An Asynchronous Incomplete Block LU Preconditioner for Computational Fluid Dynamics on Unstructured Grids

The questions related to the formulation and numerical realization of boundary conditions on a wall at the computation of turbulent flows on unstructured grids are considered. A technique is proposed for realization of weak boundary conditions assuming a non-zero value of the tangent velocity on the wall at a discretization of the Reynolds-averaged Navier — Stokes equations by the control volume method. The capabilities of the developed approach are demonstrated by the example of computing the flow in the inter-blade channel of a low-velocity compressor. The influence of the near-wall grid step on the accuracy of computations, in particular, the pressure distribution near the profile trailing edge is shown, and the solution grid dependence is investigated when using the method of near-wall functions and weak boundary conditions.

Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual (GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multigrid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization preconditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and Incomplete LU preconditioners.

The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area. Solving high‐resolution convection problems with order of magnitude 108 degrees of freedom requires solvers that scale well, with respect to both the number of degrees of freedom as well as having optimal parallel scaling characteristics on computer clusters. We investigate the use of a smoothed aggregation (SA) algebraic multigrid (AMG)‐type solution strategy to construct efficient preconditioners for the Stokes equation. We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating the convergence rate of the generalized conjugate residual method (GCR). We abbreviate this procedure as BTA‐GCR. For our experiments, we use unstructured grids with quadratic finite elements, making the model flexible with respect to geometry and topology and O(h3) accurate. We find that AMG‐type methods scale linearly (O(n)), with respect to the number of degrees of freedom, n. Although not all parts of AMG have preferred parallel scaling characteristics, we show that it is possible to tune AMG, resulting in parallel scaling characteristics that we consider optimal, for our experiments with up to 100 million degrees of freedom. Furthermore, AMG‐type methods are shown to be robust methods, allowing us to solve very ill‐conditioned systems resulting from strongly varying material properties over short distances in the model interior.

Multigrid methods with simple smoothers have been proven to be very successful for elliptic problems with no or only moderate convection. In the presence of dominant convection or anisotropies as it might appear in equations of computational fluid dynamics (e.g. in the Navier-Stokes equations), the convergence rate typically decreases. This is due to a weakened smoothing property as well as to problems in the coarse grid correction. In order to obtain a multigrid method that is robust for convection-dominated problems, we construct efficient smoothers that obtain their favorable properties through an appropriate ordering of the unknowns. We propose several ordering techniques that work on the graph associated with the (convective part of the) stiffness matrix. The ordering algorithms provide a numbering together with a block structure which can be used for block iterative methods. We provide numerical results for the Stokes equations with a convective term illustrating the improved convergence properties of the multigrid algorithm when applied with an appropriate ordering of the unknowns.

The method of discrete ordinates is commonly used to solve the Boltzmann radiation transport equation for applications ranging from simulations of fires to weapons effects. The equations are most efficiently solved by sweeping the radiation flux across the computational grid. For unstructured grids this poses several interesting challenges, particularly when implemented on distributed-memory parallel machines where the grid geometry is spread across processors. We describe a asynchronous, parallel, message-passing algorithm that performs sweeps simultaneously from many directions across unstructured grids. We identify key factors that limit the algorithm’s parallel scalability and discuss two enhancements we have made to the basic algorithm: one to prioritize the work within a processor’s subdomain and the other to better decompose the unstructured grid across processors. Performance results are give for the basic and enhanced algorithms implemented withi a radiation solver running on hundreds of processors of Sandia’s Intel Tflops machine and DEC-Alpha CPlant cluster.

SUMMARYDiscontinuous Galerkin methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids but are rather demanding in terms of computational resources. In order to improve their computational efficiency, a p‐multigrid solution strategy is here considered for the solution of the Navier–Stokes equations. In particular, a line smoother will be used to alleviate the effect of stretched grids on the convergence rate. The effectiveness and efficiency of the proposed approach in the solution of compressible shockless flow problems is demonstrated on the following 3D viscous test cases: a Delta Wing and a 3D streamlined body. Copyright © 2014 John Wiley & Sons, Ltd.

Three-dimensional aerodynamic computations on unstructured grids using a Newton–Krylov approach

A Newton–Krylov algorithm is presented for the compressible Navier–Stokes equations in three dimensions on unstructured grids. The algorithm uses a preconditioned matrix-free Krylov method to solve the linear system that arises in the Newton iterations. Incomplete factorization is used as the preconditioner, based on an approximate Jacobian matrix after the reverse Cuthill–McKee reordering of the unknowns. Several approximate viscous operators that involve only the nearest neighboring terms are studied to reduce the cost of preconditioning. The performance of the algorithm is demonstrated through numerical studies of the ONERA M6 wing and the DLR-F6 wing-body configuration. A ten-order-of-magnitude residual reduction for the wing and wing-body configurations can be obtained with a computing cost equivalent to 5500 and 8000 function evaluations, respectively, on grids with a half million nodes.

Numerical study on incomplete orthogonal factorization preconditioners

An efficient implicit high-order flux reconstruction/correction procedure via reconstruction method (FR/CPR) is developed to solve the compressible Navier-Stokes equation on unstructured hexahedral grids. The convergence rate of the proposed solver is greatly improved compared to Lower-upper Symmetry Gauss Seidel (LU-SGS) method. The restart GMRES (restarted Generalized Minimal Residual) is used to tackle the linear system by FR/CPR temporal and spatial discretization, and an automatic differential (AD) by operator overloading is proposed to construct the Jacobian matrix of the discretization system. The effects of automatic differential on the convergence rate of the developed implicit scheme have been investigated. Inviscid and viscous test cases are performed to evaluate the performance of the developed implicit solver.

Computation of 3D transonic viscous flows around the ONERA-M6 wing is performed by using the Harten-Yee TVD scheme with modification of geometrical treatment in order to seek better turbulence models. The models used here are the Jones--Launder (k-E )-model and the subgrid-scale model in the large eddy simulation (US), and for comparison the Baldwin-Lomax model. The diagonalization of flow equation system including a two-equation model necessary to perform the TVD scheme and the improvement accompanied by extending the LES to compressible flow problems are presented, and then the utilities of both models in compressible flow problems with shock waves have been j nvestigated. The (kE )-model and the LES work well in the large reverse flow problem by adjusting the coefficients compared with the Baldwin-Lomax model.

Three computer applications have been developed to solve inviscid compressible fluids problems using interactive computer graphics. The first application is a compressible flow calculator which solves for isentropic flow, normal shocks, and oblique shocks or centered expansions produced by two dimensional ramps. The second application couples the solutions generated by the first application to a more graphical presentation of the results to produce a desk top simulator of three compressible flow problems: 1) flow past a single compression ramp; 2) flow past two ramps in series; and 3) flow past two opposed ramps. The third application extends the results of the second to produce a design tool which solves for the flow through supersonic external or mixed compression inlets. The applications were originally developed to run on SGI or IBM workstations running GL graphics. They are currently being extended to solve additional types of flow problems and modified to operate on any X-based workstation.

GPU computing of compressible flow problems by a meshless method with space-filling curves

A hybrid mixed method for the compressible Navier–Stokes equations

An inverse design method for engine nacelles and wings