Homogeneous difference schemes for the coupled problems of hydrodynamics and elasticity

Finite-difference approximations of elastic forces on the staggered moving grid were constructed. For the displacement vectors at the irregular grids in which topological and geometrical structures are subjected to minimal reasonable restrictions, with regard to the finite-difference schemes of the elasticity theory problems, approximations of the vector analysis operators in plane and cylindrical geometries were constructed. Taking into account the energy balance of the medium, the families of integral consistent approximations of the vector analysis operators, which are sufficient for the discrete modeling of these processes considering the space curvature caused by the cylindrical geometry of the system, were built. The schemes, both using a stress tensor in the full form and dividing it into volumetric and deviator components, were studied. This separation is used to construct homogeneous equations that are applicable for solid body and vaporized phase. The linear theory of elasticity was used. The resulting expressions for the elastic forces were presented in the explicit form for two-dimensional flat and axisymmetric geometries for a mesh consisting of triangular and quadrangular cells. Generalization of the method for other cases (non-linear strain tensor, non-Hookean relation between strain and stress, full 3D geometry, etc.) can be performed by analogy, but this was not a subject of the current paper. Using the model problem, comparison between different temporal discretizations for the obtained ordinary differential equations system was carried out. In particular, we considered fully implicit approximation, conservative implicit approximation (Crank–Nicolson method), and explicit approximation, which is similar to the “leap-frog” method. The analysis of full energy imbalance and calculation costs showed that the latter is more advantageous. The analysis of the effectiveness of various temporal approximations was performed via numerical experiments.