Bezier

Consistent with the notion of approximating polynomials as in the preceding paragraph, finite-element approaches attempt to simplify the solution process by carefully choosing polynomials so as to minimize the number of coefficients required to determine and simplify the matrix inversion problem. One choice is the Bezier polynomials, defined by... 

Curves resulting from the choice of Bezier functions blend the values of the known boundary potentials to produce interior potential values and have the appropriate smoothness properties desired in the final solution. Further, the 5, have maximum values that distribute evenly through the mesh regions. For instance, for u between 0 and j in Eq. (15.20), the value of 5q,4 is greatest, and all other B variables approach minimum values. Thus, Bq,4 serves to sample that particular range of u values. 

Equation (15.23) displays the feature of locality that the blending functions should possess in order to be computationally advantageous that is, during the process of matrix inversion, one wishes the calculation to proceed quickly. As mentioned earlier, the use of linear approximation functions results in at most five terms on the left side of the equation analogous to (15.23), yielding a much crader approximation, but one more easily calculated. The current choice of Bezier functions, on the other hand, is rapidly convergent for methods such as relaxation, possesses excellent continuity properties (the solution is guaranteed to look and behave reasonably), and does not require substantial computation. 

Trosset J-Y, Scheraga HA (1999) Flexible docking simulations scaled collective variable Monte Carlo minimization approach using Bezier splines, and comparison with a standard Monte Carlo algorithm. J Comp Chem 20 244-252... 

D. Bezier, "Methodes Modemes d Analyse Quantitative Min rale, Masson Cie, Paris(1955) 16)W. W.Meinke, Science 121,... 

G.Chariot D.Bezier, "Methodes Modernes d Analyse Quantitative Mine rale/ Masson, Paris(1955) 26)R.Belcher C.L.Wilson, New Methods in Analytical Chemistry, Reinhold,NY(1955) 27)A.F.Dagett W.B. Meldrum, "Quantitative Analysis/1 Heath, Boston(1955) 28)M.G.MelIon, "Quantitative Analysis, 1 Crowell,NY(195 5) 29)W. Wagner, C.J.HaIl G.E.Markle, "Advanced Analytical Chemistry, Reinhold,NY(1956) 30)S.Siggia H.J.Stolten, "An Introduction... 

D.Y.Curtin, "The Systematic Identification of Organic Compounds, Wiley, NY(1956) 32)G.Charlot D.Bezier, "Quantitative Inorganic Analysis,1 translated from the French by R.C.Murray, Methuen Co,London (1957) 33)L.Meites, H.C.Thomas R.P. Bauman, "Advanced Analytical Chemistry, McGraw-Hill,NY(1958) 34)L.F.Hamilton ... 

G.Chariot 8c D.Bezier, "Quantitative Inorganic Analysis, Methuen, London (1957)... 

Gachot BA, Bezier M, Cherrier JF, Daubeze J. Cibenzoline and hypoglycaemia. Lancet 1988 2(8605) 280. 

Global Minimum in Docking Simulations A Monte Carlo Energy Minimization Approach Using Bezier Splines. 

D. Bezier, "Methodes Modemes d Analyse Quantitative Min6rale, Masson Cie, Paris(1955) l6)W.W.Meinke, Science 121, 177-8 1955) and Rus translation by N.G. Polianskii, UspekhiKhimii 15,770-80(1956) (Trace element sensitivity, comparison of activation analysis with other methods)... 

Bezier, A., Lambert, B., Baillieul, R (2002). Study of defense-related gene expression in grapevine leaves and berries infected with Botrytis cinerea. Eur. J. Plant Pathol., 108, 111-120 Blade, W.H., Boulton, R. (1988). Adsorption of protein by bentonite in a model wine solution. Am. J. Enol. Vitic., 39, 193-199... 

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