4 research outputs found
A micro-macro hybrid model with application for material and pedestrian flow
In this paper, a hybrid modeling approach for granular flow-like applications is presented. The approach allows to switch for a priori fixed points in time between the different levels of description which are the microscopic and macroscopic scale, respectively. Based on the numerical discretization of the models, the switching procedure is able to interpret information on individual objects as density distributions and vice versa. In particular, the reverse direction, i.e. from a macroscopic to a microscopic perspective, requires the solution of a nonlinear least squares problem subject to further constraints. Simulation results are given and demonstrate the good performance of the algorithm in the case of material and pedestrian flow models
Congestion in many-particle systems with volume exclusion constraints: algorithms and applications to modelling in biology
Many-particle systems with congestion are widely found in biology, for example, in cell
tissues or herds. Mathematical modelling constitutes an important tool in their study. In
contrast to common approaches, we propose two new modelling frameworks that rely on
the exact treatment of the contacts between particles: a particle-based and a continuum
framework. Both frameworks are based on the same behavioural rules, namely 1) two
particles cannot overlap with each other and 2) the particles seek a minimum of a given
confining potential at all times. The dynamics is driven by the evolution of the potential
and changes in particle characteristics, such as size.
In the first part, the static equilibria of the particle-based model are obtained as solutions
to a minimization problem. This leads to non-convex optimization under volume
exclusion constraints. Classical tools are either not applicable or not efficient. We develop
and study a new and efficient minimization algorithm to approximate a solution.
The second part concerns the time-evolution of the particle-based framework. We
develop new time-stepping schemes involving the resolution of a minimization problem at
each time-step, which is tackled with the minimization algorithm developed in the first
part. The study of these schemes is performed in the case of a system of hard-spheres
undergoing ballistic aggregation on a torus and it succeeds to simulate up to one million
particles. These new tools are applied to the study of the mechanics of a cell tissue, which
has allowed to validate them in practice.
In the third part, we develop a continuum modelling framework describing the evolution
of particle density. Our approach differs from previous ones by relying on different
modelling assumptions that are more appropriate to biological systems. We show that
this novel approach leads to a free-boundary problem and we characterize the dynamics
of the boundary.Open Acces