Cutting planes extended

Westerlund, T. H. Skrifvars I. Harjunkoski, and R. Pom. An Extended Cutting Plane Method for a Class of Non-convex MINLP Problems. Comput Chem Eng 22 357-365 (1998). 

Westerliind, T. and Lundqvist, K. (2003) Alpha-ECP, Version 5.04. An interactive MINLP-solver based on the extended cutting plane method Report 01-178-A, Process Design Laboratory, Abo Akademi University Abo, Finlande. 

To solve the problem above the branch and bound method (see e.g. [106]), generalized Benders Decomposition [108], Outer Approximation [109, 110], LP/NLP branch and bound [111] and Extended Cutting Plane Method [112] are in use. Grossmann and Kravanja [113] give an extensive compilation of literature on MINLP problems. 

The methods discussed in this section can be extended to systems that have more than three components. The problem is to convert each system to a pseudobinary system. For a quaternary system, the properties of an equilateral tetrahedron may be used to depict the composition of the system. The composition axes would be four lines drawn from the four apexes perpendicular to the opposite faces. Planes cutting the tetrahedron parallel to the bases would represent pseudoternary systems for which one composition variable would be constant. Pseudobinary systems would be depicted by the intersections of two of the pseudoternary planes. Indeed, the experimental measurements and calculations would be extensive. 

Figure 2.10 Cylindrically symmetric hydrodynamical model of accretion flow with rotation during the early collapse phase, showing the inflow of matter in the meridional plane and the build-up of a flat rotating disk structure after about 1.05 free-fall times. Arrows indicate matter flow direction and velocity, gray lines indicate cuts of isodensity surfaces with meridional plane. Dark crosses outline locations of supersonic to subsonic transition of inflow velocity this corresponds to the position of the accretion shock. Matter falling along the polar axis and within the equatorial plane arrive within 1600 yr almost simultaneously, which results in an almost instantaneous formation of an extended initial accretion disk [new model calculation following the methods in Tscharnuter (1987), figure kindly contributed by W. M. Tscharnuter],...

The symmetry treatment of incommensurate structures is beyond the scope of this chapter. From Equation (33) it is readily seen that for indexing, whatever the reflection of the diffraction pattern of an incommensurately modulated structure, we need to specify 3 + d integers (h, k, I, m, m2... m fl. It can be demonstrated that the observed 3D structure can be considered as a projection of a periodic structure m3 + d dimensions over the real 3D space, which is a hyper-plane not cutting the points of the 3 + d lattice except the origin. The superspace approach of de Wolff, Janssen and Janner is now well established and has become the routine way of treating the symmetry of the displacive incommensurate structures. The same approach has been extended to study general quasiperiodic structures (composite structures and quasicrystals). 

Given a tube, axis Oz (unit vector e ) in which a volume V is cut by a fixed cross section plane over area, A, as sketched in Fig. 1.4. The lateral dimensions of the control volume extend to the conduit walls. In this notation, n, is the outward directed unit vector normal to the surface of the control volume, n is the outward directed unit vector normal to the closed curve, lw t,z),... 

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