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.

Figures (22)

• Figure S1: An excitation pulse, generated at the left endpoint of a finite-size one space dimensional Hookean sample, regularly arrives at the right endpoint. First row: the results from the scheme that was introduced in 1Dcikk for two different pulse lengths. Second and third rows: the corresponding results obtained by the finite element software COMSOL 1Dcikk, at various settings: left column second row: Backward Differentiation Formula (BDF) Maximum order 2, third row: BDF Maximum order 5; right column second row: Dormand--Prince (DP) 5, third row: Runge--Kutta (RK) 34. In each finite element solution, dissipation error (decrease of the amplitude) and dispersion error (artificial oscillations) are both observable, even during the first three bounces. Meanwhile, the pulses in the first row keep their shape even after many bounces 1Dcikk.
• Figure S2: Spatial arrangement of the discretized quantities (two-dimensional projection). The circles stand for diagonal tensor components, squares for offdiagonal ones, and triangles for vector components, different components with differently oriented triangles. Void quantities are prescribed by boundary condition (in the case stress boundary conditions are considered, like here.)
• Figure S3: The spatial distribution of normal stress as the boundary condition on one side of a square cross-sectioned beam that is infinitely long in the $z$ direction. The excitation is a single cosine-shaped 'bump' in time, too.
• Figure S4: Distribution of a stress component (left column) and of a velocity component (right column) at various instants, in the Hooke case. From top to bottom: snapshots at instants $(1/2) \tau_\text{b}$, $\tau_\text{b}$, $(3/2) \tau_\text{b}$, $2 \tau_\text{b}$, respectively.
• Figure S5: Continuation of Figure \ref{['Hs']}: distribution of a stress component (left column) and of a velocity component (right column) at various instants, in the Hooke case. From top to bottom: snapshots at instants $(5/2) \tau_\text{b}$, $3 \tau_\text{b}$, $(7/2) \tau_\text{b}$, $4 \tau_\text{b}$, respectively.