With the increased availability of computational resources, the past decade has seen a rise in the use of computational fluid dynamics (CFD) for medical applications. There has been an increase in the application of CFD to attempt to predict the rupture of intracranial aneurysms, however, while many hemodynamic parameters can be obtained from these computations, to date, no consistent methodology for the prediction of the rupture has been identified. One particular challenge to CFD is that many factors contribute to its accuracy; the mesh resolution and spatial/temporal discretization can alone contribute to a variation in accuracy. This failure to identify the importance of these factors and identify a methodology for the prediction of ruptures has limited the acceptance of CFD among physicians for rupture prediction. The International CFD Rupture Challenge 2013 seeks to comment on the sensitivity of these various CFD assumptions to predict the rupture by undertaking a comparison of the rupture and blood-flow predictions from a wide range of independent participants utilizing a range of CFD approaches. Twenty-six groups from 15 countries took part in the challenge. Participants were provided with surface models of two intracranial aneurysms and asked to carry out the corresponding hemodynamics simulations, free to choose their own mesh, solver, and temporal discretization. They were requested to submit velocity and pressure predictions along the centerline and on specified planes. The first phase of the challenge, described in a separate paper, was aimed at predicting which of the two aneurysms had previously ruptured and where the rupture site was located. The second phase, described in this paper, aims to assess the variability of the solutions and the sensitivity to the modeling assumptions. Participants were free to choose boundary conditions in the first phase, whereas they were prescribed in the second phase but all other CFD modeling parameters were not prescribed. In order to compare the computational results of one representative group with experimental results, steady-flow measurements using particle image velocimetry (PIV) were carried out in a silicone model of one of the provided aneurysms. Approximately 80% of the participating groups generated similar results. Both velocity and pressure computations were in good agreement with each other for cycle-averaged and peak-systolic predictions. Most apparent "outliers" (results that stand out of the collective) were observed to have underestimated velocity levels compared to the majority of solutions, but nevertheless identified comparable flow structures. In only two cases, the results deviate by over 35% from the mean solution of all the participants. Results of steady CFD simulations of the representative group and PIV experiments were in good agreement. The study demonstrated that while a range of numerical schemes, mesh resolution, and solvers was used, similar flow predictions were observed in the majority of cases. To further validate the computati...

SUMMARYSix different preconditioning methods to accelerate the convergence rate of Krylov-subspace iterative methods are described, implemented and compared in the context of matrix-free techniques. The acceleration techniques comprehend Krylov-subspace iterative methods; invariant subspace-based methods and matrix approximations: Jacobi, LU-SGS, Deflated GMRES; Augmented GMRES; polynomial preconditioner and FGMRES/Krylov.The relative behaviour of the methods is explained in terms of the spectral properties of the resulting iterative matrix. The employed code uses a Newton-Krylov approach to iteratively solve the Euler or Navier-Stokes equations, for a supersonic ramp or a viscous compressible double-throat flow. The linear system approximate solver is the GMRES method, in either the restarted or FGMRES variants. The results show the better performance of the methods that approximate the iterative matrix, such as Jacobi, LU-SGS and FGMRES/Krylov.

SUMMARYThis paper reports an intercomparison of three second-order accurate reconstruction schemes to predict 2D steady-state compressible Euler and Navier-Stokes ows on unstructured meshes. The schemes comprise one monotone slope limiter (Barth and Jespersen, A1AA Paper 89-0366, 1989) and two approximately monotone methods: the slope limiter due to Venkatakrishnan and a data-dependent weighting least-squares procedure (Gooch, Journal of Computational Physics, 1997; 133:6 -17). In addition to the 1D scalar wave problem, comparisons were performed under two inviscid test cases: a supersonic 10• ramp and a supersonic bump; and two viscous laminar compressible ow cases: the Blasius boundary layer and a double-throated nozzle. The data-dependent oscillatory behaviour is found to be dependent on a user-supplied constant. The three schemes are compared in terms of accuracy and computational e ciency. The results show that the data-dependent procedure always returns a numerical steady-state solution, more accurate than the ones returned by the slope limiters. Its use for Navier-Stokes ow calculations is recommended.

A second-order accurate finite volume method for solving compressible 2D flows on hybrid structured-unstructured grids is presented. Separate reconstruction and evolution steps are taken to discretize the convective terms. For the reconstruction step, a data-dependent Least-Squares procedure is used, while for the evolution step two recent flux functions are included: the HLLC approximate Riemann solver and the AUSM+ flux vector splitting. Steady-state solutions are obtained with an implicit backward Euler scheme. The assembled system is solved by iterative means (BiCGSTAB, GMRES) with ILU pre-conditioning. Two internal, steady, 2D flow test cases are presented to validate the code: a supersonic 10° ramp inside a channel and a laminar flow through a double-throated nozzle. The code proved accurate with the use of both flux functions when comparing the computed results with both an analytical (ramp) and a reference solution (nozzle). The GMRES solver generally required less CPU time until convergence for the inviscid test-case while the BiCGSTAB solver got the edge for the viscous calculations.

The concept of a representative spectrum is introduced in the context of Newton-Krylov methods. This concept allows a better understanding of convergence rate accelerating techniques for Krylov-subspace iterative methods in the context of CFD applications of the Newton-Krylov approach to iteratively solve sets of non-linear equations. The dependence of the representative spectrum on several parameters such as mesh, Mach number or discretization techniques is studied and analyzed.