Dissipation Function Research Articles

ON THE HEAT CONDUCTION EQUATION AND THE HEAT DISSIPATION FUNCTION IN ANISOTROPIC REACTING FLUID MIXTURES WITH MAGNETIC RELAXATION

In some previous papers, within the framework of the thermody namics of irreversible processes with internal variables, a linear theory for magnetic relaxation phenomena in anisotropic mixtures, consisting of n reacting fluid components, was developed. In particular, assuming that the macroscopic magnetization m can be split in two irreversible parts m = m(0) + m(1) a generalized Snoek equation was derived. In this paper we derive for these reacting anisotropic mixtures the heat conduction equation. We show that the heat dissipation function is due to the chemical reactions, the magnetic relaxation, the electric conduc tion, the viscous, magnetic, temperature fields and the diffusion and the concentrations of the n fluid components. Also, the Snoek and De Groot special cases are studied. The obtained results find applications in nuclear resonance, in biology, in medicine and other fields, where different species of molecules have different magnetic susceptibilities and relaxation times and contribute to the total magnetization.

Theoretical and Numerical Modeling of Optical Switching of Epitaxial Nanostructures Based on Iron-Garnet Films

The paper presents a theoretical analysis of magnetization switching in a gadolinium ferrite garnet film due to the demagnetizing effect of a femtosecond laser pulse. Using the Lagrange formalism for a two-sublattice ferrimagnet, the effective Lagrangian, thermodynamic potential, and Rayleigh dissipative function are obtained. The phase diagram of the ferrite film is analyzed, and the main states of the system are identified. Magnetization switching diagrams and trajectories of the order parameter dynamics of the magnet are constructed. The ranges of magnetic fields, temperatures, and demagnetization values for the most efficient magnetization switching are analyzed.

A thermodynamics-consistent spatiotemporally-nonlocal model for microstructure-dependent heat conduction

The spatiotemporally-nonlocal phenomena in heat conduction become significant but challenging for metamaterials with artificial microstructures. However, the microstructure-dependent heat conduction phenomena are captured under the hypothesis of spatiotemporally local equilibrium. To capture the microstructure-dependent heat conduction phenomena, a generalized nonlocal irreversible thermodynamics is proposed by removing both the temporally-local and spatially-local equilibrium hypotheses from the classical irreversible thermodynamics. The generalized nonlocal irreversible thermodynamics has intrinsic length and time parameters and thus can provide a thermodynamics basis for the spatiotemporally-nonlocal law of heat conduction. To remove the temporally-local equilibrium hypothesis, the generalized entropy is assumed to depend not only on the internal energy but also on its first-order and high-order time derivatives. To remove the spatially local equilibrium hypothesis, the thermodynamics flux field in the dissipation function is assumed to relate not only to the thermodynamics force at the reference point but also to the thermodynamics force of the neighboring points. With the developed theoretical framework, the thermodynamics-consistent spatiotemporally-nonlocal models can then be developed for heat transfer problems. Two examples are provided to illustrate the applications of steady-state and transient heat conduction problems.

Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

Geometrical description and Faddeev-Jackiw quantization of electrical networks

In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.

A generalized phase-field cohesive zone model ([formula omitted]PF-CZM) for fracture

In this work a generalized phase-field cohesive zone model (μPF-CZM) is proposed within the framework of the unified phase-field theory for brittle and cohesive fracture. With the introduction of an extra dissipation function for the crack driving force, in addition to the geometric function for the phase-field regularization and the degradation function for the constitutive relation, theoretical and application scopes of the original PF-CZM are broadened greatly. These characteristic functions are analytically determined from the conditions for the length scale insensitivity and a non-shrinking crack band in a universal, optimal and rationalized manner, for almost any specific traction–separation law. In particular, with an optimal geometric function, the crack irreversibility can be considered without affecting the target traction–separation softening law. Not only concave softening behavior but also high-order cohesive traction, both being limitations of the previous works, can be properly dealt with. The global fracture responses are insensitive not only to the phase-field length scale but also to the traction order parameter, though the crack bandwidth might be affected by both. Despite the loss of variational consistency in general cases, the resulting μPF-CZM is still thermodynamically consistent. Moreover, the existing numerical implementation can be adopted straightforwardly with minor modifications. Representative numerical examples are presented to validate the proposed μPF-CZM and to demonstrate its capabilities in capturing brittle and cohesive fracture with general softening behavior. The insensitivity to both the phase-field length scale and the traction order parameter is also sufficiently verified.

Dissipative quintessential cosmic inflation

In this paper we construct a dissipative quintessential cosmic inflation. For this purpose, we add a multiplicative dissipative term in the standard quintessence field Lagrangian. We consider the specific form of dissipation as the time integral including the Hubble parameter and an arbitrary function that describes the dissipative properties of the quintessential scalar field. Inflation parameters and observables are calculated under slow-roll approximations and a detailed calculation of the cosmological perturbations is performed in this setup. We consider different forms of potentials and calculate the scalar spectral index and tensor-to-scalar ratio for a constant as well as variable dissipation function. To check the reliability of this model, a numerical analysis on the model parameters space is done in confrontation with recent observational data. By comparing the results with observational joint datasets at 68% and 95% confidence levels, we obtain some constraints on the model parameters space, specially the dissipation factor with e-folds numbers N=55 and N=60. As some specific results, we show that the power-law potential with a constant dissipation factor and N=60 is mildly consistent with observational data in some restricted domains of the model parameter space with very small and negative dissipation factor and a negligible tensor-to-scalar ratio. But this case with N=55 is consistent with observation considerably. For power-law potential and variable dissipation factor as Q=αϕn, the consistency with observation is also considerable with a reliable tensor-to-scalar ratio. The quadratic and quartic potentials with variable dissipation function as Q=αϕn are consistent with Planck2018 TT, TE, EE+lowE+lensing data at the 68% and 95% levels of confidence for some intervals of the parameter n.

Turbulence scaling from deep learning diffusion generative models

Complex spatial and temporal structures are inherent characteristics of turbulent fluid flows and comprehending them poses a major challenge. This comprehension necessitates an understanding of the space of turbulent fluid flow configurations. We employ a diffusion-based generative model to learn the distribution of turbulent vorticity profiles and generate snapshots of turbulent solutions to the incompressible Navier-Stokes equations. We consider the inverse cascade in two spatial dimensions and generate diverse turbulent solutions that differ from those in the training dataset. We analyze the statistical scaling properties of the new turbulent profiles, calculate their structure functions, energy power spectrum, velocity probability distribution function and moments of local energy dissipation. All the learnt scaling exponents are consistent with the expected Kolmogorov scaling This agreement with established turbulence characteristics provides strong evidence of the model's capability to capture essential features of real-world turbulence.

Toward a Unified Understanding of Estimating Evapotranspiration: The Linkage Between Three Effective Parsimonious Models

AbstractThe maximum information entropy production model (MaxEnt), the relative humidity at equilibrium approach (ETRHEQ), and the Surface Flux Equilibrium model (SFE) are three recently developed models to estimate evapotranspiration. Although the connection between ETRHEQ and SFE is evident, no attempts have been made to investigate the congruence, distinctions, or potential complementarity between the two models and MaxEnt. Our mathematical analysis demonstrates that minimizing the vertical variance of RH in ETRHEQ is equivalent to minimizing the dissipation function of energy fluxes in MaxEnt, under the assumption of the same eddy diffusivity of heat and water vapor and with a specific expression for the ratio between the thermal inertia terms for H and LE. The connection between ETRHEQ, SFE, and MaxEnt is independent of Monin‐Obukhov similarity theory (MOST)’s extremum solution, and MOST's extreme solution can be viewed as equivalent to introducing a constant correction factor to account for atmospheric stability. While ETRHEQ and MaxEnt can be united within a single hydrometeorological framework, they diverge in their approaches to modeling evapotranspiration, particularly in how they address the roles of vegetation and land surface heterogeneity. More importantly, the unified framework suggests that turbulence fluxes within the atmospheric boundary layer adhere to the principles of maximum information entropy production. The way in which dissipation, along with its associated entropy production, is established using information entropy theory deviates from traditional thermodynamic entropy formulations. Exploring the connection between thermodynamic and information entropy and developing proper formulations of dissipation for energy fluxes presents an appealing avenue for prospective research.

Three-dimensional model for cyclic, rate-independent and compressible response of aluminium

A three-dimensional rate-independent framework consistent with thermodynamics is presented to study the dissipative response of metals. The entropy inequality is transformed into equality by introducing a non-negative, continuous rate of dissipation function. The constitutive relation that relates the Hencky strain and Cauchy stress is parametrized by replacement stress, instead of the plastic strain, for reasons discussed. The evolution equation for the replacement stress is obtained such that among the possible processes, the one that maximizes the rate of dissipation is realized so that thermodynamic equilibrium is achieved in the shortest possible time. Appropriate 3D constitutive functions to model aluminium are prescribed for the dissipation function and a Gibbs-like potential. The variation of the transverse strain as a function of the uniaxial strain differs between the present formulation and classical plasticity. Consistent with some of the experimental observations, the material tends to be compressible in the present formulation during plastic deformations. Thus, further experimental investigations are required to choose the appropriate constitutive relation.

A multiaxial multiscale compressive thermo-mechanical damage model for Fiber Reinforced Cementitious Composites (FRCC)

In the current study, a multiaxial multiscale compressive stress-strain model for Fiber Reinforced Cementitious Composites (FRCC) was innovatively developed. The proposed model was on the basis of the theory of thermodynamics, multiscale theory and internal variable theory. The free energy function and dissipation function were established to characterize the elastic and plastic deformation of FRCC, respectively. The multiscale behaviors of FRCC in meso-scale and macro-scale was taken into account. In meso-scale, the energy dissipation functions of fiber in tension and bond-slip deformation in the interface transition zone (ITZ) were established. The resistance to cracking of cementitious matrix through the fiber bond stress were taken into account. In macro-scale, the thermo-mechanical coupling contributions of plastic damage, thermal damage, yield surface and hardening evolution were creatively considered in the potential function within thermodynamics framework. The influence of types and volume fractions of fibers on the reinforcement mechanism was discussed. When the temperature is higher than the melting point of fiber, the fiber influence coefficients was employed. The sensitivity of yield surface was discussed. The numerical predictions of this developed model were carried out for validation. The nonlinear stress-strain responses of concrete were accurately captured at temperatures ranges from 20 °C to 800 °C compared with the experimental data, which indicates the accuracy and applicability of the proposed model.

Dynamic modeling and analysis of viscoelastic hard-magnetic soft actuators with thermal effects

Hard-magnetic soft materials (HMSMs) are a class of magnetoactive smart polymers capable of sustaining a high residual magnetic flux density and can undergo large actuation strains under an external magnetic excitation. Due to these exceptional characteristics, soft actuators based on HMSMs hold enormous potential for remote-controlled applications. The temperature and viscoelasticity considerably influence the performance of these materials during the operation. This work aims to develop an analytical framework for modeling the dynamic behavior of a planar hard-magnetic soft actuators (HMSA) considering temperature and viscoelastic effects. The constitutive behavior of the viscoelastic HMSA is described by employing an incompressible neo-Hookean model in conjunction with a Zener rheological model and the Rayleigh dissipation function. The dynamic governing differential equations of motion are derived by utilizing the non-conservative form of the Euler–Lagrange equation. This study delves into the collective influence of temperature and viscoelastic properties on the stability, periodicity, and resonance characteristics of nonlinear vibrations exhibited by HMSM-based planar actuator subjected to dynamic magnetic loading, presenting the findings through time-history responses, Poincaré maps, and phase-plane plots. The presented results can help in the efficient and robust design of HMSM-based actuators and can also serve as an initial step toward the development of advanced actuators exposed to dynamic loading under variable temperatures for diverse applications in the fields of engineering and medicine.

The Stationary Navier–Stokes–Boussinesq System with a Regularized Dissipation Function

The Stationary Navier–Stokes–Boussinesq System with a Regularized Dissipation Function

Decomposition of the mechanical stress tensor: from the compressible Navier–Stokes equation to a turbulent potential flow model

We frame the mechanical stress tensor decomposition in a general procedure which involves the Helmholtz–Hodge decomposition. We highlight the impact of the mechanical stress tensor decomposition on the Navier–Stokes equation, with emphasis on the dissipation function. For fluids with low compressibility, we draw some insights on the Reynolds Averaged Navier–Stokes equations, and on the Reynolds stress tensor decomposition. We derive a turbulent potential flow model, and investigate the transition from viscous potential flow to turbulent potential flow. Under low Mach number approximation, we apply the turbulent potential flow model to one-dimensional propagation of large amplitude pressure waves in liquid-filled pipe.

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

A thermodynamically consistent phase transformation model for multiphase alloys: application to Ti$$_6$$Al$$_4$$V in laser powder bed fusion processes

A thermodynamically consistent phase transformation model for multiphase alloys: application to Ti$$_6$$Al$$_4$$V in laser powder bed fusion processes

Finite and fixed‐time stabilization of discrete‐time systems using passivity‐based control

AbstractIn this paper, we propose a new passivity‐based controller design technique for discrete‐time fully actuated systems. The controller establishes finite‐time and fixed‐time convergence of dynamical system trajectories to an equilibrium state. A new form of a dissipation function, selected a priori, is introduced to design a feedback control rule to achieve such convergence properties. An energy shaping and damping injection methodology is extended to achieve non‐asymptotic stabilization. A numerical example to validate the proposed methods is provided.

Stochastic models in the under-resolved simulations of spray formation during high-speed liquid injection

In the under-resolved simulation of the high-speed liquid injection into stagnant air, the attention in subgrid-scale models is focused on events of intense gradients of the velocity in turbulent flow, i.e., on effects of intermittency. Three typical flows are considered—the in-nozzle flow, primary atomization zone, and secondary atomization of spray droplets. In the simulation of the first two flows, the filtered Navier–Stokes equations are forced by stochastic processes with properties which incorporate the statistical physics of fluid acceleration at the high Reynolds number—in this way we update the under-resolved acceleration. In the case of in-nozzle flow, the proposed stochastic subgrid acceleration model is combined with wall-damping function, and the ability in prediction of the velocity statistics is demonstrated. In the simulation of primary atomization, the approach with stochastic subgrid acceleration is combined with the volume of fluid method. This leads to intensification of the interface dynamics, resulting in additional corrugation, with more intense shearing and stretching of liquid structures are observed. Thereby, the experimental profiles of the time-averaged liquid volume fraction distribution for four different axial locations are rather well predicted. The secondary atomization of droplets is simulated by a new stochastic model for the breakup rate along the droplet path. To this end, the breakup rate is expressed as a function of turbulent viscous dissipation which evolves along the droplet path according to the proposed stochastic process. Preliminary assessment performed against the recent experiments shows the correct predictability of this model and its low sensibility to the grid density.

Determination of Self-Heating in Silicon Photomultipliers.

The main consequence of radiation damage on a silicon photomultiplier (SiPM) is a significant increase in the dark current. If the SiPM is not adequately cooled, the power dissipation causes it to heat up, which alters its performance parameters. To investigate this heating effect, a measurement cycle was developed and performed with a KETEK SiPM glued to an Al2O3 substrate and with HPK SiPMs glued to either an Al2O3 substrate or a flexible PCB. The assemblies were connected either directly to a temperature-controlled chuck on a probe station, or through layers of materials with defined thermal resistance. An LED operated in DC mode was used to illuminate the SiPM and to tune the power dissipated in a measurement cycle. The SiPM current was used to determine the steady-state temperature reached by the SiPM via a calibration curve. The increase in SiPM temperature due to self-heating is analyzed as a function of the power dissipation in the SiPM and the thermal resistance. This information can be used to adjust the operating voltage of the SiPMs, taking into account the effects of self-heating. Similarly, this approach can be applied to investigate the unknown thermal contact of packaged SiPMs.

A Theory of Maximum Entropy Production and Its Application to Microwave Remote Sensing—Simultaneous Retrieval of Soil Moisture and Vegetation Water Content

AbstractA theory of maximum entropy production (MEP) for electromagnetic wave propagation in dielectric materials is proposed and applied to simultaneously retrieving soil moisture (SM) and vegetation water content (VWC) from L‐band microwave brightness temperature (TB). One representation of the MEP principle states that a non‐equilibrium system corresponds to such a configuration of energy fluxes that minimizes a dissipation function under the constraint of energy conservation. The dissipation function for radiative transfer is formulated as an analogy of that for heat transfer. A new physical parameter, radiative inertia as an analogy of thermal inertia, is introduced to characterize radiative attenuation in dielectric media. The radiative inertia is parameterized in terms of the penetration depth of electromagnetic waves as a function of the complex dielectric constant. The MEP based retrieval algorithm predicts SM and VWC by minimizing the dissipation function under the constraint of the conservation of radiative energy. The retrievals of SM and VWC based on the MEP theory were validated against field observations in tropical and temperate forested regions of the Amazon and North America. The proof‐of‐concept analysis demonstrates the capability of the MEP algorithm for simultaneous retrievals of SM and VWC even for dense canopy (e.g., VWC > 5 kg m−2). The MEP method is a new theoretical framework for developing innovative remote sensing algorithms of the Earth system not limited to microwave observations.