arXiv subjects

Vishnu Raveendran

Publications and source records attributed to Vishnu Raveendran.

7 recordsLinked to original sources

arXiv2026

Interface Conditions for Wave Propagation Through a Time-varying Metasurface

We study wave propagation through a time-modulated thin heterogeneous layer. The layer is assumed to have a thickness of order $\es\ll 1$ and its material properties exhibit rapid oscillations on multiple spatial and temporal scales. We aim to rigorously derive the effective model and the corresponding interface conditions for wave propagation through the limiting interface. To perform the homogenization together with dimension reduction, we generalize the notion of two-scale convergence for thin layer introduced by Neuss-Radu and J\"ager (2007) to a multiple space-time scale framework. One of the main analytical difficulties arising in the homogenization analysis is that, in general, a uniform energy estimate cannot be obtained for a wave equation with time-varying coefficients. Therefore, we identify two physically relevant classes of coefficients for which we can derive a uniform energy bound. These include coefficients with traveling wave-type modulations of their properties. Using the energy estimates and the multiscale convergence for thin layer concept, we derive the effective model, which consists of linear wave equations in the bulk domains coupled through a nonstandard jump condition at the interface. This jump condition is governed by a wave-type dynamical equation with effective macroscopic coefficients determined by suitable cell problems of elliptic and hyperbolic type. Finally, we discuss the uniqueness of the effective model.

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arXiv2026

Optimization-Based Identification of Effective Coefficients for Wave Equations in Spatio-Temporal Metamaterials

We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.

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arXiv2024

Numerical Exploration of Nonlinear Dispersion Effects via a Strongly Coupled Two-scale System

The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.

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arXiv2023

Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation

We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems. The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term. This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). We prove the existence and uniqueness of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor. Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.

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arXiv2023

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

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arXiv2022

Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder's fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.

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arXiv2021

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) process for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer. This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces -- a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

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