Patrick M. Knupp

Algebraic Mesh Quality Metrics

Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. The singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. The condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Two combined metrics, shape-volume and shape-volume orientation, are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to nonsimplicial elements. A series of numerical tests verifies the theoretical properties of the metrics defined.

Algebraic mesh quality metrics for unstructured initial meshes

Explicit formulas for size, shape, skew, and combined mesh quality metrics are given for triangular, tetrahedral, quadrilateral, and hexahedral finite elements. The formulas are examples of abstractly defined metrics whose essential properties serve to guide the formulation of effective metrics. Effectiveness is demonstrated via comparisons with other metrics, contour plots, and analysis of particular meshes.

Winslow Smoothing on Two-Dimensional Unstructured Meshes

Abstract. The Winslow equations from structured elliptic grid generation are adapted to smoothing of two-dimensional unstructured meshes using a finite difference approach. We use a local mapping from a uniform N-valent logical mesh to a local physical subdomain. Taylor Series expansions are then applied to compute the derivatives which appear in the Winslow equations. The resulting algorithm for Winslow smoothing on unstructured triangular and quadrilateral meshes gives generally superior qualilty than traditional Laplacian smoothing, while retaining the resistance to mesh folding on structured quadrilateral meshes.

Efficient maximal poisson-disk sampling

We solve the problem of generating a uniform Poisson-disk sampling that is both maximal and unbiased over bounded non-convex domains. To our knowledge this is the first provably correct algorithm with time and space dependent only on the number of points produced. Our method has two phases, both based on classical dart-throwing. The first phase uses a background grid of square cells to rapidly create an unbiased, near-maximal covering of the domain. The second phase completes the maximal covering by calculating the connected components of the remaining uncovered voids, and by using their geometry to efficiently place unbiased samples that cover them. The second phase converges quickly, overcoming a common difficulty in dart-throwing methods. The deterministic memory is O(n) and the expected running time is O(n log n), where n is the output size, the number of points in the final sample. Our serial implementation verifies that the log n dependence is minor, and nearly O(n) performance for both time and memory is achieved in practice. We also present a parallel implementation on GPUs to demonstrate the parallel-friendly nature of our method, which achieves 2.4x the performance of our serial version.

Hexahedral and Tetrahedral Mesh Untangling

Abstract.We investigate a well-motivated mesh untangling objective function whose optimization automatically produces non-inverted elements when possible. Examples show the procedure is highly effective on tetrahedral meshes and on many hexahedral meshes constructed via mapping or sweeping algorithms.

A comparison of two optimization methods for mesh quality improvement

We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.

Jacobian-Weighted Elliptic Grid Generation

Variational grid generation techniques are used to derive and analyze a weighted elliptic grid generator that controls the Jacobian of the underlying transformation in a least-squares sense. The Euler--Lagrange equations for the area and volume generators are weighted forms of the well-known Laplace generator. Weights are restricted to the class of P-matrices to help achieve global invertibility of the map. Connecting the weights to the Jacobian of the map results in an intuitive means of controlling grid spacing, area, orthogonality, and grid-line directions. Examples are given on the unit square to demonstrate point attraction, local refinement, directional alignment, and adaption to a shock.

A Framework for Variational Grid Generation: Conditioning the Jacobian Matrix with Matrix Norms

Functionals for variational grid generation, in which the Jacobian matrix and matrix norms play a central role, are presented. All first-order geometric qualities of a structured grid (length, area, volume, angle between sides, aspect ratio, grid alignment) are determined by the Jacobian matrix of the induced mapping between the logical and physical regions. Grids with desirable quality can be generated by requiring the Jacobian matrix or the corresponding metric tensor to have certain properties. Functionals can be obtained by integrating over the logical or physical domain a power of the norm of a matrix derived from the Jacobian matrix. A judicious choice of this derived matrix results in a functional whose minima are mappings with desired features. Many well-known grid generators fit in this framework; for example, the harmonic generator arises when one drives the metric tensor toward the identity. The framework provides insight into what grid qualities these functionals describe. Three new directional control functionals are proposed and tested.

Manufactured Solution for Computational Fluid Dynamics Boundary Condition Verification

Order-of-accuracy verification is necessary to ensure that software correctly solves a given set of equations. One method for verifying the order of accuracy of a code is the method of manufactured solutions. This study documents the development of a manufactured solution that allows verification of not only the Euler, Navier-Stokes, and Reynolds-averaged Navier-Stokes equation sets, but also some of their associated boundary conditions: slip, no-slip (adiabatic and isothermal), and outflow (subsonic, supersonic, and mixed). To demonstrate the usefulness of this manufactured solution, it has been used for order-of-accuracy verification in a compressible computational fluid dynamics code. All of the results shown are on skewed, nonuniform, three-dimensional meshes. The manufactured solution and sequence of meshes are designed to allow asymptotic results to be obtained with reasonable computational cost. In addition to the order of accuracy of the full code for various equation sets and boundary conditions, the order of accuracy of code portions used to calculate solution gradients has been measured as well.

Efficient and good Delaunay meshes from random points

We present a Conforming Delaunay Triangulation (CDT) algorithm based on maximal Poisson disk sampling. Points are unbiased, meaning the probability of introducing a vertex in a disk-free subregion is proportional to its area, except in a neighborhood of the domain boundary. In contrast, Delaunay refinement CDT algorithms place points dependent on the geometry of empty circles in intermediate triangulations, usually near the circle centers. Unconstrained angles in our mesh are between 30? and 120?, matching some biased CDT methods. Points are placed on the boundary using a one-dimensional maximal Poisson disk sampling. Any triangulation method producing angles bounded away from 0? and 180? must have some bias near the domain boundary to avoid placing vertices infinitesimally close to the boundary.Random meshes are preferred for some simulations, such as fracture simulations where cracks must follow mesh edges, because deterministic meshes may introduce non-physical phenomena. An ensemble of random meshes aids simulation validation. Poisson-disk triangulations also avoid some graphics rendering artifacts, and have the blue-noise property.We mesh two-dimensional domains that may be non-convex with holes, required points, and multiple regions in contact. Our algorithm is also fast and uses little memory. We have recently developed a method for generating a maximal Poisson distribution of n output points, where n = ? ( Area / r 2 ) and r is the sampling radius. It takes O ( n ) memory and O ( n log n ) expected time; in practice the time is nearly linear. This, or a similar subroutine, generates our random points. Except for this subroutine, we provably use O ( n ) time and space. The subroutine gives the location of points in a square background mesh. Given this, the neighborhood of each point can be meshed independently in constant time. These features facilitate parallel and GPU implementations. Our implementation works well in practice as illustrated by several examples and comparison to Triangle. Highlights? Conforming Delaunay triangulation algorithm based on maximal Poisson-disk sampling. ? Angles between 30? and 120?. ? Two-dimensional non-convex domains with holes, planar straight-line graphs. ? O ( n ) space, E ( n log n ) time; efficient in practice. Background squares ensure all computations are local.