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Bring it together

Bring it together. Computational Methods EML3041. 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!.

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Bring it together

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  1. Bring it together Computational Methods EML3041

  2. Why do we use numerical methods?

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

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

  5. Assembling a fulcrum for a bascule bridge Girder Trunnion Hub

  6. What model should I use to calculate contraction of trunnion?

  7. Finding the fluid temperature to get enough contraction

  8. Finding the expression for thermal expansion coefficient

  9. What is the temperature of the trunnion after half an hour? http://numericalmethods.eng.usf.edu

  10. How long does it take a trunnion to cool down? http://numericalmethods.eng.usf.edu

  11. Example A solid steel shaft at room temperature of 27°C is needed to be contracted so that it can be shrunk-fit into a hollow hub. It is placed in a refrigerated chamber that is maintained at −33°C. The rate of change of temperature of the solid shaft q is given by Using Euler’s method, find the temperature of the steel shaft after 86400 seconds. Take a step size of h = 43200 seconds. http://numericalmethods.eng.usf.edu

  12. Temperature vs. Time http://numericalmethods.eng.usf.edu

  13. What is the rate of change of heat stored in the cylinder? http://numericalmethods.eng.usf.edu

  14. Can you identify? • We talked about 7 mathematical procedures in this course. Write them down. Now connect the problem just discussed to these 7 mathematical procedures.

  15. THE END

  16. To ensure convergence in solving a nonlinear equation f(x)=0, you would use • secant method because we choose two initial guesses for the root. • bisection method because the root is bracketed. • graphical method because one can visualize the function f(x). • Newton-Raphson method because its convergence rate is fast.

  17. As water is drained out of a bathtub, the volume of water, V and its depth in the tub, h change with time, t. Which of the following is true of dV/dh and dV/dt? • They must be equal. • They must have opposite signs. • Both must be negative. • Both must be positive

  18. A ninth order polynomial with all nonzero real coefficients can have the following type of zeros? • Only one real zero. • At least one real zero. • No real zeros. • Odd number of complex roots.

  19. The Newton-Raphson method of finding real roots of the nonlinear equation f(x)=0 can be derived by using the first two terms of the Taylor series. One would then expect that adding one more term of the Taylor series to the Newton-Raphson method would result in higher accuracy of the results for the same number of iterations as the Newton-Raphson method. However, this new approach of adding one more term of Taylor series is not feasible because • one has to solve a quadratic equation to get the new estimate of the root of the equation. • the estimates of the root of the equation may turn out to be complex numbers • one needs to find the first derivative of f(x). • one needs to find the second derivative of f(x).

  20. It is desired to compute the following matrix product, P×Q×R×S. The sizes of the four matrices are: P is 1×4, Q is 4×20, R is 20×5, and S is 5×1. Amongst the choices given below, the most efficient (the least number of floating point operations of add, subtract, multiplication and divide) way to compute the matrix product is • (P × Q) × (R × S) • ((P × Q) × R) × S • P × ((Q × R) × S) • (P × (Q × R)) × S

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