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**Computational Modeling for Engineering MECN 6040**

Professor: Dr. Omar E. Meza Castillo Department of Mechanical Engineering

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**Best known numerical method of approximation**

Finite differences Best known numerical method of approximation

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**FINITE DIFFERENCE FORMULATION OF 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 big question: How good are the FD approximations?**

This leads us to Taylor series....

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**EXPASION OF TAYLOR SERIES**

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

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General Taylor Series 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=∆x What does this mean in plain English? 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”

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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,

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**The Taylor Series (xi+1-xi)= h step size (define first)**

Reminder term, Rn, accounts for all terms from (n+1) to infinity.

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**Zero-order approximation**

First-order approximation Second-order approximation

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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

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**Zero-order approximation**

First-order approximation

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**Second-order approximation**

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**Third-order approximation**

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**Fourth-order approximation**

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**Taylor Series to Estimate Truncation Errors**

If we truncate the series after the first derivative term Truncation Error First-order approximation

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**Numerical Differentiation**

Forward Difference Approximation

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**Numerical Differentiation**

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). 18

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**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). 19

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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). 20

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**Numerical Differentiation**

Forward Difference Approximation

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**Backward Difference Approximation**

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**Centered Difference Approximation**

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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 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

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**Forward Difference Approximation**

Backward Difference Approximation

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**Centered Difference Approximation**

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h=0.25 xi-1=0.25 - f(xi-1)= xi=0.5 - f(xi)=0.925 Xi+1=0.75 - f(xi+1)= Forward Difference Approximation

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**Backward Difference Approximation**

Centered Difference Approximation

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**FINITE DIFFERENCE APPROXIMATION OF HIGHER DERIVATIVE**

The forward Taylor series expansion for f(xi+2) in terms of f(xi) is Combine equations: 29

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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 30

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

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**High-Accuracy Differentiation Formulas**

High accurate estimates can be obtained by retaining more terms of the Taylor series. High-Accuracy Differentiation Formulas The forward Taylor series expansion is: From this, we can write 32

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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. 33

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Forward Formulas 34

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Backward Formulas 35

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Centered Formulas 36

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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 From Tables 37

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**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 39

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**Introduction to Finite Difference**

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. (i,j)

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**y y u y x x x Discretization: PDE FDE Explicit Methods**

Simple No stable Implicit Methods More complex Stables ∆x y n+1 ∆y y n u m,n y n-1 x x x m-1 m m+1

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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 THE Finite difference EQUATIONS**

Heat Transfer Solved Problems

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**The Matrix Inversion Method**

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**jacobi ITERATION method**

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**GAUSS-SEIDEL ITERATION**

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**Pre-specified % tolerance based on the knowledge of your solution**

Error Definitions Use absolute value. Computations are repeated until stopping criterion is satisfied. If the following Scarborough criterion is met Pre-specified % tolerance based on the knowledge of your solution

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**Matrix Inversion Method**

USIG EXCEL Matrix Inversion Method =MMULT(A7:C9,E2:E4) =MINVERSE(A2:C4)

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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. (1,3) (2,3) (3,3) (1,2) (2,2) (3,2) (1,1) (2,1) (3,1) Ts=300 K

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**System of Linear Equations**

-4 1 -800 -500 -1000 -300 = 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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PART 7 Ordinary Differential Equations ODEs

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