The theory of elasticity is a very important discipline which has a lot of applications in science and engineering. In this paper we are interested in elastic materials with different properties between interfaces implicated the discontinuous coefficients in the governing elasticity equations. The main aim is to develop a practical numerical scheme for modeling the behaviour of a simplified piecewise homogeneous medium subjected to an external action in 2D domains. Therefore, the discontinuous Galerkin method is used for the simulation of elastic waves in such elastic materials. The special attention is also paid to treatment of boundary and interface conditions. For the treatment of the time dependency the implicit Euler method is employed. Moreover, the limiting procedure is incorporated in the resulting numerical scheme in order to overcome nonphysical spurious overshoots and undershoots in the vicinity of discontinuities in discrete solutions. Finally, we present computational results for two-component material, representing a planar elastic body subjected to a mechanical hit or mechanical loading.
In this article, we deal with a numerical solution of the issue concerning one-dimensional longitudinal mechanical wave propagation in linear elastic neural weakly heterogeneous media. The crucial idea is based on the discretization of the wave equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the Crank-Nicolson scheme for the time discretization. The linearity of the second-order hyperbolic problem leads to a solution of a sequence of linear algebraic systems at each time level. The numerical experiments performed for the single traveling wave and Gauss initial impact demonstrate the high-resolution properties of the presented numerical scheme. Moreover, a well-known linear stress-strain relationship enables us to analyze a high-frequency regime for the initial excitation impact with respect to strain-frequency dependency.
The contribution deals with a theoretical and experimental investigation of fluid flow in a thin layer when a flow has nearly horizontal nature and negligible vertical acceleration compared with a gravitational one. Under these conditions the full set of Navier-Stokes equations are simplified to the Saint-Venant equations. Several variants of the equations differing by a level of geometry simplification or fluid properties are discussed. A discontinuous Galerkin method was applied for numerical simulations. This method originates from the classical methods of finite volumes but it assumes that inside the elements the searching functions are approximated by the functions of higher orders. Results of numerical simulations are compared with an experimental investigation. The numerical simulations agree very well with the experimental data in the cases when the assumptions made using the derivation of the Saint-Venant equations are fulfilled. and Příspěvek se zabývá teoretickým a experimentálním řešením proudění v tenkých vrstvách kapaliny v případech, kdy hraje významnou roli gravitační síla a kdy lze úplné Navier - Stokesovy rovnice proudění zjednodušit na Saint-Venantovy rovnice. Jsou diskutovány různé varianty rovnic lišící se stupněm zjednodušení geometrie úlohy i vlastností proudící tekutiny. Pro matematické řešení byla použita nespojitá Galerkinova metoda. Uvnitř elementů je řešení aproximováno polynomy vyššího stupně. Výsledky teoretického řešení jsou porovnány s daty získanými experimentálně. Vypočtené výsledky ukazují dobrou shodu s experimentem v případech, kdy jsou dostatečně splněny předpoklady použité při odvození SaintVenantových rovnic.
We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier-Stokes equations by the backward difference formula - discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss the choice of the preconditioner, stopping criterion and the choice of the time step and propose a new strategy which leads to an efficient and accurate numerical scheme.