Michael J. Aftosmis

Robust and Efficient Cartesian Mesh Generation for Component-Based Geometry

This work documents a new method for rapid and robust Cartesian mesh generation for component-based geometry. The new algorithm adopts a novel strategy that first intersects the components to extract the wetted surface before proceeding with volume mesh generation in a second phase. The intersection scheme is based on a robust geometry engine that uses adaptive precision arithmetic and automatically and consistently handles geometric degenerades with an algorithmic tie-breaking routine. The intersection procedure has worst-case computational complexity of O(N log N) and is demonstrated on test cases with up to 121 overlapping and intersecting components, including a variety of geometric degeneracies. The volume mesh generation takes the intersected surface triangulation as input and generates the mesh through cell division of an initially uniform coarse grid. In refining hexagonal cells to resolve the geometry, the new approach preserves the ability to directionally divide cells that are well aligned with local geometry. The mesh generation scheme has linear asymptotic complexity with memory requirements that total approximately 14-17 words/cell. The mesh generation speed is approximately 10 6 cells/minute on a typical engineering workstation

A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries

. Asrecently as five to ten years ago, mesh generation was fre-quently the most time consuming task in a typical CFD cycle.Adaptive Cartesian mesh generation methods are capable ofproducing millions of cells around complex geometries inminutes and have substantially removed this bottleneck.Why write yet another Euler solver? With robust mesh gener-ation largely in-hand, solution time resurfaces as the pacingitem in the CFD cycle. The current work attacks this issue bydesigning a scalable, accurate Cartesian solver with robustmultigrid convergence acceleration. Our primary motivationis to gain efficiency by capitalizing on the simplifications andspecialized data structures available on Cartesian grids. Sig-nificant savings in both CPU time and storage may be real-ized by taking advantage of the fact that cell faces arecoordinate aligned. In addition, higher-order methods withgood limiters are generally easier to design and perform morerobustly on uniform Cartesian meshes.Secondly, in any embedded-boundary Cartesian solver, thebody-intersectingcut-cellsdemand special attention. Thesecells can impose a substantial burden on the numerical dis-cretization since the arbitrary nature of geometric intersec-tion implies that a cut-cell may be orders of magnitudesmaller than its neighboring cells. This fact contrasts sharplywith the comparatively smooth meshes that are generallyfound on a good quality structured or unstructured mesh.Substantial research into these cut-cell issues have been stud-ied by references [9],[10],[6],[8], and [12] (among others)and we hope to take advantage of this investment.Thirdly, this work investigates a multigrid strategy that isspecialized for adaptively refined Cartesian meshes. In ourapproach, all grids in the multigrid hierarchy cover the entiredomain and include cells at many refinement levels. Thesmoother therefore iterates over the entire domain when it isinvoked on any grid in the hierarchy. In this respect, theapproach shares more with agglomeration or algebraic multi-grid techniques than with many other Cartesian or AMRmethods which iterate over only cells at the same level ofrefinement

Adjoint-Based Adaptive Mesh Refinement for Complex Geometries

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.

The Behavior of Linear Reconstruction Techniques on Unstructured Meshes

Abstract : 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. Unstructured, Upwind, Inviscid, Reconstruction, Limiters, Riemann problems.

Analysis of slope limiters on irregular grids

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 proper - ties 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 programming 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.

Adjoint Error Estimation and Adaptive Refinement for Embedded-Boundary Cartesian Meshes

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.

Applications of Space-Filling-Curves to Cartesian Methods for CFD

This paper presents a variety of novel uses of space-filling-curves (SFCs) for Cartesian mesh methods in CFD. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, many are applicable on general body-fitted meshes-both structured and unstructured. We demonstrate the use of single theta(N log N) SFC-based reordering to produce single-pass (theta(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 640 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 15% of ideal even with only around 50,000 cells in each sub-domain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with theta(M + N) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for control surface deflection or finite-difference-based gradient design methods.

Multilevel Error Estimation and Adaptive h-Refinement for Cartesian Meshes with Embedded Boundaries

This paper presents the development of a mesh adaptation module for a multilevel Cartesian solver. While the module allows mesh refinement to be driven by a variety of different refinement parameters, a central feature in its design is the incorporation of a multilevel error estimator based upon direct estimates of the local truncation error using tau-extrapolation. This error indicator exploits the fact that in regions of uniform Cartesian mesh, the spatial operator is exactly the same on the fine and coarse grids, and local truncation error estimates can be constructed by evaluating the residual on the coarse grid of the restricted solution from the fine grid. A new strategy for adaptive h-refinement is also developed to prevent errors in smooth regions of the flow from being masked by shocks and other discontinuous features. For certain classes of error histograms, this strategy is optimal for achieving equidistribution of the refinement parameters on hierarchical meshes, and therefore ensures grid converged solutions will be achieved for appropriately chosen refinement parameters. The robustness and accuracy of the adaptation module is demonstrated using both simple model problems and complex three dimensional examples using meshes with from 10(exp 6), to 10(exp 7) cells.

Adjoint-Based Adaptive Mesh Refinement for Sonic Boom Prediction

Output-driven mesh adaptation is used in conjunction with an embedded-boundary Cartesian meshing scheme for sonic-boom simulations. The approach automatically refines the volume mesh in order to minimize discretization errors in pressure signals located several body-lengths away from the surface geometry. Techniques and strategies used to improve accuracy of the propagated signal while decreasing total number of cells are described. Investigations include examination of cell aspect ratio and comparisons with manual mesh adaptation. The effectiveness of this approach is demonstrated in three dimensions using axisymmetric bodies, a lifting wing-body configuration, and both the F-5E and Shaped Sonic Boom Demonstration flight-test aircraft. Results are validated with available experimental data for a variety of signal forms. These comparisons show that accurate pressure signatures can be produced for three-dimensional geometry in just over an hour using a conventional desktop PC.

Inviscid Analysis of Extended-Formation Flight

Flying airplanes in extended formations, with separation distances of tens of wingspans, significantly improves safety while maintaining most of the fuel savings achieved in close formations. The p...