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Computational Modeling for Engineering MECN 6040. Professor: Dr. Omar E. Meza Castillo [email protected] http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering. Finite differences. Best known numerical method of approximation. FINITE DIFFERENCE FORMULATION - PowerPoint PPT Presentation
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COMPUTATIONAL MODELING FOR ENGINEERINGMECN 6040
Professor: Dr. Omar E. Meza [email protected]
http://facultad.bayamon.inter.edu/omezaDepartment of Mechanical Engineering
FINITE DIFFERENCESBest known numerical
method of approximation
FINITE DIFFERENCE FORMULATIONOF DIFFERENTIAL EQUATIONS
finite difference form of the first derivative
Taylor series expansion of the function f about the point x,
The smaller the x, the smaller the error, and thus the more accurate the approximation.
3
THE BIG QUESTION:
How good are the FD approximations?
This leads us to Taylor series....
4
▪ Numerical Methods express functions in an approximate fashion: The Taylor Series.
▪ What is a Taylor Series?Some examples of Taylor series which you must have seen
EXPASION OF TAYLOR SERIES
!6!4!2
1)cos(642 xxxx
!7!5!3
)sin(753 xxxxx
!3!2
132 xxxe x
5
▪ The general form of the Taylor series is given by
provided that all derivatives of f(x) are continuous and exist in the interval [x,x+h], where h=∆xWhat does this mean in plain English?
GENERAL TAYLOR SERIES
32
!3!2hxfhxfhxfxfhxf
As Archimedes would have said, “Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives at that single point, and I can give you the value of the function at any other point”6
▪ Example: Find the value of f(6) given that f(4)=125, f’(4)=74, f’’(4)=30, f’’’(4)=6 and all other higher order derivatives of f(x) at x=4 are zero.
▪ Solution: x=4, x+h=6 h=6-x=2▪ Since the higher order derivatives are
zero,
!324
!22424424
32
fffff
!326
!22302741256
32
f
860148125 341
7
THE TAYLOR SERIES
▪ (xi+1-xi)= h step size (define first)▪ Reminder term, Rn, accounts for all terms
from (n+1) to infinity.
nn
ii
n
iiiiiii
Rxxnf
xxfxxxfxfxf
)(!
)(!2
))(()()(
1
)(
2111
)1()1(
)!1()(
n
n
n hnfR
8
▪ Zero-order approximation
▪ First-order approximation
▪ Second-order approximation
)x(f)x(f i1i
)xx)(x(f)x(f)x(f i1iii1i
2i1ii1iii1i )xx(
!2f)xx)(x(f)x(f)x(f
9
▪ Example: Taylor Series Approximation of a polynomial Use zero- through fourth-order Taylor Series approximation to approximate the function:
▪ From xi=0 with h=1. That is, predict the function’s value at xi+1=1
▪ f(0)=1.2▪ f(1)=0.2 - True value
2.1x25.0x5.0x15.0x1.0xf 234
10
▪ Zero-order approximation
▪ First-order approximation
0.12.12.0EionapproximatvalueTrueE
2.1)1(f2.1)0(fxfxf
t
t
i1i
75.095.02.0E
95.0)1(25.02.1h)0('f0f1f25.0)0('f
25.0x1x45.0x4.0)x('f
h)x('fxfxf
t
23i
ii1i
11
▪ Second-order approximation
12
25.045.02.0E
45.0121)1(25.02.1
!2h)0(''fh)0('f0f1f
1)0(''f1x9.0x2.1)x(''f
!2h)x(''fh)x('fxfxf
t
22
2i
2i
ii1i
12
▪ Third-order approximation
1.03.02.0E
3.0619.01
21)1(25.02.1
!3h)0('''f
!2h)0(''fh)0('f0f1f
9.0)0(''f9.0x4.2)x('''f
!3h)x('''f
!2h)x(''fh)x('fxfxf
t
32
32
i
3i
2i
ii1i
13
▪ Fourth-order approximation
02.02.0E
2.02414.2
619.01
21)1(25.02.1
!4h)0(f
!3h)0('''f
!2h)0(''fh)0('f0f1f
4.2)0(f
4.2)x(f!4h)x(f
!3h)x('''f
!2h)x(''fh)x('fxfxf
t
432
4iv32
iv
iiv
4i
iv3i
2i
ii1i
14
15
▪ If we truncate the series after the first derivative term
TAYLOR SERIES TO ESTIMATE TRUNCATION ERRORS
nn
i1i
)n(
2i1ii1iii1i
R)tt(!n
v
)tt(!2''v)tt)(t('v)t(v)t(v
1i1iii1i R)tt)(t('v)t(v)t(v
)tt(R
)tt()t(v)t(v)t('v
i1i
1
i1i
i1ii
First-order approximation
Truncation Error
)tt(O i1i
16
▪ Forward Difference Approximation
NUMERICAL DIFFERENTIATION
)xx(O)xx()x(f)x(f)x('f i1i
i1i
i1ii
17
• The Taylor series expansion of f(x) about xi is
• From this:
• This formula is called the first forward divided difference formula and the error is of order O(h).
hxfxf
xxxfxfxf
xxxfxfxf
ii
ii
iii
iiiii
)()()()()(
))(()()(
1
1
1
11
NUMERICAL DIFFERENTIATION
18
• Or equivalently, the Taylor series expansion of f(x) about xi can be written as
• From this:
• This formula is called the first backward divided difference formula and the error is of order O(h).
hxfxf
xxxfxfxf
xxxfxfxf
ii
ii
iii
iiiii
)()()()()(
))(()()(
1
1
1
11
19
• A third way to approximate the first derivative is to subtract the backward from the forward Taylor series expansions:
• This yields to
• This formula is called the centered divided difference formula and the error is of order O(h2).
hxfxfxf
hxfxfxf
hxfxfxf
hxfxfxf
iii
iii
iii
iii
2)()()(
)(2)()(_________________________)()()(
)()()(
11
11
1
1
20
▪ Forward Difference Approximation
NUMERICAL DIFFERENTIATION
)xx(O)xx()x(f)x(f)x('f i1i
i1i
i1ii
21
▪ Backward Difference Approximation
)h(Oh
)x(f)x(f)x('f 1iii
22
▪ Centered Difference Approximation
)h(Oh2
)x(f)x(f)x('f 21i1ii
23
▪ Example: To find the forward, backward and centered difference approximation for f(x) at x=0.5 using step size of h=0.5, repeat using h=0.25. The true value is -0.9125
▪ h=0.5▪ xi-1=0 - f(xi-1)=1.2▪ xi=0.5 - f(xi)=0.925▪ Xi+1=1 - f(xi+1)=0.2
25.0x1x45.0x4.0x'f
2.1x25.0x5.0x15.0x1.0xf23
234
24
▪ Forward Difference Approximation
▪ Backward Difference Approximation
%9.58
45.15.0925.02.0)()()5.0(' 1
t
ii
hxfxff
%7.39
55.05.0
2.1925.0)()()5.0(' 1
t
ii
hxfxff
25
▪ Centered Difference Approximation
%6.9
11
2.12.02
)()()5.0(' 11
t
ii
hxfxff
26
▪ h=0.25▪ xi-1=0.25 - f(xi-1)=1.10351563▪ xi=0.5 - f(xi)=0.925▪ Xi+1=0.75 - f(xi+1)=0.63632813
▪ Forward Difference Approximation
%5.26
155.125.0
925.063632813.0h
)x(f)x(f)5.0('f
t
i1i
27
▪ Backward Difference Approximation
▪ Centered Difference Approximation
%4.2
934.05.010351563.163632813.0
2)()()5.0(' 11
t
ii
hxfxff
%7.21
714.025.010351563.1925.0
h)x(f)x(f)5.0('f
t
1ii
28
• The forward Taylor series expansion for f(xi+2) in terms of f(xi) is
• Combine equations:
212
21
22
22
)()()(2)(
_______________________________________2
)()()()(2
)2(2
)()2)(()()(
)2(2
)()2)(()()(
hxfxfxfxf
hxfhxfxfxf
hxfhxfxfxf
hxfhxfxfxf
iiii
iiii
iiii
iiii
FINITE DIFFERENCE APPROXIMATION OF HIGHER DERIVATIVE
29
• Solve for f ''(xi):
• This formula is called the second forward finite divided difference and the error of order O(h).
• The second backward finite divided difference which has an error of order O(h) is
221
212
)()(2)()(
)()(2)()(
hxfxfxfxf
hxfxfxfxf
iiii
iiii
30
• The second centered finite divided difference which has an error of order O(h2) is
211 )()(2)()(
hxfxfxfxf iii
i
31
• High accurate estimates can be obtained by retaining more terms of the Taylor series.
hxfh
xfxfxf
hxfxxxfxfxf
iiii
iiiiii
2)('')()()(
2)(''))(()()(
1
211
• The forward Taylor series expansion is:
• From this, we can write
High-Accuracy Differentiation Formulas
32
• Substitute the second derivative approximation into the formula to yield:
• By collecting terms:
• Inclusion of the 2nd derivative term has improved the accuracy to O(h2).
• This is the forward divided difference formula for the first derivative.
hxfxfxfxf
hhxfxfxf
hxfxfxf
iiii
iii
iii
2)(3)(4)()(
2
)()(2)()()()(
12
212
1
33
Forward Formulas
34
Backward Formulas
35
Centered Formulas
36
ExampleEstimate f '(1) for f(x) = ex + x using the centered formula of O(h4) with h = 0.25.
Solution
5.15.01225.125.01
175.025.015.05.012
12)()(8)(8)()(
2
1
1
2
2112
hxxhxx
xhxxhxx
hxfxfxfxfxf
ii
ii
i
ii
ii
iiiii
•From Tables
37
717.33
)149.2()867.2(8)740.4(8982.5)25.0(12
)5.0()75.0(8)25.1(8)5.1()(
ffffxf i
•In substituting the values:
38
ERROR▪ Truncation Error: introduced in the
solution by the approximation of the derivative▪ Local Error: from each term of the
equation▪ Global Error: from the
accumulation of local error▪ Roundoff Error: introduced in the
computation by the finite number of digits used by the computer
39
▪ Numerical solutions can give answers at only discrete points in the domain, called grid points.
▪ If the PDEs are totally replaced by a system of algebraic equations which can be solved for the values of the flow-field variables at the discrete points only, in this sense, the original PDEs have been discretized. Moreover, this method of discretization is called the method of finite differences.40
INTRODUCTION TO FINITE DIFFERENCE
(i,j)
x
Discretization: PDE FDE
Explicit Methods Simple No stable
Implicit Methods More complex Stables
¬∆x®
xm-1 xm m+1
yn+1
yn
yn-1
∆ym,nu
SUMMARY OF NODAL FINITE-DIFFERENCE RELATIONS FOR VARIOUS
CONFIGURATIONS:Case 1: Interior Node
Case 2: Node at an Internal Corner with Convection
Case 3: Node at Plane Surface with Convection
Case 4: Node at an External Corner with Convection
Case 5: Node at Plane Surface with Uniform Heat Flux
SOLVING THE FINITE DIFFERENCE EQUATIONS
Heat Transfer Solved Problems
THE MATRIX INVERSION METHOD
JACOBI ITERATION METHOD
GAUSS-SEIDEL ITERATION
ERROR DEFINITIONS▪ Use absolute value.▪ Computations are repeated until stopping
criterion is satisfied.
▪ If the following Scarborough criterion is met
63
sa Pre-specified % tolerance based on the knowledge of
your solution
)%n)-(2s 10 (0.5
USIG EXCEL
64
=MINVERSE(A2:C4)
=MMULT(A7:C9,E2:E4)
Matrix Inversion Method
65
Jacobi Iteration Method using Excel
66
Gauss-Seidel Iteration Method using Excel
A large industrial furnace is supported on a long column of fireclay brick, which is 1 m by 1 m on a side. During steady-state operation is such that three surfaces of the column are maintained at 500 K while the remaining surface is exposed to 300 K. Using a grid of ∆x=∆y=0.25 m, determine the two-dimensional temperature distribution in the column.
Ts=300 K
(1,1) (2,1) (3,1)
(1,2) (2,2) (3,2)
(1,3) (2,3) (3,3)
T11 T12 T13 T21 T22 T23 T31 T32 T33
-4 1 0 1 0 0 0 0 0 T11 -8001 -4 1 0 1 0 0 0 0 T12 -500
0 1 -4 0 0 1 0 0 0 T13
-1000
1 0 0 -4 1 0 1 0 0 T21 -3000 1 0 1 -4 1 0 1 0 T22 = 00 0 1 0 1 -4 0 0 1 T23 -5000 0 0 1 0 0 -4 1 0 T31 -8000 0 0 0 1 0 1 -4 1 T32 -500
0 0 0 0 0 1 0 1 -4 T33
-1000System of Linear Equations
69
Matrix Inversion Method
70
Iteration Method using Excel
71
Jacobi Iteration Method using Excel
72
Error Iteration Method using Excel
73
Gauss-Seidel Iteration Method using Excel
74
Error Iteration Method using Excel
78
Iteration Method using Excel