Wavelet BEM on molecular surfaces: parametrization and implementation

The present paper is dedicated to the rapid solution of boundary integral equations arising from solvation continuum models. We apply a fully discrete wavelet Galerkin scheme for the computation of the apparent surface charge on van der Waals or solvent accessible surfaces. The molecular surface is described in parametric form as a set of four-sided spherical patches. Each patch is exactly represented as rational Bézier surface. Numerical results are reported to illustrate the approach.     

Developable Bézier patches: properties and design

Geometric design of quadratic and cubic developable Bézier patches from two boundary curves is studied in this paper. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bézier control points. This set of equations allows us to infer important properties of developable Bézier patches that provide useful parameters and simplify the solution process for the patch design. With one boundary curve freely specified, five more degrees of freedom are available for a second boundary curve of the same degree. Various methods are introduced that fully utilize these five degrees of freedom for the design of general quadratic and cubic developable Bézier patches in 3D space. A more restricted generalized conical model or cylindrical model provides simple solutions for higher-order developable patches.     

Biquartic C-surface splines over irregular meshes

C¹-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C¹-surface splines in a previous paper (Peters, J SLAM J. Numer. Anal. Vol 32 No 2 (1993) 645–666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the control mesh for very sharp blends where three patches join. Biquartic C¹-surface splines do as well as possible by guaranteeing the property whenever more than three patches join and whenever the blend exceeds a certain small threshold.     

Wavelet BEM on molecular surfaces: solvent excluded surfaces

The present paper is concerned with the rapid solution of a boundary integral equation for the apparent surface charge which arises from solvation continuum models. In order to apply the wavelet Galerkin scheme the molecular surface needs to be represented as a parametric surface consisting of smooth four-sided patches. We develop an algorithm which decomposes a solvent excluded surface into a set of globally continuous four-sided NURBS patches. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.     

Generation of bi-monotone patches from quadrilateral mesh for reverse engineering

Thanks to recent improvements, computational methods can now be used to convert triangular meshes into quadrilateral meshes so that the quadrilateral elements capture well the principal curvature directional fields of surfaces and intrinsically have surface parametric values. In this study, a quadrilateral mesh generated using the mixed integer quadrangulation technique of Bommes et al. is used for input. We first segment a quadrilateral mesh into four-sided patches. The feature curves inside these patches are then detected and are constrained to act as the patch boundaries. Finally, the patch configuration is improved to generate large patches. The proposed method produces bi-monotone patches, which are appropriate for use in reverse engineering to capture the surface details of an object. A shape control parameter that can be adjusted by the user during the patch generation process is also provided to support the creation of patches with good bi-monotone shapes. This study mainly targets shape models of mechanical parts consisting of major smooth surfaces with feature curves between them.     

The Cut&Glue mesh generation algorithm

This paper describes a technique for generating quadrilateral finite element meshes on convex, four-sided patches, given an arbitrary number of elements along each side of the patch. The technique first generates a subdivision with the correct topological structure and smoothes the subdivision to obtain elements of acceptance shape for finite element analysis. The correct mesh topology is obtained from a regular subdivision by cutting rectangular corners of appropriate size and interconnecting the sides introduced by the cuts.The method can also be applied on three-dimensional patches producing meshes of brick elements with gradations in all directions.     

几何偏微分方程和离散曲面设计

使用若干个几何本质的曲率驱动的偏微分方程来构造符合指定C0或C1边界条件的三边曲面片和四边曲面片,这些方程的数值解由所涉及的微分几何算子的离散化来得到,微分几何算子的离散化则源于参数逼近.所构造的曲面片满足某些特定的几何偏微分方程,故具有理想的形状,将这些曲面片组装起来便构造出复杂的几何模型.通过反复的子分和演化,得到几何模型的多尺度表示.     

Generating strictly non-self-overlapping structured quadrilateral grids

In this paper, we present a BPM (Bézier patch mapping) algorithm which generates a strictly non-self-overlapping structured quadrilateral grid in a given four-sided planar region. Given four pieces of polynomial curves which enclose a simple region in the plane, the algorithm first constructs a Bézier patch which interpolates the four curves (as its four boundary curves), while the inner control points of its control grid remain unknown. In this paper, we show that, for the bijective condition to be satisfied, it is sufficient that the interior points satisfy a set of quadratic inequality equations. Exploiting this key result, we formulate the mapping algorithm as an optimization problem where the constraints are the bijective condition of the Bézier patch mapping (BPM), and the objective is to find out the best from all of the non-self-overlapping grids. Thus, commercial optimization solvers can be used to find the bijective mapping. If a solution to the optimization problems exists, then so does a solution to the mapping problem, and vice-versa. The BPM method is simple and intuitive, and some examples presented in this paper demonstrate its effectiveness.     

Gregory-type patches bounded by low degree intergral curves for G continuity

G² continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G² continuity to connect two Bézier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the Bézierpatch formulation. Gregory patch is a kind of free-form surface patch which is extended from Bézier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for intergral, rational, and NURBS boundary curves. In this paper, we propose some intergral boundary Gregorytype patches bounded by cubic and quartic curves for G² continuity.     

Nuclear norm regularized coding with local position-patch and nonlocal similarity for face hallucination

Learning-based face hallucination methods have received much attention and progress in past few decades. Specially, position-patch based approaches have been proposed to replace the probabilistic graph-based or manifold learning-based ones. As opposed to the existing patch based methods, where the input image patch matrix is converted into vectors before combination coefficients calculation, in this paper, we propose to directly use the image matrix based regression model for combination coefficients computation to preserve the essential structural information of the input patch matrix. For each input low-resolution (LR) patch matrix, its combination coefficients over the training image patch matrices at the same position can be computed. Then the corresponding high-resolution (HR) patch matrix can be obtained with the LR training patches replaced by the corresponding HR ones. The nonlocal self-similarities are finally utilized to further improve the hallucination performance. Various experimental results on standard face databases indicate that our proposed method outperforms some state-of-the-art algorithms in terms of both visual quantity and objective metrics.     

NURBS曲面G~1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的实用造型系统得到发展。对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一定光滑程度的合成曲面，满足局部设计和修改的目的。本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ１光滑拼接条件，得到了相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一个ＮＵＲＢＳ曲面，使其达到Ｇ１拼接是简单易行的。     

Approximate conversion of a rational boundary gregory patch to a nonuniform B-spline surface

A rational boundary Gregory patch is characterized by the facts that anyn-sided loop can be smoothly interpolated and that it can be smoothly connected to an adjacent patch. Thus, it is well-suited to interpolate complicated wire frames in shape modeling. Although a rational boundary Gregory patch can be exactly converted to a rational Bézier patch to enable the exchange of data, problems of high degree and singularity tend to arise as a result of conversion. This paper presents an algorithm that can approximately convert a rational boundary Gregory patch to a bicubic nonuniform B-spline surface. The approximating surface hasC ¹ continuity between its inner patches.     

The analysis of solids without mesh generation using trimmed patch boundary elements

The tools available to the mechanical engineer—for example, geometric modeling and computer-aided analysis—are individually quite powerful, but they are based on different geometric representations. Hence, they do not always work well together. In this paper an analysis method is presented that operates directly on the geometric modeling representation. Therefore, the time-consuming and error-prone procedure of generating a mesh is skipped. The method is based on boundary integral equations, but unlike previously published methods, the boundary elements aren-sided trimmed patches, the same patches that are used by modern geometric modelers to represent complex solids. The method is made practical by defining shape functions over the trimmed patches in such a way that the number of degrees of freedom can be controlled. This is done by using a concept called virtual nodes. The paper begins by deriving the trimmed patch boundary element. Then its properties are discussed in comparison with existing boundary element and finite element methods, and several examples are given.     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

G1连续几何偏微分方程Bézier曲面的构造

基于三角形和四边形网格上Laplace-Beltrami算子、高斯曲率和平均曲率的离散及其收敛性分析,提出了一种使用四阶几何流构造几何偏微分方程Bézier曲面的方法.使用该方法构造出的Bézier曲面既具有几何偏微分方程曲面的最优性质,同时又满足G1连续性.算法收敛性的数值实验表明该方法是有效的.     

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme—Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of self-overlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.     

From Computer Aided Design to wavelet BEM

This paper develops an interface between Computer Aided Design (CAD) and the wavelet Galerkin scheme for boundary integral equations. The key issue is an algorithm that decomposes a technical surface which was generated by CAD tools into a regular collection of parameterized four-sided patches. By a postprocessing step the global continuity of the parametrization is guaranteed. Numerical results are reported to illustrate the approach. In particular, the decomposition techniques are applied to real CAD data which come from IGES files.     

分类图斑概括的一种自适应方法——以遥感图像土地利用分类为例

从遥感图像监督分类结果到矢量对象的转换是遥感技术领域的一个瓶颈问题。提出了一种"分类图斑自适应概括"(Self-adaptive Generalization of Classified patch,SGCP)的方法,是针对这一问题的新尝试。SGCP能实现从破碎图斑到完整图像对象的自动转换,它由如下运算组成:(1)以形态学开启和形状系数分离道路与其他不透水表面;(2)以面积过滤和数学形态学操作去除噪声,以使图斑完整;(3)以递归凸残差回补简化图斑边界;(4)以膨胀和面积占优方法消除图斑裂隙;(5)以凸节点减少率评估图斑概括度,并同时以面积保持和分类精度保持评估概括精度;直至形成指定概括度的对象。概括运算的主要参数(如结构元素尺寸、递归次数、邻域窗口尺寸等)均由计算机自适应确定,同时预留部分用户调节参数,在自动概括的同时,允许人工干预概括程度。经Matlab仿真测试,该方法可以在保持分类精度与获取概括对象之间取得较好平衡。当图斑简化度上升22.9%时,面积平均变化仅为2.7%,分类精度仅平均下降0.72%。     

三角形域上C1连续的四次插值曲面

提出了一种在三角形域上构造C^1曲面的方法，该方法构造的曲面片由4个曲面加权平均产生，在三角形的边界上满足给定的边界曲线和一阶跨界导数．所构造的曲面可看作由一张基本曲面和三张过渡曲面构成．用三条曲线相交于一点且在交点处共面作为约束条件构造基本曲面，在三角形的内部具有较好形状和逼近精度．同边点法相比，文中方法产生的曲面形状更好；且该方法产生的曲面对四次多项式曲面是精确的，因而比Nielson的点边方法具有更高的插值精度．     

3维图像中边界曲面的分类追踪及抽取

3维图像分析中,边界曲面的检测与重构是一个非常重要的问题。已有的连续隐边界曲面的抽取及逼近计算技术存在着把某些零交叉曲面片错误地识别为边界曲面片的缺陷。为此,提出一个新的边界曲面的追踪及抽取的方法。该方法首先将包含边界曲面的全部立方体分为两类:包含一个连通零交叉曲面片的立方体叫第1类边缘立方体,包含两个及其以上不连通零交叉曲面片的立方体叫第2类边缘立方体;然后根据边界曲面的连续性连通性,便可追踪出两类边缘立方体;对于追踪出的第1类边缘立方体直接提取边界曲面片,对于追踪出的第2类边缘立方体的边界曲面片通过其相邻的第1类边缘立方体来提取。实验结果表明本文方法是可行有效的,而且可以有效地克服已有技术的缺陷。