arXiv subjects

Omar Richardson

Publications and source records attributed to Omar Richardson.

11 recordsLinked to original sources

arXiv2024

Parallel two-scale finite element implementation of a system with varying microstructures

We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying PDE system with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed-ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two-scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

math.NA↗

arXiv2021

Deterministic model of battery, uphill currents and non-equilibrium phase transitions

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce--back mechanism acts when the number of particles flowing in one direction exceeds a given threshold $T$. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first, but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities, of the channels and of $T$, non--equilibrium phase transitions take place in the $N\rightarrow \infty$ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant.

cond-mat.stat-mech↗

arXiv2020

Deterministic reversible model of non-equilibrium phase transitions and stochastic counterpart

N point particles move within a billiard table made of two circular cavities connected by a straight channel. The usual billiard dynamics is modified so that it remains deterministic, phase space volumes preserving and time reversal invariant. Particles move in straight lines and are elastically reflected at the boundary of the table, as usual, but those in a channel that are moving away from a cavity invert their motion (rebound), if their number exceeds a given threshold T. When the geometrical parameters of the billiard table are fixed, this mechanism gives rise to non--equilibrium phase transitions in the large N limit: letting T/N decrease, the homogeneous particle distribution abruptly turns into a stationary inhomogeneous one. The equivalence with a modified Ehrenfest two urn model, motivated by the ergodicity of the billiard with no rebound, allows us to obtain analytical results that accurately describe the numerical billiard simulation results. Thus, a stochastic exactly solvable model that exhibits non-equilibrium phase transitions is also introduced.

cond-mat.stat-mech↗

arXiv2019

A semidiscrete Galerkin scheme for a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates

In this paper, we study the numerical approximation of a coupled system of elliptic-parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulation results illustrating the expected behaviour of the system.

math.AP↗

arXiv2019

Modelling, Simulation and Parameter Identification of Active Pollution Reduction with Photovoltaic Asphalt

We develop and implement a numerical model to simulate the effect of photovoltaic asphalt on reducing the concentration of nitrogen monoxide due to the presence of heavy traffic in an urban environment. The contributions in this paper are threefold: we model and simulate the spread and breakdown of pollution in an urban environment, we provide a parameter estimation process that can be used to find missing parameters, and finally, we train and compare this simulation with different data sets. We analyze the results and provide an outlook on further research.

math.NA↗

arXiv2018

The effect of environment knowledge in evacuation scenarios involving fire and smoke - a multiscale modelling and simulation approach

We study the evacuation dynamics of a crowd evacuating from a complex geometry in the presence of a fire as well as of a slowly spreading smoke curtain. The crowd is composed of two kinds of individuals: those who know the layout of the building, and those who do not and rely exclusively on potentially informed neighbors to identify a path towards the exit. We aim to capture the effect the knowledge of the environment has on the interaction between evacuees and their residence time in the presence of fire and evolving smoke. Our approach is genuinely multiscale - we employ a two-scale model that is able to distinguish between compressible and incompressible pedestrian flow regimes and allows for micro and macro pedestrian dynamics. Simulations illustrate the expected qualitative behavior of the model. We finish with observations on how mixing evacuees with different levels of knowledge impacts important evacuation aspects.

physics.soc-ph↗

arXiv2018

Modelling interactions between active and passive agents moving through heterogeneous environments

We study the dynamics of interacting agents from two distinct inter-mixed populations: One population includes active agents that follow a predetermined velocity field, while the second population contains exclusively passive agents, i.e. agents that have no preferred direction of motion. The orientation of their local velocity is affected by repulsive interactions with the neighboring agents and environment. We present two models that allow for a qualitative analysis of these mixed systems. We show that the residence times of this type of systems containing mixed populations is strongly affected by the interplay between these two populations. After showing our modeling and simulation results, we conclude with a couple of mathematical aspects concerning the well-posedness of our models.

math.NA↗

arXiv2018

Driven particle flux through a membrane: Two-scale asymptotics of a diffusion equation with polynomial drift

Diffusion of particles through an heterogenous obstacle line is modeled as a two-dimensional diffusion problem with a one--directional nonlinear convective drift and is examined using two-scale asymptotic analysis. At the scale where the structure of heterogeneities is observable the obstacle line has an inherent thickness. Assuming the heterogeneity to be made of an array of periodically arranged microstructures (e.g. impenetrable solid rectangles), two scaling regimes are identified: the characteristic size of the microstructure is either significantly smaller than the thickness of the obstacle line or it is of the same order of magnitude. We scale up the convection-diffusion model and compute the effective diffusion and drift tensorial coefficients for both scaling regimes. The upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We numerically illustrate the behavior of the upscaled membrane in the finite thickness regime and apply it to describe the transport of {\rm CO}$_2$ through paperboard.

math.AP↗

arXiv2017

Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system

We establish the well-posedness of a coupled micro-macro parabolic-elliptic system modeling the interplay between two pressures in a gas-liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro-macro Robin problem, potentially useful in identifying quantitatively a micro-macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness arguments cast in the Schauder's fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.

math.AP↗

arXiv2017

Discrete and continuum links to a nonlinear coupled transport problem of interacting populations

We are interested in exploring interacting particle systems that can be seen as microscopic models for a particular structure of coupled transport flux arising when different populations are jointly evolving. The scenarios we have in mind are inspired by the dynamics of pedestrian flows in open spaces and are intimately connected to cross-diffusion and thermo-diffusion problems holding a variational structure. The tools we use include a suitable structure of the relative entropy controlling TV-norms, the construction of Lyapunov functionals and particular closed-form solutions to nonlinear transport equations, a hydrodynamics limiting procedure due to Philipowski, as well as the construction of numerical approximates to both the continuum limit problem in 2D and to the original interacting particle systems.

math.AP↗

arXiv2016

Large-scale multiscale particle models in inhomogeneous domains: modelling and implementation

In this thesis, we develop multiscale models for particle simulations in population dynamics. These models are characterised by prescribing particle motion on two spatial scales: microscopic and macroscopic. At the microscopic level, each particle has its own mass, position and velocity, while at the macroscopic level the particles are interpolated to a continuum quantity whose evolution is governed by a system of transport equations. This way, one can prescribe various types of interactions on a global scale, whilst still maintaining high simulation speed for a large number of particles. In addition, the interplay between particle motion and interaction is well tuned in both regions of low and high densities. We analyse links between models on these two scales and prove that under certain conditions, a system of interacting particles converges to a nonlinear coupled system of transport equations. We use this as a motivation to derive a model defined on both modelling scales and prescribe the intercommunication between them. Simulation takes place in inhomogeneous domains with arbitrary conditions at inflow and outflow boundaries. We realise this by modelling obstacles, sources and sinks. Integrating these aspects into the simulation requires a route planning algorithm for the particles. Several algorithms are considered and evaluated on accuracy, robustness and efficiency. All aspects mentioned above are combined in a novel open source prototyping simulation framework called Mercurial. This computational framework allows the design of geometries and is built for high performance when large numbers of particles are involved. Mercurial supports various types of inhomogeneities and global systems of equations. We apply our framework to simulate scenarios in crowd dynamics. We compare our results with test cases from literature to assess the quality of the simulations.

cs.CE↗