Phase trajectory portrait of the vibro-impact forced dynamics of two heavy mass particles motions along rough circle

Forced vibro-impact dynamics of the two heavy mass particle motions, in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is considered in combinations of applied analytical and numerical methods. System of two differential double equations, each for one of two heavy mass particle motions along same rough circle are composed with corresponding initial conditions as well as impact conditions. By use software package tools differential double equations are numerically integrated for obtaining phase portrait of phase trajectory branches for different mass particles initial kinetic states. By series of the phase trajectory branches for each mass particle motion between two impacts or between impact and alternation of the Coulomb’s friction force direction, two phase trajectory graphs of the system vibro-impact non-linear dynamics are composed. Different software tools are used as helping tools for calculate time moments of the series of the impacts between mass particles, as well as positions of the impacts, necessary for calculations of the impact velocities of the mass particles before and after impacts. Some comparison between forced and free vibro-impact dynamics of the two heavy mass particles in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is done. Trigger of coupled one side singularities in phase portraits are identified.     

Scheduling jobs on parallel machines with sequence-dependent setup times

Consider a number of jobs to be processed on a number of identical machines in parallel. A job has a processing time, a weight and a due date. If a job is followed by another job, a setup time independent of the machine is incurred. A three phase heuristic is presented for minimizing the sum of the weighted tardinesses. In the first phase, as a pre-processing procedure, factors or statistics which characterize an instance are computed. The second phase consists of constructing a sequence by a dispatching rule which is controlled through parameters determined by the factors. In the third phase, as a post-processing procedure, a simulated annealing method is applied starting from a seed solution which is the result of the second phase. In the dispatching rule of the second phase there are two parameters of which the values are dependent on the particular problem instance at hand. Through extensive experiments rules are developed for determining the values of the two parameters which make the priority rule work effectively. The performance of the simulated annealing procedure in the third phase is evaluated for various values of the factors.     

The propagation of weak concentration discontinuities in the isothermal flow of a multicomponent mixture in a porous medium with phase transitions

It is shown that for the general case of a system of non-linear equations, describing multicomponent isothermal flow in a porous medium with phase transitions, as in hyperbolic systems, weak concentration discontinuities propagate with finite velocities, which are determined by solving an eigenvalue problem. If the seeping phases are incompressible and there are no phase transitions, the results obtained for weak discontinuities transfer into the well-known formulae for the Buckley – Leverett model. The results are demonstrated for the case of two-component seepage with phase transitions.     

Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

Global solution to the full one-dimensional Frémond model for shape memory alloys

In this paper, we prove the existence and uniqueness of the solution to the one-dimensional initial-boundary value problem resulting from the Frémond thermomechanical model of structural phase transitions in shape memory materials. In this model, the free energy is assumed to depend on temperature, macroscopic deformation and phase fractions. The resulting equilibrium equations are the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. Fourth-order regularizing terms in the quasi-stationary momentum balance equation are not necessary, and, as far as we know for the first time, all the non-linear terms of the energy balance equation are taken into account.     

A multiphase finite element simulation of biological conversion processes in landfills

Worldwide, landfills are the most common way to dispose of waste, but have an impact on the environment as a result of harmful gas and leachate production. Estimating the long-term behaviour of a landfill in regard to this gas production and organic degrading, as well as to settlement and waste water production, is of high importance. Therefore, a model has been developed to simulate these processes. This constitutive model is based on the multiphase Theory of Porous Media. The body under investigation consists of an organic and an inorganic phase as well as a liquid and a gas phase. The equations of the model are developed on the basis of a consistent thermo-mechanical approach including the momentum balance for the solid phase and the mixture, the energy balance for the mixture and the mass balance for the gas phase. All interactions between the constituents such as mass transfers, interaction forces and energy fluxes are taken into consideration. The strongly coupled set of partial differential equations is implemented in the finite element code FEAP. The theoretical framework and the results of meantime successfully performed simulation of a real landfill body will be shown. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Riemann problem for thermoelastic materials with phase change

We consider the Riemann problem for a system of conservation laws related to a phase transition problem. The system is nonisentropic and we treat the case where the latent heat is not zero. We study the cases where the initial data are given in the same phase and in the different phases. The role of the entropy condition is studied as well as the kinetic relation and the entropy rate admissibility criterion. We confine our attention to the case where the speeds of phase boundaries are close to zero. This is one interesting case in physics. We discuss the number of phase boundaries consistent with the above criteria and the uniqueness and nonuniqueness issue of the solution to the Riemann problem.     

Inexact-Restoration Algorithm for Constrained Optimization1

We introduce a new model algorithm for solving nonlinear programming problems. No slack variables are introduced for dealing with inequality constraints. Each iteration of the method proceeds in two phases. In the first phase, feasibility of the current iterate is improved; in second phase, the objective function value is reduced in an approximate feasible set. The point that results from the second phase is compared with the current point using a nonsmooth merit function that combines feasibility and optimality. This merit function includes a penalty parameter that changes between consecutive iterations. A suitable updating procedure for this penalty parameter is included by means of which it can be increased or decreased along consecutive iterations. The conditions for feasibility improvement at the first phase and for optimality improvement at the second phase are mild, and large-scale implementation of the resulting method is possible. We prove that, under suitable conditions, which do not include regularity or existence of second derivatives, all the limit points of an infinite sequence generated by the algorithm are feasible, and that a suitable optimality measure can be made as small as desired. The algorithm is implemented and tested against the LANCELOT algorithm using a set of hard-spheres problems.     

On a Conserved Penrose-Fife Type System

We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to +∞ are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well. The authors would like to acknowledge financial support from MIUR through COFIN grants and from the IMATI of the CNR, Pavia, Italy.     

A continuum model for transport phenomena in convective flow of solid–liquid phase change material suspensions

Heat and mass transport is modeled in convective flow of a dilute binary mixture of a continuous fluid with mono-dispersed particles (PCM suspensions), in which solid–liquid phase change can take place. The model is based on the mixture continuum approach together with an approximate enthalpy formulation, in which the temporal and spatial variations of phase change fraction in the particles are considered explicitly. Derivations are given for a set of equations governing conservation of mass, momentum, species, and energy of the suspensions, as well as the evolution of phase change fraction of the dispersed particles.     

Gaussian distributions and phase space Weyl–Heisenberg frames

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a “phase space frame” associated with a Weyl–Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.     

Phase shifts and scattering cross-sections for constant potentials wells or barriers

Applying the commonly used partial wave method in the quantum theory of collisions, the scattering cross-sections of the two- and three-dimensional attractive and repulsive potentials are expressed as sums of phase shifts. With the help of the Maple software program, the phase shifts and cross-sections are plotted as functions of the kinetic energy and potential height, and it is demonstrated that the systematic convergence of the cross-sections as higher angular-momentum phase shifts are included in the summations. It is found that the cross-sections of the attractive potential oscillate with both the kinetic energy and the potential strength, even though the phase shifts are monotonic. It is also shown that in the case of the repulsive potential, both the phase shifts and cross-sections change profoundly when the kinetic energy of the incident particle is lower than the potential height.     

Parameter estimation with model order reduction and global measurements

We present a new method for parameter estimation for elliptic partial differential equations. Parameter estimation requires the evaluation of the partial differential equation for many different parameter sets. Therefore, model order reduction is reasonable. Model order reduction is composed of an offline phase and an online phase. In the offline phase the reduced model is constructed using snapshots. In this paper we use the given measurement as only snapshot. Hence, the computational costs of the offline phase are reduced. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energetic modelling for carbon fibre reinforced carbon

In this contribution an energetic model for multi-phase materials is developed describing the influence of microstructure on different length scales as well as the evolution of phase changes. Restrictions on the energy functional are discussed. In such a non-convex framework, interfacial contributions serve for relaxing the total energy. Such models can be applied to describe the macroscopic material properties of carbon fibre reinforced carbon where phase transitions between regions of different texture of the carbon matrix are observed on nanoscale as well as columnar microstructures on microscale [2]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behaviour of solutions for second order nonlinear autonomous differential equations

For a class of second order nonlinear autonomous differential equations asymptotic properties of the solutions as are studied. The investigation is done in the phase plane and the phase portrait of the equivalent system is obtained by using the zero-isocline function. Some suffcient conditions are presented which for the case of Liénard equation become also necessary. Received December 21, 1995     

Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations

Parametric identification for a class of nonlinear objects with lumped parameters described by systems of ordinary differential equations is studied. The problem is to recover the coefficients of a dynamical system depending on the phase state. For that purpose, the phase space is subdivided into a finite set of subsets or zones in which the coefficients are assumed to be constant or linear functions of state. Once the coefficients in such a form are obtained, interpolation and approximation can be used to represent the coefficients as functions of the phase variables.     

Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements

Microscopic movements are responsible for the phase transition at the macroscopic level. The power of the microscopic accelerations of these motions is not neglected, as opposed to some previous works, in the derivation of phase transition models accounting for strong dissipation or irreversible phenomena. Such models lead to nonlinear parabolic–hyperbolic systems. Some existence and uniqueness results are established, through fixed point and regularization arguments, for related Cauchy–Neumann problems.     

A digital model of coupled oscillators

A new model of coupled oscillators is proposed and investigated. All phase variables and parameters are integer-valued. The model is shown to exhibit two types of motions, those which involve periodic phase differences, and those which involve drift. Traditional dynamical concepts such as stability, bifurcation and chaos are examined for this class of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.