Authors:Ferdinando Auricchio, Elena Bonetti, Massimo Carraturo, Dietmar Hömberg, Alessandro Reali, Elisabetta Rocca Pages: 1461 - 1483 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 08, Page 1461-1483, July 2020. In this paper, a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraints and potentially multiple materials or multiscales is analyzed. First-order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible workflow to obtain a 3D-printed object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-26T07:00:00Z DOI: 10.1142/S0218202520500281 Issue No:Vol. 30, No. 08 (2020)
Authors:Laurent Boudin, David Michel, Ayman Moussa Pages: 1485 - 1515 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 08, Page 1485-1515, July 2020. We study the existence of global weak solutions in a three-dimensional time-dependent bounded domain for the incompressible Vlasov–Navier–Stokes system which is coupled with two convection–diffusion equations describing the air temperature and its water vapor mass fraction. This newly introduced model describes respiratory aerosols in the human aiways when one takes into account the hygroscopic effects, also inducing the presence of extra variables in the aerosol distribution function, temperature and size. The mathematical description of these phenomena leads us to make the assumption that the initial distribution of particles does not contain arbitrarily small particles. The proof is based on a regularization and approximation strategy that we solve by deriving several energy estimates, including ones with temperature and size. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-07-20T07:00:00Z DOI: 10.1142/S0218202520500293 Issue No:Vol. 30, No. 08 (2020)
Authors:Young-Sam Kwon, Antonin Novotny, C. H. Arthur Cheng Pages: 1517 - 1553 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 08, Page 1517-1553, July 2020. In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-21T07:00:00Z DOI: 10.1142/S021820252050030X Issue No:Vol. 30, No. 08 (2020)
Authors:L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo Pages: 1555 - 1590 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 08, Page 1555-1590, July 2020. In this paper, we tackle the problem of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test. Moreover, unlike standard isoparametric FEM, this approach allows to deal with curved edges at an intermediate scale, between the small scale (treatable by homogenization) and the bigger one (where a finer mesh would make the curve flatter and flatter). The proposed method is supported by theoretical analysis and numerical tests. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-26T07:00:00Z DOI: 10.1142/S0218202520500311 Issue No:Vol. 30, No. 08 (2020)
Authors:Nicola Bellomo, Richard Bingham, Mark A. J. Chaplain, Giovanni Dosi, Guido Forni, Damian A. Knopoff, John Lowengrub, Reidun Twarock, Maria Enrica Virgillito Pages: 1591 - 1651 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 08, Page 1591-1651, July 2020. This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS–CoV–[math] Severe Acute Respiratory Syndrome, corona virus n.[math]. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-19T07:00:00Z DOI: 10.1142/S0218202520500323 Issue No:Vol. 30, No. 08 (2020)
Authors:Nancy Rodríguez, Michael Winkler Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-14T07:00:00Z DOI: 10.1142/S0218202520500396
Authors:Alexander Mielke, Artur Stephan Pages: 1 - 43 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-12T07:00:00Z DOI: 10.1142/S0218202520500360
Authors:Young-Pil Choi, Jaeseung Lee Pages: 1 - 53 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption. For the proposed system, we first establish local-in-time existence and uniqueness of classical solutions. For the case of identical natural frequencies, we provide synchronization estimates under suitable assumptions on the initial configurations. We also analyze critical thresholds leading to finite-time blow-up or global-in-time existence of classical solutions. In particular, our proposed model exhibits the finite-time blow-up phenomenon, which is not observed in the classical Kuramoto models, even with a smooth distribution function for natural frequencies. Finally, we numerically investigate synchronization, finite-time blow-up, phase transitions, and hysteresis phenomena. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-11T07:00:00Z DOI: 10.1142/S0218202520500414
Authors:Ralf Hiptmair, Carolina Urzúa-Torres Pages: 1 - 22 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider the electric field integral equation (EFIE) modeling the scattering of time-harmonic electromagnetic waves at a perfectly conducting screen. When discretizing the EFIE by means of low-order Galerkin boundary methods (BEM), one obtains linear systems that are ill-conditioned on fine meshes and for low wave numbers [math]. This makes iterative solvers perform poorly and entails the use of preconditioning. In order to construct optimal preconditioners for the EFIE on screens, the authors recently derived compact equivalent inverses of the EFIE operator on simple Lipschitz screens in [R. Hiptmair and C. Urzúa-Torres, Compact equivalent inverse of the electric field integral operator on screens, Integral Equations Operator Theory 92 (2020) 9]. This paper elaborates how to use this result to build an optimal operator preconditioner for the EFIE on screens that can be discretized in a stable fashion. Furthermore, the stability of the preconditioner relies only on the stability of the discrete [math] duality pairing for scalar functions, instead of the vectorial one. Therefore, this novel approach not only offers [math]-independent and [math]-robust condition numbers, but it is also easier to implement and accommodates non-uniform meshes without additional computational effort. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-05T07:00:00Z DOI: 10.1142/S0218202520500347
Authors:Daewa Kim, Annalisa Quaini Pages: 1 - 23 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. The goal of this work is to study an infectious disease spreading in a medium size population occupying a confined environment. For this purpose, we consider a kinetic theory approach to model crowd dynamics in bounded domains and couple it to a kinetic equation to model contagion. The interactions of a person with other pedestrians and the environment are modeled by using tools of game theory. The pedestrian dynamics model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. Each person in the system has a contagion level that is affected by the people in their neighborhood. For the numerical solution of the coupled problem, we propose a numerical algorithm that at every time step solves one crowd dynamics problem and one contagion problem, i.e. with no subiterations between the two. We test our coupled model on a problem involving a small crowd walking through a corridor. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-03T07:00:00Z DOI: 10.1142/S0218202520400126
Authors:R. Eftimie Pages: 1 - 18 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Collective behaviours in animal communities are the result of inter-individual communication. However, communication signals are not fixed; they evolve to ensure more effective interactions between the emitter and receiver of these signals. In this study, we use a mathematical approach and investigate the effect of changes in communication signals (at both receiver and emitter levels) on the aggregation patterns displayed by these animal communities. We use simple linear stability analysis to study the impact that the loss/gain in signals strength has on the formation of stationary and moving animal aggregations. We then use numerical simulations to study the impact of these signal strengths on the long-term persistence of some stationary and moving aggregations. We show that a reduction in the strength of such communication signals can stop the movement of some aggregations. Moreover, for very weak signals, one can obtain a variety of standing wave patterns characterised by left-moving and right-moving waves of individuals passing through each other, with or without some individuals joining the opposite-moving group. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-28T07:00:00Z DOI: 10.1142/S0218202520400138
Authors:Daniele A. Di Pietro, Jérôme Droniou, Francesca Rapetti Pages: 1 - 47 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element (FE) spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete [math]-products completes the exposition. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-26T07:00:00Z DOI: 10.1142/S0218202520500372
Authors:N. Bellomo, F. Brezzi, J. Soler Pages: 1 - 6 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This editorial paper is devoted to present the papers published in a special issue focused on modeling, qualitative analysis and simulation of the collective dynamics of living, self-propelled particles. A critical analysis of the overall contents of the issue is proposed, thus leading to a forward look to research perspectives. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-19T07:00:00Z DOI: 10.1142/S0218202520020030
Authors:Linfeng Mei, Juncheng Wei Pages: 1 - 38 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci 107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst. 9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-19T07:00:00Z DOI: 10.1142/S0218202520500359
Authors:Mihaï Bostan, José Antonio Carrillo Pages: 1 - 43 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0218202520400163
Authors:Seung-Yeal Ha, Zhuchun Li, Xiongtao Zhang Pages: 1 - 51 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker–Smale (C–S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we show that the C–S model on a line exhibits a unique critical exponent for unconditional flocking on a general network so that this improves an earlier result [S.-Y. Ha and J.-G. Liu, A simple proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297–325.] on the all-to-all network. Our result also resolves the critical exponent conjecture posed in Cucker–Dong’s work [On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci. 19 (2009) 1391–1404] for 1D setting. Emergent behavior of the C–S model is independent of the special structure of the underlying network, as long as it contains a spanning tree. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0218202520500335
Authors:D. Benedetto, P. Buttà, E. Caglioti Pages: 1 - 36 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a nonlinear Fokker–Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-07-25T07:00:00Z DOI: 10.1142/S0218202520400151
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