Catálogo de publicaciones - libros

Polygonal meshes are used to model smooth surfaces in many applications. Often these meshes need to be remeshed for improving the quality, density or gradedness. We apply the Delaunay refinement paradigm to design a provable algorithm for isotropic remeshing of a polygonal mesh that approximates a smooth surface. The proofs provide new insights and our experimental results corroborate the theory.

We develop an automated all-hex meshing strategy for bifurcation geometries arising in subject-specific computational hemodynamics modeling. The key components of our approach are the use of a natural coordinate system, derived from solutions to Laplace’s equation, that follows the tubular vessels (arteries, veins, or grafts) and the use of a tripartitioned-based mesh topology that leads to balanced high-quality meshes in each of the branches. The method is designed for situations where the required number of hexahedral elements is relatively small (∼ 1000-4000), as is the case when spectral elements are employed in simulations at transitional Reynolds numbers or when finite elements are employed in viscous dominated regimes.

This paper describes an interior surface generation method and a strategy for all-hexahedral mesh generation. It is well known that a solid homeomorphic to a ball with even number of quadrilaterals bounding the surface should be able to be partitioned into a compatible hex mesh, where each associated hex element corresponds to the intersection point of three interior surfaces. However, no practical interior surface generation method has been revealed yet for generating hexahedral meshes of quadrilateral- bounded volumes. We have deduced that a simple interior surface with at most one pair of self-intersecting points can be generated as an orientable regular homotopy, or more definitively a sweep, if the self-intersecting point types are identical, while the surface can be generated as a non-orientable one (i.e. a Möbius band) if the self-intersecting point types are distinct. A complex interior surface can be composed of simple interior surfaces generated sequentially from adjacent circuits, i.e. non-self-intersecting partial dual cycles partitioned at a self-intersecting point. We demonstrate an arrangement of interior surfaces for Schneiders’ open problem, and show that for our interior surface arrangement Schneiders’ pyramid can be filled with 146 hexahedral elements. We also discuss a possible strategy for practical hexahedral mesh generation.

Unconstrained Plastering is a new algorithm with the goal of generating a conformal all-hexahedral mesh on any solid geometry assembly. Paving[1] has proven reliable for quadrilateral meshing on arbitrary surfaces. However, the 3D corollary, Plastering [2][3][4][5], is unable to resolve the unmeshed center voids due to being over-constrained by a pre-existing boundary mesh. Unconstrained Plastering attempts to leverage the benefits of Paving and Plastering, without the over-constrained nature of Plastering. Unconstrained Plastering uses advancing fronts to inwardly project unconstrained hexahedral layers from an unmeshed boundary. Only when three layers cross, is a hex element formed. Resolving the final voids is easier since closely spaced, randomly oriented quadrilaterals do not over-constrain the problem. Implementation has begun on Unconstrained Plastering, however, proof of its reliability is still forthcoming.

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.

This paper describes an approach to smooth the surface and improve the quality of quadrilateral/hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For hexahedral meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of quadrilateral/hexahedral meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations.

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes certain objective function. This objective function is derived from an algebraic quality measures of the local mesh (the set of triangles connected to the adjustable or free node ). The optimization is done in the parametric mesh , where the presence of barriers in the objective function maintains the free node inside the feasible region . In this way, the original problem on the surface is transformed into a two-dimensional one on the parametric space . In our case, the parametric space is a plane, chosen in terms of the local mesh, in such a way that this mesh can be optimally projected performing a valid mesh, that is, without inverted elements. In order to show the efficiency of this smoothing procedure, its application is presented.

In this paper, we present simple and efficient array-based mesh data structures, including a compact representation of the half-edge data structure for surface meshes, and its generalization—a half-face data structure —for volume meshes. These array-based structures provide comprehensive and efficient support for querying incidence, adjacency, and boundary classification, but require substantially less memory than pointer-based mesh representations. In addition, they are easy to implement in traditional programming languages (such as in C or Fortran 90) and convenient to exchange across different software packages or different storage media. In a parallel setting, they also support partitioned meshes and hence are particularly appealing for large-scale scientific and engineering applications. We demonstrate the construction and usage of these data structures for various operations, and compare their space and time complexities with alternative structures.

We develop a theoretical framework for constructing guaranteed quality Delaunay meshes in parallel for general two-dimensional geometries. This paper presents a new approach for constructing graded meshes, i.e., meshes with element size controlled by a user-defined criterion. The sequential Delaunay refinement algorithms are based on inserting points at the circumcenters of triangles of poor quality or unacceptable size. We call two points Delaunay-independent if they can be inserted concurrently without destroying the conformity and Delaunay properties of the mesh. The contribution of this paper is three-fold. First, we present a number of local conditions of point Delaunay-independence, which do not rely on any global mesh metrics. Our sufficient conditions of point Delaunay-independence allow to select points for concurrent insertion in such a way that the standard sequential guaranteed quality Delaunay refinement procedures can be applied in parallel to attain the required element quality constraints. Second, we prove that a quadtree, constructed in a specific way, can be used to guide the parallel refinement, so that the points, simultaneously inserted in multiple leaves, are Delaunay-independent. Third, by experimental comparison with the well-known guaranteed quality sequential meshing software, we show that our method does not lead to overrefinement, while matching its quality and allowing for code re-use.