CERFACS CERFACS Scientific Activity Report Jan. 2010 â Dec. 2011

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2 Integral equations for electromagnetism scattering 2.1 Numerical simulation of a reflectarray antenna. A. Bendali, F. Collino and M. Fares The object of this study was to explore some modeling techniques for an efficient numerical simulation of reflectarray antenna. A reflectarray antenna is a skew finite grating of shallow rectangular waveguides enlighten by a small horn antenna. A electronic device inside each waveguide is designed to set up a phase shift locally for the reflected field. In this way, the direction of the main lobe of the antenna can be modified electronically without any mechanical move of the emitting device. We developed an impedance model based on an unimodal propagation in each cell taking into account the complicate phase shift process. All the remaining parts of the radiating system are dealt with using a direct numerical simulation approach. We derived a formulation of the problem in terms of a system of integral equations next discretized by means of the method of moments. We showed that this formulation provides a reliable method for computing the field radiated by the antenna. Moreover, it is well-adapted for providing a fast method for computing the variation in the direction of the main lobe resulting from a modification of the phase shift on each cell. The numerical method has been implemented within the CERFACS Electromagnetic Solver Code (CESC). Computed radiation pattern for a reflectarray designed by TSA-CNES showed an excellent agreement with experiments hence validating the accuracy of the approach as depicted in figure 2.2 and figure 2.3. 2.2 Extension to non-conforming meshes and stabilization of the combined current and charge integral equation. A. Bendali, F. Collino, M. Fares and B. Steif An important issue in industrial applications in electromagnetism requiring the solution of a large scale boundary integral equation concerns the possibility of using meshes on different zones obtained independently each from the other, which thus do not comply with the usual matching requirement of finite element approximations. By bringing out some mathematical properties of the Combined Current and Charge Integral Equation (shortly C3IE) introduced by Taskinen and Yl-Oijala [2] when it is posed on a surface without geometrical singularities, we established that this equation can be solved by a Boundary Element Method (BEM) that requires no interelement continuity. This property is crucial when using meshes on different parts of the surface obtained independently each from the other. We showed how the C3IE can be implemented by slightly modifying a usual BEM electromagnetic solver code and that the numerical behavior of this method is very similar to the usual Combined Field Integral Equation (CFIE) when dealing with smooth surfaces. The extension to singular geometries showed that acute dihedral angles can lead to inaccuracies in the results. By considering a two-dimensional version of the formulation, we have brought out that the wrong results are due to spurious oscillations concentrating around the singular points of the geometry. Noticing that the system linking the current and the charge is a saddle-point problem, we have adapted a general procedure used for stabilizing the numerical approximation of mixed formulations 48 Jan. 2010 – Dec. 2011
ELECTROMAGNETISM AND ACOUSTICS TEAM as the Stokes system [1], consisting here in augmenting the approximation of the charge. This stabilization procedure, when coupled with a refinement of the mesh in the proximity of the geometrical singularities, obtained by a simple subdivision of the triangles, greatly reduces the effect of the spurious oscillations. This topic is the subject of Steif’s PhD thesis which is planned to be defended before June 2012. [1] F. Brezzi and M. Fortin, (1991.), Mixed and Hybrid Finite Element Methods., Springer-Verlag. [2] M. Taskinen and P. Y. Oijala, (2000), Current and Charge Integral Equation Formulation, IEEE Transactions on Antennas and Propagation, 54,1, 58–67. 2.3 Model reduction for acoustic multi-perforated walls in aircraft engines. A. Bendali, M. Fares, S. Laurens and S. Tordeux To prevent them from being destroyed, the walls of the combustion chambers of aircraft engines are protected by injection of relatively cool air through a regular array of sub-millimeter holes. The designers of these engines are also faced with combustion instabilities due to excessive levels of noise which can destroy the engine. The properties of sound attenuation by perforated walls are reinforced when they are crossed by a bias flow. An additional advantage of cooling the walls is hence to avoid combustion instabilities due to the acoustics. Predicting these effects is of paramount importance in the process of developing new engines thus inducing a crucial need in numerical simulation. However, the submillimeter size and the number of perforations make prohibitive an approach by a direct numerical simulation which must be preceded by a model reduction. An example of such a perforated wall is depicted in figure 2.7. A prohibitive discretization at the level of the perforations can be avoided by using a reduced model that brings about the effect of perforations by means an effective condition on the plate whose effective numerical simulation is thus affordable. These model reductions depend strongly on the porosity of the plate, defined as the open to total area ratio. Less than 4%, the porosity of the walls is considered as low. Usual walls involved in aircraft engines are of about 2% porosity. The model reductions for a low porosity plate are based on the calculation of the Rayleigh conductivity of an isolated perforation. This conductivity is defined as the ratio of the volume flux to the pressure across the perforation. This approach has a significant advantage : it makes it possible to couple the equations governing the propagation of acoustic waves outside a boundary layer near the plate with models of flows taking into account the dissipation by viscosity at the perforations. The models are developed assuming that the size of the aperture of the perforation are small compared with the wavelength and that they constitute an infinite doubly periodic array. A systematic study, based on the availability in CERFACS of codes tailored to take advantage of massively parallel plateformes was conducted to accurately bring out the effect of the size of the orifices and the finite size of the lattice perforations. Figure 2.10, plotting the reflection coefficient for an infinite array of perforations, shows an agreement of the model with a direct simulation yielding an error less than 1%. The same simulation, but this once for a finite size array shown in figure 2.8, yielded an error of about 10% as reported in Figure 2.9. Several studies were also conducted in this framework. An attempt to give a mathematical justification to these models, based on intuitive approximations, led to the construction of second-order conditions for approximating the acoustic field. Figure 2.10 depictes the improvement gained by using these second-order conditions. Some progresses have been also done in another direction. The thickness of the plate is usually taken into account by adding an inertial mass impedance term to the Rayleigh conductivity of the plate assued to be infinitely thin. This inertial impedance was formely determined by means of an intuitive reasoning only. A rigorous mathematical justification to these calculations made it possible to obtain it for an inclined perforation, which is the actual shape involved in realistic aircraft engines. This is an unprecedented result since no determination, even intuitive, was formely available in this case. CERFACS ACTIVITY REPORT 49
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CERFACS Scientific Activity Report
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Table des matières 1 Foreword xiii
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9 Publications 40 9.1 Books . . . .
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3 The OpenPALM coupler 104 3.1 Intr
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Table des figures CERFACS chart as
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TABLE DES FIGURES 6 Computational F
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Foreword Welcome to the 2010-2011 S
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CERFACS STRUCTURE CERFACS organizat
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CERFACS STAFF NAME POSITION PERIOD
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CERFACS STAFF SILVA GARZON Ph.D stu
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Page 97 and 98: 1 Introduction Data assimilation is
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2 The OASIS coupler The OASIS coupl
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THE OASIS COUPLER 2.3 OASIS4 Since
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3 The OpenPALM coupler 3.1 Introduc
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THE OPENPALM COUPLER In the NitroEu
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4 HPC Climate modelling 4.1 General
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5 Publications 5.1 Conference Proce
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2 Climate variability and predictab
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CLIMATE CHANGE AND IMPACT STUDIES a
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PUBLICATIONS [CM15] Y. Kwon, C. Des
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1 Introduction The CFD (Computation
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2 Combustion The research activity
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COMPUTATIONAL FLUID DYNAMICS rate-o
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COMPUTATIONAL FLUID DYNAMICS FIG. 2
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7 Aviation and Environment
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INTRODUCTION instead of regular ker
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SIMULATIONS OF AIRCRAFT WAKES AND R
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LARGE SCALE ATMOSPHERIC COMPOSITION
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LARGE SCALE ATMOSPHERIC COMPOSITION
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PUBLICATIONS 4.2 Technical Reports
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CERFACS CERFACS Scientific Activity Report Jan. 2010 â Dec. 2011

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CERFACS CERFACS Scientific Activity Report Jan. 2010 â Dec. 2011