Numerical solutions of partial differential equations in three dimensions often use hexahedral cells composing a computational grid. After the grid is constructed, it is first of all verified whether the grid is unfolded or not. For such a test in this paper, conditions of nondegeneracy are found for hexahedral cells which are given by eight corner points and generated by the trilinear map from a unit cube to a region defined by these points. The conditions include necessary conditions and two sets of sufficient conditions. They are found as conditions of positivity of the Jacobian of the trilinear map. Thus, the conditions which guarantee the invertibility of the trilinear map from a unit cube to a hexahedron are given. How general the nondegeneracy conditions are is shown by a numerical experiment. Formulas of the Jacobian of the trilinear map are obtained, and following from them, as a separate result, a formula of a volume of a cell is obtained. Key words. grid generation, hexahedral cells, positivity of the Jacobian of the trilinear map, conditions of nondegeneracy of cells, formula of a volume of a cell AMS subject classifications. 65M50, 65M06, 65M60PII. S1064827500380702

The experimental evidence on depressurization foaming of the amorphous D,L-polylactide, which is plasticized by subcritical (initial pressures below the critical value) or supercritical (initial pressures above the critical value) carbon dioxide at a temperature above the critical value, relates to two extreme cases: a slow quasi-isothermal foam expansion, and a rapid quasi-adiabatic expansion. Under certain conditions, the quasi-isothermal mode is characterized by the non-monotonic dependencies of the foam volume on the external pressure that are associated with the expansion-to-shrinkage transition. The quasi-adiabatic and quasi-isothermal expansions are characterized by a significant increase in the degree of foam expansion under conditions where the CO2 initial pressure approaches the critical value. The observed features are interpreted in terms of the energy balance in the foam volume and the phenomenological model based on the equation of the foam state. The expansion-to-shrinkage condition is based on the relationship between the average bubble radius and the pressure derivative of the surface tension for the plasticized polylactide. The maximum expansion ratio of the rapidly foamed polylactide in the vicinity of the critical point is interpreted in terms of the maximum decrement of the specific internal energy of the foaming agent (carbon dioxide) in the course of depressurization.

No abstract

No abstract

No abstract