Solution Adaptive Mesh Refinement Using Adjoint Error Analysis

A solution adaptive mesh refinement method for unstructured meshes is presented. The sensor is based on a smoothed estimate of the residual error, weighted by the the adjoint variables corresponding to the integral functional of most engineering interest. The difficulties in obtaining a smooth representation of the residual are discussed, and preliminary results are presented for two-dimensional inviscid subsonic and transonic testcases. Solution adaptive mesh refinement methods refine or derefine the mesh locally according to some error criterion. Many practical algorithms have been developed for the Euler and Navier-Stokes equations on a variety of grid types.3'4'11"13'15'24 However, adaptation sensors are most often still based on first or second derivatives of one or more flow variables. These simple sensors have various problems. Firstly, the selected sensor may not be a suitable choice for the flowfield or the integral functional one is interested in. E.g. a velocity difference sensor may exhibit large values due to the gradients of a boundary layer, even though it is sufficiently resolved, while the velocity differences in a more interesting vortical separation structure are less pronounced. Secondly, the fact that these sensors are not weighted, or are only crudely limited geometrically, leads to wasted refinement in areas where errors are present but not relevant. E.g. a shear layer shed from an airfoil will be convected downstream of its trailing edge. At one or two chord lengths from the trailing edge a poor resolution of the shear layer does not influence the flow around the airfoil significantly and need not be resolved. Thirdly, these sensors sometimes do not converge for discontinuities. E.g. a shock will become steeper and steeper with refinement and require all the refinement resources available, while there may be more important features to resolve. This can actually lead to seemingly grid-converged solutions that are incorrect.23 Recently, great progress has been made in using adjoint methods for a posteriori error estimation of hyperbolic, convection-diffusion, incompressible NavierStokes and compressible Euler equations.2'5'6'10'17"22 The use of adjoint methods is based on the fact that in many calculations it is the error in integral out

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