A discrete mathematical model for the dynamics of a crowd of gazing pedestrians with and without an evolving environmental awareness

In this article, we present a microscopic-discrete mathematical model describing crowd dynamics in no panic conditions. More specifically, pedestrians are set to move in order to reach a target destination and their movement is influenced by both behavioral strategies and physical forces. Behavioral strategies include individual desire to remain sufficiently far from structural elements (walls and obstacles) and from other walkers, while physical forces account for interpersonal collisions. The resulting pedestrian behavior emerges therefore from non-local, anisotropic and short/long-range interactions. Relevant improvements of our mathematical model with respect to similar microscopic-discrete approaches present in the literature are: (i) each pedestrian has his/her own dynamic gazing direction, which is regarded to as an independent degree of freedom and (ii) each walker is allowed to take dynamic strategic decisions according to his/her environmental awareness, which increases due to new information acquired on the surrounding space through their visual region. The resulting mathematical modeling environment is then applied to specific scenarios that, although simplified, resemble real-word situations. In particular, we focus on pedestrian flow in twodimensional buildings with several structural elements (i.e., corridors, divisors and columns, and exit doors). The noticeable heterogeneity of possible applications demonstrates the potential of our mathematical model in addressing different engineering problems, allowing for optimization issues as well

A hybrid method for hydrodynamic-kinetic flow - Part I -A particle-gridmethod for reducing stochastic noise in kinetic regimes

In this work we present a hybrid particle-grid Monte Carlo method for the Boltzmann equation, which is characterized by a significant reduction of the stochastic noise in the kinetic regime. The hybrid method is based on a first order splitting in time to separate the transport from the relaxation step. The transport step is solved by a deterministic scheme, while a hybrid DSMC-based method is used to solve the collision step. Such a hybrid scheme is based on splitting the solution in a collisional and a non-collisional part at the beginning of the collision step, and the DSMC method is used to solve the relaxation step for the collisional part of the solution only. This is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution, performing collisions as in a standard DSMC method, and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution. The latter is added to a piecewise constant reconstruction of the non-collisional part of the solution, which in fact remains unchanged during the relaxation step. Numerical results show that the stochastic noise is significantly reduced at large Knudsen numbers with respect to the standard DSMC method. Indeed in this algorithm, the particle scheme is applied only on the collisional part of the solution, so only this fraction of the solution is affected by stochastic fluctuations. But since the collisional part of the solution reduces as the Knudsen number increases, stochastic noise reduces as well at large Knudsen number

A hybrid particle-grid method for hydrodynamic-kinetic flow -Part I- Reduction of DSMC noise in kinetic regimes

in: La matematica e le sue applicazioni, 14-2009, Dip. di Matematica, Politecnico di Torino. ISSN 1974-305

Velocity discretization in numerical schemes for BGK equations

The need for accurate numerical solutions to kinetic equations has sharply increased in recent times due to the fact that the dynamics of gas in micro structures largely occurs in the kinetic regime, when the Knudsen number Kn, representing the ratio between the mean free path of molecules and the physical dimensions of the computational domain, cannot be neglected. In particular much interest has focused on models approximating the Boltzmann equations for small to moderate Knudsen numbers. One such model is the BGK model introduced in [2

A hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models

In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows. In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms. Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen number