Computational Methods Autar Kaw University of South Florida

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Presentation transcript:

Computational Methods Autar Kaw University of South Florida Bring it together Computational Methods Autar Kaw University of South Florida

Why Do We Use Numerical Methods?

Why use Numerical Methods? To solve problems that cannot be solved exactly

Why use Numerical Methods? To solve problems that have exact solutions but are otherwise intractable!

A Bascule Bridge Opening

Bascule Bridge THG

Bascule Bridge THG Hub THG picture Trunnion Girder

Trunnion-Hub-Girder Assembly Procedure Step1. Trunnion immersed in dry-ice/alcohol Step2. Trunnion warm-up in hub Step3. Trunnion-Hub immersed in dry-ice/alcohol Step4. Trunnion-Hub warm-up into girder

After Cooling, the Trunnion Got Stuck in Hub Problem After Cooling, the Trunnion Got Stuck in Hub

Why did it get stuck? Magnitude of contraction needed in the trunnion was 0.015” or more. Did it contract enough?

Video of Assembly Process http://nm.MathForCollege.com

What model should I use to calculate contraction of trunnion?

Finding the fluid temperature to get enough contraction

Finding the expression for thermal expansion coefficient

What is the temperature of the trunnion after half an hour?

How long does it take a trunnion to cool down?

What is the rate of change of heat stored in the cylinder?

Using central divided difference, the true error in the calculation of a derivative of a function is 32.0 for a step size of 0.4. If the step size is reduced to 0.1, the true error will be approximately 2.0 4.0 8.0 16.0

Given the f (x) vs x curve, and the magnitude of the areas as shown, the value of y x a 5 7 2 b c -2 2 12 Cannot be determined

A scientist finds that regressing y vs x data given below to straight-line y=a0+a1x results in the coefficient of determination, r2 for the straight-line model to be one. x 1 3 11 17 y 2 6 22 ? The missing value for y at x=17 most nearly is -2.444 2.000 6.889 34.00

The distance covered by a rocket from t=8 to t=34 seconds is calculated using multiple segment trapezoidal rule by integrating a velocity function. Below is given the estimated distance for different number of segments, n. n 1 2 3 4 5 Value 16520 15421 15212 15138 15104 The number of significant digits at least correct in the answer for n=5 is 1 2 3 4

Polynomials are most commonly used functions for interpolation because they are easy to evaluate differentiate integrate all of the above

Using 3 significant digit with chopping at all stages, the result for the following calculation is -0.0988 -0.0978 -0.0969 -0.0962

The velocity of a body is given by Then the distance covered by the body from t=0 to t=10 can be calculated by solving the differential equation for x(10) for A) B) C) D)

The root of equation f (x)=0 is found by using Newton-Raphson method The root of equation f (x)=0 is found by using Newton-Raphson method. The initial estimate of the root is xo=3, f (3)=5. The angle the tangent to the function f (x) at x=3 makes with the x-axis is 57o. The next estimate of the root, x1 most nearly is -3.2470 -0.2470 3.2470 6.2470 http://nm.mathforcollege.com

Resources http://nm. MathForCollege. com http://YouTube Resources http://nm.MathForCollege.com http://YouTube.com/numericalguy http://AutarKaw.com Contact: kaw@usf.edu

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