Implementation of Generalized Minimum Residual Krylov Subspace Method for Chemically Reacting Flows

MacLean, Matthew; White, Todd R.

doi:10.2514/6.2012-441

Cited by 5 publications

(2 citation statements)

References 30 publications

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“…In general, 10 subiterations may not be sufficiently general (e.g., Ref. [21]). An alternative strategy pursued in the StMG solver is to specify targets for reduction in the L 2 -norm of the residual of the linear system.…”

Section: Iiie Relaxation Schemementioning

confidence: 99%

Recent Advances in Agglomerated Multigrid

Nishikawa

Diskin

Thomas

et al. 2013

51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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We report recent advancements of the agglomerated multigrid methodology for complex flow simulations on fully unstructured grids. An agglomerated multigrid solver is applied to a wide range of test problems from simple two-dimensional geometries to realistic threedimensional configurations. The solver is evaluated against a single-grid solver and, in some cases, against a structured-grid multigrid solver. Grid and solver issues are identified and overcome, leading to significant improvements over single-grid solvers. I. Nomenclature R Global residual vector II. Introduction Multigrid techniques1, 2 are used to accelerate convergence of current Reynolds-Averaged Navier-Stokes (RANS) solvers for both steady and unsteady flow solutions, particularly for structured-grid applications. Lallemand et al. 3 and Smith 4 introduced Agglomerated Multigrid (AgMG) methods for unstructured-grid applications, and Mavriplis et al. [5][6][7][8] developed AgMG methods for large-scale unstructured-grid applications. The AgMG methods have been implemented in several practical unstructured grid codes and applied to configurations used for the Drag and High Lift Prediction Workshops.9, 10 The AgMG methods accelerated convergence in many instances but problems were encountered, such as grid-dependent convergence to smaller residual levels, discrepancies between single-grid and multigrid solutions, and convergence problems for cases at the edges of the flight envelope. In fact, analysis and systematic computations with these techniques showed convergence degradation on highly-refined grids even for simple model problems. To overcome the difficulty, we critically studied AgMG techniques 11, 12 for isotropic and highly-stretched grids, and developed quantitative analysis methods and computational techniques to achieve fast grid-independent convergence for a model equation representing laminar diffusion in the incompressible limit. It was found that consistent coarse-grid discretizations are essential for * Senior Research Scientist (hiro@nianet.org), National Institute of Aerospace, 100 Exploration Way, Hampton grid-independent multigrid convergence on isotropic grids, 11 and that prismatic cells, line-agglomeration, and line-implicit subiterations are essential for grid-independent multigrid convergence on highly-stretched grids. 12Building upon these fundamental studies, we extended and demonstrated these techniques for inviscid, viscous, and turbulent flow equations over complex geometries using a serial code 13 and a parallel code. 14The developed AgMG method has been shown to provide a significant speedup over a state-of-the-art singlegrid (SG) solver for the complex flows considered in the previous papers. However, several difficulties were encountered when the AgMG method was applied to some other problems. For example, multigrid convergence degraded in the presence of nearly-degenerate stencils on surface grids, grids with a polar singularity, disjoint grids in grid partitioning, etc. In this paper, we probe these and other issue...

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Section: Iiie Relaxation Schemementioning

confidence: 99%

Recent Advances in Agglomerated Multigrid

Nishikawa

Diskin

Thomas

et al. 2013

51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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“…In a review of factorisation methods for CFD applications, Saad et al [134] found that no single method was superior to the rest. For hypersonic flows, a zero-fill Incomplete Lower-Upper (ILU) factorisation of an approximate first-order flow Jacobian has shown good results for inviscid, laminar and turbulent flows [135][136][137]. An ILU factorisation of a matrix is a sparse approximation of the LU factorisation; zero-fill denotes that the sparsity pattern of the ILU factorisation is that of the original matrix.…”

Section: Preconditioningmentioning

confidence: 99%

Adjoint-based aerodynamic design optimisation in hypersonic flow

Damm¹

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The drive for improved flexibility and reusability in satellite launch systems has brought a resurgence in the popularity of scramjet-powered access-to-space launch vehicle concepts. A challenge in scramjet-powered accelerator vehicles is achieving positive thrust margins at the high-speed end of their flight envelopes. As a consequence, scramjet-powered vehicles typically rely on highly integrated airframe-engine configurations. However, due to the tight integration, the geometries and the flow physics are complex. An automated optimisation method seems an excellent candidate approach to explore the complex design space of hypersonic vehicles. This thesis focuses on the development and application of a particular optimisation method, adjoint-based optimisation, to aid in efficient aerodynamic design in hypersonic flows. Shape optimisation of scramjet-powered vehicles requires many design parameters to capture the geometric detail, and, since the flow physics is complex, the fidelity of Reynolds-Averaged Navier-Stokes (RANS) analyses is desirable. The combination of many design parameters and an expensive objective function evaluation drives the need for gradient evaluation methods that scale well with the number of design variables. An advantage of the adjoint method is that all shape sensitivities for an objective function are evaluated at the cost of only one flow solution and one adjoint solution. As a result of its efficiency, the adjoint method has become widely used for aircraft design, evolving to the design optimisation of full configurations. Despite the wide use of adjoint methods in aircraft optimisation, there has been very little application to hypersonic vehicle design. Several high-speed adjoint solvers have been reported in the literature, however, a majority of these works have only verified the adjoint sensitivities, few have followed on to demonstrate the method for use in design optimisation. The contribution of this work is the description and application of a discrete adjoint solver in high-speed compressible flow optimisation. What is unique in this work is a demonstration that complex-step differentiation works well to linearise a second-order spatially accurate unstructured RANS solver. In particular, the approach presented in this work utilises the k − ω turbulence model in high-speed ducted flow configurations. As part of this work, to provide flow analysis with a rapid turnaround , an unstructured steadystate RANS solver driven by a Jacobian-Free Newton-Krylov method was developed. Turbulence is modelled using the two-equation k − ω turbulence model. The Newton method is globalised by using the pseudo-transient approach. A restarted GMRES method is used to solve the system of linear equations arising when solving for the Newton steps. Evaluation of the matrix-vector products required in the GMRES algorithm is accomplished by Fréchet derivatives using imaginary perturbations in the complex plane. This is necessary to achieve robust convergence of the types of turbulent hypersonic flows...

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