Séminaire

14:00 - Maria Kazakova

Conditions aux limites transparentes discrètes pour le modèle de Green-Naghdi linéarisé.

Le modèle dispersif de Green-Naghdi communément utilisé pour décrire la propagation des vagues dispersives est initialement posé sur un domaine non borné. Pour les simulations numériques une procédure particulière est donc nécessaire pour imposer les conditions aux limites. L'objectif de ce travail est de construire des conditions aux limites transparentes pour le modèle de Green-Naghdi linéarisé. Les conditions obtenues directement au niveau discret ont été incorporées à un schéma numérique sur maillage colocalisé (les valeurs discrètes des inconnues du système sont évaluées aux mêmes points du maillage), puis sur mailles décalées (chaque inconnue est définie sur son maillage propre avec un décalage d'une demi-maille), tous deux d'intérêt pratique. L'algorithme numérique ainsi construit a permis la validation des conditions obtenues. La technique proposée permet de simuler proprement les ondes sortantes et rentrantes. Une analyse de stabilité est proposée, garantissant que la méthode introduite mène à des problèmes bien posés.

14:40 - Florian Monteghetti

Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics.

Je présenterai mes travaux de thèse, dont l'abstract est recopié ci-dessous. In computational aeroacoustics, time-domain impedance boundary conditions (TDIBCs) can be employed to model a locally reacting sound absorbing material. They enable to compute the effect of a material on the sound field after a homogenization distance and have proven effective in noise level predictions. The broad objective of this work is to study the physical, mathematical, and computational aspects of TDIBCs, starting from the physical literature. The first part of this dissertation defines admissibility conditions for nonlinear TDIBCs under the impedance, admittance, and scattering formulations. It then shows that linear physical models, whose Laplace transforms are irrational, admit in the time domain a time-delayed oscillatory-diffusive representation and gives its physical interpretation. This analysis enables to derive the discrete TDIBC best suited to a particular physical model, by contrast with a one-size-fits-all approach, and suggests elementary ways of computing the poles and weights. The proposed time-local formulation consists in composing a set of ordinary differential equations with a transport equation. The main contribution of the second part is the proof of the asymptotic stability of the multidimensional wave equation coupled with various classes of admissible TDIBCs, whose Laplace transforms are positive-real functions. The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a realization of the impedance, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup of contractions is then obtained by verifying the sufficient spectral conditions of the Arendt-Batty-Lyubich-Vu theorem. The third and last part of the dissertation tackles the discretization of the linearized Euler equations with TDIBCs. It demonstrates the computational advantage of using the scattering operator over the impedance and admittance operators, even for nonlinear TDIBCs. This is achieved by a systematic semi-discrete energy analysis of the weak enforcement of a generic nonlinear TDIBC in a discontinuous Galerkin finite element method. In particular, the analysis highlights that the sole definition of a discrete model is not enough to fully define a TDIBC. To support the analysis, an elementary physical nonlinear scattering operator is derived and its computational properties are investigated in an impedance tube. Then, the derivation of time-delayed broadband TDIBCs from physical reflection coefficient models is carried out for single degree of freedom acoustical liners. A high-order discretization of the derived time-local formulation, which consists in composing a set of ordinary differential equations with a transport equation, is applied to two flow ducts.

15:50 - Luiz Faria

Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

I will present a density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and a local expansion of the density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands of increased regularity. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. Numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions. Finally I will discuss a generalization of the method for different kernels.