Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows

Error estimation and control are critical ingredients for improving the reliability of computational simulations Adjoint-based techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjoint-weighted residual method for error estimation Assumptions and approximations made in the calculations are discussed Presentation of the discrete and variational formulations enables a side-by-side comparison of recent work in output-error estimation using the finite volume method and the finite element method Techniques for adapting meshes using output-error indicators are also reviewed Recent adaptive results from a variety of laminar and Reynolds-averaged Navier-Stokes applications show the power of output-based adaptive methods for improving the robustness of computational fluid dynamics computations However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics

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Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

The objective of this article is to review some developments in the use of adjoint equations in hydrodynamic stability theory. Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined. It can be used to identify optimal perturbations, pinpoint the most receptive path to break down, select the most destabilizing base-flow defect in a nominally stable configuration, and map the structural sensitivity of an oscillator. We focus on two flow cases more closely: the noise-amplifying instability of a boundary layer and the global mode occurring in the wake of a cylinder. For both cases, the clever interpretation and use of direct and adjoint modes provide key insight into the process of the transition to turbulence.

Adjoint Equations in Stability Analysis

Recent improvements in an unstructured-grid method for large-scale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.

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Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes

This paper examines the robustness and efficiency of an adjoint-based mesh adaptation method for problems with complicated geometries. The method is used to drive cell refinement in an embedded-boundary Cartesian mesh approach for the solution of the three-dimensional Euler equations. Detailed studies of error distributions and the evolution of cell-wise error histograms with mesh refinement are used to formulate an adaptation strategy that minimizes the run-time of the flow simulation. The effectiveness of this methodology for controlling discretization errors in engineering functionals of nonsmooth problems is demonstrated using several test cases in two and three dimensions. The test cases include a model problem for sonic-boom applications and parametric studies of launch-vehicle configurations over a wide range of flight conditions. The results show that the method is well-suited for the generation of aerodynamic databases of prescribed quality without user intervention.

Adjoint-Based Adaptive Mesh Refinement for Complex Geometries

Engineering computational fluid dynamics (CFD) analysis and design applications focus on output functions (e.g., lift, drag). Errors in these output functions are generally unknown and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solving the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user specified error tolerance. This procedure for estimating and adapting to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a Computer Aided Design (CAD) surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with ten times as many grid points.

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Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation

Engineering computational fluid dynamics analysis and design applications often focus on output functions, such as lift or drag. Errors in these output functions are generally unknown, and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solution of the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user-specified error tolerance. This procedure for estimation and adaptation to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a CAD surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with many more grid points

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Michael Andrew Park

Papers

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation