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Showing papers in "Multiscale Modeling & Simulation in 2013"


Journal ArticleDOI
TL;DR: In this paper, a variational principle based on the maximization of a Rayleigh coefficient is derived for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator.
Abstract: The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due to the sampling problems. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena.

281 citations


Journal ArticleDOI
TL;DR: In this paper, a review of standard oversampling strategies as performed in the multiscale finite element method (MsFEM) is presented, where the oversample is performed in a full space.
Abstract: This paper reviews standard oversampling strategies as performed in the multiscale finite element method (MsFEM). Common to those approaches is that the oversampling is performed in the full space ...

130 citations


Journal ArticleDOI
TL;DR: This paper deals with the kinetic theory modeling crowd dynamics with the aim of showing how the dynamics at the microscale is transferred to the dynamics of collective behaviors.
Abstract: This paper deals with the kinetic theory modeling crowd dynamics with the aim of showing how the dynamics at the microscale is transferred to the dynamics of collective behaviors. The derivation of a new model is followed by a qualitative analysis of the initial value problem. Existence of solutions is proved for arbitrary large times, while simulations are developed by computational schemes based on splitting methods, where the transport equations are treated by finite difference methods for hyperbolic equations. Some preliminary reasoning toward the modeling of panic conditions is proposed.

102 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a stochastic method for computing approximate solutions as functions of a small scaling parameter at a reduced complexity of $O(N)$ operations.
Abstract: Microscopic models of flocking and swarming take into account large numbers of interacting individuals. Numerical resolution of large flocks implies huge computational costs. Typically for $N$ interacting individuals we have a cost of $O(N^2)$. We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits us to compute approximate solutions as functions of a small scaling parameter $\varepsilon$ at a reduced complexity of $O(N)$ operations. Several numerical results show the efficiency of the algorithms proposed.

73 citations


Journal ArticleDOI
TL;DR: This paper considers the case where the random field may take its values in some subset of the set of real symmetric positive-definite matrices presenting sparsity and invariance with respect to given orthogonal transformations.
Abstract: This paper is concerned with the construction of a new class of generalized nonparametric probabilistic models for matrix-valued non-Gaussian random fields which may take its values in some subset of the set of real symmetric positive-definite matrices presenting sparsity and invariance with respect to given orthogonal transformations. Within the context of linear elasticity, this situation is typically faced in the multiscale analysis of heterogeneous microstructures, where the constitutive elasticity matrices may exhibit some material symmetry properties. The representation introduced involves a parametrization which offers some flexibility for forward simulations and inverse identification by uncoupling the level of statistical fluctuations of the random field and the level of fluctuations associated with a stochastic measure of anisotropy. A novel numerical strategy for random generation is subsequently proposed and consists of solving a family of Ito stochastic differential equations. A Stormer-Verlet algorithm is used for the discretization of the stochastic differential equation.

70 citations


Journal ArticleDOI
TL;DR: This paper introduces a new type of multiscale model describing the process of cancer invasion of tissue, a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics ofThe MDEs, produced at the level of the individual cancer cells.
Abstract: Cancer invasion of tissue is a key aspect of the growth and spread of cancer and is crucial in the process of metastatic spread, i.e., the growth of secondary cancers. Invasion consists in cancer cells secreting various matrix degrading enzymes (MDEs) which destroy the surrounding tissue or extracellular matrix (ECM). Through a combination of proliferation and migration, the cancer cells then actively spread locally into the surrounding tissue. Thus processes occurring at the level of individual cells eventually give rise to processes occurring at the tissue level. In this paper we introduce a new type of multiscale model describing the process of cancer invasion of tissue. Our multiscale model is a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics of the MDEs, produced at the level of the individual cancer cells. These microscale dynamics take place at the interface of the cancer cells...

58 citations


Journal ArticleDOI
TL;DR: The concept of wave propagation is made mathematically rigorous in this situation by proving the existence of entire solutions that approach traveling waves as time approaches negative infinity.
Abstract: We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity, first introduced in [Berestycki and Nadal, J. Appl. Math., 21 (...

56 citations


Journal ArticleDOI
TL;DR: This work builds a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem.
Abstract: Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic microstructures can be prohibitively expensive, as many local cell problems must be solved for different macroscopic points. In addition, the local problems may also be computationally expensive. When the microstructure varies slowly, we develop an efficient numerical method for two scales that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem. In this method, we build a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces. Essentially, solutions computed in high accuracy approximation spaces at select points in the the hierarchy are used as corrections for the error of the lower accuracy approximation spaces ...

45 citations


Journal ArticleDOI
TL;DR: It is shown that for first order finite elements in two space dimensions, the multilevel Monte Carlo finite element method converges at the same rate as the corresponding single-level Monte Carlo infinite element method, despite the majority of samples being underresolved in the multilesvel Monte CAR finite element estimator.
Abstract: In this paper Monte Carlo finite element approximations for elliptic homogenization problems with random coefficients, which oscillate on $n\in\mathbb{N}$ a priori known, separated microscopic length scales, are considered. The convergence of multilevel Monte Carlo finite element discretizations is analyzed. In particular, it is considered that the multilevel finite element discretization resolves the finest physical length scale, but the coarsest finite element mesh does not, so that the so-called resonance case occurs at intermediate multilevel Monte Carlo sampling levels. It is shown that for first order finite elements in two space dimensions, the multilevel Monte Carlo finite element method converges at the same rate as the corresponding single-level Monte Carlo finite element method, despite the majority of samples being underresolved in the multilevel Monte Carlo finite element estimator. It is proved that switching to a hierarchic multiscale finite element method such as the finite element heterog...

42 citations


Journal ArticleDOI
TL;DR: A multiscale heart simulation method that integrates the three-scale phenomena from the microscopic stochastic sarcomere kinetics to the macroscopic heartbeat via the mesoscopic myocardial cell assembly through the homogenization method is introduced.
Abstract: In this paper, we introduce a multiscale heart simulation method that integrates the three-scale phenomena from the microscopic stochastic sarcomere kinetics to the macroscopic heartbeat via the mesoscopic myocardial cell assembly. In sarcomere kinetics, the stochastic behavior of the myosin heads in the cooperative cross-bridge formation and the strain-dependent head rotations are directly simulated using a Monte Carlo (MC) method. This stochastic sarcomere model is coupled with contractile myofibril elements in a finite element (FE) cell assembly model. The cell assembly model is further coupled with a macroscopic FE heart model by means of the homogenization method. We bridge the large gap in the time step sizes between the MC and FE models with an idea based on impulse equilibrium between the sum of the stretches over the whole myosin arms and the contractile stress of the myofibril element. In the application of the homogenization method to the meso-macro coupling, a novel approach is introduced to d...

41 citations


Journal ArticleDOI
TL;DR: This work formalizes and investigates the geometrical multiscale problem, where heterogeneous fluid-structure interaction models for arteries are implicitly coupled, and introduces new coupling algorithms, describe their implementation and investigates on simple geometries the numerical reflections that occur at the interface between the heterogeneous models.
Abstract: Simulating arterial trees in the cardiovascular system can be made by the help of different models, depending on the outputs of interest and the desired degree of accuracy. In particular, one-dimensional fluid-structure interaction models for arteries are very effective in reproducing the physiological pressure wave propagation and in providing quantities like pressure and velocity, averaged on the cross section of the arterial lumen. In locations where one-dimensional models cannot capture the complete flow dynamics, e.g., in presence of stenoses and aneurysms, three-dimensional coupled fluid-structure interaction models are necessary to evaluate, for instance, critical factors responsible for pathologies which are associated to hemodynamics. In this work we formalize and investigate the geometrical multiscale problem, where heterogeneous fluid-structure interaction models for arteries are implicitly coupled. We introduce new coupling algorithms, describe their implementation and investigate on simple geometries the numerical reflections that occur at the interface between the heterogeneous models. We also simulate on a supercomputer a three-dimensional abdominal aorta under physiological conditions, coupled with up to six one-dimensional models representing the surrounding arterial branches. Finally, we compare CPU times and number of coupling iterations for different algorithms and time discretizations.

Journal ArticleDOI
TL;DR: This work addresses the problem of the consistency of different measures of the hydrodynamic radius of solid point and composite solute particles incorporated into the hybrid lattice Boltzmann--molecular dynamics (LBMD) multiscale method by demanding consistency of all these measures.
Abstract: We address the problem of the consistency of different measures of the hydrodynamic radius of solid point and composite solute particles incorporated into the hybrid lattice Boltzmann--molecular dynamics (LBMD) multiscale method. The coupling between the fluid and the particle phase is naturally implemented through a Stokesian type of frictional force proportional to the local velocity difference between the two. Using deterministic flow tests such as measuring the Stokes drag, hydrodynamic torques, and forces we first demonstrate that in this case the hydrodynamic size of the particles is ill-defined in the existing LBMD schemes. We then show how it is possible to effectively achieve the no-slip limit in a discrete simulation with a finite coefficient of the frictional force by demanding consistency of all these measures, but this requires a somewhat modified LB algorithm for numerical stability. Having fulfilled the criteria, we further show that in our consistent coupling scheme particles also obey the...

Journal ArticleDOI
TL;DR: A hybrid method for simulating kinetic equations with multiscale phenomena in the context of linear transport provides a level of accuracy that is comparable to a uniform high-fidelity treatment of the entire system.
Abstract: We present a hybrid method for simulating kinetic equations with multiscale phenomena in the context of linear transport. The method consists of (i) partitioning the kinetic equation into collisional and noncollisional components, (ii) applying a different numerical method to each component, and (iii) repartitioning the kinetic distribution after each time step in the algorithm. Preliminary results show that, for a wide range of test problems, the combination of a low-fidelity method for the collisional component and a high-fidelity method for the noncollisional component provides a level of accuracy that is comparable to a uniform high-fidelity treatment of the entire system.

Journal ArticleDOI
TL;DR: In this article, the diffusive limit of run-and-tumble kinetic models for cell motion due to chemotaxis was studied by means of asymptotic preserving schemes.
Abstract: In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well known that the diffusive limit of these models leads to the classical Patlak--Keller--Segel macroscopic model for chemotaxis. We will show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. Moreover, the numerical results indicate that the global solutions of the kinetic models stabilize for long times to steady states for all the analyzed parameter range. We demonstrate an aggregative behavior from small to a large unique aggregate for the kinetic solutions after the blow-up time in the Patlak--Keller--Segel model. We also generalize these asymptotic preserving schemes to two dimensional kinetic models in the radial case. The blow-up of solutions is numerically investigated in all these cases.

Journal ArticleDOI
TL;DR: A data-driven hybrid numerical integrator is introduced to exploit the formation of nonlinear coherent structures that often appear in nonlinear PDEs and significantly reduces the computational cost of solved equations even when bifurcations occur.
Abstract: A data-driven hybrid numerical integrator is introduced to exploit, numerically, the formation of nonlinear coherent structures that often appear in nonlinear PDEs. Full simulations of the PDE allow model reduction algorithms such as the proper orthogonal decomposition and dynamic mode decomposition to generate reduced order models in an “online” manner. Criteria based on the comparison of these two independent reduction techniques, similar to model predictive control, determine whether the reduced model is accurate without direct evaluation of the underlying PDE. The method is implemented and explored for two prototypical PDE example models and significantly reduces the computational cost of solving those equations even when bifurcations occur.

Journal ArticleDOI
TL;DR: In this article, a dual-scale modeling approach is presented for simulating the drying of a wet hygroscopic porous material that couples the porous medium (macroscale) with the underlying pore structure (microscale).
Abstract: A new dual-scale modeling approach is presented for simulating the drying of a wet hygroscopic porous material that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of wood at low temperatures and is valid in the so-called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapor in the pores. Coupling between scales is achieved by imposing the macroscopic gradients of moisture content and temperature on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic mass and thermal fluxes to be defined as averages of the microscopic fluxes over the unit cell. This novel formulation accounts for the intricate coupling of heat and mass transfer at the microscopic scale but reduces to a classical homogenization approach if a linear relationship is assumed between the microscopic gradient and flux. Simulation result...

Journal ArticleDOI
TL;DR: The a posteriori error bound is used within an adaptive algorithm to tune the critical parameters, i.e., the refinement level and the size of the different patches on which the fine scale constituent problems are solved.
Abstract: An adaptive discontinuous Galerkin multiscale method driven by an energy norm a posteriori error bound is proposed. The method is based on splitting the problem into a coarse and fine scale. Localized fine scale constituent problems are solved on patches of the domain and are used to obtain a modified coarse scale equation. The coarse scale equation has considerably less degrees of freedom than the original problem. The a posteriori error bound is used within an adaptive algorithm to tune the critical parameters, i.e., the refinement level and the size of the different patches on which the fine scale constituent problems are solved. The fine scale computations are completely parallelizable, since no communication between different processors is required for solving the constituent fine scale problems. The convergence of the method, the performance of the adaptive strategy, and the computational effort involved are investigated through a series of numerical experiments.

Journal ArticleDOI
TL;DR: A new probabilistic method to quantify parametric uncertainty in the Buckley--Leverett model is developed based on the concept of a fine-grained cumulative density function (CDF), which enables one to obtain not only average system response but also the probability of rare events, which is critical for risk assessment.
Abstract: The Buckley--Leverett (nonlinear advection) equation is often used to describe two-phase flow in porous media. We develop a new probabilistic method to quantify parametric uncertainty in the Buckley--Leverett model. Our approach is based on the concept of a fine-grained cumulative density function (CDF) and provides a full statistical description of the system states. Hence, it enables one to obtain not only average system response but also the probability of rare events, which is critical for risk assessment. We obtain a closed-form, semianalytical solution for the CDF of the state variable (fluid saturation) and test it against the results from Monte Carlo simulations.

Journal ArticleDOI
TL;DR: This paper considers the Fourier domain Kalman filter for filtering regularly spaced sparse observations of the large-scale mean variables and finds high filtering and statistical prediction skill with superpara-superparameterization.
Abstract: Superparameterization is a fast numerical algorithm to mitigate implicit scale separation of dynamical systems with large-scale, slowly varying “mean” and smaller-scale, rapidly fluctuating “eddy” term. The main idea of superparameterization is to embed parallel highly resolved simulations of small-scale eddies on each grid cell of coarsely resolved large-scale dynamics. In this paper, we study the effect of model errors in using superparameterization for filtering multiscale turbulent dynamical systems. In particular, we use a simple test model, designed to mimic typical multiscale turbulent dynamics with small-scale intermittencies without local statistical equilibriation conditional to the large-scale mean dynamics, and simultaneously force the large-scale dynamics through eddy flux terms. In this paper, we consider the Fourier domain Kalman filter for filtering regularly spaced sparse observations of the large-scale mean variables. We find high filtering and statistical prediction skill with superpara...

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales, and propose a novel algorithm for estimating both the drift and the diffusion coefficients in the effective dynamic equation.
Abstract: We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e., in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space dependent and depend on multiple unknown parameters. It is known that classical estimators, such as maximum likelihood and quadratic variation of the path estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and the diffusion coefficients in the effective dynamics based on a semiparametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the a...

Journal ArticleDOI
TL;DR: In this article, the authors developed a simple algorithmic framework to solve large-scale symmetric positive definite linear systems, which relies on two components: (1) a norm-convergent iterative method (i.e., smoother) and (2) a preconditioner.
Abstract: We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e., smoother) and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods. We prove that the combined preconditioner is positive definite and show estimates on the condition number of the preconditioned system. We combine an algebraic multigrid method and an incomplete factorization preconditioner to test the proposed framework on problems in petroleum reservoir simulation. Our numerical experiments demonstrate noticeable speed-up when we compare our combined method with the stand-alone algebraic multigrid method or the incomplete factorization preconditioner.

Journal ArticleDOI
TL;DR: This paper examines the kinetic theory associated with the coupled PDE/ODE system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles with Lennard-Jones--type repulsion.
Abstract: Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated with the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal spheroids representing self-propelled particles) with Lennard-Jones--type repulsion. A planar shear background flow is imposed on the suspension through the novel use of Lees--Edwards quasi-periodic boundary conditions applied to a representative volume. We s...

Journal ArticleDOI
TL;DR: The string method is devised to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space and derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations.
Abstract: We focus on the nucleation rate calculation for diblock copolymers by studying the two-dimensional stochastic Cahn--Hilliard dynamics with a Landau--Brazovskii energy functional. To do this, we devise the string method to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space. Both methods are combined with the semi-implicit spectral method and hence are very effective. We derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations. The computation of the determinant ratio is also discussed for obtaining the rate. The algorithm is successfully applied to investigate the nucleation from the lamellar phase to the cylinder phase in the mean field theory for diblock copolymer melts. The comparison with projected stochastic Allen--Cahn dynamics is also discussed.

Journal ArticleDOI
TL;DR: The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology.
Abstract: We present a $\Gamma$-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defect-free crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with $\Gamma$-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a ...

Journal ArticleDOI
TL;DR: This work develops asynchronous stochastic approximation algorithms for networked systems with multiagents and regime-switching topologies to achieve consensus control in an asynchronous fashion without using a global clock.
Abstract: This work develops asynchronous stochastic approximation (SA) algorithms for networked systems with multiagents and regime-switching topologies to achieve consensus control. There are several distinct features of the algorithms: (1) In contrast to most of the existing consensus algorithms, the participating agents compute and communicate in an asynchronous fashion without using a global clock. (2) The agents compute and communicate at random times. (3) The regime-switching process is modeled as a discrete-time Markov chain with a finite state space. (4) The functions involved are allowed to vary with respect to time; hence, nonstationarity can be handled. (5) Multiscale formulation enriches the applicability of the algorithms. In the setup, the switching process contains a rate parameter $\varepsilon>0$ in the transition probability matrix that characterizes how frequently the topology switches. The algorithm uses a step-size $\mu$ that defines how fast the network states are updated. Depending on their r...

Journal ArticleDOI
TL;DR: A new model in the analysis of car traffic flow phenomena which in particular refers to a bus route, namely, a closed path embedded in the urban network of roads, which promises to give an appropriate description for the car flow as well as for the buses.
Abstract: We propose a new model in the analysis of car traffic flow phenomena which in particular refers to a bus route, namely, a closed path embedded in the urban network of roads. The model consists in a scalar balance law for the (macroscopic) density of cars, with source terms representing incoming and outgoing roads (with respect to the bus route), and a coupled system of ODEs describing the dynamics of (microscopic) buses along the route. The coupling takes into account both the buses as moving bottlenecks in the car flow and the effect of the car flow on the buses' ODEs as microscopic vehicles. The resulting fully coupled micro--macro model promises to give an appropriate description for the car flow as well as for the buses. Finally, we study the model numerically.

Journal ArticleDOI
TL;DR: A numerical framework for clustering of time series via a Markov chain Monte Carlo (MCMC) method that combines concepts from recently introduced variational time series analysis and regularized clustering functional minimization to address the two main problems of the existent clustering methods.
Abstract: A numerical framework for clustering of time series via a Markov chain Monte Carlo (MCMC) method is presented. It combines concepts from recently introduced variational time series analysis and regularized clustering functional minimization [I. Horenko, SIAM J. Sci. Comput., 32 (2010), pp. 62--83] with MCMC. A conceptual advantage of the presented combined framework is that it allows us to address the two main problems of the existent clustering methods, e.g., the nonconvexity and the ill-posedness of the respective functionals, in a unified way. Clustering of the time series and minimization of the regularized clustering functional are based on the generation of samples from an appropriately chosen Boltzmann distribution in the space of cluster affiliation paths using simulated annealing and the Metropolis algorithm. The presented method is applied to sets of generic ill-posed clustering problems, and the results are compared to those obtained by the standard methods. As demonstrated in numerical example...

Journal ArticleDOI
TL;DR: High-frequency homogenization is applied herein to develop asymptotics for waves propagating along line defects in lattices; the approaches developed are anticipated to be of wide application to many other systems that exhibit surface waves created or directed by microstructure.
Abstract: High-frequency homogenization is applied herein to develop asymptotics for waves propagating along line defects in lattices; the approaches developed are anticipated to be of wide application to many other systems that exhibit surface waves created or directed by microstructure. With the aim being to create a long-scale continuum representation of the line defect that nonetheless accurately incorporates the microscale information, this development uses the microstructural information embedded within, potentially high-frequency, standing wave solutions. A two-scaled approach is utilized for a simple line defect and demonstrated versus exact solutions for quasi-periodic systems and versus numerical solutions for line defects that are themselves perturbed or altered. In particular, Rayleigh--Bloch states propagating along the line defect, and localized defect states, are identified both asymptotically and numerically. Additionally, numerical simulations of large-scale lattice systems illustrate the physics u...

Journal ArticleDOI
TL;DR: An algorithm for the computation of general Fourier integral operators associated with canonical graphs is developed based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics.
Abstract: We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists of constructing a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, thus enabling the application of wave packet--driven computation, and of constructing the associated pseudodifferential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

Journal ArticleDOI
TL;DR: The introduced notion of locally periodic two-scale convergence allows one to average a wider range of microstructures, compared to the periodic one, and is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.
Abstract: The introduced notion of locally periodic two-scale convergence allows one to average a wider range of microstructures, compared to the periodic one. The compactness theorem for locally periodic two-scale convergence and the characterization of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, with the periodicity with respect to the fast variable being dependent on the slow variable, by locally periodic functions, periodic in subdomains smaller than the considered domain but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.