Heterogeneous object modeling through direct face neighborhood alteration

Two recent advances—the use of functionally gradient materials in parts and layered manufacturing technology—have brought to the forefront the need for design and fabrication methodologies for heterogeneous objects. However, current solid modeling systems, a core component of computer-aided design and fabrication tools, are typically purely geometry based, and only after the modeling of product geometry, can a part's non-geometric attributes such as material composition be modeled. This sequential order of modeling leads to unnecessary operations and over-segmented 3D regions during heterogeneous object modeling processes.

To enable an efficient design of heterogeneous objects, we propose a novel method, direct face neighborhood operation. This approach combines the geometry and material decisions into a common computational framework as opposed to separate and sequential operations in existing modeling systems. We present theories and algorithms for direction face neighborhood alteration, which enables direct alteration of face neighborhood before 3D regions are formed. This alteration is based on set membership classification (SMC) and region material semantics. The SMC is computationally enhanced by the usage of topological characteristics of heterogeneous objects. After the SMC, boundary evaluation is performed according to the altered face neighborhood. In comparison with other solid modeling methods, the direct face neighborhood alteration method is computationally effective, allows direct B-Rep operations, and is efficient for persistent region naming. A prototype system has been implemented to validate the method and some examples are presented.     

Sweeping体的拓扑结构与精确B—Rep反算策略

本文根据五种造型法则:平行扫法则、回转扫法则、箱体法则、异形体法则和曲面立体法则的离散化原理,提出了Sweeping体的拓扑结构的生成方法。讨论了离散化多面体边界的几何属性码和CSG-索引方法,以及基于这种几何属性码和CSG-索引的精确B-Rep反算策略。     

基于几何法表示的曲面实体的分割算法

分割是实体造型的重要步骤,而边界问题是影响分割算法效率和稳定性的主要因素。工程中常用的二次曲面有自封闭的特性,在相交时交线会出现退化和自交等情况,给拓扑表示和帝体重建带来困难,另外,边界重合也是实体分割时经常要遇到的问题,该文在二次曲面几何法表示的基础上,针对二次曲面相交时交线的特性,设计了合理的分割策略,提出了有效的分割算法,并对边界重合等问题做了很好的处理,同时用实例验证了本文算法的有效性。     

Toward a Meta-Model for Computational Engineering

Geometric modeling and finite element analysis have matured in recent decades. Both methods are used extensively in engineering design. However, the link between geometric modeling, physical modeling and finite element analysis is currently cumbersome, error-prone, and ad-hoc. Topological domain modeling provides the missing link. In this paper, we propose a combined topological modeling and finite element modeling method that allows not only topological modeling, but also promotes geometric and physical modeling, by providing a topological base space for the definition of finite element meshes, fields, and the definition and solution of boundary value problems. We call the method the Constructive Topological Domain Method (CTDM). In this method, Primitive Topological Domains (PTDs), each possessing a natural coordinate space, are combined in multiple n-dimensional Cartesian coordinate spaces, called charts, using generalizations of Boolean set operations, to create Constructed Topological Domains (CTDs) capable of acting as the base spaces of fiber bundles. The charts are glued together to create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, stress, and temperature. Finite element meshes may be defined upon each of the PTDs from which the CTD is constructed, enabling the definition and solution of boundary value problems, thus avoiding the difficult and messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed PTDs, is described. The boundary conditions may be specified as analytical or as finite element-based fields upon each of the PTDs. The CTDM appears to be a promising approach to robust mathematical and computational modeling of physical objects. Simple examples are presented. ID="A1"Correspondance and offprint requests to: W. Gerstle, Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: gerstle@unm.edu     

Determining interference between pairs of solids defined constructively in computer animation

This paper presents theory and implementation of a method for detecting interference between a pair of solid objects. Often at times, when performing simulations, two solids may unwittingly interpenetrate each other. The two components of the system presented in this paper are: (1) a surface representation method to model solid objects; and (2) a method for detecting interference. Body representation of a solid in this system is based upon enveloping each solid with surfaces (called positive entities). Most computer aided design (CAD) systems use solid modeling techniques to represent solid objects. Since most solid models use Boolean operations to model complex objects, a method is presented to envelop complex objects with parametric surfaces. A method for tracing intersection curves between two surfaces is also presented. Discontinuities on surfaces are defined as negative entitics in order to extend the method to complex solids. Determining interference is based upon a numerical algorithm for computing points of intersection between boundary curves and parametrized entities. The existence of segments of these curves inside the boundary of positive and negative entities is established by computing the circulation of a function around the boundary curve. Interference between two solids is then detected. No limitations are imposed on the convexity or simplicity of the boundary curves treated.     

An application of algebraic topology to solid modeling in molecular biology

A recurring problem in solid modeling, computer graphics, and molecular modeling is the computation of the intersection of two objects. A general solution to this problem is obtained by applying two ideas of algebraic topology: (1) a chain complex, and (2) a boundary formula for the intersection of two objects. A general data structure for a chain complex made up of piecewise polynomial cells is described, as are algorithms for connectivity, containment and intersection. The basic ideas of this work are abstract, topological, and for the most part, independent of the shape and dimensionality of the objects. An application to structural molecular biology is presented. The application identifies convex and concave features of protein surfaces.     

Free-form solid modeling using deformations

One of the most important problems of available solid modeling systems is that the range of shapes generated is limited. It is not easy to model objects with free-form surfaces in a conventional solid modeling system. Such objects can be defined arbitrarily, but then operations on them are not transparent and complications occur. A method for achieving free-form effect is to define regular objects or surfaces, then deform them. This keeps various properties of the model intact while achieving the required visual appearance. This paper discusses a number of geometric modeling techniques with deformations applied to them in attempts to combine various approaches developed so far.     

一种拓扑元素的命名和辨识方法

在基于历史的参数化特征造型系统中,拓扑元素的命名和特征模型重新生成时拓扑元素的正确辨识是一个关键问题。该文提出了一种拓扑元素命名和辨识的新方法,此方法基于两个新概念－－影响特征和影响面,通过检查被引用元素和被辨识元素间的影响特征集、影响面集以及非影响集是否存在子集关系进行拓扑元素的辨识,其主要创新点在于依据导致一拓扑元素产生和产生变化的特征和特征面的信息进行拓扑元素辨识。该方法的主要优点是在特征模型编辑后物体的拓扑结构发生变异时能够更合理地辨识出设计历史中的引用拓扑元素。     

语义特征造型的拓扑元素编码机制*

针对现有拓扑元素标志与编码方法都是面向基于历程的造型系统,无法完全满足与历程无关语义特征造型的需要这一问题,提出一种以特征为基础的拓扑元素标志方法,并归纳出了一种统一格式的拓扑元素编码方法。当拓扑元素发生分裂时,给出了区分分裂面和分裂边的方法及编码格式;根据模型修改前后拓扑元素的不同变化情况,提出虚拓扑元素和子边的概念及编码格式,通过保留模型中拓扑元素间的拓扑关系,实现与历程无关的模型修改操作,并将该方法在自主开发的语义特征造型系统(HUST-CAID)中加以实现。     

Isogeometric segmentation: Construction of cutting surfaces

The objective of Isogeometric Segmentation is to generate a decomposition of a solid, given in boundary representation, into a collection of a relatively small number of base solids, which can easily be subdivided into topological hexahedra. This can be achieved by repeatedly splitting the solid. In each splitting step, one chooses a cutting loop, which is a cycle of curves around the boundary of the solid, and constructs a cutting surface that splits the solid into two simpler ones. When only hexahedra or pre-defined base solids are left this process terminates.The construction of the cutting surface must ensure that two essential properties are fulfilled: the boundary curves of the surface interpolate the previously constructed cutting loop and the surface neither intersects itself nor the boundary of the solid. A novel method for generating the cutting surface is presented in this paper. The method combines two steps: First we generate an implicit guiding surface, which is subsequently approximated by a trimmed spline surface in the second step.     

Iso‐geometric Finite Element Analysis Based on Catmull‐Clark : ubdivision Solids

We present a volumetric iso‐geometric finite element analysis based on Catmull‐Clark solids. This concept allows one to use the same representation for the modeling, the physical simulation, and the visualization, which optimizes the design process and narrows the gap between CAD and CAE. In our method the boundary of the solid model is a Catmull‐Clark surface with optional corners and creases to support the modeling phase. The crucial point in the simulation phase is the need to perform efficient integration for the elements. We propose a method similar to the standard subdivision surface evaluation technique, such that numerical quadrature can be used. Experiments show that our approach converges faster than methods based on tri‐linear and tri‐quadratic elements. However, the topological structure of Catmull‐Clark elements is as simple as the structure of linear elements. Furthermore, the Catmull‐Clark elements we use are C²‐continuous on the boundary and in the interior except for irregular vertices and edges.     

A comparison of methods for representing topological relationships

In the field of spatial information systems, a primary need is to develop a sound theory of topological relationships between spatial objects. A category of formal methods for representing topological relationships is based on point-set theory. In this paper, a high level calculus-based method is compared with such point-set methods. It is shown that the calculus-based method is able to distinguish among finer topological configurations than most of the point-set methods. The advantages of the calculus-based method are the direct use in a calculus-based spatial query language and the capability of representing topological relationships among a significant set of spatial objects by means of only five relationship names and two boundary operators.     

AN ALGORITHM ACHIEVING VALID BOOLEAN OPERATION RESULTS ON POLYHEDRAL SOLIDS

A polyhedral solid modeler that operates on boundary representations (B-reps) of ob-jects must derive topological information from numerical data.Due to finite precision of the com-puter,unavoidable numerical calculation errors may result in ambiguous or contradictory decisionof topology.These effects cause existing polyhedral modelers to fail when confronted with objectsthat nearly align or barely intersect.Based on analysing the reasons which cause the failure of Boolean operation to fail,this paperdescribes an algorithm using solid integrity to carefully design each step of Boolean operation,sothat valid polyhedral modeling results may be achieved.     

A level set based finite element algorithm for the simulation of dendritic growth

Dendritic growth is a nonlinear process, which falls into the category of self-organizing pattern formation phenomena. It is of great practical importance, since it appears frequently and, in the case of alloys, affects the engineering properties of the resulting solid. We describe a new finite element algorithm for the two–dimensional Stefan problem, where the free boundary is represented as a level set. This allows to handle topological changes of the free boundary. The accuracy of the method is verified and several numerical simulations, including topological changes of the free boundary, are presented.     

一种模糊区域的拓扑关系模型

空间区域拓扑关系建模是空间推理、地理信息系统(GIS)和计算机视觉等领域一个非常重要的主题，模糊区域的拓扑关系建模正日益受到相关领域研究者的重视，在分析现有模型的基础上，提出了一种模糊区域的拓扑关系模型，该模型利用模糊集来表示模糊区域，通过三个谓词的真值来判断区域间的拓扑关系，将分明区域作为特例统一处理，根据谓词的多种真值能够实现多层次上的拓扑关系分析．     

面向智能交通系统的仿真路网快速建模方法

提出了一种面向智能交通系统的仿真路网模型,与传统路网模型相比,增强了几 何表达和拓扑表达能力,同时其上可附着丰富的交通属性数据,能够满足精细化智能交通仿真 的需求。在此基础上,研究了一种路网数据迁移方法,从现有电子地图快速提取并构建面向智 能交通系统的道路路网。通过对几何精确性、拓扑完备性和建模高效性的测试表明,该路网模 型及其建模方法,提高了交通仿真基础路网建模的效率,降低了建模成本,为智能交通仿真的 广泛应用提供了高效、可靠的道路数据来源。     

Adaptive hexahedral mesh generation and quality optimization for solid models with thin features using a grid-based method

This paper presented an automatic inside-out grid-based hexahedral element mesh generation algorithm for various types of solid models. For the thin features with small thickness of the geometric model, corresponding treatment methods were given for successfully implementing each meshing step, containing the techniques for adaptive refinement, boundary match, topological optimization and local refinement. In order to realize the reasonable identification of refinement regions and resolve the expansion problem of refinement information fields, a thin-feature criterion and a supplementary criterion were proposed aiming at thin features of the geometry. To implement accurate boundary match for thin features, ten basic types and five complementary types of facet configurations were established, and a priority-node identification method was proposed additionally. Three topological optimization modes were newly proposed to improve the topological connections of the boundary mesh in thin features. Local refinement techniques were also built to refine the thin features of solid models. Finally, several examples were provided to demonstrate the effectiveness and reliability of the proposed algorithms.     

一种拓扑元素的标识和识别方法

给出了一种拓扑元素的标识方法，并以此方法为基础提出一种称为TRG的数据结构，同时给出了拓扑面、边、点的识别算法。系统通过TRG记录实体模型的构造历史，跟踪实体重建前后拓扑元素之间的关系。该方法能很好地记录实体的造型历史，再现原设计者的设计意图，给出的算法能正确识别面、边、点。