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# hw5_09 - (c using Gaussian quadrature over intervals of...

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03/29/09 EP 471 – Homework #5 Each problem is equally weighted: (1) It is stated without proof in Exercise 14 that Gaussian quadrature of order N produces an exact result when applied to the integration of a polynomial of order 2 N 1. Consider the following three polynomials in the interval 4 1 x : (a) 2 / 2 1 ) ( 3 2 1 x x x x f + = (b) 200 / 7 / 2 1 ) ( 5 4 3 2 x x x x x f + = (c) 1000 / 500 / 10 / 4 / 3 4 1 ) ( 7 6 5 4 3 3 x x x x x x x f + + = Integrate (a) through (c) over the interval 4 1 x using Gaussian quadrature of N = 2, 3 and 4 respectively and compare your results to the analytical values. Does the quadrature of the appropriate order produce an exact match? (2) In the previous homework assignment, we looked at a function describing a combination of background absorption and resonance behavior over the interval 10 0 x that had the form: 2 2 2 ) 4 ( 50 1 100 ) 2 ( 100 1 150 ) 1 ( 200 1 200 1 100 ) ( + + + + + + + = x x x x x a σ Integrate this function over the interval: (a) analytically (b) using a trapezoidal rule and fixed interval size
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Unformatted text preview: (c) using Gaussian quadrature over intervals of variable size (you tell me the intervals and the order of the Gaussian quadrature you’ve used) Keep track of the number of times you evaluate the integrand in parts (b) and (c). Compare the efficiency of the method of part (c) with that of part (b) by comparing the number of evaluations of the integrand required to achieve the same level of accuracy. (3) In Exercise 15, we examined the work-equivalent conversion of a uniform pressure distribution over an irregularly-shaped Q8 element. Suppose now that the load distribution over the element face is described by: 2 2 ) , ( y x y x p + = Find the work-equivalent loads at the corner and midside nodes of the irregular element given in Exercise 15 for this load distribution....
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