Dissipation Inequality Research Articles

Global entropy stability for a class of unlimited second-order schemes for 2D hyperbolic systems of conservation laws on unstructured meshes

We introduce a class of second-order accurate finite volume schemes defined on two dimensional unstructured meshes. They satisfy a global discrete entropy stability for an arbitrary strictly convex entropy provided the second-order discretization verifies a dissipative inequality. These schemes do not require usual limitation techniques. Numerical experiments are performed to confirm the accuracy and the shock capturing efficiency of the schemes.

Coupled thermo‐viscoplastic fracture model for ductile‐brittle failure of amorphous glassy polymers with phase‐field approach

AbstractAmorphous glassy polymers have an extensive use in the industrial sectors including micro‐electronics, medical industry and aerospace, therefore their design and usage have become a significant task nowadays. The fracture response of these polymers may vary from ductile to brittle depending on several factors such as entanglement density, temperature level and external loading rate. The ductile response is driven by diffuse shear zones exhibiting volume–preserving inelastic deformations while the brittle response is manifested by very small crack‐like defects composed of a sequence of fibrillar bridges separated by micro‐voids, thereby connotating void formation consisting of nucleation and propagation steps. The presents study is focused on the description of shear yielding and crazing phenomenon in terms of their respective evolution equations. In addition, an extension towards the modelling of the fracture is employed via the crack phase–field approach, considering ductile and brittle failure simultaneously. This is provided by the novel failure criterion that features a critical amount of plastic strain and void volume fraction. Since the proposed approach unitedly models the macroscopic crack initiation and propagation for ductile or brittle failure, it is asserted to be more physically grounded compared to present models in the literature. Constitutive formulations for shear yielding, crazing, and void volume fraction are derived with their specific forms starting with the local and conductive component of the dissipation inequality. The performance of model is evaluated after developing the local and global Newton–type update algorithms for the dissipative internal and primary field variables, respectively and it has been analysed by fitting of several experimental data of homogeneous and inhomogeneous tests. The findings reveal the remarkable temperature dependency on the type of failure as well as the interaction between loading rate and temperature change owing to dissipative heating in the solid.

On the time-dependent mechanics of membranes via the nonlinear finite element method

In this work, the problem of finite generalized and viscoelastic deformation of thin membranes with different geometries, made of incompressible hyperelastic materials, is formulated. The multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts, and making use of dissipation inequality, nonlinear evolution equations for the internal variables of the models are obtained. The mechanical behavior of the dampers is assumed to be linearly viscous. Therefore, the Cauchy-like stress in the dampers is similar to that in Newtonian fluids and includes terms for the hydrostatic pressure and viscosity. The implicit and second-order accurate trapezoidal method is employed for the time integration of the evolution equations. Due to the highly nonlinear governing differential equations including the effects of geometric nonlinearity and viscoelasticity, a nonlinear finite element formulation based on isoparametric elements is developed. The accuracy and performance of the developed formulation and time-dependent solutions are verified by studying several numerical examples. The obtained results are compared with theoretical and experimental data available in the literature. The proposed formulation can appropriately predict the experimental results of viscoelastic membranes for both in-plane and out-of-plane deformations.

Nonlinear Constitutive Model for Concretes at Early Ages Considering the Hydration Reaction

Hygro-thermal and hydration behaviors exist in concretes at early ages, which inevitably affect their mechanical properties. Accompanying water consumption in the concrete, there are several irreversible thermodynamic processes, including hydration heat production and strength growth. By considering coupled deformation, heat conduction, and a hydration reaction, a thermodynamically consistent continuum theory is developed for concrete at early age. At first, to satisfy the dissipation inequality, the energy dissipation caused by these irreversible processes can be tentatively described by introducing a hydration extent ℓ, so that a hydration kinetic equation considering the temperature and stress effects is established by linking the reaction rate with the concentrations of reactants. Compared to the Arrhenius linear relation between the reaction rate and chemical affinity, this nonlinear hydration kinetic equation is applicable to complex hydration reactions. Then, a fully coupled nonlinear constitutive model is constructed to interpret the thermo-chemical-mechanical interactions, whose numerical implementation is generated in commercial software ABAQUS using UEL (user-defined element) subroutines. Particularly, the model is used to predict the evolution of elastic modulus and temperature of two kinds of concretes under adiabatic testing: ordinary concrete (OC) and high performance concrete (HPC), which shows good agreement with experimental results. Furthermore, some examples are numerically studied to illustrate the interactions among mechanical deformation, hydration reaction, and heat conduction in concrete at early ages.

A cohesive phase-field fracture model for chemo-mechanical environments: Studies on degradation in battery materials

In the context of computational modeling of fracture in chemo-mechanical environments, physically-sound and strong coupling between different fields is essential. Furthermore, our knowledge of the fracture in a purely mechanical setting should be extended to the new realm adequately. In this work, we apply the cohesive phase-field (CPF) fracture models to address damage initiation and progression in a chemo-mechanical coupled environment. Since CPF models are shown to be independent of the length scale parameter, such models allow a unified simulation framework for bulk and interface damages that concurrently and competitively occur in the battery materials. First, a thermodynamical framework is discussed to obtain all the possible coupling terms consistently. Through a systematic derivation from dissipation inequality and by performing various studies, we intend to comparatively demonstrate the role of different coupling terms and their impact on the obtained results. Specifically, we focus on (1) the influence of the stress field as well as the damage variable on the flux vector, (2) concentration-dependent fracture properties, (3) advantages of cohesive phase-field formulation in the multiphysics environment, and (4) influence of phase-transformation on the cracking mechanism in solids. Finally, we present some initial studies on a simplified system of a solid-state battery system, where cracking inside a single crystalline active material surrounded by a solid electrolyte is under focus. The results of the study show the dominant crack patterns. Such investigations open up opportunities for better design of battery microstructures and enhance their lifetime and performance at the cell level.

Active Nodes for Passivity and Finite-Gain Stability of Non-autonomous Networks

This paper studies the role of a class of node systems with adjustable damping coefficients in the node dissipation inequalities, called the active nodes, for achieving passivity and finite-gain stability of network systems with external inputs and outputs. The paper first discusses the relation between the active nodes with passivity-type dissipation inequalities and those with L2-stability-type dissipation inequalities, especially focusing on how to transform one type into the other. Then, the paper proceeds to discuss the graph-theoretic condition on the locations of active nodes to achieve network passivity or finite-gain stability. Thereafter, a method for network reduction by exploiting active nodes is presented. Finally, a human-in-the-loop stabilization problem is solved for a network system by exploiting the results developed in the paper.

Dimension Analysis via Differential Lyapunov and Dissipation Inequalities⋆

Motivated by complex and diverse dynamics in engineering and nature, this paper considers Hausdorff dimension estimates for compact invariant sets of dynamical systems. Our aim is to develop a framework of dimension analysis as an extension of Lyapunov's stability theory as well as Willems’ dissipativity theory. In particular, we extend the existing result by Smith involving Lyapunov inequalities for Hausdorff dimension estimates by employing state-dependent metrics. The obtained result is then combined with the dissipativity framework for studying interconnected systems. This paper is mainly devoted to mathematics but describes several connections with recent works on the differential framework for contraction analysis.

Delay-dependent flocking dynamics of a two-group coupling system

<p style='text-indent:20px;'>A group coupling model for a system with large-scale nodes is investigated. The model is formulated as a system of functional differential equations. It incorporates two additional factors that exist in the evolution of flocking behavior, but are often ignored in modeling: (ⅰ) the diversity of interactions, including inter-group and intra-group interactions and (ⅱ) the delayed response of particles to signals from the environment or neighbors, including transmission and processing delays. Theoretically, using the divide-and-conquer method and under different delay factors, sufficient conditions for self-organizing flocking are derived by constructing a dissipative differential inequalities with continuous parameters respectively, which involve some analytical expressions of the upper bound of the delay that the system can tolerate. Results of systematic numerical simulations are presented. They not only validate the analytical results, but hint at a somehow surprising behavior of system, that is, weak flocking behavior occurs when two types of delays coexist.</p>

Emergent behaviors of thermodynamic Cucker–Smale ensemble with a unit-speed constraint

We propose the thermodynamic Cucker–Smale system with unit-speed constraints (TCSUS) and study its emergent dynamics with rigorous proofs. For this, three dissipative differential inequalities are derived in terms of $ L^\infty $-diameters of the position–velocity–temperature to verify mono-cluster flocking in the TCSUS model under suitable frameworks using two approaches. Motivated by the first approach, we provide a sufficient framework for bi-cluster flocking to the TCSUS model. Unlike to the mono-cluster flocking behavior of TCSUS, the decay rate of velocity–temperature for each cluster group is not exponential and is dependent on the decay rate of the corresponding communication weight, respectively.

Input-to-state stability of non-autonomous infinite-dimensional control systems

<p style='text-indent:20px;'>This paper addresses input-to-state stability (ISS) and integral input-to-state stability (iISS) for non-autonomous infinite-dimensional control systems. With the notion of uniformly exponential stability scalar function, ISS and iISS are considered based on indefinite Lyapunov functions. In addition, we obtain several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities. As a result, the iISS criteria of non-autonomous bilinear systems is also established. Furthermore, an illustrative example is given to show the applicability of the results.</p>

Global well-posedness and asymptotic behavior of $ BV $ solutions to a system of balance laws arising in traffic flow

<abstract><p>We establish global well-posedness and asymptotic behavior of $ BV $ solutions to a system of balance laws modeling traffic flow with nonconcave fundamental diagram. We prove the results by finding a convex entropy-entropy flux pair and verifying the Kawashima condition, the sub-characteristic condition, and the partial dissipative inequality in the framework of Dafermos. This problem is of specific interest since nonconcave fundamental diagrams arise naturally in traffic flow.</p></abstract>

Robustness Analysis of Nonlinear Systems Along Uncertain Trajectories

The paper presents a novel approach for robustness analysis of nonlinear dynamic systems in the vicinity of a reference trajectory. The approach linearizes the system with respect to a nominal trajectory and calculates a guaranteed upper bound on the worst-case gain. In contrast to existing methods rooted in linear time-varying systems analysis, the approach accurately includes perturbations that drive the system away from the reference trajectory. The approach further includes a bound for the error associated with the time-varying linearization. Hence, the results obtained in the linear framework provide a valid upper bound for the worst-case performance of the nonlinear system. The calculation of the upper bound relies on the dissipation inequalities formulated in the framework of integral quadratic constraints. It is therefore computationally much cheaper than sample-based methods such as Monte Carlo simulation. The feasibility of the approach is demonstrated on a numerical example.

Superelasticity degradation of NiTi shape memory alloy in wide ranges of temperature and loading level: Experimental observation and micromechanical constitutive model

In this work, cyclic tension-unloading tests with different peak strains (10%-18%) in the temperature range from 313 K to 393 K are performed to investigate the effects of temperature and loading level on the cyclic deformation behavior of NiTi shape memory alloy (SMA) wires. Experimental results demonstrate that the superelasticity degradation occurs during the cyclic deformation process, and such phenomenon aggravates with the rise of ambient temperature and loading level, which can be attributed to the complex interactions among the martensite transformation (MT), austenite plasticity (AP), martensite plasticity (MP) and the transformation-induced plasticity (TRIP). Then, a micromechanical cyclic constitutive model is proposed based on the framework of irreversible thermodynamics and Eshelby's inclusion theory. The martensite (M) phase is treated as mobile inclusions with alterable eigenstrain embedded in the austenite (A) phase matrix. The volume fractions of the M-phase, A-phase, and A-M interface-phase are introduced. The non-uniform stress fields in the three phases are estimated by employing the Mori-Tanaka's homogenization scheme and interfacial operator. The driving forces of the four types of the inelastic deformation mechanisms, MT, AP, MP, and TRIP are derived based on the proposed new Helmholtz free energy, instantaneous growth hypothesis of M domains, and the energy dissipation inequality. The inheritances of the plastic deformation induced by the movement of A-M interfaces during the repeated MT and its reverse are incorporated. Finally, to validate the predictive capability of the proposed model, the predicted results for the superelasticity degradation of NiTi SMA with various peak strains at different ambient temperatures are compared with the experimental ones. Moreover, the dominant plastic deformation mechanisms and the effects of AP, MP, TRIP on the cyclic deformation of NiTi SMA are discussed.

Microstructure‐based effective stress for unsaturated frozen soils

AbstractCurrent knowledge on unsaturated soils suggests that the effective stress should be related to soil microstructures, by which the immobile water attached to the soil solid in micropores and the free water mainly governed by capillary effects in relatively large pores can be distinguished. Paying attention to this knowledge, the micro‐ and macroscopic degrees of saturation to separate micro‐ and macropores occupied by the equivalent water (ice and liquid water) are used and the microstructurally based effective stress for unsaturated frozen soils is proposed using the first law of thermodynamics and the mixture theory. The formulated effective stress considers morphological and mechanical differences of ice in two porosity scales, and can reduce to familiar forms under various assumptions for both frozen–unfrozen and saturated–unsaturated states. Regarding the mixture of microscopic ice and soil particles as the solid skeleton, the elastic bulk modulus of the solid matrix for frozen soils in the regime of compressible solids has also been derived. In addition, other emerging energy‐conjugate pairs have been identified and the second law of thermodynamics is employed to obtain the dissipation inequality in the regime of inelastic deformation for unsaturated frozen soils. The theoretical work presented in this paper could provide an important insight into mechanical constitutive study for unsaturated (saturated) frozen soils in cold regions engineering.

Modeling thermoelectric effects in piezoelectric semiconductors: New fully coupled mechanisms for mechanically manipulated heat flux and refrigeration

We present a continuum theory for analyzing the interaction between the thermoelectric and mechanical fields in piezoelectric semiconductors. The balance laws and dissipation inequality are formulated in the reference configuration. Thermodynamically consistent constitutive equations are derived, including the drift-diffusion current, Fourier's law for thermal flux, Seebeck effect, and Peltier effect, by introducing a generalized Fick's law. The heat conduction equation and Joule heating generation with the semiconducting effect are derived by combing the energy balance and the second Gibbs relation. The framework is then geometrically linearized for applications in small deformation of crystal solids. Based on the newly developed framework, two new coupling mechanisms between thermoelectric and mechanical fields in crystals of class of 6mm are identified. 1) A mechanical load can block both electron and thermal fluxes in the loading area via the mechanically induced potential well. At a strain level of 1%, the current and heat flow can be reduced by as much as 80%. This effect facilitates the design of new switching devices. 2) The mechanical load can surprisingly act as a current amplifier through the induced potential barrier. Furthermore, making use of the new and unusual Joule heating generation, thermal dipoles can be created, indicating that mechanical loading can lead to local refrigeration. Via a simple numerical model, we demonstrate that the mechanical deformation can produce a temperature difference of 0.06 K at the strain level of around 1%. Based on this new and exciting cooling mechanism, we further propose a novel multi-stage pyramidal cascade device for refrigeration. The framework in this paper provides a foundation for analyzing multiple physics problems in semiconductor structures and also potential ideas for switching and refrigeration devices. Since it is based on finite deformation theory, the present framework would be helpful in analyzing the behavior of emerging flexible semiconductor materials or developing the corresponding computational methods.

Zener Model with General Fractional Calculus: Thermodynamical Restrictions

We studied a Zener-type model of a viscoelastic body within the context of general fractional calculus and derived restrictions on coefficients that follow from the dissipation inequality, which is the entropy inequality under isothermal conditions. We showed, for a stress relaxation and a wave propagation, that the restriction that follows from the entropy inequality is sufficient to guarantee the existence and uniqueness of the solution. We presented numerical data related to the solution of a wave equation for several values of parameters.

Mechanics of nonlinear visco-hyperelastic-hysteresis membranes

This contribution aims at developing a general formulation for the large deformation analysis of membranes considering both hysteresis and viscoelastic effects. Firstly, the kinematics of the deformation of thin membranes in three-dimensional space is formulated. Afterward, the visco-hyperelastic-hysteresis formulation of membranes is derived based upon the multiplicative decomposition of the deformation gradient tensor in the hysteretic and viscoelastic branches of the model. Evolution equations of the hysteresis and viscoelastic internal variables are obtained based on dissipation inequality. The formulation admits using both Newtonian and non-Newtonian fluid models to model the viscous dashpots. For the time integration of the evolution equations, the implicit Euler-backward method is employed. Variational formulation of the problem for development of a nonlinear finite element formulation, in the Total Lagrangian framework, is presented. To investigate the capability of the developed formulation, several numerical examples are provided. The results are in excellent agreement with the data available in the literature.

Partial Hölder regularity for solutions of a class of cross-diffusion systems with entropy structure

In this article we show a C0,α-partial regularity result for solutions of a certain class of cross-diffusion systems with entropy structure. Under slightly more stringent conditions on the system, we are able to obtain a C1,α-partial regularity result. Amongst others, our results yield the partial C1,α-regularity of weak solutions of the Maxwell-Stefan system, as well as the partial C1,α-regularity of bounded weak solutions of the Shigesada-Kawasaki-Teramoto model. The classical partial regularity theory for nonlinear parabolic systems as developed by Giaquinta and Struwe in the 80s proceeds by Campanato iteration which relies on energy methods. Our analysis here centers around the insight that, in the Campanato iteration strategy, we can replace the use of energy estimates by “entropy dissipation inequalities” and the use of the squared L2-distance to measure the distance between functions by the use of the “relative entropy”. In order for our strategy to work, it is necessary to regularize the entropy structure of the cross-diffusion system, thereby introducing a new technical tool, which we call the “glued entropy”.

Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation-Dissipation Inequalities.

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation–dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath’s spectral density in quantum Brownian motion studies, we formulate a “thermal” FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current–current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.

Robust attitude tracking on the Special Orthogonal Group [formula omitted] using PD-type state feedback and linear matrix inequalities

The problem of rigid-body attitude tracking in the presence of exogenous disturbances is addressed. Attitude is parameterized using the rotation matrix, an element of the Special Orthogonal Group SO(3), as it provides a singularity-free and unambiguous attitude description. The closed-loop stability and robustness properties of a PD-type state-feedback control law, proposed in literature for attitude tracking using rotation matrices, are investigated using the nonlinear H∞ control framework. Starting from a dissipation inequality, sufficient conditions are derived which ensure that the closed-loop energy gain from bounded, finite-energy exogenous disturbances to a specified error signal respects a given upper bound. Then, the sufficient conditions are reformulated using the state and input matrices for the translational double integrator, and recast as linear matrix inequalities (LMIs). Lastly, the reformulated LMIs are used to synthesize controller gains for the proportional and derivative state-feedback terms in the original SO(3) control law. The controller synthesis problem for a microsatellite is considered as a case study. The controller gains are obtained using the proposed LMI-based procedure, and the tracking and disturbance rejection capabilities of the SO(3) controller are illustrated.