# ProfessorPierreDegond

Faculty of Natural SciencesDepartment of Mathematics

Visiting Professor

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### Location

Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
to

306 results found

Degond PAA, Minakowski P, Zatorska E, 2018, Transport of congestion in two-phase compressible/incompressible flows, Nonlinear Analysis: Real World Applications, Vol: 42, Pages: 485-510, ISSN: 1468-1218

We study the existence of weak solutions to the two-phase fluid model with congestion constraint. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime achieves a threshold value that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier–Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.

Journal article

Aoki K, Degond P, Yang T, 2018, Special issue celebrating the 10th anniversary of KRM PREFACE, KINETIC AND RELATED MODELS, Vol: 11, Pages: I-I, ISSN: 1937-5093

Journal article

Degond PAA, Manhart A, Yu H, 2018, An age-structured continuum model for myxobacteria, Mathematical Models and Methods in Applied Sciences, Vol: 28, Pages: 1737-1770, ISSN: 1793-6314

Myxobacteria are social bacteria, that can glide in 2D and form counter-propagating,interacting waves. Here we present a novel age-structured, continuous macroscopic modelfor the movement of myxobacteria. The derivation is based on microscopic interactionrules that can be formulated as a particle-based model and set within the SOH (Self-Organized Hydrodynamics) framework. The strength of this combined approach is thatmicroscopic knowledge or data can be incorporated easily into the particle model, whilstthe continuous model allows for easy numerical analysis of the different effects. Howeverwe found that the derived macroscopic model lacks a diffusion term in the density equa-tions, which is necessary to control the number of waves, indicating that a higher orderapproximation during the derivation is crucial. Uponad-hocaddition of the diffusionterm, we found very good agreement between the age-structured model and the biology.In particular we analyzed the influence of a refractory (insensitivity) period following areversal of movement. Our analysis reveals that the refractory period is not necessaryfor wave formation, but essential to wave synchronization, indicating separate molecularmechanisms.

Journal article

Degond PAA, Minakowski P, Navoret L, Zatorska Eet al., 2018, Finite volume approximations of the Euler system with variable congestion, Computers and Fluids, Vol: 169, Pages: 23-39, ISSN: 0045-7930

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure (Degond et al., 2016). This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensionnal test-cases and compare it with a scheme previously proposed in Degond et al. (2016) and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.

Journal article

David I, Kohnke P, Fehrenbach J, Lopes Simoes MR, Debreuve E, Descombes X, Plouraboué F, Degond P, Druart Xet al., 2018, New objective measurements of semen wave motion are associated with fertility in sheep, Reproduction, Fertility and Development, Vol: 30, Pages: 889-896, ISSN: 1031-3613

In sheep, wave motion in semen is currently used by AI centres to select ejaculates for insemination. Despite its low cost, convenience and established ability to predict fertility, the subjectivity of this assessment is a limiting factor for its applicability. The aims of the present study were to establish an objective method for the analysis of wave motion and to assess the associations of objective parameters with fertility after cervical insemination. Collective sperm motion in undiluted semen was observed by phase contrast microscopy at low magnification in a 100-µm deep glass chamber. Images of moving dark waves over a grey background were recorded and analysed by the optic flow method, producing several velocity-related parameters. Turbulence was assessed from the motion of fluorescent polystyrene beads. Among objective parameters, optical flow entropy and the average speed of beads were both able to discriminate ejaculates suitable for insemination. Two synthetic variables of optic flow and bead motion and a global objective variable were computed from linear combinations of individual parameters and compared with the subjective motion score for their predictive value. These were as efficient as the wave motion score for assessing fertility and can be proposed for the assessment of ram semen in routine AI procedures.

Journal article

Degond PAA, Frouvelle A, Merino Aceituno S, Trescases Aet al., 2018, Quaternions in collective dynamics, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Vol: 16, Pages: 28-77, ISSN: 1540-3459

We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications.

Journal article

Blanchet A, Degond PAA, 2017, Kinetic models for topological nearest-neighbor interactions, Journal of Statistical Physics, Vol: 169, Pages: 929-950, ISSN: 1572-9613

We consider systems of agents interacting through topological interactions. Thesehave been shown to play an important part in animal and human behavior. Precisely, thesystem consists of a finite number of particles characterized by their positions and velocities.At random times a randomly chosen particle, the follower, adopts the velocity of its closestneighbor, the leader. We study the limit of a system size going to infinity and, under theassumption of propagation of chaos, show that the limit kinetic equation is a non-standardspatial diffusion equation for the particle distribution function. We also study the case whereinthe particles interact with their K closest neighbors and show that the corresponding kineticequation is the same. Finally, we prove that these models can be seen as a singular limitof the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as wellas some concentration of measure arguments.

Journal article

Barré J, Degond PAA, Zatorska E, 2017, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Modeling & Simulation, Vol: 15, Pages: 1294-1323, ISSN: 1540-3467

We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast,we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset.

Journal article

Leroy-Lerêtre M, Dimarco G, Cazalès M, Boizeau ML, Ducommun B, Lobjois V, Degond PAAet al., 2017, Are tumor cell lineages solely shaped by mechanical forces?, Bulletin of Mathematical Biology, Vol: 79, Pages: 2356-2393, ISSN: 1522-9602

This paper investigates cell proliferation dynamics in small tumor cell aggregates using an individual-based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions.

Journal article

Barré J, Carrillo de la plata J, Degond PAA, Peurichard D, Zatorska Eet al., 2017, Particle interactions mediated by dynamical networks: assessment ofmacroscopic descriptions, Journal of Nonlinear Science, Vol: 28, Pages: 235-268, ISSN: 0938-8974

We provide a numerical study of the macroscopic model of Barré et al.(Multiscale Model Simul, 2017, to appear) derived from an agent-based model for asystem of particles interacting through a dynamical network of links. Assuming thatthe network remodeling process is very fast, the macroscopic model takes the formof a single aggregation–diffusion equation for the density of particles. The theoreticalstudy of the macroscopic model gives precise criteria for the phase transitions ofthe steady states, and in the one-dimensional case, we show numerically that thestationary solutions of the microscopic model undergo the same phase transitions andbifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement withthe predicted theoretical values. This study provides a partial validation of the formalderivation of the macroscopic model from a microscopic formulation and shows thatthe former is a consistent approximation of an underlying particle dynamics, makingit a powerful tool for the modeling of dynamical networks at a large scale.

Journal article

Peurichard D, Delebecque F, Lorsignol A, Barreau C, Rouquette J, Descombes X, Casteilla L, Degond PAAet al., 2017, Simple mechanical cues could explain adipose tissue morphology, Journal of Theoretical Biology, Vol: 429, Pages: 61-81, ISSN: 1095-8541

The mechanisms by which organs acquire their functional structure and realize its maintenance (or homeostasis) over time are still largely unknown. In this paper, we investigate this question on adipose tissue. Adipose tissue can represent 20 to 50% of the body weight. Its investigation is key to overcome a large array of metabolic disorders that heavily strike populations worldwide. Adipose tissue consists of lobular clusters of adipocytes surrounded by an organized collagen fiber network. By supplying substrates needed for adipogenesis, vasculature was believed to induce the regroupment of adipocytes near capillary extremities. This paper shows that the emergence of these structures could be explained by simple mechanical interactions between the adipocytes and the collagen fibers. Our assumption is that the fiber network resists the pressure induced by the growing adipocytes and forces them to regroup into clusters. Reciprocally, cell clusters force the fibers to merge into a well-organized network. We validate this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and extra-cellular-matrix fiber elements. The model produces structures that compare quantitatively well to the experimental observations. Our model seems to indicate that cell clusters could spontaneously emerge as a result of simple mechanical interactions between cells and fibers and surprisingly, vasculature is not directly needed for these structures to emerge.

Journal article

Degond PAA, Frouvelle A, Merino-Aceituno S, 2017, A new flocking model through body attitude coordination, Mathematical Models & Methods in Applied Sciences, Vol: 27, ISSN: 1793-6314

We present a new model for multi-agent dynamics where each agent is described byits position and body attitude: agents travel at a constant speed in a given directionand their body can rotate around it adopting di erent con gurations. In this manner,the body attitude is described by three orthonormal axes giving an element inSO(3)(rotation matrix). Agents try to coordinate their body attitudes with the ones of theirneighbours. In the present paper, we give the Individual Based Model (particle model) for this dynamics and derive its corresponding kinetic and macroscopic equations.

Journal article

Degond PAA, Engel M, 2017, Numerical approximation of a coagulation-fragmentation model for animal group size statistics, Networks and Heterogeneous Media, Vol: 12, Pages: 217-243, ISSN: 1556-181X

We study numerically a coagulation-fragmentation model derivedby Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equi-librium distribution of group sizes is shown to exist in both cases of continuousand discrete group size distributions. We provide a numerical investigation ofthese equilibria using three different methods to approximate the equilibrium:a recursive algorithm based on the work of Ma et. al. [12], a Newton methodand the resolution of the time-dependent problem. All three schemes are val-idated by showing that they approximate the predicted small and large sizeasymptotic behaviour of the equilibrium accurately. The recursive algorithm isused to investigate the transition from discrete to continuous size distributionsand the time evolution scheme is exploited to show uniform convergence toequilibrium in time and to determine convergence rates.

Journal article

Degond P, Liu J-G, Pego RL, 2017, Coagulation-fragmentation model for animal group-size statistics, Journal of Nonlinear Science, Vol: 27, Pages: 379-424, ISSN: 0938-8974

We study coagulation-fragmentation equations inspired by a simple modelproposed in fisheries science to explain data for the size distribution ofschools of pelagic fish. Although the equations lack detailed balance and admitno $H$-theorem, we are able to develop a rather complete description ofequilibrium profiles and large-time behavior, based on recent developments incomplex function theory for Bernstein and Pick functions. In thelarge-population continuum limit, a scaling-invariant regime is reached inwhich all equilibria are determined by a single scaling profile. This universalprofile exhibits power-law behavior crossing over from exponent $-\frac23$ forsmall size to $-\frac32$ for large size, with an exponential cut-off.

Journal article

Degond PAA, Herty M, Liu JG, 2017, Meanfield games and model predictive control, Communications in Mathematical Sciences, Vol: 15, Pages: 1403-1422, ISSN: 1945-0796

Mean-Field Games are games with a continuum of players that incorporate the time-dimension through a control-theoretic approach. Recently, simpler approaches relying on the BestReply Strategy have been proposed. They assume that the agents navigate their strategies towardstheir goal by taking the direction of steepest descent of their cost function (i.e. the opposite of theutility function). In this paper, we explore the link between Mean-Field Games and the Best ReplyStrategy approach. This is done by introducing a Model Predictive Control framework, which consistsof setting the Mean-Field Game over a short time interval which recedes as time moves on. We showthat the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Gameapproach and the sub-optimal Best Reply Strategy.

Journal article

Degond PAA, Deluzet F, 2017, Asymptotic-Preserving methods and multiscale models for plasma physics, Journal of Computational Physics, Vol: 336, Pages: 429-457, ISSN: 0021-9991

The purpose of the present paper is to provide an overview of Asymptotic-Preserving methods for multiscale plasma simulations by addressing three sin-gular perturbation problems. First, the quasi-neutral limit of fluid and kineticmodels is investigated in the framework of non magnetized aswell as magne-tized plasmas. Second, the drift limit for fluid descriptions of thermal plasmasunder large magnetic fields is addressed. Finally efficient numerical resolutionsof anisotropic elliptic or diffusion equations arising in magnetized plasma simu-lation are reviewed.

Journal article

Degond PAA, Manhart A, Yu H, 2017, A continuum model for nematic alignment of self-propelled particles, Discrete and Continuous Dynamical Systems - Series B, Vol: 22, Pages: 1295-1327, ISSN: 1553-524X

A continuum model for a population of self-propelled particlesinteracting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematicmean direction of the ow and the densities of particles owing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

Journal article

Chertock A, Degond PAA, Neusser J, 2017, An asymptotic-preserving method for a relaxation of the NavierStokes-Korteweg equations, Journal of Computational Physics, Vol: 335, Pages: 387-4003, ISSN: 0021-9991

The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interfacemodel for compressible two-phase flows. As direct numericalsimulations basedon the NSK system are quite expensive and in some cases even impossible, weconsider a relaxation of the NSK system, for which robust numerical methodscan be designed. However, time steps for explicit numericalschemes depend onthe relaxation parameter and therefore numerical simulations in the relaxationlimit are very inefficient. To overcome this restriction, we propose an implicit-explicit asymptotic-preserving finite volume method. We prove that the newscheme provides a consistent discretization of the NSK system in the relaxationlimit and demonstrate that it is capable of accurately and efficiently computingnumerical solutions of problems with realistic density ratios and small interfacialwidths.

Journal article

Degond P, Liu J-G, Merino-Aceituno S, Tardiveau Tet al., 2017, Continuum dynamics of the intention field under weakly cohesive social interaction, Mathematical Models and Methods in Applied Sciences, Vol: 27, ISSN: 1793-6314

We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.

Journal article

Degond PAA, Peurichard D, 2017, Modelling tissue self-organization: from micro to macro models, Lecture Notes in Computational Science and Engineering, Vol: 122, Pages: 93-108, ISSN: 1439-7358

In this chapter, we present recent works concerned with the derivation of a macroscopic model for complex interconnected fiber networks from an agent-based model, with applications to, but not limited to, adipose tissue self-organization. Starting from an agent-based model for interconnected fibers interacting through alignment interactions and having the ability to create and suppress cross-links, the formal limit of large number of individuals is first investigated. It leads to a kinetic system of two equations: one for the individual fiber distribution function and one for the distribution function of connected fiber pairs. The hydrodynamic limit, in a regime of instantaneous fiber linking/unlinking then leads to a macroscopic model describing the evolution of the fiber local density and mean orientation. These works are the first attempt to derive a macroscopic model for interconnected fibers from an agent-based formulation and represent a first step towards the formulation of a large scale synthetic tissue model which will serve for the investigation of large scale effects in tissue homeostasis.

Journal article

Degond PAA, Ferreira MA, Motsch S, 2016, Damped Arrow-Hurwicz algorithm for sphere packing, Journal of Computational Physics, Vol: 332, Pages: 47-65, ISSN: 1090-2716

We consider algorithms that, from an arbitrarily sampling ofNspheres(possibly overlapping), nd a close packed con guration without overlapping.These problems can be formulated as minimization problems with non-convexconstraints. For such packing problems, we observe that the classical iterativeArrow-Hurwicz algorithm does not converge. We derive a novel algorithmfrom a multi-step variant of the Arrow-Hurwicz scheme with damping. Wecompare this algorithm with classical algorithms belonging to the class oflinearly constrained Lagrangian methods and show that it performs better.We provide an analysis of the convergence of these algorithms in the simplecase of two spheres in one spatial dimension. Finally, we investigate thebehaviour of our algorithm when the number of spheres is large in two andthree spatial dimensions.

Journal article

Degond PAA, Deluzet F, Doyen D, 2016, Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, Journal of Computational Physics, Vol: 330, Pages: 467-492, ISSN: 0021-9991

In this article, we design Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system inthe quasi-neutral limit, this limit being characterized by a Debye length negligible compared to the space scaleof the problem. These methods are consistent discretizations of the Vlasov-Maxwell system which, in thequasi-neutral limit, remain stable and are consistent with a quasi-neutral model (in this quasi-neutral model,the electric eld is computed by means of a generalized Ohm law). The derivation of Asymptotic-Preservingmethods is not straightforward since the quasi-neutral model is a singular limit of the Vlasov-Maxwell model.The key step is a reformulation of the Vlasov-Maxwell system which uni es the two models in a single setof equations with a smooth transition from one to another. As demonstrated in various and demandingnumerical simulations, the Asymptotic-Preserving methods are able to treat efficiently both quasi-neutralplasmas and non-neutral plasmas, making them particularly well suited for complex problems involving denseplasmas with localized non-neutral regions.

Journal article

Degond PAA, Filbet F, 2016, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field : formal derivation., Journal of Statistical Physics, Vol: 165, Pages: 765-784, ISSN: 0022-4715

In this paper we establish the asymptotic limit of the three dimensional Vlasov–Poissonequation with strong external magnetic field. The guiding center approximation is investigated inthe three dimensional case with a non-constant magnetic field. In the long time asymptotic limit,the motion can be split in two parts : one stationary flow along the lines of the magnetic fieldand the guiding center motion in the orthogonal plane of the magnetic field where classical driftvelocities and invariants (magnetic moment) are recovered.

Journal article

Degond PAA, Henkes S, Yu H, 2016, Self-organized Hydrodynamics with density-dependent velocity, Kinetic and Related Models, Vol: 10, Pages: 193-213, ISSN: 1937-5093

Motivated by recent experimental and computational results thatshow a motility-induced clustering transition in self-propelled particle systems,we study an individual model and its corresponding Self-Organized Hydrodynamicmodel for collective behaviour that incorporates a density-dependentvelocity, as well as inter-particle alignment. The modal analysis of the hydrodynamicmodel elucidates the relationship between the stability of the equilibriaand the changing velocity, and the formation of clusters. We find, in agreementwith earlier results for non-aligning particles, that the key criterion for stabilityis (ρv(ρ))0 ≥ 0, i.e. a nondecreasing mass flux ρv(ρ) with respect to the density.Numerical simulation for both the individual and hydrodynamic modelswith a velocity function inspired by experiment demonstrates the validity ofthe theoretical results.

Journal article

Creppy A, Plouraboué F, Praud O, Druart X, Cazin S, Yu H, Degond PAAet al., 2016, Symmetry-breaking phase-transitions in highly concentrated semen, Journal of the Royal Society Interface, Vol: 13, ISSN: 1742-5689

New experimental evidence of self-motion of aconfined active suspension is presented. Depositingfresh semen sample in an annular shaped micro-fluidic chip leads to a spontaneous vortex state ofthe fluid at sufficiently large sperm concentration.The rotation occurs unpredictably clockwise orcounterclockwise and is robust and stable. Furthermore,for highly active and concentrated semen, richerdynamics can occur such as self-sustained or dampedrotation oscillations. Experimental results obtainedwith systematic dilution provide a clear evidence of aphase transition toward collective motion associatedwith local alignment of spermatozoa akin to theVicsek model. A macroscopic theory based onpreviously derived Self-Organized Hydrodynamics(SOH) models is adapted to this context and providespredictions consistent with the observed stationarymotion.

Journal article

Barbaro ABT, Canizo JA, Carrillo de la plata J, Degond PAAet al., 2016, Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type, Multiscale Modeling & Simulation, Vol: 14, Pages: 1063-1088, ISSN: 1540-3467

We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

Journal article

Blanchet A, Degond PAA, 2016, Topological interactions in a Boltzmann-type framework, Journal of Statistical Physics, Vol: 163, Pages: 41-60, ISSN: 1572-9613

We consider a finite number of particles characterised by their positionsand velocities. At random times a randomly chosen particle, the follower,adopts the velocity of another particle, the leader. The follower choosesits leader according to the proximity rank of the latter with respect to theformer. We study the limit of a system size going to infinity and, under theassumption of propagation of chaos, show that the limit equation is akinto the Boltzmann equation. However, it exhibits a spatial non-localityinstead of the classical non-locality in velocity space. This result relies onthe approximation properties of Bernstein polynomials. We illustrate thedynamics with numerical simulations.

Journal article

Degond P, Delebecque F, Peurichard D, 2015, Continuum model for linked fibers with alignment interactions, Mathematical Models and Methods in Applied Sciences, Vol: 26, ISSN: 0218-2025

We introduce an individual-based model for fiber elements having the abilityto cross-link or unlink each other and to align with each other at the cross links.We first formally derive a kinetic model for the fiber and cross-links distributionfunctions. We then consider the fast linking/unlinking regime in which the modelcan be reduced to the fiber distribution function only and investigate its diffusionlimit. The resulting macroscopic model consists of a system of nonlinear diffusionequations for the fiber density and mean orientation. In the case of a homogeneousfiber density, we show that the model is elliptic.

Journal article

Muljadi B, Narski J, Lozinski A, Degond Pet al., 2015, Non-conforming multiscale finite element method for stokes flows in heterogeneous media. Part I: Methodologies and numerical experiments, Multiscale Modeling & Simulation, Vol: 13, Pages: 1146-1172, ISSN: 1540-3467

The Multiscale Finite Element Method (MsFEM) is developed in the vein of Crouzeix-Raviart element for solving viscous incompressible ows in genuine heterogeneous media. Such owsare relevant in many branches of engineering, often at multiple scales and at regions where analyticalrepresentations of the microscopic features of the ows are often unavailable. Full accounts to theseproblems heavily depend on the geometry of the system under consideration and are computationallyexpensive. Therefore, a method capable of solving multiscale features of the ow without con ningitself to ne scale calculations is sought after.The approximation of boundary condition on coarse element edges when computing the multiscalebasis functions critically inuences the eventual accuracy of any MsFEM approaches. The weaklyenforced continuity of Crouzeix - Raviart function space across element edges leads to a naturalboundary condition for the multiscale basis functions which relaxes the sensitivity of our method tocomplex patterns of obstacles exempt from the needs of implementing any oversampling techniques.Additionally, the application of penalization method makes it possible to avoid complex unstructureddomain and allows extensive use of simpler Cartesian meshes.Key words. Crouzeix-Raviart Element, Multiscale Finite Element Method, Stokes Equations,Penalization Method

Journal article

Degond P, Fibich G, Piccoli B, Tadmor Eet al., 2015, SPECIAL ISSUE ON MODELING AND CONTROL IN SOCIAL DYNAMICS, NETWORKS AND HETEROGENEOUS MEDIA, Vol: 10, Pages: I-II, ISSN: 1556-1801

Journal article

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