dr. jennifer parham-mocello. what is an ide? ide – integrated development environment software...
TRANSCRIPT
Dr. Jennifer Parham-Mocello
What is an IDE?IDE – Integrated Development EnvironmentSoftware application providing conveniences
to computer programmers for software development.
Consists of editor, compiler/interpreter, building tools, and a graphical debugger.
Heat Diffusion / Finite Difference Methods 2
EclipseJava, C/C++, and PHP IDEUses Java Runtime Environment (JRE)
Install JRE/JDK - http://www.oracle.com/technetwork/java/index.html
Need C/C++ compilerInstall Wascana (Windows version)
http://www.eclipselabs.org/p/wascana
Heat Diffusion / Finite Difference Methods 3
Using EclipseExample – Open HelloWorld C++ project
File -> New -> C++ ProjectEnter Project Name
Building/Compiling ProjectsProject -> Build AllRun -> Run
Console
Heat Diffusion / Finite Difference Methods 4
Heat Diffusion / Finite Difference Methods 5
Heat Diffusion / Finite Difference Methods 6
Heat Diffusion EquationDescribes the distribution of heat (or
variation in temperature) in a given region over time.
For a function u(x, t) of one spatial variables(x) and the time variable t, the heat diffusion equation is:
t
uc
x
uk
2
2
1D02
2
t
uc
x
uk
1D
or
Material Parameters – thermal conductivity (k), specific heat (c), density ()
Conceptual and theoretical basisConservation of mass, energy, momentum,
etc.Rate of flow in - Rate of flow out = Rate of
heat storage
Heat Diffusion / Finite Difference Methods 7
t
uc
x
uk
2
2
t
uc
y
u
x
uk
2
2
2
2
t
uc
z
u
y
u
x
uk
2
2
2
2
2
2
1D 2D
3D
x=0.0 x=4.0
3g/cm 7.8
Ccal/g 0.11
Ccmsec/cal 13.0
c
k
Physical Parameters
C100),0.4(
C0),0.0(
tu
tu
Boundary Conditions
C0)0.0,( xu
Initial Conditions
Wire with perfect insulation, except at ends
. and
of valuesallfor ),(for solve ,conditions
boundary and initial and ,parameters physical
, equation, reticalGiven theo2
2
tx
txu
t
uc
x
uk
Heat Diffusion / Finite Difference Methods 8
Outline of SolutionDiscretization (spatial and temporal)Transformation of theoretical equations to
approximate algebraic formSolution of algebraic equations
Heat Diffusion / Finite Difference Methods 9
DiscretizationSpatial - Partition into equally-spaced nodes
Temporal - Decide on time stepping parameters
x=0.0 x=1.0x=0.33 x=0.67
u0 u1 u2 u3
Let to = 0.0, tn = 10.0, and t = 0.1
Heat Diffusion / Finite Difference Methods 10
Approximate Theoretical with Algebra
211
2
2
)(
2
x
uuu
x
u
t
uu
t
u
iii
ttt
Finite difference approximations forfirst and second derivatives
u0 ui-1 ui+1 unu1 un-1ui
Heat Diffusion / Finite Difference Methods 11
ti
ttii
tit
it
ti
ttii
tit
it
uuuuuxc
tk
t
uuc
x
uuuk
t
uc
x
uk
112
2
11
2
2
2)(
)(
2
ti
ttii
tit
it uuuuur
xc
tkr
11
2
2
have then we,)(
Let
Heat Diffusion / Finite Difference Methods 12
Algorithm t
itt
iit
it
it uuuuur
11 2
for t=0,tn
for each node, i predict ut+t
endforendfor
• Predicting ut+t at each node• Explicit solution• Implicit solution (system of equations)
Heat Diffusion / Finite Difference Methods 13
tii
tit
ittt
i
tti
ti
ttii
tit
it
uuuuru
u
uuuuur
11
11
2
,for solvingby
2
Rearrange
Heat Diffusion / Finite Difference Methods 14
2
1
want wepurposes,stability for :Note
092.0)33.0)(0.5)(0.2(
)1.0)(0.1(
)(
33.0
1.0
0.5
2.0
.01
22
r
xc
tkr
x
t
c
k
Physical Parameters
100),0.1(
0),0.0(
tu
tu
Boundary Conditions
0)0.0,( xu
Initial Conditions
u0 u1 u2 u3
t u0 u1 u2 u30.0 0.0000 0.0000 0.0000 0.00000.1 0.0000 0.0000 9.1874 100.00000.2 0.0000 0.8432 16.6790 100.00000.3 0.0000 2.2200 22.8760 100.0000
10.0 0.0000 33.3300 66.6640 100.0000
0.0000 33.3333 66.6667 100.0000
Heat Diffusion / Finite Difference Methods 15
ExtensionTwo Dimensions
t
uuc
y
uuu
x
uuuk
t
uc
y
u
x
uk
tji
ttjiji
tji
tji
tji
tji
tji
t
,,
2
1,,1,
2
,1,,1
2
2
2
2
)(
2
)(
2
u i,j
u i,j-1
u i,j+1
u i+1,ju i-1,j
Heat Diffusion / Finite Difference Methods 16
Implement 1-D Heat DiffusionOpen New C++ ProjectName the ProjectOpen New C++ source code file
File -> New -> Source FileName C++ File (remember extension, .C, .c+
+, .cpp)
Heat Diffusion / Finite Difference Methods 17
EXTRAS
Heat Diffusion / Finite Difference Methods 18
Shorthand NotationsGradient (“Del”) Operator
z
y
x
,
z
y
x
:3D
, :2D
, :1D
u
u
u
u
y
ux
u
u
y
x
x
uu
x
Heat Diffusion / Finite Difference Methods 19
Divergence (Gradient of a vector field)
2
2
2
2
2
2
2
2
2
2
2
2
:3D
:2D
:1D
z
u
y
u
x
u
z
uy
ux
u
zyxu
y
u
x
u
y
ux
u
yxu
x
u
x
u
xu
T
T
T
Heat Diffusion / Finite Difference Methods 20
Laplacian Operator
uu
z
u
y
u
x
uu
y
u
x
uu
x
uu
T
2
2
2
2
2
2
22
2
2
2
22
2
22
2
Sometimes :Note
:3D
:2D
:1D
Heat Diffusion / Finite Difference Methods 21
Heat Diffusion Equation - rewritten
t
uc
x
uk
2
2
t
uc
y
u
x
uk
2
2
2
2
t
uc
z
u
y
u
x
uk
2
2
2
2
2
2
t
ucuk
2
•LHS represents spatial variations•RHS represents temporal variation
Heat Diffusion / Finite Difference Methods 22