The Lagrangian coordinate system and what it means for two-dimensional crowd flow models

Abstract

A continuum crowd flow model is reformulated in the Lagrangian coordinate system. The system has proven to give computational advantages over the traditional Eulerian coordinate system for (one-dimensional) road traffic flow. Our extension of the model and simulation method to (two-dimensional) crowd flow paves the way to explore whether the advantages also hold in two dimensions. This paper provides a first exploration and shows that a model and simulation method for two-dimensional crowd flow can be developed in the Lagrangian coordinate system and that is leads to promising results.keywords Crowd flow, Model, Simulation, Lagrangian coordinates ObjectiveCrowd flow models are used to describe, understand and predict collective behaviour of crowds. Roughly, two types of models exist: microscopic models in which the movements of individual pedestrians are described and traced and macroscopic (or continuum) models which are the focus of this study. Continuum models describe the dynamics of crowds as a continuum flow, in terms of average speed, velocity and density [Hughes, 2002]. It has been shown that, in contrast to previous claims, continuum models can reproduce self-organisation and certain important dynamic phenomena such as lane formation and diagonal striping [Hoogendoorn et al., 2014]. Furthermore, simulations based on continuum models have the potential to significantly reduce computation time whilst keeping high accuracy. This makes them useful for a larger range of applications, including real time state estimation and prediction for crowd management and optimisation of control strategies. For this, the solutions to the model equations need to be calculated using both fast and accurate computational methods.The Lagrangian coordinate system has been applied to traffic flow models [Leclercq et al., 2007]. Van Wageningen-Kessels et al. [2010], Yuan et al. [2012] show that the system has many advantages over the traditional Eulerian coordinate system, including more accurate reproduction of shock waves and more efficient state estimation based on trajectory data.Our main contribution is the proposal of an extension of Lagrangian simulation methods previously applied for one-dimensional traffic flow models to two-dimensional crowd flow models. We introduce the reformulation of the two-dimensional continuum crowd flow model into Lagrangian coordinates (Section 2), develop a numerical simulation method (Section 3) and show that it leads to meaningful simulation results (Section 4). Open questions and future research directions are discussed in Section 5.

A linear wave propagation‐based simulation model for dense and polarized crowds

“…26 Furthermore, Lagrangian hydrodynamics-based methods can be applied for pedestrian simulation. [27][28][29] However, they cannot be directly used for linear dense crowd simulation.…”

“…Similarly, Narain et al compute the pressures among the dense crowd with the unilateral incompressibility constraint (UIC) solver 26 . Furthermore, Lagrangian hydrodynamics‐based methods can be applied for pedestrian simulation 27‐29 . However, they cannot be directly used for linear dense crowd simulation.…”

Understanding Crowd Flow Movements Using Active-Langevin Model

“…The authors in [10] presented a physical modeling framework that describes the intelligent, non-local, and anisotropic behavior of pedestrians. In [11], the Lagrangian (moving) coordinate system has been used for simulation and modeling of crowd flows. The authors in [12] have developed a multi-class continuum modeling based on social force model to simulate bilinear crowd flows.…”

“…These keypoints are then fed to the Lucas Kanade Optical flow process [50] for tracking in the subsequent frames within the window W . The magnitudes of the keypoints are calculated using (11) and (12), respectively. The detailed implementation is explained in the Algorithm 1.…”

A dynamic network loading model for anisotropic and congested pedestrian flows

“…For artificial test scenarios with abrupt changes in demand and density, this result may however not hold. The main alternatives to an orthogonal space discretization are triangular or hexagonal grids (Guo et al, 2011;Chen et al, 2014;van Wageningen-Kessels et al, 2015b). Even irregular discretization geometries may be envisaged if they better fit the infrastructural layout.…”