson equation: bar with uniaxia oad steady state heat fow sma vertica defections of a membrane stationary heat equation Other eptic BVPs: isotropic easticity Gaerkin method Dr. Noemi Friedman PDE 2 Seite 4
5 FROM STRONG FORM TO WEAK FORM 1D: Steps of formuating the weak form (recipe) [Chapter 2] Lu(x) = f(x) 1.) Mutipy by test/weight function v(x) and integrate Lu, v p, v = v Lu x v x dx f x v x dx = v Check wether the PDE hods in the v weighted avarage sense over 2.) Reduce order of Lu, v by using the integration by parts 3.) Appy boundary conditions if it hods for a test functions from a sufficienty arge set, then the PDE must hod [Gockenbach] Dr. Noemi Friedman PDE 2 Seite 5
6 1D Exampe: bar under axia oads with inhomogenious Neumann BC. [On inhomogenous BCs see Chapter 2.2] I. Formuate the strong equation 1.) kinematica equation: (strain dispacement) p(x) t 2.) equiibrium equation: (forces in equiibrium) 3.) constitutive equation: (stress strain) 4.) Boundary conditions: u = Dirichet boundary u = Aσ = t Neumann boundary EA du /dx = t x = [, ] Dr. Noemi Friedman PDE 2 Seite 6
7 1D Exampe: bar under axia oads with inhomogenious Neumann BC. inear term f(v) : p, v = Lu, v = biinear term a(u, v): p x v x dx Lu, v p, v = v H 1 Integration by parts: EA d2 u(x) dx 2 v(x)dx a a = EA du b = ab ab b = dv dx dx Lu, v = EA du dx v Lu, v p, v = EA du dv dx dx dx = tv() + EA du dv dx dx dx v = EA du dv dx dx dx tv p x v x dx = EA du dx x= = t EA du dv dx dx dx = p x v x dx + tv() (principe of virtua work) Dr. Noemi Friedman PDE 2 Seite 7
8 Mutidimensiona stationary heat equation with inhomogeneous Neumann BC. Strong form: Exampe: Lu(x) = f(x) Δu x = f u = u n = h 1.) Mutipy by test function v and integrate Δu x v x d f x v x d = 2.) Reduce biinear term s order by using Green s identity a u, v = Lu, v : Δu x v x d = u x v x d u v x dγ n. Dr. Noemi Friedman PDE 2 Seite 8
9 Mutidimensiona stationary heat equation with inhomogeneous Neumann BC. Δu x v x d = u x v x d u v x dγ n 3.) Appy boundary conditions u n v x dγ = u Γ N n v x dγ + u Γ D n v x dγ = hv x dγ Γ N h Δu x v x d = u x v x d hv x dγ Γ N u x v x d = f x v x d + Γ N hv x dγ. Dr. Noemi Friedman PDE 2 Seite 9
10 Mutidimensiona stationary heat equation with inhomogeneous Dirichet and Neumann BC. Strong form: Exampe: Lu(x) = f(x) Δu x = f u = g u n = h 1.) Mutipy by test function v and integrate convert to homogeneous probem: u = ω + u ω:known function,ω = g on Γ D u:new function that we ook for Δu x v x d f x v x d = 2.) Reduce order of Lu, v by using divergence theoreem Δu x v x d = u x v x d u v x dγ n. Dr. Noemi Friedman PDE 2 Seite 1
11 Mutidimensiona stationary heat equation with inhomogeneous Dirichet and Neumann BC. Δu x v x d = u x v x d u v x dγ n 3.) Appy boundary conditions u n v x dγ = u Γ N n v x dγ + u Γ D n v x dγ = hv x dγ Γ N h u x v x d = f x v x d + Γ N hv x dγ ω x + u x v x d = f x v x d + Γ N hv x dγ u x v x d = f x v x d + hv x dγ Γ N ω x v x d from natura/neumann BC from essentia/dirichet BC Gaerkin method Dr. Noemi Friedman PDE 2 Seite 11
12 Existence and uniqueness of the soution of BVPs [Chapter 2.4] Strong form: Weak form: Lu(x) = f(x) Lu, v = f, v v? u? a(u, v) F(v) biinear term inear term Does the soution exists? Does it have a unique soution? In accordance to the Lax-Migram Lemma if: F( ) bounded, inear functiona a, bounded, V-eiptic biinear functiona and V a Hibert space For a specific BVP one has to find the right Hibert space (Soboev space or L2 space) where the conditions for a, and F( ) are satisfied, and then we know, that in that space we have a unique soution! Gaerkin method Dr. Noemi Friedman PDE 2 Seite 12 soution u exists unique soution u V +soution u depends continiousy on f
13 Existence and uniqueness of the soution of BVPs exampes 1. Poisson equation: Δu x = f x F( ) inear functiona, in H 1 : bounded a, biinear functiona, in H : bounded, V-eiptic 1 a, biinear functiona, in L 2 : not bounded a, biinear functiona, in H 1 : not V-eiptic 2. Pate equation F( ) inear functiona, in H E 2 : bounded a, ΔΔu x = f x biinear functiona, in H E : bounded, V-eiptic 2 F(v) = f x v x dx a, = u x v x dx F(v) = f x v x dx Gaerkin method Dr. Noemi Friedman PDE 2 Seite 13 unique soution u H 1 v H 1 a, = Δu x Δv x dx we have to narrow down the space in which we ook for the soution, because we can not prove that there is a unique soution in L2 or in H1 unique soution u H E 2 v H E 2
14 Proxi mode projection of the soution [Chapter 3] Let s say we know that there is a unique soution in V = H 1. But V is an infinite dimensiona space Let s narrow down the space, where we are trying to find the soution, to a finite dimensiona space exampe: instead of finding u(x) H 1 we try to find the coefficients α j of a proxi mode (ansatz function): u h x = j=1 Gaerkin method Dr. Noemi Friedman PDE 2 Seite 14 n α j ω j (x) where ω j x : known (ineary independent) basis or ansatz functions u h x : the approximation of the soution u(x), which is in an n-dimensiona space: u h V h = span{ω 1, ω 2, ω n }
15 Proxi mode projection of the soution Let s fix this subspace to a specific V h (that is, we fix the ansatz functions in our exampes) How do we get the best approximation of the soution in this space from the equation: Our goa is to minimize the difference in between the soution and the approximation: error = u x u h x < u x z x z x V h The best approximation u h to u from V h is the one where the error is orthogona to the space of V h, that is to a possibe z x V h. Instead of writing it for a z (as z is an n-dimensiona space) we can write ω i i = 1.. n u x u h x, ω i x = i = 1.. n Lu(x) = f(x) Best approximation to u x from V h Gaerkin method Dr. Noemi Friedman PDE 2 Seite 15
16 Proxi mode projection of the soution Pugging in the proxi mode to the orthogonaity condition we have: u n j=1 α j ω j, ω i = i = 1.. n Rearranging the equation we get: u, ω i n α j ω j, ω i = i = 1.. n j=1 n j=1 α j ω j, ω i = u, ω i i = 1.. n G ij b j n j=1 α j G ij = b j Gα = b Gaerkin method Dr. Noemi Friedman PDE 2 Seite 16
17 Proxi mode projection of the soution Gα = b where G ij = ω j, ω i : can be cacuated from the basis functions and the inner product of the given space (if the basis is orthonorma the matrix is the identity matrix b j = u, ω i :? α j : The coefficients that we are ooking for But how do we get b j = u, ω i? We know Lu, v L2 = f, v L2 If the eft hand side of the weak equation is a V-eiptic, bounded biinear functiona, that is aso symmetric, then it can be written as: Lu, v L2 = a(u, v) = u, v E Gaerkin method Dr. Noemi Friedman PDE 2 Seite 17
18 Proxi mode projection of the soution That means that with Gaerkin method we orthogonaize the projection, that is, we minimize the error in the energy space. And the matrix equation, that we can cacuate the coefficients from, wi have the form: Gα = b with G ij = ω j, ω i E = a(ω j, ω i ) b j = u, ω i E = f, ω i L2 Exampe: x =,, u =, u = Lu = EA 2 u(x) 2 x G ij = ω j (x), ω i (x) E = f = p EA dω j(x) dx dω i (x) dx dx b j = f, ω i L2 = p x ω i x dx Gaerkin method Dr. Noemi Friedman PDE 2 Seite 18
19 Gaerkin method is an orthogona projection The soution of the BVP satisfies the initia weak formuation: a u, v = F v v V and as V h V, it aso satisfies a u, v = F v v V h Simiary, the approximation of the soution u h satisfies: a u h, v = F v v V h Subtracting the ast two equations : a u, v a u h, v = v V h a u u h, v = v V h Gaerkin method gives the best approximation of the true soution in the given subspace V h in the energy norm In case a(u, u) is a symmetric biinear term Gaerkin method Dr. Noemi Friedman PDE 2 Seite 19
20 Céa s theorem Concusion from before: a, : symmetric Gaerkin method gives the best approximation in the energy norm But what about the error in other norms? According to Céa s theorem (see prove at the ecture), even without a, being symmetric, the error of the approximation of Gaerkin wi be aways bounded: u u h M δ u v v V h Where M and δ are constants from the conditions of boundedness and V-eipticity of the biinear term a, : a u, v M u v a u, u δ u 2 and u v is the norm of the difference in between the true soution and any v V h This term depends on the n-dimensiona space V h chosen, and the space where the true soution ies in. Gaerkin method Dr. Noemi Friedman PDE 2 Seite 2
Introduction to PDEs and Numerical Methods Tutorial 10. Finite Element Analysis
Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Introduction to PDEs and Numerica Methods Tutoria. Finite Eement Anaysis Dr. Noemi Friedman, 3..2 FROM STRONG FORM TO WEAK FORM inhomogeneous
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationLecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationMA 201: Partial Differential Equations Lecture - 10
MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationMultigrid Method for Elliptic Control Problems
J OHANNES KEPLER UNIVERSITÄT LINZ Netzwerk f ür Forschung, L ehre und Praxis Mutigrid Method for Eiptic Contro Probems MASTERARBEIT zur Erangung des akademischen Grades MASTER OF SCIENCE in der Studienrichtung
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationMA 201: Partial Differential Equations Lecture - 11
MA 201: Partia Differentia Equations Lecture - 11 Heat Equation Heat conduction in a thin rod The IBVP under consideration consists of: The governing equation: u t = αu xx, (1) where α is the therma diffusivity.
More informationWavelet Galerkin Solution for Boundary Value Problems
Internationa Journa of Engineering Research and Deveopment e-issn: 2278-67X, p-issn: 2278-8X, www.ijerd.com Voume, Issue 5 (May 24), PP.2-3 Waveet Gaerkin Soution for Boundary Vaue Probems D. Pate, M.K.
More informationModel Solutions (week 4)
CIV-E16 (17) Engineering Computation and Simuation 1 Home Exercise 6.3 Mode Soutions (week 4) Construct the inear Lagrange basis functions (noda vaues as degrees of freedom) of the ine segment reference
More informationIntroduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis Dr. Noemi Friedman, 25.10.201.
More informationCABLE SUPPORTED STRUCTURES
CABLE SUPPORTED STRUCTURES STATIC AND DYNAMIC ANALYSIS OF CABLES 3/22/2005 Prof. dr Stanko Brcic 1 Cabe Supported Structures Suspension bridges Cabe-Stayed Bridges Masts Roof structures etc 3/22/2005 Prof.
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationChapter 5. Wave equation. 5.1 Physical derivation
Chapter 5 Wave equation In this chapter, we discuss the wave equation u tt a 2 u = f, (5.1) where a > is a constant. We wi discover that soutions of the wave equation behave in a different way comparing
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationHILBERT? What is HILBERT? Matlab Implementation of Adaptive 2D BEM. Dirk Praetorius. Features of HILBERT
Söerhaus-Workshop 2009 October 16, 2009 What is HILBERT? HILBERT Matab Impementation of Adaptive 2D BEM joint work with M. Aurada, M. Ebner, S. Ferraz-Leite, P. Godenits, M. Karkuik, M. Mayr Hibert Is
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationVibrations of Structures
Vibrations of Structures Modue I: Vibrations of Strings and Bars Lesson : The Initia Vaue Probem Contents:. Introduction. Moda Expansion Theorem 3. Initia Vaue Probem: Exampes 4. Lapace Transform Method
More informationNumerical methods for elliptic partial differential equations Arnold Reusken
Numerica methods for eiptic partia differentia equations Arnod Reusken Preface This is a book on the numerica approximation of partia differentia equations. On the next page we give an overview of the
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationСРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS FOR BEAMS
СРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА Милко Стоянов Милошев 1, Константин Савков Казаков 2 Висше Строително Училище Л. Каравелов - София COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS
More information3.10 Implications of Redundancy
118 IB Structures 2008-9 3.10 Impications of Redundancy An important aspect of redundant structures is that it is possibe to have interna forces within the structure, with no externa oading being appied.
More informationUI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE
UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641,
More informationWork and energy method. Exercise 1 : Beam with a couple. Exercise 1 : Non-linear loaddisplacement. Exercise 2 : Horizontally loaded frame
Work and energy method EI EI T x-axis Exercise 1 : Beam with a coupe Determine the rotation at the right support of the construction dispayed on the right, caused by the coupe T using Castigiano s nd theorem.
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationVTU-NPTEL-NMEICT Project
MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Progress Report The Project on Deveopment of Remaining Three Quadrants to NPTE Phase-I under grant in aid
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More informationNonlinear Analysis of Spatial Trusses
Noninear Anaysis of Spatia Trusses João Barrigó October 14 Abstract The present work addresses the noninear behavior of space trusses A formuation for geometrica noninear anaysis is presented, which incudes
More informationNumerical solution of one dimensional contaminant transport equation with variable coefficient (temporal) by using Haar wavelet
Goba Journa of Pure and Appied Mathematics. ISSN 973-1768 Voume 1, Number (16), pp. 183-19 Research India Pubications http://www.ripubication.com Numerica soution of one dimensiona contaminant transport
More informationM. Aurada 1,M.Feischl 1, J. Kemetmüller 1,M.Page 1 and D. Praetorius 1
ESAIM: M2AN 47 (2013) 1207 1235 DOI: 10.1051/m2an/2013069 ESAIM: Mathematica Modeing and Numerica Anaysis www.esaim-m2an.org EACH H 1/2 STABLE PROJECTION YIELDS CONVERGENCE AND QUASI OPTIMALITY OF ADAPTIVE
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationPlatzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 6: Numerical solution of the heat equation with FD method: method of lines, Euler
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationPlatzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen PARAMetric UNCertainties, Budapest STOCHASTIC PROCESSES AND FIELDS Noémi Friedman Institut für Wissenschaftliches Rechnen, wire@tu-bs.de
More informationMeshfree Particle Methods for Thin Plates
Meshfree Partice Methods for Thin Pates Hae-Soo Oh, Christopher Davis Department of Mathematics and Statistics, University of North Caroina at Charotte, Charotte, NC 28223 Jae Woo Jeong Department of Mathematics,
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More informationWMS. MA250 Introduction to Partial Differential Equations. Revision Guide. Written by Matthew Hutton and David McCormick
WMS MA25 Introduction to Partia Differentia Equations Revision Guide Written by Matthew Hutton and David McCormick WMS ii MA25 Introduction to Partia Differentia Equations Contents Introduction 1 1 First-Order
More informationFurther analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
Further anaysis of mutieve Monte Caro methods for eiptic PDEs with random coefficients A. L. Teckentrup, R. Scheich, M. B. Gies, and E. Umann Abstract We consider the appication of mutieve Monte Caro methods
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
More informationIntroduction to PDEs and Numerical Methods Lecture 1: Introduction
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 1: Introduction Dr. Noemi Friedman, 28.10.2015. Basic information on the course Course
More informationc 1999 Society for Industrial and Applied Mathematics
SIAM J. NUMER. ANAL. Vo. 36, No. 2, pp. 58 606 c 999 Society for Industria and Appied Mathematics OVERLAPPING NONMATCHING GRID MORTAR ELEMENT METHODS FOR ELLIPTIC PROBLEMS XIAO-CHUAN CAI, MAKSYMILIAN DRYJA,
More informationPublished in: Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics
Aaborg Universitet An Efficient Formuation of the Easto-pastic Constitutive Matrix on Yied Surface Corners Causen, Johan Christian; Andersen, Lars Vabbersgaard; Damkide, Lars Pubished in: Proceedings of
More informationIntroduction to PDEs and Numerical Methods Lecture 7. Solving linear systems
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 7. Solving linear systems Dr. Noemi Friedman, 09.2.205. Reminder: Instationary heat
More information1 Heat Equation Dirichlet Boundary Conditions
Chapter 3 Heat Exampes in Rectanges Heat Equation Dirichet Boundary Conditions u t (x, t) = ku xx (x, t), < x (.) u(, t) =, u(, t) = u(x, ) = f(x). Separate Variabes Look for simpe soutions in the
More informationOn a geometrical approach in contact mechanics
Institut für Mechanik On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof Universität Karsruhe, Institut für Mechanik Institut für Mechanik Kaiserstr. 12, Geb. 20.30 76128
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationAn approximate method for solving the inverse scattering problem with fixed-energy data
J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999
More informationA very short introduction to the Finite Element Method
A very short introduction to the Finite Element Method Till Mathis Wagner Technical University of Munich JASS 2004, St Petersburg May 4, 2004 1 Introduction This is a short introduction to the finite element
More information4 NON-LINEAR ANALYSIS
4 NON-INEAR ANAYSIS arge displacement elasticity theory, principle of virtual work arge displacement FEA with solid, thin slab, and bar models Virtual work density of internal forces revisited 4-1 SOURCES
More informationA Fictitious Time Integration Method for a One-Dimensional Hyperbolic Boundary Value Problem
Journa o mathematics and computer science 14 (15) 87-96 A Fictitious ime Integration Method or a One-Dimensiona Hyperboic Boundary Vaue Probem Mir Saad Hashemi 1,*, Maryam Sariri 1 1 Department o Mathematics,
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationHomework #04 Answers and Hints (MATH4052 Partial Differential Equations)
Homework #4 Answers and Hints (MATH452 Partia Differentia Equations) Probem 1 (Page 89, Q2) Consider a meta rod ( < x < ), insuated aong its sides but not at its ends, which is initiay at temperature =
More informationGeneral Decay of Solutions in a Viscoelastic Equation with Nonlinear Localized Damping
Journa of Mathematica Research with Appications Jan.,, Vo. 3, No., pp. 53 6 DOI:.377/j.issn:95-65...7 Http://jmre.dut.edu.cn Genera Decay of Soutions in a Viscoeastic Equation with Noninear Locaized Damping
More informationC1B Stress Analysis Lecture 1: Minimum Energy Principles in Mechanics Prof Alexander M. Korsunsky Hilary Term (January 08)
CB Stress Anaysis ecture : Minimum Energy Principes in Mechanics Prof Aexander M. Korsunsky Hiary Term (January 8) http://users.ox.ac.uk/~engs6/4me6.htm This course introduces seected chapters in Stress
More informationReliability: Theory & Applications No.3, September 2006
REDUNDANCY AND RENEWAL OF SERVERS IN OPENED QUEUING NETWORKS G. Sh. Tsitsiashvii M.A. Osipova Vadivosto, Russia 1 An opened queuing networ with a redundancy and a renewa of servers is considered. To cacuate
More information1 Equations of Motion 3: Equivalent System Method
8 Mechanica Vibrations Equations of Motion : Equivaent System Method In systems in which masses are joined by rigid ins, evers, or gears and in some distributed systems, various springs, dampers, and masses
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationIntroduction to PDEs and Numerical Methods Tutorial 5. Finite difference methods equilibrium equation and iterative solvers
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 5. Finite difference methods equilibrium equation and iterative solvers Dr. Noemi
More informationVibrations of beams with a variable cross-section fixed on rotational rigid disks
1(13) 39 57 Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks Abstract The work is focused on the probem of vibrating beams with a variabe cross-section fixed on a rotationa
More informationA nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency
INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Probems 17 (001) 39 359 www.iop.org/journas/ip PII: S066-5611(01)14169-9 A noninear mutigrid for imaging eectrica conductivity and permittivity
More informationNumerische Mathematik
Numer. Math. (1999) 84: 97 119 Digita Object Identifier (DOI) 10.1007/s002119900096 Numerische Mathematik c Springer-Verag 1999 Mutigrid methods for a parameter dependent probem in prima variabes Joachim
More informationBourgain s Theorem. Computational and Metric Geometry. Instructor: Yury Makarychev. d(s 1, s 2 ).
Bourgain s Theorem Computationa and Metric Geometry Instructor: Yury Makarychev 1 Notation Given a metric space (X, d) and S X, the distance from x X to S equas d(x, S) = inf d(x, s). s S The distance
More informationFourier series. Part - A
Fourier series Part - A 1.Define Dirichet s conditions Ans: A function defined in c x c + can be expanded as an infinite trigonometric series of the form a + a n cos nx n 1 + b n sin nx, provided i) f
More informationHomotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order
Int Journa of Math Anaysis, Vo 6, 2012, no 49, 2431-2448 Homotopy Perturbation Method for Soving Partia Differentia Equations of Fractiona Order A A Hemeda Department of Mathematics, Facuty of Science
More informationVolume 13, MAIN ARTICLES
Voume 13, 2009 1 MAIN ARTICLES THE BASIC BVPs OF THE THEORY OF ELASTIC BINARY MIXTURES FOR A HALF-PLANE WITH CURVILINEAR CUTS Bitsadze L. I. Vekua Institute of Appied Mathematics of Iv. Javakhishvii Tbiisi
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationDYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE
3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses
More informationCE601-Structura Anaysis I UNIT-IV SOPE-DEFECTION METHOD 1. What are the assumptions made in sope-defection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More information830. Nonlinear dynamic characteristics of SMA simply supported beam in axial stochastic excitation
8. Noninear dynamic characteristics of SMA simpy supported beam in axia stochastic excitation Zhi-Wen Zhu 1, Wen-Ya Xie, Jia Xu 1 Schoo of Mechanica Engineering, Tianjin University, 9 Weijin Road, Tianjin
More informationFFTs in Graphics and Vision. Spherical Convolution and Axial Symmetry Detection
FFTs in Graphics and Vision Spherica Convoution and Axia Symmetry Detection Outine Math Review Symmetry Genera Convoution Spherica Convoution Axia Symmetry Detection Math Review Symmetry: Given a unitary
More informationFundamental Solutions and Green s functions. Simulation Methods in Acoustics
Fundamental Solutions and Green s functions Simulation Methods in Acoustics Definitions Fundamental solution The solution F (x, x 0 ) of the linear PDE L {F (x, x 0 )} = δ(x x 0 ) x R d Is called the fundamental
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationFinite element method for structural dynamic and stability analyses
Finite eement method for structura dynamic and stabiity anayses Modue-9 Structura stabiity anaysis Lecture-33 Dynamic anaysis of stabiity and anaysis of time varying systems Prof C S Manohar Department
More informationHigh Accuracy Split-Step Finite Difference Method for Schrödinger-KdV Equations
Commun. Theor. Phys. 70 208 43 422 Vo. 70, No. 4, October, 208 High Accuracy Spit-Step Finite Difference Method for Schrödinger-KdV Equations Feng Liao 廖锋, and Lu-Ming Zhang 张鲁明 2 Schoo of Mathematics
More informationReflection principles and kernels in R n _+ for the biharmonic and Stokes operators. Solutions in a large class of weighted Sobolev spaces
Refection principes and kernes in R n _+ for the biharmonic and Stokes operators. Soutions in a arge cass of weighted Soboev spaces Chérif Amrouche, Yves Raudin To cite this version: Chérif Amrouche, Yves
More informationTorsion and shear stresses due to shear centre eccentricity in SCIA Engineer Delft University of Technology. Marijn Drillenburg
Torsion and shear stresses due to shear centre eccentricity in SCIA Engineer Deft University of Technoogy Marijn Drienburg October 2017 Contents 1 Introduction 2 1.1 Hand Cacuation....................................
More informationThe Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation
The Symmetric Antipersymmetric Soutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B 2 + + A X B C Its Optima Approximation Ying Zhang Member IAENG Abstract A matrix A (a ij) R n n is said to be symmetric
More informationIntroduction to PDEs and Numerical Methods Tutorial 4. Finite difference methods stability, concsistency, convergence
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 4. Finite difference methods stability, concsistency, convergence Dr. Noemi Friedman,
More informationMalaysian Journal of Civil Engineering 30(2): (2018)
Maaysian Journa of Ci Engineering 3():331-346 (18) BUBNOV-GALERKIN METHOD FOR THE ELASTIC BUCKLING OF EULER COLUMNS Ofondu I.O. 1, Ikwueze E. U. & Ike C. C. * 1 Dept. of Mechanica and Production Engineering,
More informationThe Bending of Rectangular Deep Beams with Fixed at Both Ends under Uniform Load
Engineering,,, 8-9 doi:.6/eng..7 Pubised Onine December (ttp://.scirp.org/journa/eng) Te Bending of Rectanguar Deep Beams it Fied at Bot Ends under Uniform Load Abstract Ying-Jie Cen, Bao-Lian Fu, Gang
More informationA. Distribution of the test statistic
A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch
More informationCoupling of LWR and phase transition models at boundary
Couping of LW and phase transition modes at boundary Mauro Garaveo Dipartimento di Matematica e Appicazioni, Università di Miano Bicocca, via. Cozzi 53, 20125 Miano Itay. Benedetto Piccoi Department of
More informationIntroduction to PDEs and Numerical Methods Tutorial 11. 2D elliptic equations
Platzalter für Bild, Bild auf Titelfolie inter das Logo einsetzen Introduction to PDEs and Numerical Metods Tutorial 11. 2D elliptic equations Dr. Noemi Friedman, 3. 1. 215. Overview Introduction (classification
More informationBCCS TECHNICAL REPORT SERIES
BCCS TCHNICAL RPORT SRIS The Crouzeix-Raviart F on Nonmatching Grids with an Approximate Mortar Condition Taa Rahman, Petter Bjørstad and Xuejun Xu RPORT No. 17 March 006 UNIFOB the University of Bergen
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
odue 2 naysis of Staticay ndeterminate Structures by the atri Force ethod Version 2 E T, Kharagpur esson 12 The Three-oment Equations- Version 2 E T, Kharagpur nstructiona Objectives fter reading this
More informationFRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA)
1 FRST 531 -- Mutivariate Statistics Mutivariate Discriminant Anaysis (MDA) Purpose: 1. To predict which group (Y) an observation beongs to based on the characteristics of p predictor (X) variabes, using
More informationThe Dimensional Reduction Approach for 2D Non-prismatic Beam Modelling: a Solution Based on Hellinger-Reissner Principle
The Dimensiona Reduction Approach for 2D Non-prismatic Beam Modeing: a Soution Based on Heinger-Reissner Principe Ferdinando Auricchio a, Giuseppe Baduzzi a,, Caro Lovadina b a Dipartimento di Ingengeria
More informationUNCOMPLICATED TORSION AND BENDING THEORIES FOR MICROPOLAR ELASTIC BEAMS
11th Word Congress on Computationa Mechanics WCCM XI 5th European Conference on Computationa Mechanics ECCM V 6th European Conference on Computationa Fuid Dynamics ECFD VI E. Oñate J. Oiver and. Huerta
More informationSmoothers for ecient multigrid methods in IGA
Smoothers for ecient mutigrid methods in IGA Cemens Hofreither, Stefan Takacs, Water Zuehner DD23, Juy 2015 supported by The work was funded by the Austrian Science Fund (FWF): NFN S117 (rst and third
More information1 Discretizing BVP with Finite Element Methods.
1 Discretizing BVP with Finite Element Methods In this section, we will discuss a process for solving boundary value problems numerically, the Finite Element Method (FEM) We note that such method is a
More information