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Introduction to PDEs and Numerical Methods Lecture 1 ... · Introduction to PDEs and Numerical...

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Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Dr. Noemi Friedman, 28.10.2015. Introduction to PDEs and Numerical Methods Lecture 1: Introduction
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  • Platzhalter fr Bild, Bild auf Titelfolie hinter das Logo einsetzen

    Dr. Noemi Friedman, 28.10.2015.

    Introduction to PDEs and Numerical Methods

    Lecture 1:

    Introduction

  • 28. 10. 2015. | PDE lecture | Seite 2

    Basic information on the course

    Course Title:

    Introduction to PDEs and Numerical Methods

    Lecturer:

    Nomi Friedman

    [email protected]

    Mhlenpfordtstr. 23, 8th floor

    Room: 819

    Assistant (exercises):

    Jaroslav Vondejc

    [email protected]

    Mhlenpfordtstr. 23, 8th floor

    Room: 822

    Assistant2 (small tutorials):

    Stephan Lenz (CSE student)

    mailto:[email protected]:[email protected]

  • 28. 10. 2015. | PDE lecture | Seite 3

    Basic information on the course

    Credits and work load:

    5 credits: 6-7 hours/week

    Pre-requisits:

    Differential operators,

    elementary knowledge of PDEs,

    basics of linear algebra,

    basic MATLAB coding skills

    Requisits:

    Weekly assignments in group of two or three (min 50%)

    Written exam: 29.2.2016., 10:30 - 12:00, room ZI 24.1

    Script, recommended literature:

    See webpage: https://www.tu-braunschweig.de/wire/lehre/ws15/pde1

    Software used:

    MATLAB, FEniCS (Python interface)

  • 28. 10. 2015. | PDE lecture | Seite 4

    Information about the assignments

    Homework assignments in groups (max. group of three)

    Submission of homework

    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:

    e-mail: [email protected]

    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

    Consultation:

    Noemi Friedman (after the lecture, office hours by arrangement, please, take

    appointment first by e-mail: [email protected])

    Jaroslav Vondejc (will be assigned on the tutorial

    mailto:[email protected]:[email protected]

  • 28. 10. 2015. | PDE lecture | Seite 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:

    , , , , , , ,

    , , , ,

    ,

    2 , , ,

    , = (, , , )

    Ordinary differential equation:

    Equation specifying a relation between the derivative(s) of an unkown univariable

    function and maybe the function itself:

    ,

    ,

    2

    2, = ()

    Boundary Value Problem (BVP), Initial Boundary Value Problem (IBVP):

    PDE with initial/boundary conditions

  • 28. 10. 2015. | PDE lecture | Seite 6

    Motivation simulation of planets

  • 28. 10. 2015. | PDE lecture | Seite 7

    Motivation heat convection

    Source:

    ANSYS http://www.ansys.com/staticassets/ANSYS/staticassets/product/16-highlights/underhood-simulation-surface-temps-heat-

    transfer-manifold-2.jpg

    computed surface temperatures due to

    convective and radiative heat transfer from

    the exhaust manifold to surrounding

    objects

  • 28. 10. 2015. | PDE lecture | Seite 8

    Motivation structural analysis

    Source: ANSYS

    http://wildeanalysis.co.uk/system/photos/838/preview/ansys_ex

    plicit_str.png?1273430962

  • 28. 10. 2015. | PDE lecture | Seite 9

    Motivation flow problems

    The Stokes equation

    Source:

    FENICS documentation:

    http://fenicsproject.org/documentation/dolfin/1.6.0/python/demo/documented/stokes-taylor-hood/python/documentation.html

    + = in = 0 in

    (x)(x)

  • Motivation flow problems

    28. 10. 2015. | PDE lecture | Seite 10

    Source: https://33463d8ba37cd0a930f1-

    eb07ed6f28ab61e35047cec42359baf1.ssl.cf5.rackcdn.com/ugc/entry/5

    44867a78ef07-133981577813387476428_0129_fast15.jpg

    Source: TU Braunschweig SFB 880

  • 28. 10. 2015. | PDE lecture | Seite 11

    Motivation highly coupled systems

  • 28. 10. 2015. | PDE lecture | Seite 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, Greens function)

    Numerical solutions of PDEs:

    Finite difference method

    Finite element method

  • 28. 10. 2015. | PDE lecture | Seite 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

  • 28. 10. 2015. | PDE lecture | Seite 14

    Differential operators partial derivative

    Partial derivative:

    (, , , )

    , , ,

    , , , (

    )

    (Image source: Wikipedia)

    = (, ) = 2 + + 2

    = 2 +

    =1,=1

    = 3

    =

    Example:

  • 28. 10. 2015. | PDE lecture | Seite 15

    Differential operators mixed derivative

    Mixed derivative:

    (, )

    2

    =

    2

    (, , ) = 2cos()

    = 2cos()

    = 2 cos()

    2

    = 2cos()

    2

    = 2cos()

    Example:

  • 28. 10. 2015. | PDE lecture | Seite 16

    Differential operators total derivative

    Total derivative:

    Total differential (differential change of f):

    =

    +

    +

    +

    = , , ,

    =

    +

    +

    +

    , = 2 + 2

    =

    +

    = 2 + 2 =

    = 2

    total derivative

    partial derivative

    Example:

  • 28. 10. 2015. | PDE lecture | Seite 17

    Differential operators gradient

    Nabla operator:

    Gradient:

    = , , : 3 (vector-scalar function)

    =

    Example:

    direction: greatest rate of increase of

    the function

    magnitude: the slope of the function in

    that direction

  • 28. 10. 2015. | PDE lecture | Seite 18

    Differential operators directional derivative

    = , , : 3 (vector-scalar function)

    = lim0

    +

    = = =

    , ,

    , ,

    , ,

    Directional derivative:

    (normalised):

    = lim0

    +

    =

    What is the differential equation to define a wave

    traveling with speed ?In the direction is constant directionalderivative is zero:

    , =

    , = 0 ( , = 0)

    Example 1:

  • 28. 10. 2015. | PDE lecture | Seite 19

    Differential operators directional derivative

    , = , = 1

    = 0

    =1 , = 0

    Lets suppose = sin( ) is a solution of the transport equation.What is its directional derivative in the direction:

    + = 0Transport equation:

    =1

    , = , = 1cos( )

    cos( = 0

    Example 2:

  • 28. 10. 2015. | PDE lecture | Seite 20

    Differential operators - divergence

    of (, , ): 3 3((of a vector field):

    , , =

    (, , )(, , )

    (, , )

    =

    , , = , , =

    =

    +

    +

    Example:

    Divergence:

  • 28. 10. 2015. | PDE lecture | Seite 21

    Differential operators - Laplace

    Example:

    Laplace operator:

  • 28. 10. 2015. | PDE lecture | Seite 22

    Differential operators rotation (curl)

    of (, , ): 3 3((of a vector field):

    , , =

    (, , )(, , )

    (, , )

    =

    , , = , , =

    =

    Example:

    Rotation (curl):

    direction: axis of rotation

    magnitude: magnitude of rotation

  • 28. 10. 2015. | PDE lecture | Seite 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: = 0 is a solution of the equation elliptic/parabolic/hyperbolic (only for second order PDEs)

    + 2 + + lower order derivatives = 0

    2 = 0 parabolic 2 < 0 hyperbolic 2 > 0 elliptic

  • 28. 10. 2015. | PDE lecture | Seite 24

    Classification of PDEs, examples of PDEs

    Wave equation

    = 0

    Laplace equation

    + = 0Heat equation

    = 0

    Order 2 2 2

    Constant

    coefficient?

    yes yes yes

    Linear? yes yes yes

    Homogenous? yes yes yes

    Class A=1, B=0, C=

    2 =

    Hyperbolic

    A=1, B=0, C=1 2 =1

    Elliptic

    A=-1, B=0, C=0 2 = 0

    Parabolic

  • 28. 10. 2015. | PDE lecture | Seite 25

    Classification of PDEs, examples of PDEs

    Linearity:

  • 28. 10. 2015. | PDE lecture | Seite 26

    Classification of PDEs, examples of PDEs

  • 28. 10. 2015. | PDE lecture | Seite 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

    : cpecific heat capacity Energy []

    energy is required to raise the tempreture by 1 of 1 materialHomogenous material properties along the bar ( and are constants along the bar)

    [

    3]: density of the material of the bar

    Problem description , =? (temparature)

    0 +

    (0, ) (, )

  • 28. 10. 2015. | PDE lecture | Seite 28

    Introductory example: heat flow in a bar

    0 +

    (0, ) (, )

    Total energy in the bar section of length is:

    = 0 +

    +

    ( , 0)) =

    0 +

    +

    ,

    +

    0

    0But Im only interested in:

    +

    , =

    +

    ,

  • 28. 10. 2015. | PDE lecture | Seite 29

    Introductory example: heat flow in a bar

    +

    , =

    +

    ,

    Change of heat energy by time in the section of length

    (, ) ( + , )

    Change of energy by time in the section of length from heat flux:

    ,

    2: heat flux

    + , , =

    +

    ,

    Fundamental

    theorem of calculus:

    Conservation of energy:

    +

    ,

    =

    +

    ,

    +

    ,

    +

    ,

    = 0

  • 28. 10. 2015. | PDE lecture | Seite 30

    Introductory example: heat flow in a bar

    (, ) ( + , )

    +

    ,

    +

    ,

    = 0

    ,

    +

    ,

    = 0 0 < <

    ,

    : change of temperature with increasing

    Assumption: Fouriers law of heat conduction: , = ,

    ,

    2 ,

    2= 0Heat equation

    ()() ,

    ()

    ,

    = 0 (for inhomogenous material

    properties)

    (for homogenous material properties)

  • 28. 10. 2015. | PDE lecture | Seite 31

    Introductory example: heat flow in a bar

    (, ) ( + , )

    +

    ,

    +

    ,

    =

    +

    (, )

    ,

    +

    ,

    = (, ) 0 < <

    ,

    2 ,

    2= (, )

    Heat equation

    ()() ,

    ()

    ,

    = (, ) (for inhomogenous material properties)

    (for homogenous material properties)

    The heat equation with source or sink (inhomogenous heat equation)

  • 28. 10. 2015. | PDE lecture | Seite 32

    Introductory example: heat flow in a bar

    0

    (0, ) (, )

    ,

    2 ,

    2= (, )

    Boundary conditions:

    a) Perfect isolation at the end (flux across the boundaries is zero):

    0,

    = 0

    ,

    = 0

    b) Perfect thermal contact:

    0, = 0 , = 0


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