Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves

This paper is devoted to comparing the asymptotics of a solution, describing the wave motion of a discrete lattice and its continuous approximations. The transition from a discrete medium to a continuous one changes the symmetry of the system. The influence of this change on the asymptotic behavior of waves is of great interest. For the discrete case, Schrödinger’s analytical solution of the initial-value problem for the Lagrange lattice is used. Various continuous approximations are proposed to approximate the lattice. They are based on Debye’s concept of quasicontinuum. The asymptotics of the initial motion and the behavior of the systems in the vicinity of the quasifront and at large times are compared. The approximations of phase and group velocities is analyzed. The merits and limitations of the described approaches are discussed.

STATIC CONDITIONS OF A LOOSE ENVIRONMENT IN SOLVING THE TASKS OF ITS VIBROMOVEMENT

The method of formalizing a granular medium in the form of a continuous deformable medium provides that the granular medium is continuous. In this case, the methods of mechanics of continuous deformable media in Eulerian or Lagrangian coordinates are used. The method allows you to take into account the interaction between particles in the form of elastic, viscous or plastic components of internal resistance. With this formalization, the most widespread application of the Navier-Stokes equation. Unfortunately, the exact correct solution of such an equation in an analytical form does not exist. It should be noted that the most common method is formalization in the form of a loose discrete medium. It can be used under conditions when the dimensions of the element, in which the changes in the kinematic and dynamic parameters are determined, exceed by at least an order of magnitude the maximum size of a fraction of a granular medium. With this formalization, you can use the methods of mechanics of granular discrete media. When solving the problems of dynamics of motion of a loose discrete medium, it becomes necessary to compose the equations of its statics, which are necessary to solve the problems of dynamics. This is because the static equations allow one to supplement the boundary conditions and provide nonzero solutions for the problems of oscillations of a loose discrete medium, since during oscillations of such a medium zero values of quantities determined (density and velocity components of displacements) can occur at points where their functions intersect the zero line relative to which oscillating motion occurs. The article describes the conditions of the static state of a granular medium in a working chamber, which are an addition to the boundary conditions when solving the problems of oscillations of a granular discrete medium, which is formalized as continuous with a significant manifestation of dry friction.

A Macroscale Hydrogeological Numerical Model of the Suio Hydrothermal System (Central Italy)

The complex behaviour of the Suio hydrothermal system (central Italy) and its potential exploitation as a renewable energy source are still unclear. To quantitatively evaluate the geothermal resource, the Suio hydrothermal system has been investigated with a hydrogeological numerical model that couples fluid flow, thermal convection, and transport of diluted species inside a hybrid continuum-discrete medium. The numerical model, calibrated and validated with available and new experimental data, unveiled the complex behaviour of the hydrothermal system. The normal tectonic displacements, the fracturing of the karst hydrostructure, and the aquitard distribution strongly influence the hydrothermal basin. In particular, a dual fluid circulation, sustained by steady-state thermal and pressure gradients, modulates the hydrothermalism at the several springs and wells. The presence of a medium to a low-temperature reservoir allows for potential exploitation of the geothermal resource.

Continuous and discrete models in the mechanics of deformable solid bodies

The study provides an overview of modelling possibilities for the mechanical behaviour of media. The discrete, continuous or differential geometric as well as the discrete nature and continuous description grid continuum model in particular are highlighted. We point out that the differential geometric model is based on the concept of continuity and interprets a continuous medium model. We reveal that the grid continuum model is based on the application of numerical method and interprets a discrete medium model.

The Path of the Shortest Time

The principal of least action is one of the fundamental ideas in physics. The path of the shortest time of a particle in the presence of gravity is an example of this principal. In this paper some methods are introduced to teach the optimal path in introductory physics courses. The optimal path (path of the shortest time) is calculated for a few families of paths. Finally a numerical method according to Snell’s law in a discrete medium is used to find the general optimal path and is compared with the brachistochrone path.