mecn 3500 inter - bayamon lecture 9 numerical methods for engineering mecn 3500 professor: dr. omar...
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MEC
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Lecture
9Numerical Methods for Engineering
MECN 3500
Professor: Dr. Omar E. Meza Castilloomeza@bayamon.inter.edu
http://www.bc.inter.edu/facultad/omezaDepartment of Mechanical EngineeringInter American University of Puerto Rico
Bayamon Campus
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Topic Lecture
Mathematical Modeling and Engineering Problem Solving 1
Introduction to Matlab 2
Numerical Error 3
Root Finding 4-5-6
System of Linear Equations 7-8
Finite Difference 9
Least Square Curve Fitting
Polynomial Interpolation
Numerical Integration
Ordinary Differential Equations
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Best known numerical method of approximation
Finite Difference
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To understand the theory of finite differences.
To apply FD to the solution of specific problems as a function of accuracy, condition matrix, and performance of iterative methods.
Course Objectives
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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.
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• 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
hxf
hxfxfxf
hxf
hxfxfxf
hxf
hxfxfxf
iiii
iiii
iiii
iiii
FINITE DIFFERENCE APPROXIMATION OF HIGHER
DERIVATIVE
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• 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)()(
h
xfxfxfxf
h
xfxfxfxf
iiii
iiii
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• The second centered finite divided difference which has an error of order O(h2) is
211 )()(2)(
)(h
xfxfxfxf iiii
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• High accurate estimates can be obtained by retaining more terms of the Taylor series.
hxf
h
xfxfxf
hxf
xxxfxfxf
iiii
iiiiii
2
)('')()()(
2
)(''))(()()(
1
211
• The forward Taylor series expansion is:
• From this, we can write
High-Accuracy Differentiation Formulas
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• 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.
h
xfxfxfxf
hh
xfxfxf
h
xfxfxf
iiii
iii
iii
2
)(3)(4)()(
2
)()(2)()()(
)(
12
212
1
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Forward Formulas
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Backward Formulas
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Centered Formulas
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Example
Estimate f '(1) for f(x) = ex + x using the centered formula of O(h4) with h = 0.25.
Solution
5.15.012
25.125.01
1
75.025.01
5.05.012
12
)()(8)(8)()(
2
1
1
2
2112
hxx
hxx
x
hxx
hxx
h
xfxfxfxfxf
ii
ii
i
ii
ii
iiiii
•From Tables
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717.33
)149.2()867.2(8)740.4(8982.5
)25.0(12
)5.0()75.0(8)25.1(8)5.1()(
ffff
xf i
•In substituting the values:
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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
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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.
Introduction to Finite Difference
(i,j)
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x
Discretization: PDE FDE
Explicit Methods Simple No stable
Implicit Methods More complex Stables
¬∆x®
xm-1 x m m+1
yn+1
yn
yn-1
∆y
m,nu
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Summary of nodal finite-difference relations for various configurations:
Case 1: Interior Node
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Case 2: Node at an Internal Corner with Convection
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Case 3: Node at Plane Surface with Convection
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Case 4: Node at an External Corner with Convection
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Case 5: Node at Plane Surface with Uniform Heat Flux
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Solving Finite Difference Equations
Heat Transfer Solved Problem
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Method
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Jacobi Iteration Method
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Gauss-Seidel Iteration
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Error Definitions Use absolute value. Computations are repeated until stopping
criterion is satisfied.
If the following Scarborough criterion is met
sa Pre-specified % tolerance
based on the knowledge of your solution
)%n)-(2s 10 (0.5
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Using Excel
=MINVERSE(A2:C4)
=MMULT(A7:C9,E2:E4)
Matrix Inversion Method
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Jacobi Iteration Method using Excel
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Gauss-Seidel Iteration Method using Excel
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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)
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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
-1000
System of Linear Equations
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Matrix Inversion Method
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Iteration Method using Excel
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Jacobi Iteration Method using Excel
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Error Iteration Method using Excel
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Gauss-Seidel Iteration Method using Excel
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Error Iteration Method using Excel
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Iteration Method using Excel
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Lecture 9MEC
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Fit the data with multiple linear regression
x1 x2 y0 0 5
2 1 10
2.5 2 9
7 3 0
4 6 3
5 2 27
100
5.243
54
544814
4825.765.16
145.166
2
1
0
a
a
a
3,4,5 210 aaa
21 345 xxy
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Use the polyfit function
Regression in Excel
Use Add Trendline
Regression in Matlab
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Homework7 www.bc.inter.edu/facultad/omeza
Omar E. Meza Castillo Ph.D.
61
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