Chemical Engineering Hand Written_Notes_Part_114

Chemical Engineering Hand Written_Notes_Part_114 - 230 5....

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230 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 9 At he boundary points of the element, we have b u i ( z i 1 )= b u i 1 = a i + b i z i 1 (10.49) b u i ( z i b u i = a i + b i z i (10.50) This implies (10.51) a i = b u i 1 z i b u i z i 1 z ; b i = b u i b u i 1 z Thus, the polynomial on the i’th segment can be written as b u i ( z b u i 1 z i b u i z i 1 z + μ b u i b u i 1 z z (10.52) z i 1 z z i for i =1 , 2 ,...n 1 andnoww ecanw o rkint e rm so funknownv a lu e s { b u 0 , b u 1 , .... b u n } instead of parameters a i and b i. A more elegant and useful form of equation (10.52) can be found by de f ning shape functions (see Figure 9) (10.53) M i ( z z z i z ; N i ( z z z i 1 z The graphs of these shape functions are straight lines and they have fundamental properties M i ( z ( 1; z = z i 1 0; z = z i ) (10.54) N i ( z ( z = z i 1 z = z i ) (10.55) This allows us to express b u i ( z ) as b u i ( z b u i 1 M i ( z )+ b u i N i ( z ) i , 2 1
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10. NUMERICAL METHODS BASED ON OPTIMIZATION FORMULATION 231 Figure 10 Note that the coe cient b u i appears in polynomials b u i ( z ) and b u i +1 ( z ) , i.e. b u i ( z )= b u i 1 M i ( z )+ b u i N i ( z ) b u i +1 ( z b u i M i +1 ( z b u i +1 N i +1 ( z ) Thus, we can de f ne a continuous trial function by combining N i ( z ) and M i +1 ( z ) as follows v ( i ) ( z N i ( z z z i 1
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_114 - 230 5....

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