Showing papers in "Mathematical Models and Methods in Applied Sciences in 2008"
TL;DR: A critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field.
Abstract: This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.
424 citations
TL;DR: This work focuses on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker.
Abstract: Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady-state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker. The dynamics of criminals and of the attractiveness field are coupled to each other via specific biasing and feedback mechanisms. Depending on parameter choices, we observe and describe several regimes of aggregation, including hotspots of high criminal activity. On the basis of the discrete system, we also derive a continuum model; the two are in good quantitative agreement for large system sizes. By means of a linear stability analysis we are able to determine the parameter values that will lead to the creation of stable hotspots. We discuss our model and results in the context of established crim...
367 citations
TL;DR: In this article, a kinetic mean-field version of the Couzin-Vicsek algorithm is proposed and its formal macroscopic limit is provided, which is proved to be hyperbolic.
Abstract: The discrete Couzin–Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools is considered. In this paper, a kinetic (mean-field) version of the CVA model is proposed and its formal macroscopic limit is provided. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.
344 citations
TL;DR: This paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions, and a qualitative analysis of the models is conducted with the aim of analyzing their properties.
Abstract: This paper, that deals with the modelling of crowd dynamics, is the first one of a project finalized to develop a mathematical theory refereing to the modelling of the complex systems constituted by several interacting individuals in bounded and unbounded domains. The first part of the paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions. The second part of the paper deals with a qualitative analysis of the models with the aim of analyzing their properties. Finally, a critical analysis is proposed in view of further development of the modelling approach. Additional reasonings are devoted to understanding the conceptual differences between crowd and swarm modelling.
203 citations
TL;DR: In this article, the authors considered the adaptive solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of saddle point type, and gave a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one.
Abstract: We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of "saddle point" type. For the adaptive algorithm we assume the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf–sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dorfler's strategy, but also by the maximum strategy and the equidistribution strategy.
177 citations
TL;DR: In this article, the authors considered the coupling between two diffusion-reaction problems, one taking place in a three-dimensional domain Ω, the other in a one-dimensional subdomain Λ.
Abstract: In this paper we consider the coupling between two diffusion-reaction problems, one taking place in a three-dimensional domain Ω, the other in a one-dimensional subdomain Λ. This coupled problem is the simplest model of fluid flow in a three-dimensional porous medium featuring fractures that can be described by one-dimensional manifolds. In particular this model can provide the basis for a multiscale analysis of blood flow through tissues, in which the capillary network is represented as a porous matrix, while the major blood vessels are described by thin tubular structures embedded into it: in this case, the model allows the computation of the 3D and 1D blood pressures respectively in the tissue and in the vessels. The mathematical analysis of the problem requires non-standard tools, since the mass conservation condition at the interface between the porous medium and the one-dimensional manifold has to be taken into account by means of a measure term in the 3D equation. In particular, the 3D solution is singular on Λ. In this work, suitable weighted Sobolev spaces are introduced to handle this singularity: the well-posedness of the coupled problem is established in the proposed functional setting. An advantage of such an approach is that it provides a Hilbertian framework which may be used for the convergence analysis of finite element approximation schemes. The investigation of the numerical approximation will be the subject of a forthcoming work.
166 citations
TL;DR: In this paper, the decay property of the dissipative Timoshenko system in the one-dimensional whole space is studied and the decay structure is of the regularity-loss type.
Abstract: We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2 decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.
141 citations
TL;DR: In this paper, the authors deal with the mathematical modelling of crowd dynamics within the framework of continuum mechanics, using the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients.
Abstract: This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.
129 citations
TL;DR: In this paper, the authors study the evolution of a single crack in an elastic body and assume that the crack path is known in advance, and they construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.
Abstract: We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.
126 citations
TL;DR: It is proved that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize.
Abstract: This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the BPX preconditioner and multigrid V-cycle preconditioner, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or meshsize, and we prove that all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and meshsize. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize. We also present some numerical experiments to demonstrate the theoretical results.
97 citations
TL;DR: In this paper, the authors considered the initial value problem for a nonlinear version of the dissipative Timoshenko system and employed the time weighed L2 energy method which is combined with the optimal L2 decay estimates for lower order derivatives of solutions.
Abstract: We consider the initial value problem for a nonlinear version of the dissipative Timoshenko system. This syetem verifies the decay property of regularity-loss type. To overcome this difficulty caused by the regularity-loss property, we employ the time weighed L2 energy method which is combined with the optimal L2 decay estimates for lower order derivatives of solutions. Then we show the global existence and asymptotic decay of solutions under smallness and enough regularity conditions on the initial data. Moreover, we show that the solution approaches the linear diffusion wave expressed in terms of the superposition of the heat kernels as time tends to infinity.
TL;DR: In this paper, a scaled asymptotic expansion with respect to the skin depth is proposed to derive Generalized Impedance Boundary Conditions (GIBCs) for electromagnetic scattering problems from imperfect conductors with smooth boundaries.
Abstract: This paper is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) for electromagnetic scattering problems from imperfect conductors with smooth boundaries. These boundary conditions can be seen as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth.
TL;DR: In this paper, the authors considered the propagation of a crack in a brittle material along a prescribed crack path and defined a quasi-static evolution by means of stationary points of the free energy.
Abstract: We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasi-static evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith's criterion in a suitable form which takes into account both stable and unstable propagations, as well as an energy balance formula which accounts for dissipation in the unstable regime. If the load is monotonically increasing, this solution is explicit and almost everywhere unique. For more general loads we construct a solution via time discretization. Finally, we consider a finite element discretization of the problem and prove convergence of the discrete solutions.
TL;DR: In this article, a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor is considered, and weak solutions are provided in generalized Orlicz spaces.
Abstract: The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.
TL;DR: A new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks is introduced, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions.
Abstract: We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 - d inviscid Burgers equation. We first prove the existence of minimizers and, by a Γ-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so-called one-sided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms.
TL;DR: In this paper, the existence of crack evolutions based on critical points of the energy functional is proved, in the case of a cohesive zone model with prescribed crack path, and it turns out that evolutions of this type satisfy a maximum stress criterion for the crack initiation.
Abstract: The existence of crack evolutions based on critical points of the energy functional is proved, in the case of a cohesive zone model with prescribed crack path. It turns out that evolutions of this type satisfy a maximum stress criterion for the crack initiation. With an explicit example, it is shown that evolutions based on the absolute minimization of the energy functional do not enjoy this property.
TL;DR: A more realistic model namely the Second Order Model with Constraints (in short SOMC) is proposed, derived from the Aw and Rascle model,1 which takes into account this feature.
Abstract: Recently, Berthelin et al.,4 introduced a traffic-flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper, a more realistic model namely the Second Order Model with Constraints (in short SOMC) is proposed, derived from the Aw and Rascle model,1 which takes into account this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill and Whitham model.20 An existence result of weak solutions for this model by means of cluster dynamics is proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely.
TL;DR: In this paper, the authors construct spaces of differential forms which form a complex under the exterior derivative of a cellular complex, which is isomorphic to the cochain complex of the cellular complex.
Abstract: Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.
TL;DR: In this article, the existence of global-in-time weak solutions to a coupled microscopic-macroscopic bead-spring model with microscopic cut-off has been studied, which arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains.
Abstract: We study the existence of global-in-time weak solutions to a coupled microscopic–macroscopic bead-spring model with microscopic cut-off, which arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function ψ that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term and a cut-off function βL(ψ) = min(ψ,L) in the drag term, where L ≫ 1. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force potentials including, in particular, the widely used finitely extensible nonlinear elastic potential. A key ingredient of the argument is a special testing procedure in the weak formulation of the Fokker–Planck equation, based on the convex entropy function . In the case of a corotational drag term, passage to the limit as L → ∞ recovers the Navier–Stokes–Fokker–Planck model with centre-of-mass diffusion, without cut-off.
TL;DR: In this article, a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials is presented, where the problem is treated within the framework of the energetic formulation of rate-independent processes and existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level.
Abstract: This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.
TL;DR: In this article, the evolution of a multi-phase system where the motion of the interfaces is driven by anisotropic curvature and some of the phases are subject to volume constraints is considered.
Abstract: We consider the evolution of a multi-phase system where the motion of the interfaces is driven by anisotropic curvature and some of the phases are subject to volume constraints. The dynamics of the phase boundaries is modeled by a system of Allen–Cahn type equations for phase field variables resulting from a gradient flow of an appropriate Ginzburg–Landau type energy. Several ideas are presented in order to guarantee the additional volume constraints. Numerical algorithms based on explicit finite difference methods are developed, and simulations are performed in order to study local minima of the system energy. Wulff shapes can be recovered, i.e. energy minimizing forms for anisotropic surface energies enclosing a given volume. Further applications range from foam structures or bubble clusters to tessellation problems in two and three space dimensions.
TL;DR: In this paper, a model of opinion formation in a population of interacting individuals under the influence of external leaders or persuaders, which act in a time periodic fashion, is presented.
Abstract: This paper concerns a model of opinion formation in a population of interacting individuals under the influence of external leaders or persuaders, which act in a time periodic fashion. The model is formulated within a general framework inspired to a discrete generalized kinetic approach, which has been developed in Ref. 6. It is expressed by a system of non-autonomous nonlinear ordinary differential equations. The dynamics of such a system is investigated and the existence of a globally asymptotically stable periodic solution is analytically proved in three example cases, each one corresponding to a different quantitative choice of the actions of the persuaders. Equivalently, in three particular cases a time periodic asymptotic trend of the opinions evolution is established. Several computational simulations are described and discussed, suggesting that for the model under investigation analogous qualitative results hold true more generally, also in cases involving quantitatively different persuaders actions.
TL;DR: In this paper, the convex envelope was discretized using a finite difference method and the resulting scheme yields an explicit local method to compute convex envelopes with smooth and nonsmooth data.
Abstract: A fully nonlinear partial differential equation for the convex envelope was recently introduced by the author. In this paper, the equation is discretized using a finite difference method. The resulting scheme yields an explicit local method to compute the convex envelope. The scheme is shown to converge. Computational results are presented for smooth and nonsmooth data. Extensions to higher dimensions and unstructured grids are discussed.
TL;DR: In this paper, a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains) is introduced.
Abstract: We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutually orthogonal families of curves. The method is illustrated on a specific example where uniqueness can be proved by studying an unusual system of hyperbolic PDEs. The questions we address here are of interest in elasticity theory, optimal designs, as well as in functional analysis.
TL;DR: In this article, the existence and uniqueness of a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term was proved based on the Banach fixed point theorem.
Abstract: We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.
TL;DR: In this paper, the authors considered the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic-elliptic coupled system of the radiating gas in two dimensions.
Abstract: In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.
TL;DR: In this paper, the exponential decay and polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space were established.
Abstract: We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ.
TL;DR: In this paper, the authors considered the time-harmonic Maxwell equations with opposite sign dielectric and/or magnetic coefficients and proved that the space of electric fields is compactly embedded in L 2.
Abstract: We consider the time-harmonic Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic coefficients. We prove that, in the case of sign-shifting dielectric coefficients, the space of electric fields is compactly embedded in L 2. We build a three-field variational formulation equivalent to Maxwell system for sign-shifting magnetic coefficients and show that, under some suitable conditions, the formulation fits into the coercive plus compact framework. © 2008 World Scientific Publishing Company.
TL;DR: In this paper, the stationary, isothermal rotational spinning process of fibers is considered and analytical bounds for the initial viscous stress of the fiber are obtained for the inviscid case δ > 0 numerical simulations are carried out.
Abstract: The stationary, isothermal rotational spinning process of fibers is considered. The investigations are concerned with the case of large Reynolds (δ = 3/Re ≪ 1) and small Rossby numbers (e ≪ 1). Modelling the fibers as a Newtonian fluid and applying slender body approximations, the process is described by a two-point boundary value problem of ODEs. The involved quantities are the coordinates of the fiber's centerline, the fluid velocity and viscous stress. The inviscid case δ = 0 is discussed as a reference case. For the viscous case δ > 0 numerical simulations are carried out. Transfering some properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for δ > 3e2 no physical relevant stationary solution can exist.
TL;DR: In this article, the authors determine the relaxation of some transversally-isotropic energy densities, i.e. functions W : ℝ3×3 → [0,∞] with the property W(QFR) = W(F) for all Q ∈ SO(3) and all R ∈ So(3), such that Rn0 = n0, where n0 is a fixed unit vector.
Abstract: We determine the relaxation of some transversally-isotropic energy densities, i.e. functions W : ℝ3×3 → [0,∞] with the property W(QFR) = W(F) for all Q ∈ SO(3) and all R ∈ SO(3) such that Rn0 = n0, where n0 is a fixed unit vector. One physically relevant example is a model for smectic A elastomers. We discuss the implications of our result for the computation of macroscopic stress–strain curves for this material and compare with experiment.