Adaptive Skin Meshes Coarsening for Biomolecular Simulation

In this paper, we present efficient algorithms for generating hierarchical molecular skin meshes with decreasing size and guaranteed quality. Our algorithms generate a sequence of coarse meshes for both the surfaces and the bounded volumes. Each coarser surface mesh is adaptive to the surface curvature and maintains the topology of the skin surface with guaranteed mesh quality. The corresponding tetrahedral mesh is conforming to the interface surface mesh and contains high quality tetrahedra that decompose both the interior of the molecule and the surrounding region (enclosed in a sphere). Our hierarchical tetrahedral meshes have a number of advantages that will facilitate fast and accurate multigrid PDE solvers. Firstly, the quality of both the surface triangulations and tetrahedral meshes is guaranteed. Secondly, the interface in the tetrahedral mesh is an accurate approximation of the molecular boundary. In particular, all the boundary points lie on the skin surface. Thirdly, our meshes are Delaunay meshes. Finally, the meshes are adaptive to the geometry.     

基于黎曼度量的复杂参数曲面有限元网格生成方法

给出了三维空间的黎曼度量和曲面自身的黎曼度量相结合的三维复杂参数曲面自适应网格生成的改进波前推进算法.详细阐述了曲面参数域上任意一点的黎曼度量的计算和插值方法;采用可细化的栅格作为背景网格,在降低了程序实现的难度的同时提高了网格生成的速度;提出按层推进和按最短边推进相结合的方法,在保证边界网格质量的同时,提高曲面内部网格的质量.三维自适应黎曼度量的引入,提高了算法剖分复杂曲面的自适应性.算例表明,该算法对复杂曲面能够生成高质量的网格,而且整个算法具有很好的时间特性和可靠性.     

Molecular Surface Modeling and Meshing

An increasingly important part of quantum chemistry is devoted to molecular surfaces. To model such a surface, each constituting atom is idealized by a simple sphere. Surface mesh generation techniques are then used either for visualization or for simulation, where mesh quality has a strong influence on solution accuracy. This paper presents our new approach to generating quality meshes on arbitrary molecular surfaces (van der Waals, solvent-accessible or solvent-excluded surfaces). First, a boundary representation (B-rep) of the surface is obtained, i.e. a set of patches and the topological relations between them. Second, an appropriate parameterization and a metric map are computed for each patch. Third, meshes of the parametric domains are generated with respect to an induced metric map, using a combined advancing-front – generalized-Delaunay approach. Finally these meshes are mapped onto the entire surface. Several application examples illustrate various capabilities of our method. ID="A1" Correspondence and offprint requests to: P. Laug, GAMMA project, INRIA, BP 105, 78153 Rocquencourt Cedex, France. E-mail: Patrick.Laug@inria.fr     

A weighted mean approach to smooth parametric representation of polygon meshes

A method for representing genus-zero polygon meshes by smooth parametric surfaces is described. A surface is defined by a weighted sum of linear functions, each describing a polygon face in parametric form. Rational Gaussian blending functions that adapt to the size and shape of mesh faces are used as the weights. The proposed representation has a very high degree of continuity everywhere and provides a smoothness parameter that can be varied to produce surfaces at varying resolutions. It is shown that the representation facilitates geometry processing of meshes. The use of locally supported weight functions as an alternative to rational Gaussian weights is also discussed.     

Simple and effective variational optimization of surface and volume triangulations

Optimizing surface and volume triangulations is critical for many advanced numerical simulation applications. We present a variational approach for smoothing triangulated surface and volume meshes to improve their overall mesh qualities. Our method seeks to reduce the discrepancies between the actual elements and ideal reference elements by minimizing two energy functions based on conformal and isometric mappings. We derive simple, closed-form formulas for the values, gradients, and Hessians of these energy functions, which reveal important connections of our method with some well-known concepts and methods in mesh generation and surface parameterization. We then introduce a simple and efficient iterative algorithm for minimizing the energy functions, including a novel asynchronous step-size control scheme. We demonstrate the effectiveness of our method experimentally and compare it against Laplacian smoothing and some other mesh smoothing techniques.     

Polynomial surfaces interpolating arbitrary triangulations

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.     

Anisotropic quadrilateral mesh generation: An indirect approach

In this paper a new indirect approach is presented for anisotropic quadrilateral mesh generation based on discrete surfaces. The ability to generate grids automatically had a pervasive influence on many application areas in particularly in the field of Computational Fluid Dynamics. In spite of considerable advances in automatic grid generation there is still potential for better performance and higher element quality. The aim is to generate meshes with less elements which fit some anisotropy criterion to satisfy numerical accuracy while reducing processing times remarkably. The generation of high quality volume meshes using an advancing front algorithm relies heavily on a well designed surface mesh. For this reason this paper presents a new technique for the generation of high quality surface meshes containing a significantly reduced number of elements. This is achieved by creating quadrilateral meshes that include anisotropic elements along a source of anisotropy.     

GEOMPACK — a software package for the generation of meshes using geometric algorithms

We describe the mathematical software package GEOMPACK, which contains standard Fortran 77 routines for the generation of two-dimensional triangular and three-dimensional tetrahedral finite element meshes using efficient geometric algorithms. This package results from our research into mesh generation and geometric algorithms. It contains routines for constructing two- and three-dimensional Delaunay triangulations, decomposing a general polygonal region into simple or convex polygons, constructing the visibility polygon of a simple polygon from a viewpoint, and other geometric algorithms, from which our mesh generation method is built and others can be implemented. Our method generates meshes in polygonal or polyhedral regions specified by their boundary representation and possible interfaces between subregions.     

General Object Reconstruction Based on Simplex Meshes

In this paper, we propose a general tridimensional reconstruction algorithm of range and volumetric images, based on deformable simplex meshes. Simplex meshes are topologically dual of triangulations and have the advantage of permitting smooth deformations in a simple and efficient manner. Our reconstruction algorithm can handle surfaces without any restriction on their shape or topology. The different tasks performed during the reconstruction include the segmentation of given objects in the scene, the extrapolation of missing data, and the control of smoothness, density, and geometric quality of the reconstructed meshes. The reconstruction takes place in two stages. First, the initialization stage creates a simplex mesh in the vicinity of the data model either manually or using an automatic procedure. Then, after a few iterations, the mesh topology can be modified by creating holes or by increasing its genus. Finally, an iterative refinement algorithm decreases the distance of the mesh from the data while preserving high geometric and topological quality. Several reconstruction examples are provided with quantitative and qualitative results.     

不规则网格上的曲面设计方法

提出一种在不规则网格上构造曲面的方法.其基本思想是,通过均匀双三次B样条基函数的分解和子基函数的分类,将B样条曲面方法推广到任意四边形网格.给定一个任意四边形控制网格,首先对每个控制点构造一个基函数;所有控制点加权组合形成整体曲面.构造的曲面是分片双三次有理参数多项式曲面.此方法可以看成是均匀B样条曲面构造方法的扩展,如果控制网格是规则四边形网格,那么构造得到的曲面与均匀双三次B样条曲面是一致的.最后,实例证明此方法能够有效地构造曲面.     

基于黎曼度量参数曲面四边形网格生成方法及UG 实现

论文给出了基于黎曼度量的参数曲面网格生成的改进铺砖算法。阐述了曲 面自身的黎曼度量,并且运用黎曼度量计算二维参数域上单元节点的位置,从而使映射到三 维物理空间的四边形网格形状良好。文中对原有铺砖法相交处理进行了改进,在运用铺砖法 的同时调用UG-NX 强大的二次开发库函数获取相应的信息,直接在UG-NX 模型的表面生 成四边形网格。算例表明,该法能在曲面上生成质量好的网格。     

High quality refinable G‐splines for locally quad‐dominant meshes with T‐gons

Polyhedral modeling and re‐meshing algorithms use T‐junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor‐product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad‐dominant mesh as the control net of a hierarchical spline – and so associate a smooth surface with the mesh as in the popular tensor‐product spline paradigm. While G‐splines for multi‐sided holes or generalized subdivision can, in principle, convert quad‐dominant meshes with T‐junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n‐gons with T‐junctions (T‐gons) in unstructured quad‐dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad‐dominant mesh as quad‐meshes including τ‐nets, i.e. T‐gons surrounded by quads; and presents the first high‐quality G‐spline construction that can use τ‐nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T‐gons can be neighbors, separated by only one quad, both of T‐gons and of points where many quads meet. A τ‐net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi‐3 cap is not formally smooth, but achieves the same high‐quality highlight line distribution.     

High quality compatible triangulations

Compatible meshes are isomorphic meshings of the interiors of two polygons having a correspondence between their vertices. Compatible meshing may be used for constructing sweeps, suitable for finite element analysis, between two base polygons. They may also be used for meshing a given sequence of polygons forming a sweep. We present a method to compute compatible triangulations of planar polygons, sometimes requiring extra (Steiner) vertices. Experimental results show that for typical real-life inputs, the number of Steiner vertices introduced is very small. However, having a small number of Steiner vertices, these compatible triangulations are usually not of high quality, i.e. they do not have well-shaped triangles. We show how to increase the quality of these triangulations by adding Steiner vertices in a compatible manner, using remeshing and mesh smoothing techniques. The total scheme results in high-quality compatible meshes with a small number of triangles. These meshes may then be morphed to obtain the intermediate triangulated sections of a sweep, if needed.     

Surface Maps via Adaptive Triangulations

We present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus-0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete-continuous optimization algorithm performs both energy-based remeshing as well as global second-order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre-computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface-to-surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay-based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi-resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state-of-the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.     

Geometric accuracy analysis for discrete surface approximation

In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density. This paper gives explicit formulae to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formulae can be directly applied to design sampling density for data acquisitions and surface reconstructions. Furthermore, we prove that the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace–Beltrami operators on the meshes are also convergent to those on the smooth surfaces. These theoretical results lay down the foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.Practical algorithms for approximating surface Delaunay triangulations are introduced based on global conformal surface parameterizations and planar Delaunay triangulations. Thorough experiments are conducted to support the theoretical results.     

Template-based quadrilateral mesh generation from imaging data

This paper describes a novel template-based meshing approach for generating good quality quadrilateral meshes from 2D digital images. This approach builds upon an existing image-based mesh generation technique called Imeshp, which enables us to create a segmented triangle mesh from an image without the need for an image segmentation step. Our approach generates a quadrilateral mesh using an indirect scheme, which converts the segmented triangle mesh created by the initial steps of the Imesh technique into a quadrilateral one. The triangle-to-quadrilateral conversion makes use of template meshes of triangles. To ensure good element quality, the conversion step is followed by a smoothing step, which is based on a new optimization-based procedure. We show several examples of meshes generated by our approach, and present a thorough experimental evaluation of the quality of the meshes given as examples.     

A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion

Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions.     

Efficient tetrahedral mesh generation based on sampling optimization

We present a heuristic approach to tetrahedral mesh generation for implicit closed surfaces. It consists of a surface sampling step and a volume sampling step that both work in a unified optimization framework. First, high‐quality isotropic samplings as well as a triangular mesh on the surface are generated. Then uniform volume samplings are determined by optimizing the point distribution inside the closed surface domain. Finally, the tetrahedral mesh is easily obtained by constrained Delaunay triangulation. Experimental results show that the new method can generate ideal tetrahedral meshes for closed implicit surfaces efficiently that are Delaunay based. Our method has the advantage of high efficiency and nice performance at surface boundaries. Copyright © 2015 John Wiley & Sons, Ltd.     

All-hexahedral mesh generation via inside-out advancing front based on harmonic fields

The generation of hexahedral meshes is an open problem that has undergone significant research. This paper deals with a novel inside-out advancing front method to generate unstructured all-hexahedral meshes for given volumes. Two orthogonal harmonic fields, principal and radial harmonic fields, are generated to guide the inside-out advancing front process based on a few user interactions. Starting from an initial hexahedral mesh inside the given volume, we advance the boundary quadrilateral mesh along the streamlines of radial field and construct layers of hexahedral elements. To ensure high quality and uniform size of the hexahedral mesh, quadrilateral elements are decomposed in such a way that no non-hexahedral element is produced. For complex volume with branch structures, we segment the complex volume into simple sub-volumes that are suitable for our method. Experimental results show that our method generates high quality all-hexahedral meshes for the given volumes.     

Comparison of the meccano method with standard mesh generation techniques

The meccano method is a novel and promising mesh generation technique for simultaneously creating adaptive tetrahedral meshes and volume parameterizations of a complex solid. The method combines several former procedures: a mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. In this paper we present the main advantages of our method against other standard mesh generation techniques. We show that our method constructs meshes that can be locally refined using the Kossaczky bisection rule and maintaining a high mesh quality. Finally, we generate volume T-mesh for isogeometric analysis, based on the volume parameterization obtained by the method.