computational modeling for engineering mecn 6040 professor: dr. omar e. meza castillo...

COMPUTATIONAL MODELING FOR ENGINEERINGMECN 6040

Professor: Dr. Omar E. Meza Castilloomeza@bayamon.inter.edu

http://facultad.bayamon.inter.edu/omezaDepartment of Mechanical Engineering

INTRODUCTION TO THE THEORY

OF PDEs

LEARNING OBJECTIVES

1. Be able to distinguish between the 3 classes of 2nd order, linear PDE's. Know the physical problems each class represents and the physical/mathematical characteristics of each.

2. Be able to describe the differences between finite-difference and finite-element methods for solving PDEs.

3. Be able to solve Elliptical (Laplace/Poisson) PDEs using finite differences.

4. Be able to solve Parabolic (Heat/Diffusion) PDEs using finite differences.

DEFINITIONS AND TERMINOLOGY

DIFFERENTIAL EQUATIONAn equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE).

DEFINITIONS AND TERMINOLOGY

h

xfhxf

dx

dyh

)()(lim

0

DEFINITION OF A DERIVATIVEIf y=f(x), the derivative of y or f(x) with respect to x is defined as

The derivative is also denoted by y’, dy/dx or f’(x)

THE EXPONENTIAL FUNCTION

dependent variable: y independent variable: x

xexfy 2)(

yedx

xde

dx

ed

dx

dy xxx

22)2()( 22

2

DEFINITIONS AND TERMINOLOGY

Differential Equations are CLASSIFIED by type, order and linearity.

TYPEThere are two main types of differential equation: “ordinary” and “partial”.

DEFINITIONS AND TERMINOLOGY

Ordinary differential equation (ODE) Differential equations that involve only ONE independent variable are called ordinary differential equations.Examples: ,

only ordinary (or total) derivatives

xeydx

dy5 06

2

2

ydx

dy

dx

ydyx

dt

dy

dt

dx 2

DEFINITIONS AND TERMINOLOGY

Partial differential equation (PDE) Differential equations that involve two or more independent variables are called partial differential equations.Examples:

only partial derivatives

t

u

t

u

x

u

22

2

2

2

x

v

y

u

DEFINITIONS AND TERMINOLOGY

ORDERThe order of a differential equation is the order of the highest derivative found in the DE.

second order first order

xeydx

dy

dx

yd

45

3

2

2

DEFINITIONS AND TERMINOLOGY

xeyxxy 2'

3'' xy

first order

second order

DEFINITIONS AND TERMINOLOGY

DEFINITIONS AND TERMINOLOGY

LINEAR OR NONLINEARAn n-th order differential equation is said to be linear if the function

is linear in the variables)1(' ,..., nyyy

0),......,,( )(' nyyyxf

DEFINITIONS AND TERMINOLOGY

there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.

)()()(...)()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

or

linear first-order ordinary differential equation

linear second-order ordinary differential equation

linear third-order ordinary differential equation

0)(4 xydx

dyx

02 ''' yyy

04)( xdydxxy

xeydx

dyx

dx

yd 53

3

3

PDE'S DESCRIBE THE BEHAVIOR OF MANY

ENGINEERING PHENOMENA:

▪ Wave propagation

▪ Fluid flow (air or liquid)▪ Air around wings, helicopter blade, atmosphere

▪ Water in pipes or porous media

▪ Material transport and diffusion in air or water

▪ Weather: large system of coupled PDE's for momentum, pressure, moisture, heat, …

▪ Vibration

▪ Mechanics of solids: ▪ stress-strain in material, machine part, structure

▪ Heat flow and distribution

▪ Electric fields and potentials

▪ Diffusion of chemicals in air or water

▪ Electromagnetism and quantum mechanics

CLASIFIQUE LAS SIGUIENTES ECUACIONES:

Solución (a)

032

2

2

2

2

2

2

2

2

2

y

u

x

u

y

u

x

u

y

u

x

u(c) (b) (a)

parabólicaACB

C,BAy

u

x

u

:04

;0,030;3

2

2

2

elípticaACB

CBAy

u

x

u

ahiperbólicACB

CBAy

u

x

u

:04

;1,0,1;0

:04

;1,0,1

2

2

2

2

2

2

2

2

2

2

0;

Solución (b)

Solución (c)