Boris Diskin

Discrete Adjoint-Based Design for Unsteady Turbulent Flows on Dynamic Overset Unstructured Grids

A discrete adjoint-based design methodology for unsteady turbulent flows on three-dimensional dynamic overset unstructured grids is formulated, implemented, and verified. The methodology supports both compressible and incompressible flows and is amenable to massively parallel computing environments. The approach provides a general framework for performing highly efficient and discretely consistent sensitivity analysis for problems involving arbitrary combinations of overset unstructured grids that may be static, undergoing rigid or deforming motions, or any combination thereof. General parent–child motions are also accommodated, and the accuracy of the implementation is established using an independent verification based on a complex-variable approach. The methodology is used to demonstrate aerodynamic optimizations of a wind-turbine geometry, a biologically inspired flapping wing, and a complex helicopter configuration subject to trimming constraints. The objective function for each problem is successful...

Towards Verification of Unstructured-Grid Solvers

New methodology for verification of finite-volume computational methods using unstructured grids is presented. The discretization order properties are studied in computational windows, easily constructed within a collection of grids or a single grid. Tests are performed within each window and address a combination of problem-, solution-, and discretization/grid-related features affecting discretization error convergence. The windows can be adjusted to isolate particular elements of the computational scheme, such as the interior discretization, the boundary discretization, or singularities. Studies can use traditional grid-refinement computations within a fixed window or downscaling, a recently-introduced technique in which computations are made within windows contracting toward a focal point of interest. Grids within the windows are constrained to be consistently refined, allowing a meaningful assessment of asymptotic error convergence on unstructured grids. Demonstrations of the method are shown, including a comparative accuracy assessment of commonly-used schemes on general mixed grids and the identification of local accuracy deterioration at boundary intersections. Recommendations to enable attainment of design-order discretization errors for large-scale computational simulations are given.

Sonic-Boom Mitigation Through Aircraft Design and Adjoint Methodology

This paper presents a novel approach to design of the supersonic aircraft outer mold line (OML) by optimizing the A-weighted loudness of sonic boom signature predicted on the ground. The optimization process uses the sensitivity information obtained by coupling the discrete adjoint formulations for the augmented Burgers Equation and Computational Fluid Dynamics (CFD) equations. This coupled formulation links the loudness of the ground boom signature to the aircraft geometry thus allowing ecient shape optimization for the purpose of minimizing the impact of loudness. The accuracy of the adjoint-based sensitivities is veried against sensitivities obtained using an independent complex-variable approach. The adjoint based optimization methodology is applied to a conguration previously optimized using alternative state of the art optimization methods and produces additional loudness reduction. The results of the optimizations are reported and discussed.

Distributed Relaxation Multigrid and Defect Correction Applied to the Compressible Navier-Stokes Equations

The distributed-relaxation multigrid and defect- correction methods are applied to the two- dimensional compressible Navier-Stokes equations. The formulation is intended for high Reynolds number applications and several applications are made at a laminar Reynolds number of 10,000. A staggered- grid arrangement of variables is used; the coupled pressure and internal energy equations are solved together with multigrid, requiring a block 2x2 matrix solution. Textbook multigrid efficiencies are attained for incompressible and slightly compressible simulations of the boundary layer on a flat plate. Textbook efficiencies are obtained for compressible simulations up to Mach numbers of 0.7 for a viscous wake simulation.

A Critical Study of Agglomerated Multigrid Methods for Diffusion

Agglomerated multigrid techniques used in unstructured-grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit. The studied target-grid discretizations and discretizations used on agglomerated grids are typical of current node-centered formulations. Agglomerated multigrid convergence rates are presented using a range of two- and three-dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids. Two agglomeration techniques are used within an overall topology-preserving agglomeration framework. The results show that multigrid with an inconsistent coarse-grid scheme using only the edge terms (also referred to in the literature as a thin-layer formulation) provides considerable speedup over single-grid methods but its convergence deteriorates on finer grids. Multigrid with a Galerkin coarse-grid discretization using piecewise-constant prolongation and a heuristic correction factor is slower and also grid-dependent. In contrast, grid-independent convergence rates are demonstrated for multigrid with consistent coarse-grid discretizations. Convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two-grid cycle are replaced by their idealized counterparts.

Recent Advances in Agglomerated Multigrid

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 three- dimensional 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.

Textbook multigrid efficiency for the incompressible Navier-Stokes equations: High Reynolds number wakes and boundary layers

Textbook multigrid efficiencies for high Reynolds number simulations based on the incompressible Navier-Stokes equations are attained for a model problem of flow past a finite flat plate. Elements of the Full Approximation Scheme multigrid algorithm, including distributed relaxation, defect correction, and boundary treatment, are presented for the three main physical aspects encountered: entering flow, wake flow, and boundary layer flow. Textbook efficiencies, i.e., reduction of algebraic errors below discretization errors in one full multigrid cycle, are attained for second order accurate simulations at a laminar Reynolds number of 10,000.

Half-Space Analysis of the Defect-Correction Method for Fromm Discretization of Convection

A novel, comprehensive, discrete, half-space analysis for the defect-correction method has been developed. This analysis plays the same role for nonelliptic-problem solvers as the full-space Fourier mode analysis plays for elliptic-problem solvers. Numerical simulations confirm the accuracy of the half-space analysis. The following important findings about the defect-correction method applied to the Fromm discretization of the two-dimensional convection equation are reported: nThe initial convergence rate of the defect-correction method is principally a function of the relative accuracy of the operators involved in the defect-correction iterations. The asymptotic convergence rate is about 0.5 per defect-correction iteration. If the driver operator is first-order accurate, then the initial convergence rates may be slow. The number of iterations required to get into the asymptotic convergence regime or/and to converge the algebraic error below the discretization-error level can be proportional to h -1/3. This h-dependent delay is a multidimensional phenomenon---it cannot be observed in one-dimensional problems, and it disappears in the case of close alignment between the grid and the convection equation characteristic. If the driver operator is second-order accurate, the defect-correction solver demonstrates the asymptotic convergence rate from the very beginning. Only one defect-correction iteration is required to converge algebraic error substantially below the discretization-error level.

Development and Application of Parallel Agglomerated Multigrid Methods for Complex Geometries

We extend previous serial developments of agglomerated multigrid techniques for fully unstructured grids in three dimensions to parallel computations. We demonstrate a robust parallel fully-coarsened agglomerated multigrid technique for the Euler, the Navier-Stokes, and the RANS equations for 3D complex geometries, incorporating the following key developments: consistent and stable coarse-grid discretizations, a hierarchical agglomeration scheme, and line-agglomeration/relaxation using prismatic-cell discretizations in the highly-stretched grid regions. A signicant speed-up in computer time over state-of-art large-scale computations is demonstrated.

Multigrid Solvers for Nonaligned Sonic Flows

We investigate an approach to the solution of nonelliptic equations on a rectangular grid. The multigrid algorithms presented here demonstrate the textbook multigrid efficiency even in the case that the equation characteristics do not align with the grid. To serve as a model problem, the two-dimensional (2D) and three-dimensional (3D) linearized sonic flow equations have been chosen. Efficient full-multigrid (FMG) solvers for the problems are demonstrated.