Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation
01 Sep 2004-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 42, Iss: 9, pp 1854-1862
TL;DR: In this paper, an adaptive mesh procedure that links to a CAD surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations for three-dimensional Euler problems.
Abstract: 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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TL;DR: Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined as mentioned in this paper, and 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.
Abstract: 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.
303 citations
07 Jan 2008
TL;DR: The results show that the adjoint-based mesh adaptation method used for controlling discretization errors in engineering functionals of nonsmooth problems is well-suited for the generation of aerodynamic databases of prescribed quality without user intervention.
Abstract: 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.
181 citations
TL;DR: An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier–Stokes equations is presented, constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system.
174 citations
25 Jun 2007
TL;DR: The approach relies on the solution of an adjoint equation and provides error estimates that can be used to both improve the accuracy of the functional and guide a mesh refinement procedure, a significant step in research toward automating the simulation process for flows in complex geometries.
Abstract: We present an approach for the computation of error estimates in output functionals such as lift or drag for an embedded-boundary Cartesian mesh method. The approach relies on the solution of an adjoint equation and provides error estimates that can be used to both improve the accuracy of the functional and guide a mesh refinement procedure. This is a significant step in our research toward automating the simulation process for flows in complex geometries. The accuracy of the approach is verified on an analytic model problem and validated against common results in the literature. The robustness of the approach is examined for two test cases in three dimensions, namely, an isolated wing in transonic flow and a canard-controlled missile in supersonic flow. The results demonstrate that the approach is tolerant of coarse initial meshes. A practical advantage of the approach is that the adaptive mesh refinement may be performed with a fixed surface triangulation. In all cases considered, the approach provided reliable estimates of the output functional on computationally affordable meshes.
144 citations
09 Jan 2006
98 citations
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TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
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Abstract: This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.
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TL;DR: An adaptive mesh procedure for improving the quality of steady state solutions of the Euler equations in two dimensions is described, implemented in conjunction with a finite element solution algorithm, using linear triangular elements, and an explicit time-stepping scheme.
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TL;DR: An implicit, Navier-Stokes solution algorithm is presented for the computation of turbulent flow on unstructured grids using an upwind algorithm and a backward-Euler time-stepping scheme.
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TL;DR: In this article, an anisotropic, unstructured grid adaptive method is presented for improving the accuracy of functional outputs of viscous, compressible flow simulations for general discretizations.
418 citations