9b-Quadrature

9b-Quadrature - William Klug Quadrature Weights and...

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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
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© 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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© William Klug Quadrature Weights and Quadrature Points • For n = 1 Linear
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© William Klug Quadrature Weights and Quadrature Points • For n = 2 Cubic
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© William Klug Quadrature Weights and Quadrature Points
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Unformatted text preview: William Klug Quadrature Weights and Quadrature Points William Klug Quadrature Weights and Quadrature Points William Klug Quadrature Weights and Quadrature Points 1 0 2 1 2 -0.57735027 0.57735027 1.0 1.0 3 3 -0.77459667 0.0 0.77459667 0.55555555 0.88888889 0.55555555 5 4 -0.86113631 -0.33998104 0.33998104 0.86113631 0.34785485 0.65214515 0.65214515 0.34785485 7 5 -0.90617975 -0.53846931 0.0 0.53846931 0.90617975 0.23692689 0.47862867 0.56888889 0.47862867 0.23692689 9 k 2 n 1 William Klug Example: Quadratic Polynomial This result is exact William Klug Summary and Review I = f dX X 1 X 2 = d 1 1 fdX = f X ( ) ( ) Jd = ( ) d Idea:...
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9b-Quadrature - William Klug Quadrature Weights and...

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