The multiscale hybrid-mixed method for the maxwell equations in heterogeneous media

International audienceIn this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a " divide-and-conquer " framework. Several numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solving heterogeneous media problems on coarse meshes

An upscaled DGTD method for time-domain electromagnetics

International audienceIn this work, we address time-dependent electromagnetic wave propagation problems with strong multiscale features with application to nanophotonics, where problems usually involve complex mul-tiscale geometries, heterogeneous materials, and intense, localized electromagnetic fields. Nanopho-tonics simulations require very fine meshes to incorporate the influence of geometries as well as high order polynomial interpolations to minimize dispersion. Our goal is to design a family of innovative high performance numerical methods perfectly adapted for the simulation of such multiscale problems. For that purpose we extend the Multiscale Hybrid-Mixed (MHM) finite element method, originally proposed for the Laplace problem in [1], to the solution of 2d and 3d transient Maxwell equations with heterogeneous media. The MHM method arises from the decomposition of the exact electromagnetic fields in terms of the solutions of locally independent Maxwell problems. Those problems are tied together with an one field formulation on top of a coarse mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are also driven by local Maxwell problems. A high order Discontinuous Galerkin method (see [2]) in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and parallelizable, and yields a staggered algorithm within a divide-and-conquer framework. In this talk we will present the general formulation of this MHM-DGTD method and presen some preliminary numerical results in 2d. REFERENCES 1. C. Harder, D. Paredes, and F. Valentin. A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys., 245:107-130, 2013. 2. S. Descombes, C. Durochat, S. Lanteri, L. Moya, C. Scheid, and J. Viquerat. Recent advances on a DGTD method for time-domain electromagnetics. Photonics and Nanostructures-Fundamentals and Applications, 11(4):291-302, 2013

A machine learning approach for efficient uncertainty quantification using multiscale methods

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.Comment: Journal of Computational Physics (2017

The heterogeneous multiscale method for dispersive Maxwell systems

In this work, we apply the finite element heterogeneous multiscale method to a class of dispersive first-order time-dependent Maxwell systems. For this purpose, we use an analytic homogenization result, which shows that the effective system contains additional dispersive effects. We provide a careful study of the (time-dependent) micro problems, including $H^2$ and micro errors estimates. Eventually, we prove a semi-discrete error estimate for the method

The Heterogeneous Multiscale Method for dispersive Maxwell systems

In this work, we apply the finite element heterogeneous multiscale method to a class of dispersive first-order time-dependent Maxwell systems. For this purpose, we use an analytic homogenization result, which shows that the effective system contains additional dispersive effects. We provide a careful study of the (time-dependent) micro problems, including $\mathrm{H}^2$ and micro errors estimates. Eventually, we prove a semi-discrete error estimate for the method.Comment: 26 page

Une méthode hybride multi-échelles pour les problèmes de Darcy utilisant des solveurs locaux à éléments finis mixtes

International audienceMultiscaled Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flows, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small details are resolved by local problems, using fine representations inside the macro elements, setting the multiplier as Neumann boundary conditions (downscaling). In this work a variant of the method is developed, denoted by MHM-H(div), adopting mixed finite elements at the dowscaling stage, instead of continuous finite elements used in all previous publications of the method. Thus, this alternative MHM method inherits improvements typical of mixed methods, as better flux accuracy, and local mass conservation at the mi-cro scale level inside the macro elements, which are important properties for multi-phase flows in rough heterogeneous media. Different two-scale stable space settings are considered. Vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries. In each macro element, the internal flux components, with vanishing normal traces, and the potential approximations, may be enriched in different extents: with respect to internal mesh size, internal polynomial degree, or both, the choice being determined by the problem at hands. A unified general error analysis of the MHM-H(div) method is presented for all these two-scale space scenarios. Both MHM versions are compared for 2D test problems, with smooth solutions, for convergence rates verification, and for a Darcy's flow in heterogeneous media. MHM-H(div) 3D simulations are presented for a known singular Darcy's solution, using adaptive macro partitions, and for an oscillatory permeability scenario

On pore-scale modeling and simulation of reactive transport in 3D geometries

Pore-scale modeling and simulation of reactive flow in porous media has a range of diverse applications, and poses a number of research challenges. It is known that the morphology of a porous medium has significant influence on the local flow rate, which can have a substantial impact on the rate of chemical reactions. While there are a large number of papers and software tools dedicated to simulating either fluid flow in 3D computerized tomography (CT) images or reactive flow using pore-network models, little attention to date has been focused on the pore-scale simulation of sorptive transport in 3D CT images, which is the specific focus of this paper. Here we first present an algorithm for the simulation of such reactive flows directly on images, which is implemented in a sophisticated software package. We then use this software to present numerical results in two resolved geometries, illustrating the importance of pore-scale simulation and the flexibility of our software package.Comment: 15 pages, 6 figure

Coupled DEM-LBM method for the free-surface simulation of heterogeneous suspensions

The complexity of the interactions between the constituent granular and liquid phases of a suspension requires an adequate treatment of the constituents themselves. A promising way for numerical simulations of such systems is given by hybrid computational frameworks. This is naturally done, when the Lagrangian description of particle dynamics of the granular phase finds a correspondence in the fluid description. In this work we employ extensions of the Lattice-Boltzmann Method for non-Newtonian rheology, free surfaces, and moving boundaries. The models allows for a full coupling of the phases, but in a simplified way. An experimental validation is given by an example of gravity driven flow of a particle suspension

Numerical homogenization of time-dependent Maxwell\u27s equations with dispersion effects

This thesis studies the propagation of electromagnetic waves in heterogeneous structures such as metamaterials. The governing equations for this problem are Maxwell\u27s equations with highly oscillatory parameters. We use an analytic homogenization result which yields an effective Maxwell system that involves a convolution integral. This convolution represents dispersive effects that result from the interaction of the wave with the (locally) periodic microscopic structure. We discretize in space using the Finite Element Heterogeneous Multiscale Method (FE-HMM) and provide a semi-discrete error estimate. The rigorous error analysis in space is supplemented by a rather standard time discretization at the end of which an efficient, fully discrete method is proposed. This method uses a recursive approximation of the convolution that relies on the assumption that the convolution kernel is an exponential function. Eventually, we present numerical experiments both for the microscopic and the macroscopic scale