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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 9b Numerical Integration: Gaussian Quadrature

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© William Klug Numerical Integration • Quadrature: – Approximate integrals by sampling. • Recall: Midpoint rule, Trapezoidal rule, Simpson’s rule
© William Klug Generic Quadrature Formula I = f dX X 1 X 2 = φ d ξ 1 1 fdX = f X ( ) ( ) Jd = φξ ( ) d I = d 1 1 w 1 1 ( ) + w 2 2 ( ) + + w n n ( ) = w l l ( ) l = 1 n Idea:

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© William Klug Gauss Quadrature • A method to exactly integrate polynomials of order k , denoted , where – Example: cubic or less gets computed exactly P k ( ξ ) = c 0 + c 1 + + c k k d 1 1 d 1 1 2 d 1 1 3 d 1 1

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