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Multiscale models and simulations for diffusion and interaction in heterogeneous domains
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). (Applied Analysis)ORCID iD: 0000-0002-2185-641x
2021 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains, with a focus on multiscale diffusions. Although diffusion is often studied in terms of continuum observables, it is a consequence of the motion of individual particles. Incorporating interactions between constituents and geometry often runs into complications, since interactions typically act on multiple length scales. We address this issue by studying different types of multiscale models and by applying them to a variety of scenarios known for their inherent complexity.

Our contributions can be grouped in two parts. In the first part, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct finite element schemes with desirable approximation properties that resolve the microscopic domain variations and support parallel execution. In the second part of the thesis, we investigate certain interacting particle systems and their links to families of partial differential equations. In this spirit, we analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within an interacting particle system in an extension of the classical Ehrenfest model.

Our overall focus is two-fold. On the one hand, we increase the theoretical understanding of multiscale models by providing modeling, analysis and simulation of specific two-scale couplings. On the other hand, we design computational frameworks and tailored implementations to improve the application of multiscale modeling to complex scenarios and large-scale systems. In this way, our contributions aim to expand the capacity of mathematical modeling to numerically approximate the rich and complex physical world.

Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains. Our contributions can be grouped in two parts. First, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct convergent and efficient finite element schemes that resolve the microscopic domain variations. Second, we investigate certain interacting particle systems and their links to a family of partial differential equations. We analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We also construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within a particle system extending the classical Ehrenfest model.

Our focus is two-fold: we increase the theoretical understanding of certain two-scale couplings, while on the other hand, we develop computational multiscale frameworks for a variety of scenarios known for their inherent complexity.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2021. , p. 212
Series
Karlstad University Studies, ISSN 1403-8099 ; 2021:10
Keywords [en]
multiscale modeling, finite element methods, interacting particle sytems, population dynamics, non-equilibrium dynamics
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-83568ISBN: 978-91-7867-195-3 (print)ISBN: 978-91-7867-205-9 (electronic)OAI: oai:DiVA.org:kau-83568DiVA, id: diva2:1541363
Public defence
2021-05-19, Via Zoom, 15:00 (English)
Opponent
Supervisors
Available from: 2021-04-28 Created: 2021-03-31 Last updated: 2021-05-11Bibliographically approved
List of papers
1. Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system
Open this publication in new window or tab >>Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system
2018 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 97, no 1, p. 89-106Article in journal (Refereed) Published
Abstract [en]

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 twoscale 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.

Place, publisher, year, edition, pages
Taylor & Francis, 2018
Keywords
Upscaled porous media, two-scale PDE, inverse micro–macro Robin problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-62809 (URN)10.1080/00036811.2017.1364366 (DOI)000417831700007 ()
Available from: 2017-08-25 Created: 2017-08-25 Last updated: 2021-03-31Bibliographically approved
2. A semidiscrete Galerkin scheme for a two-scale coupled elliptic-parabolic system: Well-posedness and convergence approximation rates
Open this publication in new window or tab >>A semidiscrete Galerkin scheme for a two-scale coupled elliptic-parabolic system: Well-posedness and convergence approximation rates
2020 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 60, no 4, p. 999-1031Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2020
Keywords
Elliptic–parabolic system. Weak solutions, Galerkin approximations, Distributed microstructures, Error analysis
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-76764 (URN)10.1007/s10543-020-00805-4 (DOI)000518299500001 ()2-s2.0-85081600486 (Scopus ID)
Available from: 2020-02-15 Created: 2020-02-15 Last updated: 2022-10-18Bibliographically approved
3. Parallel two-scale finite element implementation of a system with varying microstructures
Open this publication in new window or tab >>Parallel two-scale finite element implementation of a system with varying microstructures
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

Keywords
multiscale modeling, varying microstructures, finite elements, computational effiency
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-83566 (URN)
Available from: 2021-03-31 Created: 2021-03-31 Last updated: 2021-06-17Bibliographically approved
4. Discrete and continuum links to a nonlinear coupled transport problem of interacting populations
Open this publication in new window or tab >>Discrete and continuum links to a nonlinear coupled transport problem of interacting populations
2017 (English)In: The European Physical Journal Special Topics, ISSN 1951-6355, E-ISSN 1951-6401, Vol. 226, no 10, p. 2345-2357Article in journal (Refereed) Published
Abstract [en]

We are interested in exploring interacting particle systemsthat can be seen as microscopic models for a particular structure ofcoupled transport flux arising when different populations are jointlyevolving. The scenarios we have in mind are inspired by the dynamicsof pedestrian flows in open spaces and are intimately connectedto cross-diffusion and thermo-diffusion problems holding a variationalstructure. The tools we use include a suitable structure of the relativeentropy controlling TV-norms, the construction of Lyapunov functionalsand particular closed-form solutions to nonlinear transport equations,a hydrodynamics limiting procedure due to Philipowski, as wellas the construction of numerical approximates to both the continuumlimit problem in 2D and to the original interacting particle systems.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
interacting-particle systems, population dynamics
National Category
Physical Sciences
Research subject
Mathematics; Biology
Identifiers
urn:nbn:se:kau:diva-47564 (URN)10.1140/epjst/e2017-70009-y (DOI)000404711400012 ()
Available from: 2017-01-01 Created: 2017-01-01 Last updated: 2021-03-31Bibliographically approved
5. Effects of Environment Knowledge in Evacuation Scenarios Involving Fire and Smoke: A Multiscale Modelling and Simulation Approach
Open this publication in new window or tab >>Effects of Environment Knowledge in Evacuation Scenarios Involving Fire and Smoke: A Multiscale Modelling and Simulation Approach
2018 (English)In: Fire technology, ISSN 0015-2684, E-ISSN 1572-8099, Vol. 55, no 2, p. 415-436Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2018
Keywords
Crowd dynamics, Environment knowledge, Evacuation, Fire and smoke dynamics, Particle methods, Transport processes, Incompressible flow, Smoke, Transport process, Fires
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-68390 (URN)10.1007/s10694-018-0743-x (DOI)000460581600003 ()2-s2.0-85048553425 (Scopus ID)
Available from: 2018-07-04 Created: 2018-07-04 Last updated: 2021-03-31Bibliographically approved
6. Deterministic reversible model of non-equilibrium phase transitions and stochastic counterpart
Open this publication in new window or tab >>Deterministic reversible model of non-equilibrium phase transitions and stochastic counterpart
Show others...
2020 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 53, no 30, article id 305001Article in journal (Refereed) Published
Abstract [en]

Npoint 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 thresholdT. When the geometrical parameters of the billiard table are fixed, this mechanism gives rise to non-equilibrium phase transitions in the largeNlimit: lettingT/Ndecrease, 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.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2020
Keywords
billiards; Ehrenfest urn model; non-equilibrium states; phase transitions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-77788 (URN)10.1088/1751-8121/ab94ec (DOI)000553030100001 ()2-s2.0-85088575439 (Scopus ID)
Available from: 2020-05-18 Created: 2020-05-18 Last updated: 2021-03-31Bibliographically approved
7. Deterministic model of battery, uphill currents and non-equilibrium phase transitions
Open this publication in new window or tab >>Deterministic model of battery, uphill currents and non-equilibrium phase transitions
2021 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 103, no 3Article in journal (Refereed) Published
Abstract [en]

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, nonequilibrium phase transitions take place in the N→∞ 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.

Place, publisher, year, edition, pages
American Physical Society, 2021
Keywords
classical transport, nonequilibrium statistical mechanics, phase transitions, statistical physics
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-83552 (URN)10.1103/PhysRevE.103.032119 (DOI)000650938800004 ()2-s2.0-85102924881 (Scopus ID)
Available from: 2021-03-29 Created: 2021-03-29 Last updated: 2021-06-07Bibliographically approved

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  • ieee
  • modern-language-association-8th-edition
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  • en-GB
  • en-US
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  • nn-NO
  • nn-NB
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