Steven E. Benzley

GENERALIZED 3‐D PAVING: AN AUTOMATED QUADRILATERAL SURFACE MESH GENERATION ALGORITHM

This paper discusses the extension of the paving algorithm for all quadrilateral mesh generation to arbitrary three-dimensional trimmed surfaces. Methods of calculating angles, projecting elements, and detecting collisions between paving boundaries, for general surfaces are presented. Extensions of the smoothing algorithms for three dimensions are set forth. Advances in the use of scalar sizing functions are presented. These functions can be used to better approximate internal mesh density from boundary densities and surface characteristics.

The spatial twist continuum: a connectivity based method for representing all-hexahedral finite element meshes

Abstract This paper introduces the spatial twist continuum (STC), a powerful extension of the dual of a hexahedral mesh. The STC captures the global connectivity constraints inherent in hexahedral meshing. We begin by describing the two-dimensional analog of the representation for quadrilateral meshes: The STC of a quadrilateral mesh is an arrangement of curves called chords. Chords pass through opposite quadrilateral edges and intersect at quadrilateral centroids. The power of the STC is displayed in the three-dimensional representation, where continuous surfaces called twist planes pass through layers of hexahedra. Pairs of twist planes intersect to form chords that pass through opposite faces of hexahedra. Triples of twist planes orthogonally intersect at the centroids of hexahedra. The continuity of the twist planes and chords, and how twist planes and chords twist through space, are the basis of the spatial twist continuum.

Automated Hexahedral Mesh Generation by Generalized Multiple Source to Multiple Target Sweeping

This paper presents an enhanced sweeping mesh generation algorithm. Traditional sweeping techniques create all hexahedral element meshes by projecting an existing single-surface mesh along a specified trajectory to a specified single target surface. The work reported here enhances the traditional method by creating all hexahedral element meshes between multiple source surfaces and multiple target surfaces. The new algorithm is based on an innovated application of Boolean operations. The core of this process is the rebuilding of the boundary representations of source surfaces by Boolean operations between the boundary loops of both target and source to incorporate a virtual geometry decomposition. Published in 2000 by John Wiley & Sons, Ltd.

Two and three-quarter dimensional meshing facilitators

This paper presents generated enhancements for robust ‘two and three-quarter dimensional meshing’, including: (1) automated interval assignment by integer programming for submapped surfaces and volumes, (2) surface submapping, and (3) volume submapping. An introduction to the simplex method, an optimization technique of integer programming, is presented. Simplification of complex geometry is required for the formulation of the integer programming problem. A method of ‘i-j unfolding’ is defined which explains how irregular geometry can be realigned into a simplified form that is suitable for submap interval assignment solutions. Also presented is the processes by which submapping eliminates the decomposition of surface geometry, through a pseudodecomposition process, producing suitable mapped meshes. The process of submapping involves the creation of ‘interpolated virtual edges’, user defined ‘vertex types’ and ‘i-j-k space’ traversals. The creation of ‘interpolated virtual edges’ is the method by which submapping automatically subdivides surface geometry. The ‘interpolated virtual edge’ is formulated according to an interpolation scheme using the node discretization of curves on the surface. User defined ‘vertex types’ allow direct user control of surface decomposition and interval assignment by modifying ‘i-j-k space’ traversals. Volume submapping takes the geometry decomposition to a higher level by using ‘mapped virtual surfaces’ to eliminate decomposition of complex volumes.

Adaptive sweeping techniques

This paper presents an adaptive approach to sweeping one-to-one and many-to-one geometry. The automatic decomposition of many-to-one geometry into one-to-one “blocks” and the selection of an appropriate node projection scheme are vital steps in the efficient generation of high-quality swept meshes. This paper identifies two node projection schemes which are used in tandem to robustly sweep each block of a one-to-one geometry. Methods are also presented for the characterization of one-to-one geometry and the automatic assignment of the most appropriate node projection scheme. These capabilities allow the sweeper to adapt to the requirements of the sweep block being processed. The identification of the two node projection schemes was made after an extensive analysis of existing schemes was completed. One of the node projection schemes implemented in this work, BoundaryError, was selected from traditional node placement algorithms. The second node projection scheme, SmartAffine, is an extension of simple affine transformations and is capable of efficiently sweeping geometry with source and/or target curvature while approximating the speed of a simple transform. These two schemes, when used in this adaptive setting, optimize mesh quality and the speed that swept meshes can be generated while minimizing required user interaction.

A Selective Approach to Conformal Refinement of Unstructured Hexahedral Finite Element Meshes

Hexahedral refinement increases the density of an all-hexahedral mesh in a specified region, improving numerical accuracy. Previous research using solely sheet refinement theory made the implementation computationally expensive and unable to effectively handle concave refinement regions and self-intersecting hex sheets. The Selective Approach method is a new procedure that combines two diverse methodologies to create an efficient and robust algorithm able to handle the above stated problems. These two refinement methods are: 1) element by element refinement and 2) directional refinement. In element by element refinement, the three inherent directions of a Hex are refined in one step using one of seven templates. Because of its computational superiority over directional refinement, but its inability to handle concavities, element by element refinement is used in all areas of the specified region except regions local to concavities. The directional refinement scheme refines the three inherent directions of a hexahedron separately on a hex by hex basis. This differs from sheet refinement which refines hexahedra using hex sheets. Directional refinement is able to correctly handle concave refinement regions. A ranking system and propagation scheme allow directional refinement to work within the confines of the Selective Approach Algorithm.

Localized coarsening of conforming all-hexahedral meshes

Finite element mesh adaptation methods can be used to improve the efficiency and accuracy of solutions to computational modeling problems. In many applications involving hexahedral meshes, localized modifications which preserve a conforming all-hexahedral mesh are desired. Effective hexahedral refinement methods that satisfy these criteria have recently become available; however, due to hexahedral mesh topology constraints, little progress has been made in the area of hexahedral coarsening. This paper presents a new method to locally coarsen conforming all-hexahedral meshes. The method works on both structured and unstructured meshes and is not based on undoing previous refinement. Building upon recent developments in quadrilateral coarsening, the method utilizes hexahedral sheet and column operations, including pillowing, column collapsing, and sheet extraction. A general algorithm for automated coarsening is presented and examples of models that have been coarsened with this new algorithm are shown. While results are promising, further work is needed to improve the automated process.

Automatic All Quadrilateral Mesh Adaption through Refinement and Coarsening

This work presents a new approach to conformal all-quadrilateral mesh adaptation. Most current quadrilateral adaptivity techniques rely on mesh refinement or a complete remesh of the domain. In contrast, we introduce a new method that incorporates both conformal refinement and coarsening strategies on an existing mesh of any density or configuration. Given a sizing function, this method modifies the mesh by combining template-based quadrilateral refinement methods with recent developments in localized quadrilateral coarsening and quality improvement into an automated mesh adaptation routine. Implementation details and examples are included.

A methodology for quadrilateral finite element mesh coarsening

High fidelity finite element modeling of continuum mechanics problems often requires using all quadrilateral or all hexahedral meshes. The efficiency of such models is often dependent upon the ability to adapt a mesh to the physics of the phenomena. Adapting a mesh requires the ability to both refine and/or coarsen the mesh. The algorithms available to refine and coarsen triangular and tetrahedral meshes are very robust and efficient. However, the ability to locally and conformally refine or coarsen all quadrilateral and all hexahedral meshes presents many difficulties. Some research has been done on localized conformal refinement of quadrilateral and hexahedral meshes. However, little work has been done on localized conformal coarsening of quadrilateral and hexahedral meshes. A general method which provides both localized conformal coarsening and refinement for quadrilateral meshes is presented in this paper. This method is based on restructuring the mesh with simplex manipulations to the dual of the mesh. In addition, this method appears to be extensible to hexahedral meshes in three dimensions.

Conformal Refinement and Coarsening of Unstructured Hexahedral Meshes

This paper describes recently developed procedures for local conformal refinement and coarsening of all-hexahedral unstructured meshes. Both refinement and coarsening procedures take advantage of properties found in the dual or “twist planes” of the mesh. A twist plane manifests itself as a conformal layer or sheet of hex elements within the global mesh. We suggest coarsening techniques that will identify and remove sheets to satisfy local mesh density criteria while not seriously degrading element quality after deletion. A two-dimensional local coarsening algorithm is introduced. We also explain local hexahedral refinement procedures that involve both the placement of new sheets, either between existing hex layers or within an individual layer. Hex elements earmarked for refinement may be defined to be as small as a single node or as large as a major group of existing elements. Combining both refinement and coarsening techniques allows for significant control over the density and quality of the resulting modified mesh.