Arbitrary Dissipation Research Articles

Well-posedness of the water-wave with viscosity problem

In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for the Dias, Dyachenko & Zakharov system (Physics Letters A 2008) modeling gravity waves with viscosity when surface tension is not taken into account.

Technology of self-healing road pavements: current status and development prospects

Introduction. Presently, there exists no unified methodology that could control self-healing properties of materials due to the lack of criteria characterizing the ability of materials to respond to impacts and to cope with adverse effects produced on their properties or structure. All assessment methods applied to self-healing properties of materials are reduced to the calculation of the ratio of relative changes in the value of parameters to be measured. A “self-healing” material is a material capable of initiating liquidation of structural defects, so that the rate of this process could exceed the one of spontaneous propagation of defects, while the resistance of the new structure to operating loads is not lower than the initial one. Development of solutions, capable of implementing this principle, will ensure longer service lives of materials.
 Materials and methods. A set of general methods of logical research was applied. These methods are based on the theoretical analysis of process solutions presented in professional editions, information resources maintained by their developers and in mass media, patents, research articles and reports.
 Results. The self-healing technology is applicable to asphalt concrete with regard for its structural features and thermoplastic properties. Self-healing means restoration of the structural integrity of physical and chemical bonds by moistening crack surfaces, diffusion of molecules between them, and arbitrary dissipation of molecules to ensure reinforcement. Self-healing process uniformity is attainable by applying the technology of functional agent encapsulation so that it could be added to the mix along with other components at the stage of the asphalt concrete mix preparation.
 Conclusions. Existing technological solutions can generate capsules that contain the recovery agent in the amount of up to 90…94 % of the total volume. General requirements and quality standards that apply to self-healing materials, used in construction, must be developed to identify their efficiency depending on application conditions. Provisions of the adhesion theory can be applied to the self-healing technology in the course of its implementation.
 The study was funded by the grant from the Russian Science Foundation (Project No. 19-79-00262).

Accurate dissipative forces in optimization integrators

We propose a method for accurately simulating dissipative forces in deformable bodies when using optimization-based integrators. We represent such forces using dissipation functions which may be nonlinear in both positions and velocities, enabling us to model a range of dissipative effects including Coulomb friction, Rayleigh damping, and power-law dissipation. We propose a general method for incorporating dissipative forces into optimization-based time integration schemes, which hitherto have been applied almost exclusively to systems with only conservative forces. To improve accuracy and minimize artificial damping, we provide an optimization-based version of the second-order accurate TR-BDF2 integrator. Finally, we present a method for modifying arbitrary dissipation functions to conserve linear and angular momentum, allowing us to eliminate the artificial angular momentum loss caused by Rayleigh damping.

Ambient Temperature and Self-Heating Scaling Laws for Materials With Temperature-Dependent Thermal Conductivity

Two of the primary variables affecting junction temperature of semiconductor devices are the self-heating due to internal power dissipation within the device and the device's base (or ambient) temperature. For materials with temperature-independent material properties, the junction temperature is a linear function of these two variables, which allows for simple “scaling” of the junction temperature for arbitrary dissipation and/or base temperatures. In materials with temperature-dependent material properties, however, the relationship between junction temperature and either variable is nonlinear. The scaling law between junction temperature and dissipated power and base temperature for materials with temperature-dependent material properties are developed in this work. This scaling law allows for fast computation of junction temperature for any values of power dissipation and/or base temperature given the junction temperature for one specific instance of power dissipation and base temperature and hence may find applicability in fast electrothermal solvers.

Magnetoplasmon spectrum for realistic off-plane structure of dissipative 2D system

The rigorous analysis of the textbook result (Chiu and Quinn, 1974) gives unexpectedly the dramatic change of the magnetoplasmon spectrum taking into account both the arbitrary dissipation and asymmetric off-plane structure of 2D system. For given wave vector the dissipation enhancement leads to decrease(increase) of magnetoplasmon frequency at low(high) magnetic field. At certain range of disorder the purely relaxational mode appears in magnetoplasmon spectrum. In strong magnetic fields the magnetoplasmon frequency falls to cyclotron resonance line even in presence of finite dissipation. The observation of nonlinearity and, moreover, the mysterious zig-zag behavior 2D magnetoplasmon spectrum is consistent with our findings.

An Analytical Solution for Boundary Layer Flows Over a Moving-Flat Porous Plate With Viscous Dissipation

The problem of boundary layer flow of an incompressible fluid over a moving porous flat plate is investigated, by taking into account the heat due to viscous dissipation. The governing boundary layer equations of this flow field were solved analytically using the Laplace transform technique. These new exact analytical solutions for velocity and temperature were obtained with arbitrary Prandtl number and dissipation parameter (or Eckert number Ec). The corresponding solutions for nonporous plate are discussed. Applying numerical values into the analytical expressions of the temperature and heat transfer coefficient, we also discussed the effects of the dissipation parameter in the cases of water, gas, and ammonia flow. We can finally deduce that the fluid temperature of the present problem will increase in the case of viscous dissipation with positive Ec, but this temperature will decrease with negative Ec.

Typical and rare fluctuations in nonlinear driven diffusive systems with dissipation

We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated with a path in mesoscopic phase space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in nondissipative systems. This is a consequence of the quasielasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal density and current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbative solution thereof shows that the probability distribution of small fluctuations is always Gaussian, as expected from the central limit theorem. On the other hand, strong separation from the Gaussian behavior is observed for large fluctuations, with a distribution which shows no negative branch, thus violating the Gallavotti-Cohen fluctuation theorem, as expected from the irreversibility of the dynamics. The dissipation large-deviation function exhibits simple and general scaling forms for weakly and strongly dissipative systems, with large fluctuations favored in the former case but heavily suppressed in the latter. We apply our results to a general class of diffusive lattice models for which dissipation, nonlinear diffusion, and driving are the key ingredients. The theoretical predictions are compared to extensive numerical simulations of the microscopic models, and excellent agreement is found. Interestingly, the large-deviation function is in some cases nonconvex beyond some dissipation. These results show that a suitable generalization of macroscopic fluctuation theory is capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.

An Example Exempted from Thomson–Tait–Chetayev's Theorem

An example is given of a mechanical system whose behavior does not follow Thomson–Tait–Chetayev's (TTC) theorem which states that, for a system with an unstable potential, a state stabilized by gyroscopic forces goes unstable after the addition of arbitrary dissipation. The example is brought by a system, closely related with a heavy symmetrical top, describing motion of a charged spherical pendulum subjected to the Lorentz force, in a magnetic-monopole field sitting at the sphere center, as well as the gravity force. A drag force proportional to the velocity is exerted on the pendulum. The upright state, an equilibrium stabilized by the Lorentz force, is shown to be exempted from the TTC theorem. The numerical calculation of the full nonlinear system is performed for precession. For the slow precession, the drag force acts to continuously tilt down the top axis toward vertically downward equilibrium, following the dissipation-induced instability. On the contrary, for the fast precession, the drag acts to...

A Fast Algorithm for Thermal Transient Multisource Simulation Using Interpolated Zth Functions

This paper presents a very straightforward method to compute the transient thermal response to arbitrary power dissipation profiles in electronic devices with multiple heat sources. Using cubic spline interpolation of simulated or measured unit power step response curves (Zth-functions), additional errors and efforts due to model reduction can be avoided. The simple analytic form of the interpolating splines can be exploited to evaluate the convolution integral of the Zth functions with arbitrary power profiles at low computational costs. For periodic power profiles the speed of the computation can be further enhanced by replacing early pulses with the energy conserving average. An implementation of the algorithm in a spreadsheet program (EXCEL) has demonstrated that the results are in very good agreement with temperature profiles computed by transient finite-element simulation but can be obtained in a fraction of the time.

Nontrivial velocity distributions in inelastic gases

We study freely evolving and forced inelastic gases using the Boltzmann equation. We consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient epsilon. In the freely evolving case, we find that the velocity distribution decays algebraically, P(v,t) ~ v^{-sigma} for sufficiently large velocities. We derive the exponent sigma(d,epsilon), which exhibits nontrivial dependence on both d and epsilon, exactly. In the forced case, the velocity distribution approaches a steady-state with a Gaussian large velocity tail.

A new CDM-based approach to structural deterioration

Structural deterioration often occurs without visible manifestation. Continuum damage mechanics (CDM) enables one to predict the state of damage in such situations and to estimate residual strength⧹service life of an existing structure. The accumulation of damage is modeled as a dissipative process that is governed by the laws of thermodynamics. The rate of dissipation in a deformable system, R, depends on the work done on the system and the evolution of the Helmholtz free energy, Ψ. Under certain thermodynamical conditions, the first variation of Ψ vanishes, and partial differential equations for damage growth in R prior to damage localization are obtained. This approach obviates the need of introducing arbitrary dissipation potential functions with undetermined constants in the damage growth equations. All solutions use only readily available material parameters. Assuming that damage occurs isotropically under uniaxial loading, closed-form solutions are obtained for ductile damage as a function of plastic strain, for creep damage as a function of time and for fatigue damage as function of number of cycles. The models are validated with published laboratory data.

Thermodynamics of vortex lines in layered superconductors

We study the dissipative thermodynamics of vortex lines in layered superconductors within a simple string model in the dilute limit of negligible vortex interactions and compute the specific heat $C_v$ in presence of arbitrary dissipation. The interplay of dissipation, inertia and elasticity is shown to control the qualitative thermodynamical behavior and their relative amount determines two very distinct regimes for the specific heat. In the dissipation dominated case we find a behavior $C_v \sim \sqrt{T}$ for a large interval of temperature below $T_c$.

Dynamics of the dissipative two-state system under ac modulation of bias and coupling energy.

We consider a driven dissipative two-state system subject to periodic modulation of both of its two parameters. The exact master equation that governs the dynamics is derived. Closed form analytical solutions are presented within the noninteracting-blip approximation for the stochastic forces, and for the exactly solvable case $\ensuremath{\alpha}=\frac{1}{2}$ of the Ohmic viscosity. We apply these results to a tight-binding particle with zero intrinsic bias. Selection rules are obeyed for arbitrary dissipation. For Ohmic dissipation, the effects of fast asymmetry modulation result in an overall reduction of quantum coherence and in a possible localization. On the contrary, a multiplicative modulation of the tunneling coupling leads to an exponential enhancement of tunneling.

Modeling of one-dimensional weakly nonlinear waves that propagate in media with arbitrary dissipation and dispersion mechanisms.

In his paper [J. Acoust. Soc. Am. 77, 2050 (1985)] Blackstock presented a generalized Burgers equation for the propagation of one-dimensional weakly nonlinear waves in various media. His results, and the approach he employed there, however, are limited to harmonic waves. In this paper, we present a general approach to model nonlinear waves of more general wave forms that propagate in media with arbitrary absorption and dispersion relations. The resulting equation is again called the generalized Burgers equation (to follow the terminology of the literature). It is found that steady shock solutions for various media can be described by the corresponding simplified version of the equation. An efficient numerical method by means of spectral analysis is developed for solving the generalized Burgers equation. Typical results exemplified by the case of a sinusoidal wave source are also reported in this paper.

Relationship of second harmonic absorption coefficient to nonlinear attenuation of the fundamental

Woodsum [J. Sound Vib. 69, 27–33 (1980)] proposed a method for suppressing the saturation that arises from the distortion of a finite-amplitude wave during propagation. The method is based on the introduction of an agent into the medium that selectively absorbs the second harmonic of the fundamental. While the validity of the proposed method is not in question, the method of analysis used in the original paper was based on numerical solutions of coupled nonlinear equations, which offered no simple intuitive explanation for the effect. Derived here is a simple, intuitive, analytical solution for the amplitude of the fundamental, which supports the previous result and which is valid for a medium having arbitrary dissipation and dispersion properties. The original conclusion is generalized to include saturation reduction via introduction of dispersion as described by Zablotskaya and Soluyan [“A Possible Approach to the Amplification of Sound Waves,” Soy. Phys. Acoust. 13, 254–256 (1967)].

Quantum Kramers model: Solution of the turnover problem.

The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.

NONLINEAR THEORY OF DISTORTION OF BOUNDED ULTRA- SONIC BEAM IN RELAXATING MEDIA

In this article, a nonlinear wave equation of bounded acoustic beam in relaxating media is presented and a general solution of higher harmonics resulting from the nonlinear distortion of propagation is found by using the pertubation approach. Investigation shows that for the Gaussian ultrasonic field, the solution of higher harmonics can be given analytically, moreover their corresponding radical distributions of amplitude always maintain the Gaussian profiles. Although the dispersion may affect the amplitude of various harmonics, the variation of phase velocity of those keep the same law as the small amplitude waves of corresponding frequencies do.We also show that by means of Blackstock' operator the obtained results can be applied te the case of media with arbitrary dissipation and dispersion, including some biomedia in which the relationship between the absorption and frequency may be obtained only empirically.

Analogues of potential vorticity in electrically-conducting fluids

Abstract Exploiting the (contravariant) vectorial form for vorticity and magnetic field, and co-(variant) vectorial form for the gradient of a scalar and magnetic potential, and using the geometric Lie derivative operator, conservation of various analogues of potential vorticity are discussed for a barotropic non-dissipative electrically-conducting fluid. These analogues include the potential magnetic field, helicity, magnetic helicity, and cross helicity, together with some higher order quantities. It is noted that the volume conservation of potential vorticity continues to hold in the presence of arbitrary dissipation. However, of the analogue quantities derived for the non-dissipative system, only potential magnetic field and cross helicity have invariant integrals in the presence of dissipation. We conclude that only they are true analogues of potential vorticity. Finally, a straightforward generalisation of the method for tensorial relationships is noted.

Transverse normal modes of finite vortex arrays

The normal modes of stationary vortex patterns in two dimensions are studied for arbitrary dissipation in both the unbounded and circle-bounded cases. All eigenvectors are given for the ground-state patterns of N = 2 through 12 vortices and N = 19; all frequencies are given for N = 2 through 20. The complete eigenfrequency spectrum of single-ring patterns is given for all N. The integrals of unbounded motion are generalized to include dissipation. It is shown that angular momentum is strictly confined to the breathing mode of all unbounded patterns and of all bounded, single-ring patterns. Comparisons are made with infinite-vortex lattice modes, with a continuum model, and with earlier predictions of pattern stability and mode frequencies.

Three-wave parametric-decay instability with arbitrary dissipation in homogeneous plasmas

The theory of the nonlinear decay of a steady incident pump wave into two oppositely propagating modes in a bounded homogeneous region is extended to include arbitrary dissipation on each wave. A new representation of the coupled mode equations shows that, in addition to the usual coupling nonlinearity, two transverse-wave-damping-induced nonlinearities appear, leading to enhanced saturation. A threshold analysis yields general absolute and convective instability criteria extending earlier results of Kroll, and of Porkolab and Chang. The steady-state boundary value problem is solved to obtain the eigenvalue of the stimulated backscattering problem, the anomalous reflectivity, or the convective amplification of a noise source. Good agreement is found between the results of analytical and numerical integrations.