Four-point subdivision

N.Dyn, M.S.Floater, K.Hormann A C2 four-point subdivision scheme with fourth order accuracy and its extensions. ppl45-156 in Mathematical Methods for Curves and Surfaces, (eds Daehlen, Morken and schumaker), Nashboro Press 2005 ISBN 0-9728482-4-X... 

U. H.Augsdorfer, N.A.Dodgson and M.A.Sabin Variations on the four-point subdivision scheme. CAGD 27(1), pp78-95, 2010 P.Costantini and C.Manni A geometric approach for Hermite subdivision. Numer.Math. 2010... 

Consider the four point scheme again. Its kernel is ( — 1+4z — z2) and the kernel subdivision matrix is... 

S.Daniel and P.Shunmugaraj An approximating C2 non-stationary subdivision scheme. CAGD 26(7), pp810-821, 2009 N.Dyn, M.S.Floater and K.Hormann Four-point curve subdivision based on iterated chordal and centripetal parameterizations. CAGD 26(3), pp279-286, 2009... 

Nevertheless, near the feed point, there is a difference between TMB and all SMB cases due to the fact that the internal flow rates in the TMB are smaller than in the SMB, leading to a small dilution of the feed stream. As a consequence, near the feed inlet, TMB concentrations will be higher than in the SMB operation. The raffinate and extract purities in SMB units with four (95.2 % and 89.5 %), eight (98.7 % and 95.9%) and 12 columns (99.1 % and 96.8%) are increasing towards the one obtained in the equivalent TMB unit (99.3 % and 97.7 %). The optimum degree of subdivision of the SMB unit will depend of the difficulty of the separation and the product purity requirements. Typically, systems for the pharmaceutical industry have six to 16 columns. 

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