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

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.

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

On the example of the Poynting–Thomson–Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest—elastic—level of modeling.