Vincent Mouysset

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain Ω with artificial boundary conditions set on the arbitrarily shaped boundary of Ω. These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

On the Existence and Uniqueness of a Solution for Some Frequency-Dependent Partial Differential Equations Coming from the Modeling of Metamaterials

Several systems coming from the theory of linear wave propagation are investigated, on a bounded domain, in the presence of frequency-dependent materials like metamaterials. For each system we show generic well-posedness results under assumptions that are relevant for some models in the literature. This means existence and uniqueness of a solution for all frequencies except for a discrete locally finite and possibly empty set of frequencies. Finally, some examples of materials are studied, like a periodic array of split-ring resonators (SRR), a chiral metamaterial based on the

A Discontinuous Galerkin Formalism to Solve the Maxwell-Vlasov Equations. Application to High Power Microwave Sources

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Adaptive refinement and selection process through defect localization for reconstructing an inhomogeneous refraction index

-particle resonator model, a bi-anisotropic metamaterial made from SRR, absorbing boundary conditions of perfectly matched layers type for the acoustics waves, an example of acoustic metamaterial having negative bulk modulus and an elastic metamaterial.

Accurate, Automatic and Compressed Visualization of Radiated Helmholtz Fields from Boundary Element Solutions.

In this paper, we present a Particle-In-Cell (PIC) method based on a Discontinuous Galerkin (DG) scheme to solve the Maxwell-Vlasov equations in time-domain. Comparisons with an other industrial software are given to validate the method.

Well-suited and adaptive post-processing for the visualization of hp simulation results

We consider the iterative reconstruction of both the internal geometry and the values of an inhomogeneous acoustic refraction index through a piecewise constant approximation. In this context, we propose two enhancements intended to reduce the number of parameters to reconstruct, while preserving accuracy. This is achieved through the use of geometrical informations obtained from a previously developed defect localization method. The first enhancement consists in a preliminary selection of relevant parameters, while the second one is an adaptive refinement to enhance precision with a low number of parameters. Each of them is numerically illustrated.

An optimization approach for the localization of defects in an inhomogeneous medium from acoustic far-field measurements at a fixed frequency

We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions, combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.

Highly Accurate Schemes for Wave Propagation Systems: Application in Aeroacoustics

Abstract While high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (Discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method.

Sur l'existence et l'unicité des solutions pour des équations de Drude–Born–Fedorov homogénéisées en domaine borné et applications aux métamatériaux

We are interested in the localization of defects in non-homogeneous non-absorbing media, with far-field measurements, generated by plane waves. We present here a way to handle the inhomogeneous background problem by the means of a constrained optimization approach; in the lines of the inf criterion from the factorization method for constant backgrounds. We show that both cost function and constraints present a simple form that is easy to compute. They can thus be treated by many well-known optimization methods, and we numerically illustrate four of them.

Sur une approximation des champs propagés par les équations de Maxwell instationnaires, homogènes, à l'extérieur d'un domaine borné

The Discontinuous Galerkin (DG) method is considered for computational aeroacoustic. A software has been developed to make it possible to test a large variety of configurations (non‐conform grid, variable polynomial order). To deal with instationary phenomena involved by some shear flows, a compromise between time computation and accuracy is deduced from some numerical experiments.