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We use backward error analysis for differential equations to obtain modified or distorted equations describing the behaviour of the Newmark scheme applied to the transient structural dynamics equation. Using these results, we show how to construct compensation terms from the original parameters of the system, which improve the performance of Newmark simulations without changing the time step or modifying the scheme itself. Two such compensations are given: one eliminates numerical damping, while the other achieves fourth-order accurate calculations using the traditionally second-order Newmark method., Comment: 27 pages, 14 figures

Symplectic numerical schemes for reversible dynamical systems predict the solution reliably over large times as well, and are a good starting point for extension to schemes for simulating irreversible situations like viscoelastic wave propagation and heat conduction coupled via thermal expansion occuring in rocks, plastics, biological samples etc. Dissipation error (artificial nonpreservation of energies and amplitudes) of the numerical solution should be as small as possible since it should not be confused with the real dissipation occuring in the irreversible system. In addition, the other well-known numerical artefact, dispersion error (artificial oscillations emerging at sharp changes), should also be minimal to avoid confusion with the true wavy behaviour. The continuum thermodynamical aspects (respect for balances with fluxes, systematic constitutive relationships between intensive quantities and fluxes, the second law of thermodynamics with positive definite entropy production, and the spacetime-based kinematic viewpoint) prove valuable for obtaining such extended schemes and for monitoring the solutions. Generalizing earlier works in this direction, here, we establish and investigate such a numerical scheme for one-dimensional viscoelastic wave propagation in the presence of heat conduction coupled via thermal expansion, demonstrating long-term reliability and the applicability of thermodynamics-based quantities in supervising the quality of the solution., Comment: 13 pages, 7 figures, 3 tables

An explicit compatibility condition formula is presented for the Eulerian left Cauchy-Green deformation tensor field. It is shown to be the appropriate finite-strain counterpart of Saint-Venant's compatibility condition. The difference between the Eulerian problem and the Lagrangian one is highlighted., Comment: 9 pages

Applying simultaneously the methodology of Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier's law and its various extensions -- the Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffreys type, Ginzburg-Landau (Allen-Cahn) type and ballistic-diffusive -- heat conduction equations are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction, and open new ways for numerical simulations of heat-conduction problems., Comment: 28 pages

It is a long-lasting task to understand heat conduction phenomena beyond Fourier. Besides the low-temperature experiments on extremely pure crystals, it has turned out recently that heterogeneous materials with macro-scale size can also show thermal effects that cannot be modelled by the Fourier equation. This is called over-diffusive propagation, different from low-temperature observations, and is found in numerous samples made from metal foam, rocks, and composites. The measured temperature history is indeed similar to what Fourier's law predicts but the usual evaluation cannot provide reliable thermal parameters. This paper is a report on our experiments on several rock types, each type having multiple samples with different thicknesses. We show that size-dependent thermal behaviour can occur for both Fourier and non-Fourier situations. Moreover, based on the present experimental data, we find an empirical relation between the Fourier and non-Fourier parameters, which may be helpful in later experiments to develop a more robust and reliable evaluation procedure.

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., Comment: 24 pages, 101 figures; v3: 4 typos corrected

Living systems are subject to the arrow of time; from birth, they undergo complex transformations (self-organization) in a constant battle for survival, but inevitably ageing and disease trap them to death. Can ageing be understood and eventually reversed? What tools can be employed to further our understanding of ageing? The present article is an invitation for biologists and clinicians to consider key conceptual ideas and computational tools (known to mathematicians and physicists), which potentially may help dissect some of the underlying processes of ageing and disease. Specifically, we first discuss how to classify and analyze complex systems, as well as highlight critical theoretical difficulties that make complex systems hard to study. Subsequently, we introduce Topological Data Analysis - a novel Big Data tool - which may help in the study of complex systems since it extracts knowledge from data in a holistic approach via topological considerations. These conceptual ideas and tools are discussed in a rather informal way to pave future discussions and collaborations between mathematicians and biologists studying ageing.

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 analysed 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 modelling., Comment: 24 pages, 47 figures

The internal variable methodology of nonequilibrium thermodynamics, with a symmetric tensorial internal variable, provides an important rheological model family for solids, the so-called Kluitenberg-Verh\'as model family [1]. This model family is distinguished not only from theoretical aspects but also on experimental grounds (see [2] for plastics and [3, 4, 5] for rocks). In this article, we present and discuss how the internal variable formulation of the Kluitenberg-Verh\'as model family can be presented in the nonequilibrium thermodynamical framework GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) [6, 7, 8, 9], for the benefit of both thermodynamical methodologies as well as for promising practical applications., Comment: 12 pages

In classical continuum theory, Volterra's principle [1, 2] is a long-known method to solve linear rheological (viscoelastic) problems derived from the corresponding elastic ones. Here, we introduce and present another approach that is simpler to apply (no operator inverse is required to compute but only linear ordinary differential equations to solve). Our method starts with the known elastic solution, replaces the elasticity coefficients with time dependent functions, derives differential equations on them, and determines the solution corresponding to the initial conditions. We present several examples solved via this new method, like tunnels and spherical hollows opened in various initial stress states, and pressurizing of thick-walled tubes and spherical tanks. These examples are useful for applications and, in parallel, are suitable for testing and validating numerical methods of various kinds., Comment: 49 pages, 23 figures

The non-Fourier heat conduction phenomenon on room temperature is analyzed from various aspects. The first one shows its experimental side, in what form it occurs and how we treated it. It is demonstrated that the Guyer-Krumhansl equation can be the next appropriate extension of Fourier's law for room temperature phenomena in modeling of heterogeneous materials. The second approach provides an interpretation of generalized heat conduction equations using a simple thermomechanical background. Here, Fourier heat conduction is coupled to elasticity via thermal expansion, resulting in a particular generalized heat equation for the temperature field. Both of the aforementioned approaches show the size dependency of non-Fourier heat conduction. Finally, a third approach is presented, called pseudo-temperature modeling. It is shown that non-Fourier temperature history can be produced by mixing different solutions of Fourier's law. That kind of explanation indicates the interpretation of underlying heat conduction mechanics behind non-Fourier phenomena.

We show that the antisymmetric spinor tensor representation of spin-0 relativistic quantum mechanics provides a conserved current with positive definite timelike component, interpretable as probability density. The construction runs in complete analogy to the spin-1/2 case, and provides an analogously natural one-particle Hilbert space description for spin 0. Except for the free particle, the obtained formulation proves to be inequivalent to the one based on the Klein--Gordon equation. The second quantized version may lead to new field theoretical interaction terms for zero-spin particles., Comment: 9 pages, no figures; material presented at the Zim\'anyi School, December 7-11, 2015, Budapest, Hungary

In this paper we present a thermodynamically consistent material model which is capable of modelling ductile-to brittle failure mode transition in ductile material undergoing deformations at high strain rates, and demonstrate the performance of the model in a numerical study using a fully coupled thermal-structural finite element analysis of a notched aluminium alloy specimen loaded in cyclic tension. The model is based on an objective representation of the deformation and stress measures and on a rate type constitutive equations. It does not only complies with the principles of material modelling, but it also uses constitutive equations, evolution equations and even "normality rules" during return mapping which can be expressed in terms of power conjugate stress and strain measures, or their objective rates., Comment: 30 pages, 4 figures

Rocks are important examples for solid materials where, in various engineering situations, elastic, thermal expansion, rheological/viscoelastic and plastic phenomena each may play a remarkable role. Nonequilibrium continuum thermodynamics provides a consistent way to describe all these aspects in a unified framework. This we present here in a formulation where the kinematic quantities allow arbitrary nonzero initial (e.g., in situ) stresses and such initial configurations which - as a consequence of thermal or remanent stresses - do not satisfy the kinematic compatibility condition. The various characteristic effects accounted by the obtained theory are illustrated via experimental results where loaded solid samples undergo elastic, thermal expansion and plastic deformation and exhibit rheological behaviour. From the experimental data, the rheological coefficients are determined, and the measured temperature changes are also explained by the theory., Comment: 15 pages, to appear in Period. Polytech. Civil Eng

An irreversible thermodynamical theory of solids is presented where the kinematic quantities are defined in an automatically objective way. Namely, auxiliary elements like reference frame, reference time and reference configuration are avoided by formulating the motion of the continuum on spacetime directly. Solids are distinguished from fluids by possessing not only an instantaneous metric tensor but also a relaxed metric. The elastic state variable is defined through comparing these two metrics. Thermal expansion is conceived as temperature dependence of the relaxed metric and plasticity, an irreversible change in the relaxed metric, is described via a plastic change rate tensor. Thermomechanics is built around these -- finite deformation -- kinematic quantities by starting from mechanics and adding thermodynamical requirements gradually. The obtained theory is not restricted to isotropic media., Comment: 10 pages, 2 figures

We take the viewpoint that the physically acceptable solutions of the Lorentz--Dirac equation for radiation back-reaction are actually determined by a second order equation of motion, the self-force being given as a function of spacetime location and velocity. We propose three different methods to obtain this self-force function. For two example systems, we determine the second order equation of motion exactly in the nonrelativistic regime via each of these three methods, the three methods leading to the same result. We reveal that, for both systems considered, back-reaction induces a damping proportional to velocity and, in addition, it decreases the effect of the external force., Comment: 13 pages

The reciprocity principle is that, when an emitted wave gets scattered on an object, the scattering transition amplitude does not change if we interchange the source and the detector - in other words, if incoming waves are interchanged with appropriate outgoing ones. Reciprocity is sometimes confused with time reversal invariance, or with invariance under the rotation that interchanges the location of the source and the location of the detector. Actually, reciprocity covers the former as a special case, and is fundamentally different from - but can be usefully combined with - the latter. Reciprocity can be proved as a theorem in many situations and is found violated in other cases. The paper presents a general treatment of reciprocity, discusses important examples, shows applications in the field of photon (M\"ossbauer) scattering, and establishes a fruitful connection with a recently developing area of mathematics., Comment: 20 pages, 4 figures. Three typos present in the Journal version corrected here, one sentence added. Note: Antiskewsymmetric matrices are also called skew Hamiltonian

For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain range ($x$ being a space coordinate in one or more dimensions), the corresponding Schr\"odinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian. The paper reviews and interprets known results, with the intention to provide a practical guide for all those interested in how to approach these ambiguous situations., Comment: This is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/