Nontrivial Equilibrium State Research Articles

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Primary and subharmonic resonance features of a graphene nanoplatelet strengthened beam in a thermal ambient directly attacking nonlinear integro-partial differential governing equations

The nonlinear aspects of advanced structures may be utilized for design purposes and making use of advantages and avoiding disadvantages of such structures. On the other hand, developing information from secondary resonance examinations of advanced structures is worthwhile for engineering objectives such as mass detection, energy harvesting and damage detection of systems likewise structures. Moreover, a precise solution methodology discards any possibility of qualitative errors and incorrect prejudice. Accordingly, for the first time, the nonlinear primary resonance alongside the subharmonic resonance of an advanced beam structure strengthened by means of graphene nanoplatelets in a thermal ambient is addressed in details employing the direct method of multiple scales. With the aim of defeating to the disadvantageous influences of the discretization procedure on the nonlinear dynamics of the graphene nanoplatelet strengthened beam, the nonlinear integro-partial differential equation of motion in accompanied by the in-homogeneous boundary conditions are attacked directly resorting to the method of multiple scales. It is shown that how an FGX distribution pattern of graphene nanoplatelets for a beam subjected to a subharmonic resonance excitation relative to an FGO distribution pattern effectively extends the range of excitation frequency that leads to a bistable solution. This deduction is very proficient for harvesting energy scopes. A similar finding is observed for a CC beam with respect to a CS beam. It is seen that when an FGX beam is exposed to a subharmonic resonance in a wider range of force amplitude experiences a non-trivial equilibrium state relative to an FGO beam. However, the strengthening of the beam with graphene nanoplatelets decreases the lower bound of the excitation frequency required for the activation of subharmonic resonance more significant for an FGX distribution pattern. Moreover, the primary resonance frequency response shows different trend prior to and after the situation of the external force frequency to the first natural frequency ratio equal to one for an FGX and an FGO graphene nanoplatelet strengthened beams. The presented results open a new view of the future of engineering designs for various engineering scopes such as energy harvesting, mass detection and damage detection purposes employing advanced structures.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium.

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

Mathematical Study of a Resource-Based Diffusion Model with Gilpin-Ayala Growth and Harvesting.

This paper focuses on a Gilpin-Ayala growth model with spatial diffusion and Neumann boundary condition to study single species population distribution. In our heterogeneous model, we assume that the diffusive spread of population is proportional to the gradient of population per unit resource, rather than the population density itself. We investigate global well-posedness of the mathematical model, determine conditions on harvesting rate for which non-trivial equilibrium states exist and examine their global stability. We also determine conditions on harvesting that leads to species extinction through global stability of the trivial solution. Additionally, for time periodic growth, resource, capacity and harvesting functions, we prove existence of time-periodic states with the same period. We also present numerical results on the nature of nonzero equilibrium states and their dependence on resource and capacity functions as well as on Gilpin-Ayala parameter [Formula: see text]. We conclude enhanced effects of diffusion for small [Formula: see text] which in particular disallows existence of nontrivial states even in some cases when intrinsic growth rate exceeds harvesting at some locations in space for which a logistic model allows for a nonzero equilibrium density.

Lyapunov Mappings and Analysis of a Nonlinear Spatio-temporal Epidemic Model

This paper focuses on the global asymptotic properties of the SIR diffusive model for infectious diseases. Using the analytic technique with Lyapunov functions, we developed conditions for the global attractor of a unique disease steady-state and the disease free equilibrium. The most eminent refuge to the model is the direct Lyapunov mapping. We investigate the global well-posedness of the mathematical model, determine conditions on Ro for which non-trivial equilibrium states exist, and examine their global stability. We are interested in finding the model’s basic reproductive number, which determines whether the disease dies out or persists in the population. Finally, we consider a series of computational results to verify the theoretical results. The extensive numerical simulations show the dynamics of different population groups over time. The effects of different parameters on the compartments are shown in detail. The findings allude that the dynamics of the system are entirely estimated by the deterministic value Ro. Dhaka Univ. J. Sci. 69(3): 161-170, 2022 (June)

Optimal Harvest Problem for Fish Population—Structural Stabilization

The influence of environmental conditions and fishery on a typical pelagic or semi-pelagic fish population is studied. A mathematical model of population dynamics with a size structure is constructed. The problem of the optimal harvest of a population in unstable environment conditions is investigated and an optimality system to the problem research is constructed. The solutions properties in various cases have also been investigated. Environmental conditions influence the fish population through recruitment. Modelling of recruitment rate is made by using a stochastic imitation of environmental conditions. In the case of stationary environment, a population model admits nontrivial equilibrium state. The parameters of fish population are obtained from this equilibrium condition. The variability of environment leads to large oscillations of generation size. The fluctuations of the fish population density follow the dynamics of recruitment rate fluctuations but have smaller gradients than recruitment. The dynamics of the optimal fishing effort is characterized by high variability. The population and the average size of individuals decrease under the influence of fishery. In general, the results of computer calculations indicate the stabilization of the population dynamics under influence of size structure. Optimal harvesting also contributes to stabilization.

Uncertainty quantification for hybrid random logistic models with harvesting via density functions

The so-called logistic model with harvesting, p′(t)=rp(t)(1−p(t)K)−c(t)p(t), p(t0)=p0, is a classical ecological model that has been extensively studied and applied in the deterministic setting. It has also been studied, to some extent, in the stochastic framework using the Itô Calculus by formulating a Stochastic Differential Equation whose uncertainty is driven by the Gaussian white noise. In this paper, we present a new approach, based on the so-called theory of Random Differential Equations, that permits treating all model parameters as a random vector with an arbitrary join probability distribution (so, not just Gaussian). We take extensive advantage of the Random Variable Transformation method to probabilistically solve the full randomized version of the above logistic model with harvesting. It is done by exactly computing the first probability density function of the solution assuming that all model parameters are continuous random variables with an arbitrary join probability density function. The probabilistic solution is obtained in three relevant scenarios where the harvesting or influence function is mathematically described by discontinuous parametric stochastic processes having a biological meaning. The probabilistic analysis also includes the computation of the probability density function of the nontrivial equilibrium state, as well as the probability that stability is reached. All these results are new and extend their deterministic counterpart under very general assumptions. The theoretical findings are illustrated via two numerical examples. Finally, we show a detailed example where results are applied to describe the dynamics of stock of fishes over time using real data.

Compressible viscous heat-conducting surface wave without surface tension

In this paper, we consider a three-dimensional compressible viscous heat-conducting fluid in a horizontally periodic domain, bounded above by a free surface and below by a rigid bottom. The motion of the fluid is governed by the full compressible, gravity-driven Navier–Stokes equations with appropriate boundary conditions. On the free surface, the effect of surface tension is neglected, and the temperature is assumed to satisfy the Robin boundary condition. Motivated by Y. Guo and I. Tice [Anal. PDE 6, 287–369 (2013), Arch. Ration. Mech. Anal. 207, 459–531 (2013), and Anal. PDE 6, 1429–1533 (2013)], we establish the global well-posedness of this free boundary problem, provided that the initial data are close to a nontrivial equilibrium state.

Piezoelectrically induced nonlinear resonances for dynamic morphing of lightweight panels

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads. An asymptotic approach based on the method of multiple scales is employed to study the plate dynamics and, primarily, to investigate the effect of the plate slenderness on the nonlinearity of the lowest normal modes. A bifurcation analysis is carried out to investigate the wide array of resonances induced by the harmonic axial excitation, acting in nontrivial equilibrium states, which include primary, super-harmonic, sub-harmonic, and principal parametric resonances, respectively. For selected slendernesses of the plate, corresponding to nonlinear softening, hardening and quasi-linear behaviors, respectively, comprehensive parametric studies on the effect of the PZT cross-sectional size, the plate damping, and the piezoelectric coupling coefficient are carried out to identify the threshold voltages triggering the different resonances.

Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms

This article concerns the Cauchy problem of conservation laws with nonlocal dissipation-type terms in R^3. By using Green's function and the time-frequency decomposition method, we study global classical solutions and their long time behavior including pointwise estimates for large initial data, for solutions near the nontrivial equilibrium state.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/129/abstr.html

Modelling of Spatially Distributed Predator–Prey System with Periodically Migrating Predator (Case Study of the White Sea Intertidal Harpacticoids and Benthic Microalgae)

A time-continuous, spatially discrete motile predator–immobile prey model has been developed to describe the interactions between harpacticoid copepods and benthic microalgae in the intertidal zone. Harpacticoids perform periodic and nondirectional migrations; the local intensity of these migrations is trophic-dependent, related to the ratio of real to maximum food consumption rates. The zero-dimensional (non-spatial) analog of the model has three nontrivial equilibrium states, two of which are stable: one corresponds to overgrazing (permanently low abundance of prey controlled by hungry predator); the second is “welfare” (well-fed predator and resource-limited prey). Simulations show that in the spatially distributed model with close-to-real parameter values, two regimes can be realized: either total overgrazing or a persistent heterogeneous dynamic regime with short-lived patches of both populations, similar to that observed in nature. The predator’s ability to migrate narrows the “welfare” domain in parametric space, but increases the trophic efficiency of the system: in this domain, both average prey abundance and average consumption rate for the predator are higher than without migrations. The migrating gain is larger in an environment that is spatially heterogeneous for prey. Assumption of local awareness of the predator (preferred migrations toward a high abundance of prey) has no significant effect on the state of the system.

Global well-posedness for the 2D Boussinesq equations with a velocity damping term

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $(0,x_2)$. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [ACWX] (see P.3597).

The Andronov–Hopf Bifurcation in a Biophysical Model of the Belousov Reaction

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the Belousov reaction mechanism. The process of main component interaction in such a reaction can be interpreted by a phenomenologically similar to it predator–prey model. Thereby, we consider a parabolic boundary value problem consisting of three Volterra-type equations, which is the mathematical model of this reaction. We carry out a local study of the neighborhood of the system’s non-trivial equilibrium state and define a critical parameter at which the stability is lost in this neighborhood in an oscillatory manner. Using standard replacements, we construct the normal form of the considered system and the form of its coefficients, that define the qualitative behavior of the model, and show the graphical representation of these coefficients depending on the main system parameters. The obtained normal form makes it possible to prove a theorem on the existence of an orbitally asymptotically stable limit cycle, which bifurcates from the equilibrium state, and find its asymptotics. To identify the applicability limits of the found asymptotics, we compare the oscillation amplitudes of one periodic solution component obtained on the basis of asymptotic formulas and by numerical integration of the model system. Along with the main case of Andronov–Hopf bifurcation, we consider various combinations of normal form coefficients obtained by changing the parameters of the studied system and the resulting behavior of solutions near the equilibrium state. In the second part of the paper, we consider the problem of diffusion loss of stability of the spatially homogeneous cycle obtained in the first part. We find a critical value of the diffusion parameter at which this cycle of the distributed system loses its stability.

The Andronov–Hopf Bifurcation in a Biophysical Model of the Belousov Reaction

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the Belousov reaction mechanism. The process of the main components interaction in such a reaction can be interpreted by a “predator – prey” model phenomenologically similar to it. Thereby, we consider a parabolic boundary value problem consisting of three Volterratype equations, which is a mathematical model of this reaction. We carry out a local study of the neighborhood of the system non-trivial equilibrium state, define a critical parameter, at which the stability is lost in this neighborhood in an oscillatory manner. Using standard replacements, we construct the normal form of the considering system and the form of its coefficients defining the qualitative behaviour of the model and show the graphical representation of these coefficients depending on the main system parameters. On the basis of it, we prove a theorem on the existence of an orbitally asymptotically stable limit cycle, which bifurcates from the equilibrium state, and find its asymptotics. To identificate the limits of found asymptotics applicability, we compare the oscillation amplitudes of one periodic solution component obtained on the basis of asymptotic formulas and by numerical integration of the model system. Along with the main case of Andronov–Hopf bifurcation, we consider various combinations of normal form coefficients obtained by changing the parameters of the studied system, and the corresponding to them solutions behaviour near the equilibrium state. In the second part of the paper, we consider the problem of the diffusion loss of stability of a spatially homogeneous cycle obtained in the first part. We find a critical value of diffusion parameter, at which this cycle of distributed system loses the stability.

Invariant characteristics of self-organization modes in Belousov reaction modeling

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the mechanism of Belousov reaction. The process of the main components interaction in such reaction can be interpreted by a phenomenologically similar to it “predator-prey” model. Thereby, we consider a parabolic boundary value problem consisting of three Volterra-type equations, which is a mathematical model of this reaction. We carry out a local study of the neighborhood of the system’s non-trivial equilibrium state and construct the normal form of the considering system. Finally, we do a numerical analysis of the coexisting chaotic oscillatory modes of the boundary value problem in a flat area, which have different nature and occur as the diffusion coefficient decreases.

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.

Temephos spraying and thermal fogging efficacy on Aedes aegypti in homogeneous urban residences

An optimal control model of Aedes aegypti population dynamics concerning classification of indoor-outdoor life cycles is taken into account in this paper. A dengue epidemics measure, the basic mosquito offspring, is obtained from the well-known next-generation matrix. The number is used to analyse the stability of the mosquito-free equilibrium. This mosquito-free equilibrium describes a steady-state condition in the mosquito population dynamics where there are no mosquitoes. Comprehensive analysis on the existence and stability of the positive non-trivial equilibrium state is shown as well. Further work deals with designation of the control measures and numerical implementation of the optimal control model using various scenarios. We highlight those measures expressed as the semi-discrete mass profile of the Temephos spraying and the thermal fogging.

Enumeration of all wedged equilibrium configurations in contact problem with Coulomb friction

For a linear structure subjected to the unilateral contact condition with a fixed obstacle, we refer to a nontrivial equilibrium state as a wedged configuration. Finding a wedged configuration is called a wedging problem. This paper discusses theoretical properties of solution set of the finite-dimensional wedging problem with the Coulomb friction, and presents numerical methods for computing all the wedged configurations. We propose algorithms for enumerating all the finitely many representative solutions, with which we can completely describe the solution set of the wedging problem. There exists a positive critical friction coefficient defined as the minimum value of friction coefficient with which at least one wedged configuration exists. We also propose an algorithm for computing the critical friction coefficient, which is based on the bisection method and the second-order cone program.

Critical friction for wedged configurations

A wedged configuration with Coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads. We analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3-D context. The critical friction coefficient, μ w , is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses. For friction coefficients μ with μ > μ w the wedged problem has at least a solution and for μ < μ w there exits no wedged configurations. For the in-plane problem we discuss the link between the critical friction and the smallest real eigenvalue μ s which is related to the loss of uniqueness. The wedged problem is stated in a discrete framework using a mixed finite element approach and the (discrete) critical friction coefficient μ h w is introduced as the solution of a global minimization problem involving a non differentiable and non-convex functional. The existence of Φ h * , the displacement field of a critical wedged state, is proved and a specific numerical method, based on a genetic algorithm, was developed to compute the critical wedged configurations. Some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations.

A New Regularization of Coulomb Friction

We present a new regularization of Coulomb’s law of friction that permits a straight-forward incorporation of frictional forces within existing numerical simulations. Similar to existing regularizations, the proposed modification to Coulomb friction leads to a continuous representation of friction and does not require the identification of transitions between slip and stick. However, unlike more common regularizations, the current reformulation maintains a structure at zero contact velocity that is identical to the classical, discontinuous form of Coulomb friction. The implementation of this regularization is presented through two examples in which slip-stick motion induced by sliding friction is of primary importance. The first is a simple one degree-of-freedom system and illustrates the existence of nontrivial equilibrium states. The second example is a multi-degree-of-freedom system in which the present model provides a computationally efficient scheme for simulating the dissipation arising from sliding friction. For systems in which slip-stick transitions are important the proposed regularization provides a computationally efficient scheme to obtain time-accurate simulations.