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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 LectureMathematical Modeling and Engineering Problem Solving 1Introduction to Matlab 2Numerical Error 3Root Finding 4-5-6System of Linear Equations 7-8Finite Difference 9Least Square Curve FittingPolynomial Interpolation Numerical IntegrationOrdinary 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

hxfhxfxfxf

hxfhxfxfxf

hxfhxfxfxf

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)()(

hxfxfxfxf

hxfxfxfxf

iiii

iiii

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• The second centered finite divided difference which has an error of order O(h2) is

211 )()(2)()(

hxfxfxfxf iii

i

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• 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

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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.

hxfxfxfxf

hhxfxfxf

hxfxfxf

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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Estimate 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

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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()(

ffffxf i

•In substituting the values:

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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

∆ym,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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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

-1000System of Linear Equations

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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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Fit the data with multiple linear regression

x1 x2 y0 0 52 1 102.5 2 97 3 04 6 35 2 27

1005.243

54

5448144825.765.16145.166

2

1

0

aaa

3,4,5 210 aaa

21 345 xxy

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Use the polyfit function

Regression in ExcelUse Add Trendline

Regression in Matlab

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Homework7 www.bc.inter.edu/facultad/omeza

Omar E. Meza Castillo Ph.D.

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