lecture 17

lecture 17 - CE 30125 Lecture 17 LECTURE 17 NUMERICAL...

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CE 30125 - Lecture 17 p. 17.1 LECTURE 17 NUMERICAL INTEGRATION • Find or • Often integration is required. However the form of may be such that analytical integration would be very difficult or impossible. Use numerical integration techniques. • Finite element (FE) methods are based on integrating errors over a domain. Typically we use numerical integrators. • Numerical integration methods are developed by integrating interpolating polyno- mials. If x  x d a b = x y y d ux vx x d a b = fx
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CE 30125 - Lecture 17 p. 17.2 Trapezoidal Rule • Trapezoidal rule uses a first degree Lagrange approximating polynomial ( , nodes, linear interpolation). 76 • Define the linear interpolating function • Establish the integration rule by computing N 1 = N 1 +2 = f 0 f 1 x 0 x 1 f (x) g (x) gx  f o x 1 x x 1 x o ---------------   f 1 xx o x 1 x o + = If x x d x o x 1 ex + x d x o x 1 ==
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CE 30125 - Lecture 17 p. 17.3 • Trapezoidal Rule Ig x  xE + d x o x 1 = If o x 1 x x 1 x o ---------------   f 1 xx o x 1 x o + + d x o x 1 = o x 1 x x 2 2 ---- x 1 x o -------------------  f 1 x 2 2 x o x x 1 x o + x 1 x o E + = o x 2 1 x 2 1 2 ------- x 1 x o --------------------- f 1 x 2 1 2 x o x 1 x 1 x o ------------------------ f o x 1 x o x o 2 2 x 1 x o f 1 x o 2 2 x o 2 x 1 x o ---------------- E + + = I x 1 x o 2 f o f 1 +  E + =
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CE 30125 - Lecture 17 p. 17.4 • Trapezoidal Rule integrates the area of the trapezoid between the two data or interpola- tion points. • Evaluating the error for trapezoidal rule. • The error is dependent on the integral of the difference . However integrating the dependent error approximation for the interpolating function does not work out in general since is a function of ! • We must express in terms of a series of terms expanded about in order to evaluate correctly. • An alternative strategy is to evaluate by developing Taylor series expansions for , and . • We do note that as E E ex  fx gx = x x o Ee x x d x o x 1 = Ef x x x 1 x o 2 ---------------   f o f 1 + d x o x 1 = f o f 1 x 1 x o h =
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CE 30125 - Lecture 17 p. 17.5 Evaluation of the Error for Trapezoidal Rule Evaluation of the error by integrating e(x) • We note that •How ev e r •Thu s • Recall that for Lagrange interpolation was expressed as: EI g x  x d x o x N Ef x xg x x d x o x N d x o x N ex fx gx x d x o x N x x d x o x N d x o x N = Ee x x d x o x N = xx o 1 N N 1 + ! --------------------------------------------------------------- f N 1 + =
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CE 30125 - Lecture 17 p. 17.6 •No t e s • Procedure applies to higher order integration rules as well. • In general is a function of • Neglecting the dependence of , can lead to incorrect results. e.g. for Simpson’s rule you will integrate out the dependent term and the result would be !
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lecture 17 - CE 30125 Lecture 17 LECTURE 17 NUMERICAL...

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