METHOD AND CODE FOR THE EVALUATION OF THE TOTAL INTERSECTION VOLUME OF TWO ARBITRARILY LOCATED IN SPACE HEXAHEDRA WITH NONPLANE FACES

V.I. Delov, L.V. Dmitrieva, V.V. Sadchikov
VANT. Ser.: Mat. Mod. Fiz. Proc 1997. Вып.1. С. 47.

      The report is devoted to the description of the method developed for the calculation of spatial problem of the intersection of two arbitrarily located in space; hexahedra and evaluation of their total volume of intersection, if any. The need in such algorithms and codes occurs particularly in the calculation of gas-dynamic problems in Lagrangian coordinates. It is known that in this case the numerical calculation of complicated problems necessitates one-shot grid reconfiguration and rescaling the values to the new grid because further computations are impossible. These situations are encountered, for example, in the event of strong grid distortions, pinch of physical domains, formation of small-scale jet flows.
      The calculation of the problems presented here is also addressed when creating Lagrangian-Eulerian methods for continuum mechanics problems with strong deformations. In addition, such algorithms are extremely needed to create the graphics packages and codes, oriented to the operation with 3-D geometrical objects.
      The problem solution reduces to finding the exact intersection volume of two tetrahedra arbitrarily located in space.
      The methods are considered to split the polyhedra into elementary tetrahedra together with the main program implementation features for the algorithms proposed. The most computationally cost efficient ways are reported to divide the polyhedra without additional computational errors.
      The report presented the results of test computations for the intersection of hexahedra with the faces representing second order surfaces-hyperbolic paraboloids that are used in 3-D “D” code.

      The report proposed an approach to the construction of conservative differential-difference representations of equations describing nonstationary elastic-plastic flows in Lagrangian variables. The method is the further development of 2-D method for the generation of spatial approximations to the equations of motion in gas dynamics [1,2] for elastic-plastic media.
      In this work the matrix of kinetic energy determining the approximation technique for the pressure gradient is taken in the canonical form that is traditionally used in gas-dynamic codes.
      The report presented the difference formulas for the components of deformation rate tensor and the resulting difference approximations for the evaluation of derivatives with respect to the components of the stress deviator.
      The computational results were reported obtained with difference schemes where the grid distribution of quantities in time is taken like in “D” code [3] and the time derivative is approximated with the second order accuracy. The problem describing the elastic oscillations of membrane is taken to show unquestionable advantages of the resulting difference schemes as compared to the classical Wilkins scheme.
      1. Isaev V.N., Sofronov I.D. Construction of discrete models for gas-dynamic equations based on the interconversion law of kinetic and internal energies of continuum // Voprosy Atomnoy Nauki i Tekhniki. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. N 1(15). P. 3-7.
      2. Delov V.I., Isaev V.N., Sofronov I.D. Conservative and invariant differential-difference representations of gas- dynamic equations in cixisymmetric case // Ibid. 1987. N 1. P. 3-10.
      3. Dmitriev N.A., Dmitrieva L.V., Malinovskaya E.V., Sofronov I.D. A method for the calculation of 2-D gas-dynamic problems in Lagrangian variables // Theoretical Fundamentals and Construction of Numerical Algorithms in Computational Physics / Ed. by Babenko K.I. M.: Nauka, 1979. P. 175-200.