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An existence result of smooth solutions for a complex material flow problem is provided. The considered equations are of hyperbolic type including a nonlocal interaction term. The existence proof is based on a problem-adapted linear iteration scheme exploiting the structure conditions of the nonlocal term. 35Q70, 35L65 Copyright

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... The pressureless Euler system with non-local forces has been studied recently and very limited results have been done. The local existence of classical solutions of the complex material flow dynamics which has been derived in [11], under structural condition for the interaction force has been obtained in [6]. In [3], for the 1-D model with damping and non-local interaction, a critical threshold for the existence of classical solution by using the characteristic method is presented. ...

In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.

... See [4,28,30] for threshold conditions for the pressureless systems, and [8] for isothermal pressure p(ρ) = ρ with small initial data. For the case where the communication weight is not positive, such as in the pedestrian and material flow case, an additional damping effect is required to prove the global smooth solution for small initial data in [6,32,33]. ...

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

... The pressureless Euler system with non-local forces has been studied recently and very limited results have been done. The local existence of classical solutions of the complex material flow dynamics which has been derived in [12], under structural condition for the interaction force has been obtained in [7]. In [4], for the 1-D model with damping and non-local interaction, a critical threshold for the existence of classical solution by using the characteristic method is presented. ...

In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.

... The literature ranges from the individual computation of part trajectories (microscopic models) to rescaled models describing the evolution of the part density (macroscopic models). Depending on the type of application, numerical methods (Aggarwal et al. 2015;Göttlich and Schindler 2015;Evers et al. 2015), theoretical investigations (Che et al. 2016) and model hierarchies Yu 2002, 2005) are provided. However, optimization issues of such problems are typically considered for fluid problems in general such as Allain et al. (2014), McNamara et al. (2004), Schlitzer (2000) or Wojtan et al. (2006). ...

We are interested in an optimal packing density problem for material flows on conveyor belts in two spatial dimensions. The control problem is concerned with the initial configuration of parts on the belt to ensure a high overall flow rate and to further reduce congestion. An adjoint approach is used to compare the optimization results from the microscopic model based on a system of ordinary differential equations with the corresponding macroscopic model relying on a hyperbolic conservation law. Computational results highlight similarities and differences of both optimization models and emphasize the benefits of the macroscopic approach.

The re-analysis of experimental data on mass measurements of ura- nium fission products obtained at the ESR in 2002 is discussed. State-of-the-art data analysis procedures developed for such measurements are employed.

For the simulation of material flow problems based on two-dimensional hyperbolic partial differential equations different numerical methods can be applied. Compared to the widely used finite volume schemes
we present an alternative approach, namely, the discontinuous Galerkin method, and explain how this method works
within this framework. An extended numerical study is carried out comparing the finite volume and the discontinuous
Galerkin approach concerning the quality of solutions.

Material flow simulation is in increasing need of multi-scale models. On the one hand, macroscopic flow models are used for large-scale simulations with a large number of parts. On the other hand, microscopic models are needed to describe the details of the production process. In this paper, we present a hierarchy of models for material flow problems ranging from detailed microscopic, discrete element method type, models to macroscopic models using scalar conservation laws with non-local interaction terms. Numerical simulations are presented at all levels of the hierarchy, and the results are compared to each other for several test cases.

The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad
hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t→∞, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy’s law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L
∞ weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in L
p
with decay rates, to matching Barenblatt’s profile of the porous medium equation. The density function tends to the Barenblatt’s solution of the porous medium equation while the momentum is described by Darcy’s law.

We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills. Key words. Interacting particle systems, swarming, kinetic theory, milling patterns. AMS subject classifications. 92D50, 82C40, 82C22, 92C15

The Euler-Poisson system consists of the balance laws for electron density and current density coupled to the Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well- and ill-prepared initial data on the whole space case is discussed in this paper. Although it has some relations to the incompressible limit of the Euler equations, i.e. the limit velocity satisfies the incompressible Euler equations with damping, things are more complicated due to the linear singular perturbation including the coupling with the Poisson equation. A careful analysis on the structure of the linear perturbation has been done so that we are able to show the convergence for well-prepared initial data and ill-prepared initial data where the convergence occurs away from time t = 0.

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.

A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

This book presents a view of the state of the art in multi-dimensional hyperbolic partial differential equations, with a particular emphasis on problems in which modern tools of analysis have proved useful. Ordered in sections of gradually increasing degrees of difficulty, the text first covers linear Cauchy problems and linear initial boundary value problems, before moving on to nonlinear problems, including shock waves. The book finishes with a discussion of the application of hyperbolic PDEs to gas dynamics, culminating with the shock wave analysis for real fluids. © Sylvie Benzoni-Gavage and Denis Serre, 2007. All rights reserved.

In this paper a model comparison approach based on material flow systems is investigated that is divided into a microscopic and a macroscopic model scale. On the microscopic model scale particles are simulated using a model based on Newton dynamics borrowed from the engineering literature. Phenomenological observations lead to a hyperbolic partial differential equation on the macroscopic model scale. Suitable numerical algorithms are presented and both models are compared numerically and validated against real-data test settings.

The distinct element method is a numerical model capable of describing the mechanical behavior of assemblies of discs and spheres. The method is based on the use of an explicit numerical scheme in which the interaction of the particles monitored contact by contact and the motion of the particles modelled particle by particle. The main features of the distinct element method are described. The method is validated by comparing force vector plots obtained from the computer program BALL with the corresponding plots obtained from a photoelastic analysis.

The mathematical theory of hyperbolic systems of conservation laws and the theory of shock waves presented in these lectures were started by Eberhardt Hopf in 1950, followed in a series of studies by Olga Oleinik, the author, and many others. In 1965, James Glimm introduced a number of strikingly new ideas, the possibilities of which are explored.
In addition to the mathematical work reported here there is a great deal of engineering lore about shock waves; much of that literature up to 1948 is reported in Supersonic Flow and Shock Waves by Courant and Friedrichs. Subsequent work, especially in the sixties, relies on a great deal of computation.
A series of lectures, along the lines of these notes, was delivered at a Regional Conference held at the University of California at Los Angeles in September, 1971, arranged by the Conference Board of Mathematical Sciences, and sponsored by the National Science Foundation. The notes themselves are based on lectures delivered at Oregon State University in the summer of 1970, and at Stanford University, summer of 1971. To all these institutions, my thanks, and my thanks also to the Atomic Energy Commission, for its generous support over a number of years of my research on hyperbolic conservation laws. I express my gratitude to Julian Cole and Victor Barcilon, organizers of the regional conference, for bringing together a very stimulating group of people.

The asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave.

We study the time-asymptotic behavior of solutions for the isentropic Euler equations with damping in multi-dimensions. The global existence and pointwise estimates of the solutions are obtained. Furthermore. we obtain the optimal L-p, l < p less than or equal to proportional to, Convergence rate of the solution when it is a perturbation of a constant state. Our approach is based on a detailed analysis of the Green function of the linearized system and some energy estimates. (C) 2001 Academic Press.

While the discrete element method (DEM) is attracting increasing interest for the simulation of industrial granular flow, much of the previous DEM modelling has considered two-dimensional (2D) flows and used circular particles. The inclusion of particle shape into DEM models is very important and allows many flow features, particularly in hoppers, to be more accurately reproduced than was possible when using only circular particles. Elongated particles are shown here to produce flow rates up to 30% lower than for circular particles and give flow patterns that are quite different. The yielding of the particle microstructure resembles more the tearing of a continuum solid, with large-scale quasi-stable voids being formed and large groups of particles moving together. The flow becomes increasingly concentrated in a relatively narrow funnel above the hopper opening. This encourages the hope that DEM may be able to predict important problems such as bridging and rat-holing. Increasing the blockiness or angularity of the particles is also shown to increase resistance to flow and reduces the flow rates by up to 28%, but without having perceptible effect on the nature of the flow. We also describe our methodology for constructing and modelling geometrically complex industrial applications in three dimensions and present a series of industrially important three-dimensional (3D) case studies. The charge motion in a 5 m diameter ball mill and in a Hicom nutating mill, discharge from single- and four-port cylindrical hoppers, and particle size separation by a vibrating screen are demonstrated. For each case, plausible particle size distributions (PSDs) have been used. The results obtained indicate that DEM modelling is now sufficiently advanced that it can make useful contributions to process optimisation and equipment design. Finally the parallelisation of such a DEM code is described and benchmark performance results for a large-scale 2D hopper flow are presented.

The limit of the vanishing ratio of the electron mass to the ion mass in the isentropic transient Euler–Poisson equations with periodic boundary conditions is proved. The equations consist of the balance laws for the electron density and current density for a given ion density, coupled to the Poisson equation for the electrostatic potential. The limit is related to the low-Mach-number limit of Klainerman and Majda. In particular, the limit velocity satisfies the incompressible Euler equations with damping. The difference to the zero-Mach-number limit comes from the electrostatic potential which needs to be controlled. This is done by a reformulation of the equations in terms of the enthalpy, higher-order energy estimates and a careful use of the Poisson equation.

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One of the essential questions in the area of granular matter is, how to obtain macroscopic tensorial quantities like stress and strain from ``microscopic'' quantities like the contact forces in a granular assembly. Different averaging strategies are introduced, tested, and used to obtain volume fractions, coordination numbers, and fabric properties. We derive anew the non-trivial relation for the stress tensor that allows a straightforward calculation of the mean stress from discrete element simulations and comment on the applicability. Furthermore, we derive the ``elastic'' (reversible) mean displacement gradient, based on a best-fit hypothesis. Finally, different combinations of the tensorial quantities are used to compute some material properties. The bulk modulus, i.e. the stiffness of the granulate, is a linear function of the trace of the fabric tensor which itself is proportional to the density and the coordination number. The fabric, the stress and the strain tensors are {\em not} co-linear so that a more refined analysis than a classical elasticity theory is required.