• Aucun résultat trouvé

Quadrature-based diffusive representation of the fractional derivative with applications in aeroacoustics and eigenvalue methods for stability

N/A
N/A
Protected

Academic year: 2021

Partager "Quadrature-based diffusive representation of the fractional derivative with applications in aeroacoustics and eigenvalue methods for stability"

Copied!
2
0
0

Texte intégral

(1).  

(2)  

(3)  

(4)        

(5)                 

(6) 

(7)  

(8) .  an author's 

(9)  

(10)  

(11) https://oatao.univ-toulouse.fr/21172 

(12) 

(13) .    

(14) . Monteghetti, Florian Quadrature-based diffusive representation of the fractional derivative with applications in aeroacoustics and eigenvalue methods for stability. (2018) In: 10th Workshop SDS2018 STRUCTURAL DYNAMICAL SYSTEMS: Computational Aspects, 12 June 2018 - 15 June 2018 (Capitolo (Monopoli) - Bari, Italy).   .   

(15)  

(16)  

(17)     

(18)  

(19) 

(20)  

(21)  

(22) 

(23)   .

(24) Florian Monteghetti ONERA/DMPE, Université de Toulouse, 31055 Toulouse, France ISAE-SUPAERO/DISC, Université de Toulouse, 31055 Toulouse, France florian.monteghetti@isae.fr. Quadrature-based diffusive representation of the fractional derivative with applications in aeroacoustics and eigenvalue methods for stability The Riemann-Liouville fractional derivative can be recast into an observer of an infinite-dimensional state ϕ through its so-called diffusive representation ˆ ∞ 1/2 µ(ξ)∂t ϕ(t, ξ) dξ, with ∂t ϕ(t, ξ) = −ξϕ(t, ξ) + f (t), d f (t) = 0 √. ϕ(0, ξ) = 0, and µ(ξ) = 1/π ξ. This purely time-local representation of the 1 hereditary operator d /2 is computationally relevant insofar as it can be conveniently and accurately discretized. A discretization method, which involves a linear least-squares optimization, has proven suitable for time-domain simulations,1 but it requires an a priori choice of the Nξ discrete poles ξi and leads to spectral pollution, which prevents its use in eigenvalue problems that arise in fractional ordinary (FDE) and partial differential equations (FPDE). This talk introduces a quadrature-based discretization method whose sole parameter is Nξ and illustrates its properties on two applications. It is first applied to the stability study of a fractional delay differential equation, by computing the spectrum of the associated infinitesimal generator.2 It yields accurate and convergent spectra, by contrast with the optimizationbased method. The second application is a bidimensional wave propagation problem, namely the linearized Euler equations with a fractional impedance boundary condition.3 Perspectives include the application to fractional PDEs and the extension to other hereditary operators of diffusive type. This is a joint work with Denis Matignon (ISAE-SUPAERO/DISC) and Estelle Piot (ONERA). The presenter has been financially supported by the French ministry of defense (DGA), ONERA, and ISAE-SUPAERO/DISC. 1. Lombard, B. and Matignon, D. “Diffusive Approximation of a Time-Fractional Burger’s Equation in Nonlinear Acoustics”. SIAM Journal on Applied Mathematics 76.5 (2016), pp. 1765–1791. doi: 10.1137/16M1062491.. 2. Monteghetti, F., Haine, G., and Matignon, D. “Stability of Linear Fractional Differential Equations with Delays: a coupled Parabolic-Hyperbolic PDEs formulation”. 20th World Congress of the International Federation of Automatic Control (IFAC). (Toulouse, France). July 9–14, 2017. doi: 10.1016/j.ifacol.2017.08.1966.. 3. Monteghetti, F., Matignon, D., and Piot, E. “Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations”. (Submitted).. 1.

(25)

Références

Documents relatifs

This is achieved through the interaction of several components: the panning algorithm, VBAP; the distribution of the speakers in the room (speaker setup); and the specification of

Keywords: objective fractional derivative constitutive equation; viscoelasticity; rest state stability analysis; Hadamard stability analysis; solution existence,

In this paper we study isoperimetric inequalities for the eigenvalues of the Laplace operator with constant and locally constant boundary conditions.. Existence and sta- bility

We then prove Strichartz estimates for the frac- tional Schr¨ odinger and wave equations on compact Riemannian manifolds without boundary (M, g). This result extends the

Keywords and phrases: stochastic wave equation, random field solution, spa- tially homogenous Gaussian noise, fractional Brownian motion..

Asymptotic stability of the linearised Euler equations with long-memory impedance boundary condition.. Florian Monteghetti, Denis Matignon, Estelle Piot,

This paper is organized as follows. Section 2 recalls elementary facts about diffusive representations and introduces the proposed Q β,N discretization method, where β is a

Wave propagation in a fractional viscoelastic Andrade medium: diffusive approximation and numerical modeling... Wave propagation in a fractional viscoelastic Andrade medium: