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

Fülöp, Tamás; Kovács, Róbert; Szücs, Mátyás; Fawaier, Mohammad

doi:10.3390/e22020155

Cited by 14 publications

(28 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [21], the Hooke case (τ = 0, Ê =0) was realized in the FEM software COMSOL, a promising environment for modelling as the user's own model equations can also be added. The results were disappointing.…”

Section: Hookean Waves -What Commercial Fem Software Can Offermentioning

confidence: 99%

“…This failure has motivated us to invent a novel finite difference scheme, which is a second-order accurate extension of a symplectic Euler method reinterpreted as discretized quantity values are bookept according to a pattern that is staggered with half space and time step shifts both [21], see Fig. 6.…”

Section: The Principles Behind Our Finite Difference Schemementioning

confidence: 99%

“…Well, the situation is discouraging. Even for Hookean wave propagation [21] -and also for wavelike beyond-Fourier models for heat conduction [22,23] -, the solution requires large memory demand, large computational time, and the outcome is infected with numerical artefacts: instability, dissipation error (artificial decrease of amplitudes and energies), and dispersion error (artificial oscillations near fast changes) [more details are given in Section 3.2 below]. All these hold for already the simplest settings of a homogeneous one-dimensional sample.…”

Section: Introductionmentioning

confidence: 99%

“…• the powerful symplectic methods for reversible systems, • the second law of thermodynamics, which acts not only as a consistency requirement but, being realized as the mathematical asymptotic stability property of thermodynamical models, can help in stability of the numerical scheme as well [29][30], and • spacetime aspects encoded in the basic equations of the model: e.g., the balances of mass, of momentum, and of energy are actually a spacetime divergence of a spacetime tensor. Such a finite difference scheme has been successfully built, its resistance against numerical artefacts has been ensured through accuracy and stability analysis, and test runs have proved that the scheme is precise and fast with low memory demand [21], both for Hooke and PTZ models.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Wave Propagation in Rocks – Investigating the Effect of Rheology

Fülöp

2020

Period. Polytech. Civil Eng.

Self Cite

View full text Add to dashboard Cite

Rocks exhibit beyond-Hookean, delayed and damped elastic, behaviour (creep, relaxation etc.). In many cases, the Poynting–Thomson–Zener (PTZ) rheological model proves to describe these phenomena successfully. A forecast of the PTZ model is that the dynamic elasticity coefficients are larger than the static (slow-limit) counterparts. This prediction has recently been confirmed on a large variety of rock types. Correspondingly, according to the model, the speed of wave propagation depends on frequency, the high-frequency limit being larger than the low-frequency limit. This frequency dependence can have a considerable influence on the evaluation of various wave-based measurement methods of rock mechanics. As experience shows, commercial finite element softwares are not able to properly describe wave propagation, even for the Hooke model and simple specimen geometries, the seminal numerical artefacts being instability, dissipation error and dispersion error, respectively. This has motivated research on developing reliable numerical methods, which amalgamate beneficial properties of symplectic schemes, their thermodynamically consistent generalization (including contact geometry), and spacetime aspects. The present work reports on new results obtained by such a numerical scheme, on wave propagation according to the PTZ model, in one space dimension. The simulation outcomes coincide nicely with the theoretically obtained phase velocity prediction.

show abstract

Section: Hookean Waves -What Commercial Fem Software Can Offermentioning

confidence: 99%

Section: The Principles Behind Our Finite Difference Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wave Propagation in Rocks – Investigating the Effect of Rheology

Fülöp

2020

Period. Polytech. Civil Eng.

Self Cite

View full text Add to dashboard Cite

show abstract

“…-There is also a third, very practical, point here: it is easier to solve dissipative equations than non-dissipative ones. Therefore, a dissipative extension can be a tool to design powerful numerical procedures [75,78].…”

Section: Discussionmentioning

confidence: 99%

Variational principles and nonequilibrium thermodynamics

Ván

Kovács

2020

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

Variational principles play a fundamental role in deriving the evolution equations of physics. They work well in the case of non-dissipative evolution, but for dissipative systems, the variational principles are not unique and not constructive. With the methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, in several cases, one can also reproduce the Euler–Lagrange form and symplectic structure of the evolution equations for non-dissipative processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that, instead of searching for variational principles for dissipative systems, there is another viable programme: the second law alone can be an effective tool to construct evolution equations for both dissipative and non-dissipative processes. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

show abstract