Four Spacetime Dimensional Simulation of Rheological Waves in Solids and the Merits of Thermodynamics

Abstract

The recent results attained from a thermodynamically conceived numerical scheme applied on wave propagation in viscoelastic/rheological solids are generalized here, both in the sense that the scheme is extended to four spacetime dimensions and in the aspect of the virtues of a thermodynamical approach. Regarding the scheme, the arrangement of which quantity is represented where in discretized spacetime, including the question of appropriately realizing the boundary conditions, is nontrivial. In parallel, placing the problem in the thermodynamical framework proves to be beneficial in regards to monitoring and controlling numerical artefacts—instability, dissipation error, and dispersion error. This, in addition to the observed preciseness, speed, and resource-friendliness, makes the thermodynamically extended symplectic approach that is presented here advantageous above commercial finite element software solutions.

Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

“…On that basis, a finite difference approach has been elaborated recently [8], but it is strongly limited to simple (regular) geometries. That numerical methodology is also adapted for rheological and nonlinear models [9], with carefully investigating the role of initial and boundary conditions analytically, too [1,2]. Therefore, we aim to develop a hp-version finite element method (FEM), which is able to preserve the advantageous properties of the earlier finite difference scheme to handle the initial and boundary conditions reliably, and, also, to be adaptable for complex geometries and possess high accuracy and fast convergence properties.…”

A SPH-GFDM Coupled Method for Elasticity Analysis

“…A mixed-type finite element approximation for radiation problems using fictitious domain method written in the form of pseudo-differential operator was proposed by Nasir et al [8], which is also efficiently applied to compute the solution of the radiation problem outside the computational domain and to compute the far-field pattern. The mixed finite element method utilizes the relatively simple interpolation function, which are also widely applied in thermodynamics [9,10], but the coefficient matrix of the simultaneous equations it derived is not positive definite, which limits the wide application of this method to a certain extent.…”

New Directions for Contact Integrators

“…With the range of applications of contact geometry growing rapidly, geometric numerical integrators that preserve the contact structure have gained increasing attention [5,6,13,14,17,19]. Deferring to the above literature for detailed presentations of contact systems, their properties and many of their uses, in this work we will present new applications of the contact geometric integrators introduced by the authors in [6,17,19] to two particular classes of examples inspired by celestial mechanics and cosmology.…”