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# NEW IDEAS IN THE MULTIDIMENSIONAL PHASE DIAGRAMS SIMULATION

In many cases the hypersurfaces of the multidimensional phase diagram may be depicted by a one-dimensional (1-D) linear contour using equations of skewed hyperplanes. Some of them simulate the diagram's ruled hypersurfaces with their generating simplexes parallel to the diagram's base. When liquids, solids and solvus hypersurfaces were approximated, a generating simplex of skewed hyperplane wasn't horizontal. In T-x-y diagrams this simplex is 1-D and is a second power hyperbolic paraboloid equation, which describes two ruled surfaces with the different move along four points on its contour.
In T-x-y-z diagrams two types of the skewed hyperplanes are used: 1-D and 2-D generating simplexes. The unruled hypersurfaces, depicted in the x-y-z projection as hexaedroids, are a six power equation derived when 1-D simplex slides along two skewed planes of the 4-D space. The same equation simulate ruled hypersurfaces generated by a horizontal segment of three-phase equilibrium in binary systems. As four points are on the contour of each directing skewed plane of this skewed "hexaedroidal" hyperplane, then four six power equations describe four skewed hyperplanes. A three power equation of skewed hyperplane with2-D generating simplex and three directing lines simulate other type of ruled hypersuface produced by four-phase equilibrium in ternary systems. Its x-y-z projection resembles a prism or truncated pyramid. Its side faces may be depicted by two different skewed planes with six equations of third power calculated by six points coordinates for six different skewed hyperplanes with the same generating triangle but with different directing lines.
The main stages of a heterogeneous design for quaternary systems with the eutectic type of interaction and without solubility in solids consists of 29 hypersurfaces: 4 - liquids, 12 - ruled hypersurfaces with generating segment, 12 - ruled hypersurfaces with generating triangle and a horizontal hyperplane. On the basis of explicit equations for four liquids hypersurfaces any section of heterogeneous fields are constructed. Mass fractions for any microstructure element of heterogeneous mixtures are determined. New equations of hypersurfaces are used to simulate and visualize analogous T-x-y-z diagrams but with the solubility in solids which consists of 71 hypersurfaces.