Method of numerical calculation of optimal forms of bodies of revolution at movement in soil medium

The method of calculating the optimal forms of axisymmetric projectiles during motion in soil medium on the base of direct optimization method is developed. The methods of local variations and cyclic by coordinate descent are used. Direct numerical calculations are carried out in an axisymmetric formulation. The form of a optimal body found on the basis of a model of local interaction is taken as an initial approximation. A local interaction model (LIM) is used below, based on the analytical solution of the one-dimensional problem of the expansion of a spherical cavity in a Grigoryan soil medium assuming that the medium behind the shock wave front is incompressible. The main assumptions made in solving the problem of optimizing the shape of axisymmetric bodies within the framework of the LIM, i.e., that a model that is quadratic in the velocity is applicable, the friction is proportional to the pressure and flow over the body occurs without separation, have been confirmed theoretically and experimentally earlier. The applicability of the LIM in describing the penetration of sharp cones has been demonstrated experimentally and theoretically by comparison with the results of numerical calculations in an axisymmetric formulation using a Grigoryan soil medium model. The errors in the LIM in determining the drag forces when applied to blunt bodies have been shown too. In this work the effectiveness of the developed methodology is shown in the problem determining the shape of minimum resistance projectile among the bodies of revolution having a set length and radius of the cross section. Good agreement has been reached between the results for the generatrix of a body of revolution in the form of a parametric Bezier polynomial and a piecewise linear curve. Convergence of successive approximation methods for the solution of a parametric optimization problem is studied. The essential role of two-dimensional flow effects was revealed.