Constraints algorithms

Ku, D. and De Micheli, G. (1992) Relative scheduling under timing constraints Algorithms for highlevel synthesis of digital circuits. IEEE Trans Comput Aided Des, 11 (6), 696-718. 

The intersection will also be a convex hull, as the intersection of two convex hulls is again a convex hull. Therefore, the basic operations of gamut-constraint algorithms involve computing the convex hull of a set of points and then intersecting these hulls. 

Figure 6.16 Results obtained using Forsyth s gamut-constraint algorithm. The algorithm assumes a single illuminant for the entire image, so we cannot use this algorithm if multiple illuminants are present. For both images, the RGB cube was assumed to be the canonical gamut.

Note that this set may be empty. Since several maps may be allowed, we have to choose some method that selects the map that describes the given illuminant best. The two-dimensional gamut-constraint algorithm chooses the map m... 

Figure 6.22 Results obtained using a two-dimensional gamut-constraint algorithm with the assumption that the illuminant can be modeled as a black-body radiator. The two graphs again show the possible illuminants as CIE chromaticities and the illuminant that was selected by the two-dimensional gamut-constraint algorithm.

The two-dimensional gamut-constraint algorithm first projects all colors c onto the plane at b = 1. Because both the sample and the background are achromatic, the results will be independent of the reflectance R of the sample. 

Gamut-constraint algorithm 2D, no constraint placed on the illuminant, maximum trace selected. 

Gamut-constraint algorithm 2D, illuminant assumed to be a black-body radiator. 

We find that all the elements of the symmetric traceless pressure and the antisymmetric pressure are linear functions of sin26 or cos20. One can consequently calculate all the shear viscosities and all the twist viscosities by using the director constraint algorithm to fix the director at various angles relative to the stream lines and calculating the pressure tensor elements as function of... 

They can be evaluated directly by using the director constraint algorithm to fix the director in the desired direction or one can evaluate the fluctuation relations for the various viscosity coefficients and substituting the values into the above expressions. We finally note that we have not developed any simulation algorithms for Tjy, or K. 

The director constraint algorithm makes it very easy to calculate the Mies-owicz viscosities. One simply fixes the director in the desired direction and calculates the shear stress. In a liquid crystal consisting of prolate ellipsoids one has % > It is easy to realise that T7j must be the smallest viscosity... 

There is one major technical problem that must be overcome In a liquid crystal most properties are best expressed relative a to a director based coordinate system. This is not a problem in a macroscopic system where the reorientation rate of the director is virtually zero but it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. When NEMD methods are applied the fictitious mechanical field exerts torques that twist the director and might make it impossible to reach a steady state. This problem has been solved by devising a Lagrangian constraint algorithm that fixes the director in space so that a director based coordinate system becomes an inertial frame. 

We thus conclude the section on the numerical implementation of SLLOD dynamics for two very important and useful ensembles. However, our work is not yet complete. The use of periodic boundary conditions in the presence of a shear field must be reconsidered. This is explained in detail in the next section. Furthermore, one could imagine a situation in which SLLOD dynamics is executed in conjunction with constraint algorithms for the internal degrees of freedom and electrostatic interactions. An immediate application of this extension would be the simulation of polar fluids (e.g., water) under shear. This extension has been performed, and the integrator is discussed in detail in Ref. 42. 

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