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Preliminary verification and validation of an efficient Euler solver for adaptively refined Cartesian meshes with embedded boundaries is presented. The parallel, multilevel method makes use of a new on-the-fly parallel domain decomposition strategy based upon the use of space-filling curves, and automatically generates a sequence of coarse meshes for processing by the multigrid smoother. The coarse mesh generation algorithm produces grids which completely cover the computational domain at every level in the mesh hierarchy. A series of examples on realistically complex three-dimensional configurations demonstrate that this new coarsening algorithm reliably achieves mesh coarsening ratios in excess of 7 on adaptively refined meshes. Numerical investigations of the scheme's local truncation error demonstrate an achieved order of accuracy between 1.82 and 1.88. Convergence results for the multigrid scheme are presented for both subsonic and transonic test cases and demonstrate Wcycle multigrid convergence rates between 0.84 and 0.94. Preliminary parallel scalability tests on both simple wing and complex complete aircraft geometries show a computational speedup of 52 using 64 processors with the run-time mesh partitioner.

This paper examines the behavior of flux and slope limiters on non-uniform grids in multiple dimensions. Many slope limiters in standard use do not preserve linear solutions on irregular grids impacting both accuracy and convergence. We rewrite some well-known limiters to highlight their underlying symmetry, and use this form to examine the properties of both traditional and novel limiter formulations on non-uniform meshes. A consistent method of handling stretched meshes is developed which is both linearity preserving for arbitrary mesh stretchings and reduces to common limiters on uniform meshes. In multiple dimensions we analyze the monotonicity region of the gradient vector and show that the multidimensional limiting problem may be cast as the solution of a linear progranzming problem. For some special cases we present a new directional limiting formulation that preserves linear solutions in multiple dimensions on irregular grids. Computational results using model problems and complex three-dimensional examples are presented, demonstrating accuracy, monotonicity and robustness.

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PUblic repoaang bJrdenf for this collection of inlormation is estimated to average I hour per resprose, including the tame for reviewing intristiOs, searching exisig data sourcs gathering and mantaining the data needed, and completing and reviewing the collecton of .fonmetion. ABSTRACT (Maximum 200 words)This report presents an assessment of a variety of reconstruction schemes on meshes with both quadrilateral and triangular tessellations. The investigations measure the order of accuracy, absolute error and convergence properties associated with each method. Linear reconstruction approaches using both Green-Gauss and least squares gradient estimation are evaluated against a structured MUSCL scheme wherever possible. In addition to examining the influence of polygon degree and reconstruction strategy, results with three limiters are examined and compared against unlimited results when feasible. The methods are applied on quadrilateral, right triangular, and equilateral triangular elements in order to facilitate an examination of the scheme behavior on a variety of element shapes. The numerical test cases include well known internal and external inviscid examples and also a supersonic vortex problem for which there exists a closed form solution to the 2-D compressible Euler equations. Such investigations indicate that the least squares gradient estimation provides significantly more reliable results on poor quality meshes. Furthermore, limiting only the face normal component of the gradient can significantly increase both accuracy and convergence while still preserving the integral cell average, and maintaining monoticity. The first order method performs poorly on stretched triangular meshes, and analysis shows that such meshes result in poorly aligned left and right states for the Riemann problem. The higher average valence of a vertex in the triangular tessellations does not appear to enhance the wave propagation, accuracy, or convergence properties of the method. Typically, quadrilateral elements provide superior or equivalent discrete solutions with approximately 50% fewer edges in the domain (2-D). However, on very poor quality meshes, the triangular elements routinely yield superior accuracy as a result of the trapezoidal quadrature of the Galerkin portion of the numerical flux function.

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.

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