Computing the Conditioning of the Components of a Linear Least Squares Solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the least squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities. In particular, we show that, in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right-hand side is exactly the condition number of this solution component when perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK routines to compute the variance-covariance matrix and the least squares conditioning and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace in Theorie Analytique des Probabilites, 1820, for computing the mass of Jupiter and experiments from the space industry with real physical data

Large Eddy Simulations of gaseous flames in gas turbine combustion chambers

Recent developments in numerical schemes, turbulent combustion models and the regular increase of computing power allow Large Eddy Simulation (LES) to be applied to real industrial burners. In this paper, two types of LES in complex geometry combustors and of specific interest for aeronautical gas turbine burners are reviewed: (1) laboratory-scale combustors, without compressor or turbine, in which advanced measurements are possible and (2) combustion chambers of existing engines operated in realistic operating conditions. Laboratory-scale burners are designed to assess modeling and funda- mental flow aspects in controlled configurations. They are necessary to gauge LES strategies and identify potential limitations. In specific circumstances, they even offer near model-free or DNS-like LES computations. LES in real engines illustrate the potential of the approach in the context of industrial burners but are more difficult to validate due to the limited set of available measurements. Usual approaches for turbulence and combustion sub-grid models including chemistry modeling are first recalled. Limiting cases and range of validity of the models are specifically recalled before a discussion on the numerical breakthrough which have allowed LES to be applied to these complex cases. Specific issues linked to real gas turbine chambers are discussed: multi-perforation, complex acoustic impedances at inlet and outlet, annular chambers.. Examples are provided for mean flow predictions (velocity, temperature and species) as well as unsteady mechanisms (quenching, ignition, combustion instabil- ities). Finally, potential perspectives are proposed to further improve the use of LES for real gas turbine combustor designs

Numerical investigation of fluid structure interaction between unsteady flow and vibrating liner in a combustion chamber

Numerical investigations of fluid structure interaction between unsteady flow\ud and vibrating liner in a combustion chamber are undertaken. The computational study consist of two approaches. Firstly, a partioned procedure consists of coupling the LES code AVBP for combustion modelling with the FEM code CaluliX for structural dynamic analysis. The CFD code CFX together with the FEM Ansys package are then used.\ud Results of unsteady fluid structure interaction applied to combustion system are presented and compare well with experimental results

Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure

Using LES to Study Reacting Flows and Instabilities in Annular Combustion Chambers

Great prominence is put on the design of aeronautical gas turbines due to increasingly stringent regulations and the need to tackle rising fuel prices. This drive towards innovation has resulted sometimes in new concepts being prone to combustion instabilities. In the particular field of annular combustion chambers, these instabilities often take the form of azimuthal modes. To predict these modes, one must compute the full combustion chamber, which remained out of reach until very recently and the development of massively parallel computers. Since one of the most limiting factors in performing Large Eddy Simulation (LES) of real combustors is estimating the adequate grid, the effects of mesh resolution are investigated by computing full annular LES of a realistic helicopter combustion chamber on three grids, respectively made of 38, 93 and 336 million elements. Results are compared in terms of mean and fluctuating fields. LES captures self-established azimuthal modes. The presence and structure of the modes is discussed. This study therefore highlights the potential of LES for studying combustion instabilities in annular gas turbine combustors

Securely Outsourcing Large Scale Eigen Value Problem to Public Cloud

Cloud computing enables clients with limited computational power to economically outsource their large scale computations to a public cloud with huge computational power. Cloud has the massive storage, computational power and software which can be used by clients for reducing their computational overhead and storage limitation. But in case of outsourcing, privacy of client's confidential data must be maintained. We have designed a protocol for outsourcing large scale Eigen value problem to a malicious cloud which provides input/output data security, result verifiability and client's efficiency. As the direct computation method to find all eigenvectors is computationally expensive for large dimensionality, we have used power iterative method for finding the largest Eigen value and the corresponding Eigen vector of a matrix. For protecting the privacy, some transformations are applied to the input matrix to get encrypted matrix which is sent to the cloud and then decrypting the result that is returned from the cloud for getting the correct solution of Eigen value problem. We have also proposed result verification mechanism for detecting robust cheating and provided theoretical analysis and experimental result that describes high-efficiency, correctness, security and robust cheating resistance of the proposed protocol

O-Palm: an open source dynamic parallel coupler

Since 1996 CERFACS has been developing the PALM parallel coupler, which is currently used for more than 50 research and industrial projects ranging from operational data assimilation to multi-physics modelling, from climate change impact assessment to fluid and structure interactions. It can be defined a dynamic coupler for its ability to deal with situations where the component execution scheduling and the data exchange patterns cannot be entirely defined before execution. Under the name O-PALM, it is now open source (LGPL license). This document provides some highlights on the design of PALM and on the main implementation choices and a brief description of some representative applications