Dynamics and Coarsening of Interfaces for the Viscous Cahn—Hilliard Equation in One Spatial Dimension

In one spatial dimension, the metastable dynamics and coarsening process of an n -layer pattern of internal layers is studied for the Cahn–Hilliard equation, the viscous Cahn–Hilliard equation, and the constrained Allen–Cahn equation. These models from the continuum theory of phase transitions provide a caricature of the physical process of the phase separation of a binary alloy. A homotopy parameter is used to encapsulate these three phase separation models into one parameter-dependent model. By studying a differential-algebraic system of ordinary differential equations describing the locations of the internal layers for a metastable pattern for this parameter-dependent model, we are able to provide detailed comparisons between the internal layer dynamics for the three models. Layer collapse events are studied in detail, and the analytical theory is supplemented by numerical results showing the different behaviors for the different models. Finally, an asymptotic-numerical algorithm, based on our asymptotic information of layer collapse events and the conservation of mass condition, is devised to characterize the entire coarsening process for each of these models. Numerical realizations of this algorithm are shown.     

Special solutions for an equation arising in sand ripple dynamics

We study a nonlinear fourth order evolution equation arising in the context of sand ripple dynamics. We analyse the set of stationary solutions and travelling waves in order to recover the observed phenomenology such as different wavelengths ripples, travelling waves, coarsening and time scales. Moreover, we construct an approximate solution which describes the early stages of the dynamics and which suggests the existence of coarsening and of time scales with different dynamical behaviour.     

Reduced Dynamics of the Non-holonomic Whipple Bicycle

Though the bicycle is a familiar object of everyday life, modeling its full nonlinear three-dimensional dynamics in a closed symbolic form is a difficult issue for classical mechanics. In this article, we address this issue without resorting to the usual simplifications on the bicycle kinematics nor its dynamics. To derive this model, we use a general reduction-based approach in the principal fiber bundle of configurations of the three-dimensional bicycle. This includes a geometrically exact model of the contacts between the wheels and the ground, the explicit calculation of the kernel of constraints, along with the dynamics of the system free of any external forces, and its projection onto the kernel of admissible velocities. The approach takes benefits of the intrinsic formulation of geometric mechanics. Along the path toward the final equations, we show that the exact model of the bicycle dynamics requires to cope with a set of non-symmetric constraints with respect to the structural group of its configuration fiber bundle. The final reduced dynamics are simulated on several examples representative of the bicycle. As expected the constraints imposed by the ground contacts, as well as the energy conservation, are satisfied, while the dynamics can be numerically integrated in real time.     

Mathematical analysis of a population model with an age–weight structured two-stage life history: asymptotic behavior of solutions

In this paper, we propose a model for the dynamics of a physiologically structured population of individuals whose life cycle is divided into two stages: the first stage is structured by the weight, while the second one is structured by the age, the exit from the first stage occurring when a threshold weight is attained. The model originates in a complex one dealing with a fish population and covers a large class of situations encompassing two-stage life histories with a different structuring variable for each state, one of its key features being that the maturation process is determined in terms of a weight threshold to be reached by individuals in the first stage. Mathematically, the model is based on the classical Lotka–MacKendrick linear model, which is reduced to a delayed renewal equation including a constant delay that can be viewed as the time spent by individuals in the first stage to reach the weight threshold. The influence of the growth rate and the maturation threshold on the long-term behavior of solutions is analyzed using Laplace transform methods.     

Convergence Results for a Coarsening Model Using Global Linearization

Summary. We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in repeatedly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the time-dependent distribution of interval lengths can be solved explicitly using a global linearization transformation. This allows us to derive rigorous results on the long-time asymptotics of the solutions. If the average length of the intervals is finite, we prove that all distributions approach a uniquely determined self-similar solution. We also obtain global stability results for the family of self-similar profiles which correspond to distributions with infinite expectation.     

Solving a final value fractional diffusion problem by boundary condition regularization

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing fractional derivatives provides a suitable mathematical model for describing such a process. The initial boundary value problem is hard to solve due to the nonlocal property of the fractional order derivative. We consider a final value problem in a bounded domain for fractional evolution process with respect to time, which means to recover the initial state for some slow diffusion process from its present status. For this ill-posed problem, we construct a regularizing solution using quasi-reversible method. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. Numerical examples are presented to show the validity of the proposed scheme.     

Model order reduction by projection applied to the universal Reynolds equation

The approach presented in this paper yields a reduced order solution to the universal Reynolds equation for incompressible fluids, which is valid in lubrication as well as in cavitation regions, applied to oil-film lubricated journal bearings in internal combustion engines. The extent of cavitation region poses a free boundary condition to the problem and is determined by an iterative spatial evaluation of a superposed modal solution. Using a Condensed Galerkin and Petrov–Galerkin method, the number of degrees of freedom of the original grid is reduced to obtain a fast but still accurate short-term prediction of the solution. Based on the assumption that a detailed solution of a previous combustion cycle is available, a basis and an optimal test space for the Galerkin method is generated. The resulting reduced order model is efficiently exploited in a time-saving evaluation of the Jacobian matrix describing the elastohydrodynamic coupling in a multi-body dynamics simulation using flexible components. Finally, numerical results are presented for a single crankshaft main bearing of typical dimensions.     

Can topology reshape segregation patterns?

We consider a metapopulation version of the Schelling model of segregation over several complex networks and lattices. We show that the segregation process is topology independent and hence it is intrinsic to the individual tolerance. The role of the topology is to fix the places where the segregation patterns emerge. In addition we address the question of the time evolution of the segregation clusters, resulting from different dynamical regimes of a coarsening process, as a function of the tolerance parameter. We show that the underlying topology may alter the early stage of the coarsening process, once large values of the tolerance are used, while for lower ones a different mechanism is at work and it results to be topology independent.     

Computation of the Coupled Dynamics of Fibers in a Spray

The PUR-fiber-spray molding technology is a manufacturing process which produces polyurethane-based (PUR) composites by spraying the matrix together with reinforcing fibers in a tool form or on a substrate. Thereby chopped fibers are laterally (sidewise) injected in the polyurethane-air spray cone for wetting before the entire composite is spread on the substrate, where it starts curing. To investigate and compute the fiber orientation and density distribution in the final composites manufactured by this process, a new approach simplifying the multiply coupled interaction of the three phases is presented in this paper. Hereby it is presumed that the final position and orientation of a fiber on a substrate results from its dynamics and coupled interactions with air, PUR-droplets and other fibers within the spray cone. Thus, a model of the process is built, that computes the transient behavior of the air-liquid droplets mixture by the CFD code ANSYS Fluent and its influence on the dynamics of the fibers by an extra code called FIDYST. For this multiphase problem two approaches are presented for the droplet-fiber coupling using a concept called “homogenization” of the liquid phase (droplets in the continuous phase). (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the upper bound in SA AMG convergence analysis

In [M. Brezina, P. Vaněk and P. S. Vassilevski, An improved convergence of smoothed aggregation algebraic multigrid, Numer. Linear Algebra Appl., 19 (2012), pp. 441–469], a uniform convergence bound for smoothed aggregation algebraic multigrid with aggressive coarsening and massive polynomial prolongator and multigrid smoothers is established provided that the number of smoothing steps is equal to the coarsening ratio parameter ν. The final convergence estimate needs the uniform bound for the constant C_ν ∕ (2ν + 1). In this note, we give an improved upper bound. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximate reduction of non-linear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.     

Modelling and simulation of gas dynamics in an exhaust pipe

The gas dynamics in an exhaust pipe is studied. In particular we focus on the warm up of the catalytic converter in very short times after the engine start. This is done by combustion a small unburnt part of the exhaust gas. This process is classically modelled by gas dynamic equations. Compared to the existing literature we improve the (one-dimensional) modelling approach using a small Mach number technique and a network ansatz for the full exhaust pipe. The final simplified model on one hand still describes the main features and on the other hand it is computationally a few orders of magnitude faster than the original model. Performing numerical simulations we compare the new model to the (classical) full model and to experimental results in the literature.     

A CA-based epidemic model for HIV/AIDS transmission with heterogeneity

The complex dynamics of HIV transmission and subsequent progression to AIDS make the mathematical analysis untraceable and problematic. In this paper, we develop an extended CA simulation model to study the dynamical behaviors of HIV/AIDS transmission. The model incorporates heterogeneity into agents’ behaviors. Agents have various attributes such as infectivity and susceptibility, varying degrees of influence on their neighbors and different mobilities. Additional, we divide the post-infection process of AIDS disease into several sub-stages in order to facilitate the study of the dynamics in different development stages of epidemics. These features make the dynamics more complicated. We find that the epidemic in our model can generally end up in one of the two states: extinction and persistence, which is consistent with other researchers’ work. Higher population density, higher mobility, higher number of infection source, and greater neighborhood are more likely to result in high levels of infections and in persistence. Finally, we show in four-class agent scenario, variation in susceptibility (or infectivity) and various fractions of four classes also complicates the dynamics, and some of the results are contradictory and needed for further research.     

Analytic solution and estimation of parameters on a stochastic exponential model for a technological diffusion process

In this paper we examine the behaviour of a stochastic model that describes a technological diffusion process (continuously increasing process). Furthermore we obtain a solution for the proposed model through the estimation of the volatility using three different approximations. The adjustment of real data to the final stochastic model confirms its ability of describing and forecasting real cases.     

From a cellular automaton model of tumor–immune interactions to its macroscopic dynamical equation: A drift–diffusion data analysis approach

Several models of tumor growth have been developed from various perspectives and for multiple scales. Due to the complexity of interactions, how the macroscopic dynamics formed by such interactions at the microscopic level is a difficult problem. In this paper, we focus on reconstructing a model from the output of an experimental model. This is carried out by the data analysis approach. We simulate the growth process of tumor with immune competition by using cellular automata technique adapted from previous studies. We employ an analysis of data given by the simulation output to derive an evolution equation of macroscopic dynamics of tumor growth. In a numerical example we show that the dynamics of tumor at stationary state can be described by an Ornstein–Uhlenbeck process. We show further how the result can be linked to the stochastic Gompertz model.     

The dynamics of a Bertrand duopoly with differentiated products: Synchronization,intermittency and global dynamics

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.     

Sampling and multilevel coarsening algorithms for fast matrix approximations

This paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider the column subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.     

A Microcomputer Based Solution to a Practical Scheduling Problem

This paper presents a scheduling problem with constraints imposed on the waiting time between stages. The process in which it occurs involves the preparation, cooking and chilling of meals. A maximum of 30 min is permitted for waiting between the completion of the cooking stage and the start of chilling down to temperature of <3°C. If this is not achieved, then the food has to be discarded. Clearly the scheduling of the final cooking stage and chilling facilities is crucial in this context. The development of a procedure to do this scheduling, using a microcomputer, is described and some typical results presented. Further uses of the model are briefly outlined.     

Reduced-order modelling of self-excited,time-periodic systems using the method of Proper Orthogonal Decomposition and the Floquet theory

The mathematical models of dynamical systems become more and more complex, and hence, numerical investigations are a time-consuming process. This is particularly disadvantageous if a repeated evaluation is needed, as is the case in the field of model-based design, for example, where system parameters are subject of variation. Therefore, there exists a necessity for providing compact models which allow for a fast numerical evaluation. Nonetheless, reduced models should reflect at least the principle of system dynamics of the original model.

In this contribution, the reduction of dynamical systems with time-periodic coefficients, termed as parametrically excited systems, subjected to self-excitation is addressed. For certain frequencies of the time-periodic coefficients, referred to as parametric antiresonance frequencies, vibration suppression is achieved, as it is known from the literature. It is shown in this article that by using the method of Proper Orthogonal Decomposition (POD) excitation at a parametric antiresonance frequency results in a concentration of the main system dynamics in a subspace of the original solution space. The POD method allows to identify this subspace accurately and to set up reduced models which approximate the stability behaviour of the original model in the vicinity of the antiresonance frequency in a satisfying manner. For the sake of comparison, modally reduced models are established as well.     

The concertina pattern

This is a continuation of a series of papers on the concertina pattern. The concertina pattern is a ubiquitous metastable, nearly periodic magnetization pattern in elongated thin film elements. In previous papers, a reduced variational model for this pattern was rigorously derived from 3-d micromagnetics. Numerical simulations of the reduced model reproduce the concertina pattern and show that its optimal period ${\widehat{w}_{opt}}$ is an increasing function of the applied external field ${\widehat{h}_{ext}}$ . The latter is an explanation of the experimentally observed coarsening. Domain theory, which can be heuristically derived from the reduced model, predicts and quantifies this dependence of ${\widehat{w}_{opt}}$ on ${\widehat{h}_{ext}}$ . In this paper, we rigorously extract these heuristic observations of domain theory directly from the reduced model. The main ingredient of the analysis is a new type of estimate on solutions of a perturbed Burgers equation.