Description: HV15R: 3D engine fan. F. Pacull, Lyon, France
|(bipartite graph drawing)||(graph drawing of A+A')|
|number of rows||2,017,169|
|number of columns||2,017,169|
|structural full rank?||yes|
|# of blocks from dmperm||24,683|
|# strongly connected comp.||24,683|
|explicit zero entries||0|
|nonzero pattern symmetry||84%|
|numeric value symmetry||0%|
|kind||computational fluid dynamics problem|
|Additional fields||size and type|
CFD matrices from Francois Pacull, FLUOREM, in Lyon, France We are dealing with CFD and more precisely steady flow parametrization. The equations involved are the compressible Navier-Stokes ones (RANS). These matrices are real, square and indefinite, they correspond to the Jacobian with respect the conservative fluid variables of the discretized governing equations (finite-volume discretization). Thus they have a block structure (corresponding to the mesh nodes: the block size is the number of variables per mesh node), they are not symmetric (however, their blockwise structure has a high level of symmetry) and they often show some kind of hyperbolic behavior. They have not been scaled or reordered. They are generated through automatic differentiation of the flow solver around a steady state. A right hand-side is also given for each matrix: this represents the derivative of the equations with respect to a parameter (of operation or shape). Since they are generated automatically, they may have "silent" variables: these are variables corresponding to an identity submatrix associated with a null right hand-side, for example one of the three velocity components in a 2D case, or the turbulent variables in a "frozen" turbulence case. We believe that these matrices are good test cases when studying preconditioning methods for iterative methods, such as block incomplete factorization, or when studying domain decomposition methods or deflation. They are actually being studied by a few researchers in France regarding numerical methods, through the LIBRAERO research project of the ANR (national research agency): ANR-07-TLOG-011. Francois Pacull, Lyon, France. fpacull at fluorem.com Specific problem descriptions: This is a 3D Reynolds-Averaged-Navier-Stokes case. HV15R: 3D engine fan. The flow has a low Mach number. Number of mesh nodes: 288,167 block size: 7 variables: [rho, rho*u, rho*v, rho*w, rho*E, rho*k, rho*omega] matrix order: 2,017,169 nnz: 283,073,458 In 2011, this problem took 3.5 hours to solve, using GMRES with an adaptive Schwarz preconditioner and ILU withing the subdomains, requiring about 100GB of memory. Reference: "A Study of ILU Factorization for Schwartz Preconditioners with Application to Computational Fluid Dynamics", F. Pacull, S. Aubert, M. Buisson, Proceedings of the 2nd Intl Conf on Parallel, Distributed, Grid, and Cloud Computing for Engineering, B.H.V Topping and P. Iva'nyi, Editors. Civil-Comp Press, Stirlingshire, Scotland, 2011.
|nnz(chol(P*(A+A'+s*I)*P')) with AMD||52,974,483,341|
|Cholesky flop count||3.3e+15|
|nnz(L+U), no partial pivoting, with AMD||105,946,949,513|
|nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD||54,682,928,546|
|nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD||98,098,157,490|
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Maintained by Tim Davis, last updated 12-Mar-2014.
Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.