Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne–Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.     

Macroscopic and mesoscopic modeling based on the concept of generalized stresses for cutting simulation

Based on the concept of generalized stresses proposed by GURTIN [2] and FOREST et al. [1] macro- and meso-scopic modelling are presented. For the macroscopic modelling we develop a multi-mechanism model for strain rate and temperature dependent asymmetric plastic material behavior accompanied by phase transformation with consideration of the trip-strain. Furthermore, we extend the multi-mechanism model with the gradient of phase fraction, which is considered as an extra degree of freedom. For mesoscopic modelling a phase field model is implemented for describing phase transformations. For the scenario of a cutting process we have a martensite-austenite-martensite transformation. A generalized principle of virtual power is postulated involving generalized stresses and used to derive the constitutive equations for both approaches. Furthermore, parameters of the multi-mechanism model related to visco-plasticity with SD-effect and the trip-strain are identified for the material DIN 100Cr6. In the examples a cutting simulation for testing the multi-mechanism model and a phase-transformation simulation are shown. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A relaxation-based approach to damage modeling

Common material models that take into account softening effects due to damage have the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is often used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach [2] to regularization that no longer needs a non-local level but directly provides mesh-independent results. Due to the new variational approach we are also able to improve the calculation times and convergence behavior. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep damage modeling of a polycrystalline material

A polycrystalline material is investigated under creep conditions within the framework of continuum micromechanics. Geometrical 3D model of a polycrystalline microstructure is represented as a unit cell with grains of random crystallographical orientation and shape. Thickness of the plains, separating neighboring grains in the unit cell, can have non-zero value. Obtained geometry assigns a special zone in the vicinity of grain boundaries, possessing unordered crystalline structure. A mechanical behavior of this zone should allow sliding of the adjacent grains. Within the grain interior crystalline structure is ordered, what prescribes cubic symmetry of a material. The anisotropic material model with the orthotropic symmetry is implemented in ABAQUS and used to assign elastic and creep behavior of both the grain interior and grain boundary material. Appropriate parameters set allows transition from the orthotropy to the cubic symmetry for the grain interior. Material parameters for the grain interior are identified from creep tests for single crystal copper. Model parameters for the grain boundary are set from the physical considerations and numerical model validation according to the experimental data of the grain boundary sliding in a polycrystalline copper [2]. As the result of analysis representative number of grains and grain boundary thickness in the unit cell are recommended. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fatigue damage modeling of fibrous soft tissues

Ultimate tendon failure is often caused by fatigue loading. Recent interventions revealed a three-phase progression of histological changes during cyclic loading of the tendon. It starts from localized kinked fiber deformations, continues with additional fiber delaminations and finally leads to fiber angulations and discontinuities [5, 6]. In the present contribution, we propose a physically motivated constitutive model able to describe fatigue evolution in tendon subject to cyclic loading. The damage of the collagen fibers is elucidated by a successive permanent opening of tropocollagen molecules [7], which represent the basic building blocks of collagen fibrils. The fibril strain increase is triggered by a time-force depending rupture of glycosaminoglycan sidechains of adjacent collagen fibrils. The so obtained model is in line with recent experimental findings available in literature. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress-triaxiality-dependent modeling of ductile damage and fracture behavior

The paper deals with the effect of stress triaxiality on the inelastic behavior of aluminum alloys. The proposed continuum model takes into account stress-triaxiality-dependence of the yield condition as well as of the damage criterion and the fracture condition with different branches corresponding to different micro-mechanisms. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Macroscopic models for long-range dependent network traffic

A common way to inject long-range dependence in a stochastic traffic model possessing a weak regenerative structure is to make the variance of the underlying period infinite (while keeping the mean finite). This method is supported both by physical reasoning and by experimental evidence. We exhibit the long-range dependence of such a process and, by studying its second-order properties, we asymptotically match its correlation structure to that of a fractional Brownian motion. By studying a certain distributional limit theorem associated with such a process, we explain the emergence of an extremely skewed stable Lévy motion as a macroscopic model for the aforementioned traffic. Surprisingly, long-range dependence vanishes in the limit, being “replaced” by independent increments and highly varying marginals. The marginal distribution is computed and is shown to match the one empirically obtained in practice. Results on performance of queueing systems with Lévy inputs of the aforementioned type are also reported in this paper: they are shown to be in agreement with pre-limiting models, without violating experimental queueing analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Macroscopic models for fibroproliferative disorders: A review

In this paper we review some mathematical modelling of organ reparative processes (wound healing) for both the physiological and pathological case. The natural process of healing consists in a series of overlapping phases involving cells, chemicals, extracellular matrix (ECM) and the environment surrounding the wound site. Sometimes the healing process fails and the reparative mechanism produces pathological conditions which are commonly termed fibrosis or fibroproliferative disorders. Biological insight into the pathogenesis, progression and possible regression of fibrosis is lacking and many issues are still open. Mathematical modelling can surely play its part in this field and this paper is aimed at showing what has been done so far and what has still to be done to achieve a unified framework for studying these kinds of problems. Due to the high complexity of this phenomenon, multi-scale modelling is certainly the appropriate approach that should be used for studying these kinds of problems. Unfortunately most of the mathematical literature on this topic consists of macroscopic continuous models which fail to investigate processes occurring at smaller length scales (cellular, sub-cellular). We present a review of some of the mathematical literature, showing the widely used approaches, focusing on the interpretation of results and indicating possible developments in the study of these highly complex systems.     

半导体宏观数学模型及其数学理论

本文介绍半导体宏观数学模型的一些发展近况以及有关模型的数学分析结果 .     

Experimental and theoretical modeling of shrinkage damage formation in fiber composites

The cure of a thermoset matrix in the formation of composites is always accompanied by chemical shrinkage that generates internal stresses. In composites with high fiber content, the matrix is cured under three-dimensionally constrained conditions. The results of the previous experimental and theoretical modeling of formation of shrinkage damage under these conditions in epoxy-amine systems are briefly discussed. The effect of the model geometry (tube and plate models), scale factor, cure schedule, and chemical structure of composites is analyzed. A theoretical model for predicting the possibility of formation of shrinkage damage in fiber composites is proposed. A regular square structure was considered. Analysis showed that the maximum level of shrinkage stress in the matrix at the ultimate fiber fraction ⁺ was close to the stress level ⁺ in an experimental long tube model, where the formation of shrinkage damage took place. The experimental results for the short tube model showed that the shrinkage damage in epoxy-amine systems occurred up to approximately ⁺/3. The damage development took place within the whole range of fiber content from ⁺ to ^* (where the shrinkage stress level was about ⁺/3). In the long tube model, cohesive defects always nucleated inside the matrix. The damage grew, reached the inner surface of the tube, and then extended as adhesive debondings. A similar situation is expected in composites with a high fiber content. The damage considered is local, and the total monolithic character of a composite product is conserved.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998. Riga, Latvia.Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka 142432, Moscow Region, Russia. Translated form Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 264–275, March–April, 1998.     

Macroscopic energy diffusion for a chain of anharmonic oscillators

We study the energy diffusion in a chain of anharmonic oscillators where the Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove that under diffusive rescaling of space–time, energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and some upper and lower bounds.     

Numerical damage assessment of Haghia Sophia bell tower by nonlinear FE modeling

In the present paper, the numerical damage assessment of the masonry bell tower called “Haghia Sophia” in Trabzon, Turkey is performed by nonlinear 3D finite element modeling. The behavior of bell tower is determined under several different conditions: nonlinear static analysis containing dead and wind loads and nonlinear seismic analysis. In addition to, an assessment of the tower’s stability with respect to the tilt of the tower is carried out by means of a nonlinear analysis. In the nonlinear dynamic analysis, the east–west component of 1992 Erzincan earthquake is used. Cracking and crushing of the masonry have been taken into account, as well as the influence of material nonlinearity. The numerical analysis has given a valuable picture of possible damage evolution, providing useful hints for the prosecution of structural monitoring. The displacement and stress fields, as well as the distribution of cracking have been calculated and compared to the actual distribution of fractures in the tower. It is seen from the numerical results that there is a good agreement with present damages of the bell tower.     

Macroscopic QCD and quark-gluon plasma

We propose the equations of macroscopic QCD and consider the high-energy behavior of the quark-gluon medium described by these equations. The parton currents traversing it induce the emission of Cherenkov gluons, the wake effect, and the transition radiation. Comparison with experimental data reveals a quite large value of the chromopermittivity of the medium. The dispersion equations show that the proper modes of the medium develop instability.     

Macroscopic wave propagation for 2D lattice with random masses

We consider a simple two-dimensional harmonic lattice with random, independent, and identically distributed masses. Using the methods of stochastic homogenization, we prove that solutions with initial data, which varies slowly relative to the lattice spacing, converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub-Gaussian random variables.     

Macroscopic dimension and essential manifolds

M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.