David C. Gossard

A hierarchical data structure for representing assemblies: part I

Abstract A data structure representing assemblies in a database can be divided into two parts. The first part is the data structure used to store topological and geometric information on each component in an assembly. The second part is the data structure used to store information on how all the components in an assembly are connected. Any existing ‘boundary representation’ scheme can be used to represent each component. In this paper, a winged-edge data representation with extensions to handle multiply-connected faces is recommended. A tree structure using the concept of ‘virtual link’ is created to represent the relationships between the components in an assembly.

Quantitative analysis of cryo-EM density map segmentation by watershed and scale-space filtering, and fitting of structures by alignment to regions.

Cryo-electron microscopy produces 3D density maps of molecular machines, which consist of various molecular components such as proteins and RNA. Segmentation of individual components in such maps is a challenging task, and is mostly accomplished interactively. We present an approach based on the immersive watershed method and grouping of the resulting regions using progressively smoothed maps. The method requires only three parameters: the segmentation threshold, a smoothing step size, and the number of smoothing steps. We first apply the method to maps generated from molecular structures and use a quantitative metric to measure the segmentation accuracy. The method does not attain perfect accuracy, however it produces single or small groups of regions that roughly match individual proteins or subunits. We also present two methods for fitting of structures into density maps, based on aligning the structures with single regions or small groups of regions. The first method aligns centers and principal axes, whereas the second aligns centers and then rotates the structure to find the best fit. We describe both interactive and automated ways of using these two methods. Finally, we show segmentation and fitting results for several experimentally-obtained density maps.

Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing

This paper presents a new computational method for fully automated triangular mesh generation, consistently applicable to wire-frame, surface, solid, and nonmanifold geometries. The method, called bubble rrzeshing, is based on the observation that a pattern of tightly packed spheres mimics a Voronoi diagram, from which a set of well-shaped Delaunay triangles and tetrahedral can be created by connecting the centers of the spheres. Given a domain geometry and a node-spacing function, spheres are packed on geometric entities, namely, vertices, edges, faces, and volumes, in ascending order of dimension. Once the domain is filled with spheres, mesh nodes are placed at the centers of these spheres and are then connected by constrained Delaunay triangulation and tet rahedrizat ion. To obtain a closely packed configuration of spheres, the authors devised a technique for physically based mesh relaxation with adaptive population control, The process of mesh relaxation significantly reduces the number of ill-shaped triangles and tetrahedral.

Variational geometry in computer-aided design

A system has been developed which utilizes variational geometry in the design and modification of mechanical parts. Three-dimensional constraints between characteristic points are used to define an objects geometry. Modification of geometry is accomplished by alteration of one or more constraints. A matrix method is used to determine the shape of the part by simultaneous solution of constraint equations. A method for increasing the speed and efficiency of the solution procedure is described. The method uses the relationships between the geometry and constraints to minimize the number of equations and variables to be solved.

Constraint Management in Conceptual Design

The management of constraints during conceptual design is a non-trivial task. Of particular interest are “mathematical” constraints that relate to a design’s performance (i.e. function), physical laws it must obey (i.e. physics) and its geometrical and topological properties (i.e. form). Constraints are continually being added, deleted and modified as a design develops. The constraints are often numerous, complex and contradictory. The resulting constraint set may contain conflicting and/or unrealizable requirements.

Recognizing shape features in solid models

A procedure for defining and recognizing shape features 3-D solid models is presented in which a shape feature is defined as a single face or a set of continuous faces possessing certain characteristic facts in topology and geometry. The system automatically extracts these facts from an example shape feature interactively indicated by the user. The resulting representation of the shape feature can be interactively edited and parameterized. Graph matching accomplishes feature recognition. The system searches the solid model for B-rep subgraphs with the same characteristic facts as the shape feature to be recognized. When the system recognizes a shape feature, it removes the geometry associated with the feature from the original solid model to produce a simpler solid model. It then examines the simpler solid model to determine whether additional features have been revealed. The process repeats until no additional features are found.<<ETX>>

Solid model input through orthographic views

This paper describes the results of basic studies on procedures for creating solid models of component geometry from two-dimensional orthographic projections. An interactive graphic program was developed to allow the input of three orthographic views of a component geometry by digitizing from a drawing. The views may contain straight lines and circular arcs, solid or dashed. No restrictions are placed on the order or direction of lines and arcs in any view. Using an extension of the Wesley-Markowski procedure, the program constructs a three-dimensional solid model of the object. When the projections are ambiguous, multiple solid models are produced. The solid model may contain planar, cylindrical, conical, spherical and toroidal surfaces. Topological information of the solid model is stored in a winged edge structure. Geometric information is stored as vertex coordinates and surface equations. The procedure for 2D-3D conversion provides a powerful new method for manual input of solid models, a common interface to all turnkey graphics systems, and, properly integrated with existing technology for scanning of drawings, a powerful new method for acquisition of CAD/CAM data bases from existing drawings. The procedure is described, examples of typical input and output are shown, and possible extensions are discussed.

Multidimensional curve fitting to unorganized data points by nonlinear minimization

Abstract Many papers have addressed the problem of fitting curves to data points. However, most of the approaches are subject to a restriction that the data points must be ordered. The paper presents a method for generating a piecewise continuous parametric curve from a set of unordered and error-filled data points. The resulting curve not only provides a good fit to the original data but also possesses good fairness. Excluding the endpoints of the curve, none of the connectivity information needs to be specified, thus eliminating the necessity of an initial parameterization. The standard regularization method for univariate functions is modified for multidimensional parametric functions and results in a nonlinear minimization problem. Successive quadratic programming is applied to find the optimal solution. A physical model is also supplied to facilitate an intuitive understanding of the mathematical background.

Automatic triangular mesh generation of trimmed parametric surfaces for finite element analysis

This paper describes a new computational method for fully automated triangulation of the trimmed parametric surfaces used in finite element analysis. The method takes as input the domain geometry and a node-spacing function, and then generates a mesh, or a set of connected triangles, that satisfies basic requirements such as (1) precise control over node spacing or triangle size, (2) node placement that is compatible with domain boundaries, (3) generation of well-shaped triangles, and (4) continuous remeshing and local remeshing capabilities. The approach consists of two stages: placing initial nodes using recursive spatial subdivision, and relaxing the mesh by assuming the presence of proximity-based, repulsive/attractive internode forces and then performing dynamic simulation for a force-balancing configuration of nodes. In both stages, algorithms are developed in accordance with the observation that a pattern of tightly packed spheres mimics Voronoi polygons, from which well-shaped Delaunay triangles can be created by connecting the centers of the spheres.

Modeling the assembly of compliant, non-ideal parts

All manufactured parts and tooling have unavoidable variations from their nominal shapes. During assembly, compliant parts are further deformed by the relatively rigid assembly tooling. Lack of knowledge regarding variations and deformations often results in expensive problems. Since most current computer-aided design (CAD) systems today are based on ideally sized, ideally located, and rigid geometry, they are unable to model or predict the effects of variations in parts and tooling. This paper proposes a model for the assembly of compliant, non-ideal parts.