Nonconcentration in partially rectangular billiards

In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy $E$ in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that $E$ is not resonating with the Dirichlet spectrum of the rectangular part.Comment: 28 pages, 1 figure; Typos corrected, error in final proof corrected and techniques strengthene

Spectral simplicity and asymptotic separation of variables

We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.Comment: 53 pages, 2 figure

Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

We consider a model of surface-mediated diffusion with alternating phases of pure bulk and surface diffusion. For this process, we compute the mean exit time from a disk through a hole on the circle. We develop a spectral approach to this escape problem in which the mean exit time is explicitly expressed through the eigenvalues of the related self-adjoint operator. This representation is particularly well suited to investigate the asymptotic behavior of the mean exit time in the limit of large desorption rate $\lambda$. For a point-like target, we show that the mean exit time diverges as $\sqrt{\lambda}$. For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $\lambda$ tends to infinity. We also revise the optimality regime of surface-mediated diffusion. Although the presentation is limited to the unit disk, the spectral approach can be extended to other domains such as rectangles or spheres.Comment: 21 pages, 7 figure

Existence of global strong solutions to a beam-fluid interaction system

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases

Design and Tests of 500kW RF Windows for the ITER LHCD System

In the frame of a R\&D effort conducted by CEA toward the design and the qualification of a 5 GHz LHCD system for the ITER tokamak, two 5 GHz 500 kW/5 s windows have been designed, manufactured and tested at high power in collaboration with the National Fusion Research Institute (NFRI). The window design rely on a symmetrical pill-box concept with a cylindrical beryllium oxide ceramic brazed on an actively water cooled copper skirt. The ceramic RF properties have been measured on a test sample to get realistic values for guiding the design. Low power measurements of the manufactured windows show return losses below-32 dB and insertion losses between-0.01 dB and-0.05 dB, with an optimum frequency shifted toward lower frequencies. High power tests conducted at NFRI show unexpected total power loss for both windows. The ceramic temperature during RF pulses has been found to reach unexpected high temperature, preventing these windows to be used under CW conditions. A post-mortem RF analysis of samples taken from one window shows that the dielectric properties of the ceramic were not the ones measured on the manufacturer sample, which partly explain the differences with the reference modelling

Design and RF measurements of a 5 GHz 500 kW window for the ITER LHCD system

International audienceCEA/IRFM is conducting R&D efforts in order to validate the critical RF components of the 5 GHz ITER LHCD system, which is expected to transmit 20 MW of RF power to the plasma. Two 5 GHz 500 kW BeO pill-box type window prototypes have been manufactured in 2012 by the PMB Company, in close collaboration with CEA/IRFM. Both windows have been validated at low power, showing good agreement between measured and modeling, with a return loss better than 32 dB and an insertion loss below 0.05 dB. This paper reports on the window RF design and the low power measurements. The high power tests up to 500kW have been carried out in March 2013 in collaboration with NFRI. Results of these tests are also reported. In the current ITER LHCD design, 20 MW Continuous Wave (CW) of Radio-Frequency power at 5 GHz are expected to be generated and transmitted to the plasma. In order to separate the vacuum vessel pressure from the cryostat waveguide pressure, forty eight 5 GHz 500kW CW windows are to be assembled on the waveguides at the equatorial port flange. For nuclear safety reasons, forty eight additional windows could be located in the cryostat section, to separate and monitor the cryostat waveguide pressure from the exterior transmission line pressure. These windows are identified as being one of the main critical components for the ITER LHCD system since first ITER LHCD studies [1] [2] [3] or more recently [4] [5] , and clearly require an important R&D effort. In this context and even if the LHCD system is not part of the construction baseline, the CEA/IRFM is conducting a R&D effort in order to validate a design and the performances of these RF windows. In order to begin the assessment of this need, two 5 GHz 500 kW/5 s pill-box type windows prototypes have been manufactured in 2012 by the PMB Company in close collaboration with the CEA/IRFM [6]. The section 2 of this paper reports the RF and mechanical design of a 5 GHz window. Some features of the mechanical design and the experimental RF measurements at low power are reported in section 3. High power results, made in collaboration with NFRI, are detailed in section 4. The development of CW windows is discussed in the conclusion. 2-RF AND MECHANICAL DESIGN The proposed 5 GHz RF window is based on a pill-box design [2] , i.e. a ceramic brazed in portion of a circular waveguide, connected on either side to a rectangular waveguide section. Typical design rules of thumb of such device are circular section diameter about the same size of the diagonal of the rectangular waveguide (cf. FIGURE 1). Without taking into account the ceramic, the circular section length is approximately half a guided wavelength of the circular TE 11 mode, in order for the device to act as a half-wave transformer. Once optimized, taking into account the ceramic, matching is correct only for a narrow band of frequency and is very sensitive to the device dimensions and the ceramic relative permittivity. The heat losses in the ceramic, which have to be extracted by an active water cooling, depends on the inside electric field topology and of ceramic dielectric loss (loss tangent). Undesirable modes due to parasitic resonances can be excited in the ceramic volume, raising the electric field an