Introduction to PDEs and Numerical Methods Lecture 1: Introduction

Transcription

1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 1: Introduction Dr. Noemi Friedman,

2 Basic information on the course Course Title: Lecturer: Introduction to PDEs and Numerical Methods Noémi Friedman Mühlenpfordtstr. 23, 8th floor Room: 819 Assistant (exercises): Jaroslav Vondřejc Mühlenpfordtstr. 23, 8th floor Room: 822 Assistant2 (small tutorials): Stephan Lenz (CSE student) PDE lecture Seite 2

3 Basic information on the course Credits and work load: 5 credits: 6-7 hours/week Pre-requisits: Requisits: Differential operators, elementary knowledge of PDEs, basics of linear algebra, basic MATLAB coding skills Weekly assignments in group of two or three (min 50%) Written exam: , 10:30-12:00, room ZI 24.1 Script, recommended literature: See webpage: Software used: MATLAB, FEniCS (Python interface) PDE lecture Seite 3

4 Information about the assignments Homework assignments in groups (max. group of three) Submission of homework Consultation: Written homework Submit on the beginning of the tutorial (include cover sheet with subject name (PDE1), names and matriculation number of students, assignment number) For program codes: subject: assignment# NAMES (#: number of the assignment, NAMES: names of students) (e.g.: assignment1 J. Smith, K. Park) Homework is due to the beginning of the tutorials Noemi Friedman (after the lecture, office hours by arrangement, please, take appointment first by Jaroslav Vondřejc (will be assigned on the tutorial PDE lecture Seite 4

5 Definition: ODEs PDEs Partial differential equation: Equation specifying a relation between the partial derivative(s) of an unkown multivariable function and maybe the function itself: F u x, y, z, t, u x, y, z, t, u x, y, z, t y, 2 u x, y, z, t y, = f(x, y, z, t) Ordinary differential equation: Equation specifying a relation between the derivative(s) of an unkown univariable function and maybe the function itself: F u t, du t t, d 2 u t dt 2, = f(t) Boundary Value Problem (BVP), Initial Boundary Value Problem (IBVP): PDE with initial/boundary conditions PDE lecture Seite 5

6 Motivation simulation of planets PDE lecture Seite 6

7 Motivation heat convection computed surface temperatures due to convective and radiative heat transfer from the exhaust manifold to surrounding objects Source: ANSYS PDE lecture Seite 7

8 Motivation structural analysis PDE lecture Seite 8 Source: ANSYS plicit_str.png?

9 Motivation flow problems The Stokes equation u(x) u + p I = f in Ω u = 0 in Ω p(x) Source: FENICS documentation: PDE lecture Seite 9

10 Motivation flow problems Source: TU Braunschweig SFB 880 Source: eb07ed6f28ab61e35047cec42359baf1.ssl.cf5.rackcdn.com/ugc/entry/ a78ef _0129_fast15.jpg PDE lecture Seite 10

11 Motivation highly coupled systems PDE lecture Seite 11

12 Overview of the course Introduction (definition of PDEs, classification, basic math, introductory examples of PDEs) Analitical solution of elementary PDEs (Fourier series/transform, seperation of variables, Green s function) Numerical solutions of PDEs: Finite difference method Finite element method PDE lecture Seite 12

13 Overview of this lecture Basic definitions, motivation Differential operators: basic notations, divergence, Laplace, curl, grad Classification of PDEs Introductory example: heat flow in a bar PDE lecture Seite 13

14 Differential operators partial derivative φ(x, y, z, t) Example: Partial derivative: z = f(x, y) = x 2 + xy + y 2 φ, xφ, φ x, φ, φ,x, (φ ) φ = 2x + y φ t = φ φ x=1,y=1 = 3 (Image source: Wikipedia) PDE lecture Seite 14

15 Example: Differential operators mixed derivative φ(x, y) Mixed derivative: f(x, y, z) = xy 2 cos(z) 2 φ y = 2 φ y f = y2 cos(z) 2 φ y = 2ycos(z) f y = 2xy cos(z) 2 φ y = 2ycos(z) PDE lecture Seite 15

16 Differential operators total derivative φ r = φ x, y, z, t Total derivative: dφ dt = φ dt t dt + φ dx dt + φ dy y dt + φ dz z dt Total differential (differential change of f): dφ = dφ dt + φ φ φ dx + dy + y z dz Example: φ x, y = x 2 + 2y y x = x φ = 2x dφ dx = φ dx dx + φ dy y dx = 2x + 2 partial derivative total derivative PDE lecture Seite 16

17 Differential operators gradient φ r = φ x, y, z : R 3 R (vector-scalar function) Nabla operator: Gradient: = y z direction: greatest rate of increase of the function magnitude: the slope of the function in that direction Example: PDE lecture Seite 17

18 Differential operators directional derivative φ r = φ x, y, z : R 3 R (vector-scalar function) Directional derivative: D v f r = lim h 0 φ r + hv φ r h (normalised): = v f r = v T f r = v x v y v z D v f r = lim h 0 f r + hv f r h v = v v f r φ x, y, z φ x, y, z y φ x, y, z z Example 1: x u t What is the differential equation to define a wave traveling with speed c? In the direction x ct u is constant directional derivative is zero: D v u x, t = v v u x, t = 0 (v u x, t = 0) PDE lecture Seite 18

19 Differential operators directional derivative x u v = c t 1 v u x, t = 0 v u x, t = v T u x, t = c 1 u u t = 0 Transport equation: u t + cu x = 0 Example 2: Let s suppose u = sin(x ct) is a solution of the transport equation. What is its directional derivative in the direction: v = c 1 v u x, t = v T cos(x ct) u x, t = c 1 c cos(x ct = PDE lecture Seite 19

20 Differential operators - divergence Divergence: of g(x, y, z): R 3 R 3 ( (of a vector field): g x, y, z = g x (x, y, z) g y (x, y, z) g z (x, y, z) = g x g y g z divg x, y, z = g x, y, z = y z g x g y g z = g x + g y y + g z z Example: PDE lecture Seite 20

21 Differential operators - Laplace Laplace operator: Example: PDE lecture Seite 21

22 Differential operators rotation (curl) Rotation (curl): of g(x, y, z): R 3 R 3 ( (of a vector field): g x, y, z = rotg x, y, z = g x, y, z = det direction: axis of rotation magnitude: magnitude of rotation g x (x, y, z) g y (x, y, z) g z (x, y, z) = g x g y g z i j k = y z g x g y g z g z y g y z g x z g z g y g x y Example: PDE lecture Seite 22

23 Classification of PDEs Constant/variable coefficients Stationary/instationary (not time dependent/time dependent) Linear/nonlinear linearity condition: order order of the highest derivative homogeneous/inhomogenous inhomogeneous: additive terms which do not depend on unknown function homogenous: u = 0 is a solution of the equation elliptic/parabolic/hyperbolic (only for second order PDEs) Au xx + 2Bu xy + Cu yy + lower order derivatives = 0 AC B 2 = 0 parabolic AC B 2 < 0 hyperbolic AC B 2 > 0 elliptic PDE lecture Seite 23

24 Classification of PDEs, examples of PDEs Wave equation u tt c 2 u xx = 0 Laplace equation u xx + u yy = 0 Heat equation u t u xx = 0 Order Constant coefficient? yes yes yes Linear? yes yes yes Homogenous? yes yes yes Class A=1, B=0, C= c 2 AC B 2 = c 2 Hyperbolic A=1, B=0, C=1 AC B 2 =1 Elliptic A=-1, B=0, C=0 AC B 2 = 0 Parabolic PDE lecture Seite 24

25 Classification of PDEs, examples of PDEs Linearity: PDE lecture Seite 25

26 Classification of PDEs, examples of PDEs PDE lecture Seite 26

27 Introductory example: heat flow in a bar Basic assumptions: Uniform cross-section Temperature varies only in the longitudinal direction Relationship between heat energy and temperature is linear c J gk : cpecific heat capacity Energy [J] cj energy is required to raise the tempreture by 1K of 1g material Homogenous material properties along the bar (ρ and c are constants along the bar) ρ[ g m3]: density of the material of the bar Problem description u x, t =? (temparature) q(0, t) x A q(l, t) 0 x x + x l PDE lecture Seite 27

28 Introductory example: heat flow in a bar Total energy in the bar section of length x is: E tot = E 0 + E 0 + x+ x x x x+ x Aρcu s, t ds Aρc(u s, t T 0 ))ds = x+ x x AρcT 0 ds x AρcT 0 ds But I m only interested in: x+ x Aρcu s, t ds = t x x x+ x Aρc u s, t t ds A q(0, t) q(l, t) x 0 x x + x l PDE lecture Seite 28

29 Introductory example: heat flow in a bar Change of heat energy by time in the section of length x t x x+ x Aρcu s, t ds = x x+ x Aρc u s, t t ds q(x, t) x q(x + x, t) Change of energy by time in the section of length x from heat flux: q x, t J m 2 s : heat flux Fundamental theorem of calculus: Aq x + x, t Aq x, t = Conservation of energy: x x+ x A q s, t ds x x+ x A q s, t ds = x x+ x Aρc u s, t t ds x x+ x A q s, t + Aρc u s, t t ds = PDE lecture Seite 29

30 Introductory example: heat flow in a bar x x+ x A q s, t + Aρc u s, t t ds = 0 q(x, t) x q(x + x, t) q x, t + ρc u x, t t = 0 0 < x < l u x,t : change of temperature with increasing x Assumption: Fourier s law of heat conduction: q x, t = κ u x, t Heat equation ρc u x, t t κ 2 u x, t 2 = 0 (for homogenous material properties) ρ(x)c(x) u x, t t κ(x) u x, t = 0 (for inhomogenous material properties) PDE lecture Seite 30

31 Introductory example: heat flow in a bar The heat equation with source or sink (inhomogenous heat equation) x x+ x A q s, t + Aρc u s, t t ds = x x+ x Af(s, t) ds q(x, t) x q(x + x, t) q x, t Heat equation + ρc u x, t t = f(x, t) 0 < x < l ρc u x, t t κ 2 u x, t 2 = f(x, t) (for homogenous material properties) u x, t ρ(x)c(x) t κ(x) u x, t = f(x, t) (for inhomogenous material properties) PDE lecture Seite 31

32 Introductory example: heat flow in a bar q(0, t) A q(l, t) 0 l ρc u x, t t κ 2 u x, t 2 = f(x, t) Boundary conditions: a) Perfect isolation at the end (flux across the boundaries is zero): u 0, t u l, t κ = 0 κ = 0 t b) Perfect thermal contact: u 0, t = 0 u l, t = 0 t PDE lecture Seite 32