Mathematics, Hybrid computing and HPC

International audienceHPC appears more and more as a key player in the field of numerical simulation and data processing. This trend comes of course with the desire to perform simulations that are closer and closer to real world situations, and with the development of clusters and platforms that provide access to hundreds to thousands CPU/GPU nodes. The application domains encompass many fields, from fluid mechanics to biology and nano-sciences, in academic research as well as for industrial applications. Concerning industrial applications, major groups have often already a good practice of HPC, with dedicated manpower and available in-house platforms. The access of SMEs to HPC is more problematic as they do not have the appropriate resources in hardware and manpower, and it is sometimes hard for them to have a clear idea of the gain they will obtain through HPC. In the first part of the talk, I will talk about a national initiative led by INRIA, GENCI and BPI, to promote the access of SMEs to HPC. This initiative provides support both in terms of market analysis, access to hardware and technical environment. It now involves middle-size HPC platforms that are distributed in French universities. This initiative will therefore give new opportunities to researchers, in particular mathematicians, to be connected to industrial collaborations. HPC is actually not only a question of accessing hardware and adapting existing codes to massively parallel platforms. It also raises questions about mathematical and numerical models that optimize the emerging hardware and analyze the huge amount of data associated with these simulations, and software engineering to distribute algorithms on heterogeneous clusters. Mathematicians therefore can use HPC as a mean to access challenging industrial collaborations in which they can contribute through new methods and algorithms, in both scientific computing and statistics

Numerical analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid

We present and analyze a penalization method wich extends the the method of [1] to the case of a rigid body moving freely in an incompressible fluid. The fluid-solid system is viewed as a single variable density flow with an interface captured by a level set method. The solid velocity is computed by averaging at avery time the flow velocity in the solid phase. This velocity is used to penalize the flow velocity at the fluid-solid interface and to move the interface. Numerical illustrations are provided to illustrate our convergence result. A discussion of our result in the light of existing existence results is also given. [1] Ph. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math. 81: 497--520 (1999)Comment: 23 page

Anisotropic sub-grid scale numerical schemes for Large Eddy Simulations of turbulent flows

Subgrid-scale models using anisotropic eddy-viscosity tensors are considered and used to derive new numerical schemes for Large Eddy Simulations. The derivation is based on integral representations of differential operators and directly leads to algorithms using only velocity values at grid points. The asymptotic behavior of the model near boundaries is discussed and several implementation options are described, including techniques for backscatter control which distinguish between the strain directions in the flow. A numerical validation is carried out on channel flow calculations and possible extensions of the method are outlined

Semi-Lagrangian particle methods for hyperbolic problems

International audienceParticle methods are often associated with frequent remeshing to maintain the regularity of the particle distribution and the accuracy of the method. In that case particle methods can be seen as forward, conservative, semi-lagrangian methods.In the case of systems, pressure terms are separated from advection terms and semi-lagrangian particle methods are a particular case of flux splitting methods. For CFL numbers smaller than one, they can also be interpreted as finite-volumemethods.In the linear case, the related finite-volume methods are rather classical (e.g. Lax-Wendroff or Beam-Warming schemes) but in the non-linear case one obtains less conventional methods, in particular with good entropy properties. On the other hand, the analogy with finite-volume methods allows to borrow from the finite-volume world several technics that can be translatedinto particle remeshing schemes. Among these technics, limiters allow to deriveTVD particle methods for which convergence can be proved, at least in the scalar case, whereas convergence for totally grid-free particles (e.g. the so-called SPH methods) remains an essentially open question. However, the main practical interest of particle methods lies in the possibility to use non-CFL constrained time-steps, a possibility which still remains largely unexplored for non-linear systems.In this talk I will address these questions. I will also discuss the derivation of high order methods for linear hyperbolic equations and their application for the Direct Numerical Simulation of turbulent transport

A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies

We present an accurate Lagrangian method based on vortex particles, level-sets, and immersed boundary methods, for animating the interplay between two fluids and rigid solids. We show that a vortex method is a good choice for simulating bi-phase flow, such as liquid and gas, with a good level of realism. Vortex particles are localized at the interfaces between the two fluids and within the regions of high turbulence. We gain local precision and efficiency from the stable advection permitted by the vorticity formulation. Moreover, our numerical method straightforwardly solves the two-way coupling problem between the fluids and animated rigid solids. This new approach is validated through numerical comparisons with reference experiments from the computational fluid community. We also show that the visually appealing results obtained in the CG community can be reproduced with increased efficiency and an easier implementation

Hybrid spectral-particle method for the turbulent transport of a passive scalar

International audienceThis paper describes a novel hybrid method, combining a spectral and a particle method, to simulate the turbulent transport of a passive scalar. The method is studied from the point of view of accuracy and numerical cost. It leads to a significative speed up over more conventional grid-based methods and allows to address challenging Schmidt numbers. In particular, theoretical predictions of universal scaling in forced homogeneous turbulence are recovered for a wide range of Schmidt numbers for large, intermediate and small scales of the scalar

Spatially distributed control for optimal drag reduction of the flow past a circular cylinder

We report high drag reduction in direct numerical simulations of controlled flows past circular cylinders at Reynolds numbers of 300 and 1000. The flow is controlled by the azimuthal component of the tangential velocity of the cylinder surface. Starting from a spanwise-uniform velocity profile that leads to high drag reduction, the optimization procedure identifies, for the same energy input, spanwise-varying velocity profiles that lead to higher drag reduction. The three-dimensional variations of the velocity field, corresponding to modes A and B of three-dimensional wake instabilities, are largely responsible for this drag reduction. The spanwise wall velocity variations introduce streamwise vortex braids in the wake that are responsible for reducing the drag induced by the primary spanwise vortices shed by the cylinder. The results demonstrate that extending two-dimensional controllers to three-dimensional flows is not optimal as three-dimensional control strategies can lead efficiently to higher drag reductio