Interval Methods for B-Rep Model Verification and Rectification

2000 ◽  
Author(s):  
Guoling Shen ◽  
Takis Sakkalis ◽  
Nicholas M. Patrikalakis
Keyword(s):  
Interval Arithmetic ◽  
Boundary Representation ◽  
Model Verification ◽  
Interval Methods ◽  
Topological Structures ◽  
Numerical Errors ◽  
Solid Models ◽  
Exact Boundary ◽  
Geometric Consistency ◽  
Model Conversion

Abstract Boundary representation (B-rep) models often have geometric specifications inconsistent with their topological structures due to numerical errors. In this paper, we verify the geometric consistency of B-rep models and evaluate existing inconsistencies of such models using interval arithmetic. Moreover, we convert conventional B-rep models into interval solid models to correct them. An interval solid is defined as a collection of non-degenerate boxes whose union covers the intended exact boundary and is guaranteed to be gap-free. An example illustrates our method for model conversion.

scholarly journals Intersections With Validated Error Bounds for Building Interval Solid Models

2005 ◽  
Author(s):  
Harish Mukundan ◽  
Kwang Hee Ko ◽  
Nicholas M. Patrikalakis
Keyword(s):  
Interval Arithmetic ◽  
Error Control ◽  
Model Space ◽  
Critical Issue ◽  
Boundary Representation ◽  
Solid Model ◽  
Efficient Computation ◽  
Solid Models ◽  
Bounding Surfaces ◽  
Marching Method

Interval arithmetic has been considered as a step forward to counter numerical robustness problem in geometric and solid modeling. The interval arithmetic boundary representation (B-rep) scheme was developed to tackle this problem. In constructing an interval B-rep solid, robust and efficient computation of intersections between the bounding surfaces of the solid is a critical issue. To address this problem, a marching method based on a validated interval ordinary differential equation (ODE) solver was proposed, motivated by its potential for the interval B-rep model construction. In this paper, we concentrate on the issue of error control in model space using the validated ODE solver, and further explain that the validated ODE solver can be used in the construction of an interval B-rep solid model using such an error control.

Topological and Geometric Consistency in Boundary Representations of Solid Models of Mechanical Components

1993 ◽  
Vol 115 (4) ◽  
pp. 762-769 ◽  
Author(s):  
A. G. Jablokow ◽  
J. J. Uicker ◽  
D. A. Turcic
Keyword(s):  
Data Exchange ◽  
Solid Modeling ◽  
Boundary Representation ◽  
Solid Model ◽  
Design Data ◽  
Solid Models ◽  
Geometric Consistency ◽  
Boundary Representations ◽  
Mechanical Components ◽  
Manufacturing Applications

This paper describes a method of verifying the consistency (i.e., agreement) between the topology and geometry of boundary representation (B-rep) solid models of mechanical components. This verification is well-suited for implementation as an algorithm and has been implemented as such in a polyhedral boundary representation solid modeling system (Jablokow, 1989). This technique and algorithm is important in the design of mechanical components for design documentation, integration with analysis and manufacturing applications, and design data exchange between solid modeling systems. Information regarding boundary representations has typically divided into the geometry and topology. It is important that the two are consistent for a valid solid model. In this work the genus of a solid model of an object is calculated topologically and geometrically and then compared to verify the consistency of the solid model. The genus of an object gives insight as to the geometric complexity of the object. This is equivalent to verifying the Gauss-Bonnet Theorem for the model, and is discussed in the paper.

Topological and Geometric Consistency in Boundary Representations of Solid Models

1990 ◽  
Author(s):  
Andrei G. Jablokow ◽  
John J. Uicker ◽  
David A. Turcic
Keyword(s):  
Solid Modeling ◽  
Boundary Representation ◽  
Solid Model ◽  
Geometric Complexity ◽  
And Topology ◽  
Solid Models ◽  
Geometric Consistency ◽  
Boundary Representations ◽  
Bonnet Theorem

Abstract This paper describes a method of verifying the consistency between the topology and geometry of boundary representation (B-rep) of solid models. This verification is well suited for implementation as an algorithm and has been implemented as such in a polyhedral boundary representation solid modeling system (Jablokow 1989). Information regarding boundary representations is typically divided into the geometry and topology. It is important that the two are consistent for a valid solid model. In this work the genus of an object is calculated topologically and geometrically and then compared to verify the consistency of the solid model. The genus of an object gives insight as to the geometric complexity of the object. This is equivalent to verifying the Gauss-Bonnet Theorem for the model, and is discussed in the paper.

scholarly journals An approach to automatic boundary segmentation of solid models using virtual topology: toward reconstruction of design features

2020 ◽  
Vol 7 (3) ◽  
pp. 367-385
Author(s):  
Yingzhong Zhang ◽  
Yufei Fu ◽  
Jia Jia ◽  
Xiaofang Luo
Keyword(s):  
Design Feature ◽  
Boundary Representation ◽  
Virtual Topology ◽  
Design Features ◽  
Complex Part ◽  
Novel Approach ◽  
Solid Models ◽  
Medium Level ◽  
Feature Based ◽  
Low Dimensional

Abstract Boundary segmentation of solid models is the geometric foundation to reconstruct design features. In this paper, based on the shape evolution analysis for the feature-based modeling process, a novel approach to the automatic boundary segmentation of solid models for reconstructing design features is proposed. The presented approach simulates the designer’s decomposing thinking on how to decompose an existing boundary representation model into a set of design features. First, the modeling traces of design features are formally represented as a set of feature vertex adjacent graphs that use low-dimensional vertex entities and their connection relations. Then, a set of Boolean segmentation loops is searched and extracted from the constructed feature vertex adjacent graphs, which segment the boundary of a solid model into a set of regions with different design feature semantics. In the search process, virtual topology operations are employed to simulate the topological changes resulting from Boolean operations in feature modeling processes. In addition, to realize effective search, search strategies and search algorithms are presented. The segmentation experiments and case study show that the presented approach is feasible and effective for the boundary segmentation of medium-level complex part models. The presented approach lays the foundation for the later reconstruction of design features.

Integrating Design Process Knowledge With CAD Models

2001 ◽  
Author(s):  
Erik E. Hayes ◽  
William C. Regli
Keyword(s):  
Design Process ◽  
Computer Aided Design ◽  
Search Space ◽  
Temporal Changes ◽  
Boundary Representation ◽  
Process Knowledge ◽  
Solid Models ◽  
Design And Manufacturing ◽  
Cad Models ◽  
Manufacturing Problems

Abstract Solid models are static entities, often defined by boundary representation models as sets of enclosing surfaces. Constructive Solid Geometry and feature-based computer-aided design environments create procedural descriptions of 3D objects in forms of history or CSG trees. These representations are temporally fixed, i.e., they describe the state of an object at a point in time. This paper describes a method to represent and capture temporal evolution of solid models — what we call model process history. We define process history to be all states of a model — the search space of design process. This paper presents a representational formalism we call model process graphs (MPGs). We use MPGs to integrate a model’s description with a model of temporal changes that occur during the design process. We believe that MPG representations can have valuable application for many design and manufacturing problems. The paper describes our preliminary results to use MPGs to (1) create a record of design process; (2) store process-based design rationale; (3) represent in-process shapes for machined artifacts. We anticipate that similar structures will find application in other design and manufacturing problems where important process knowledge is embodied by temporal changes occurring in model evolution.

Interval Method for Interrogation of Implicit Solid Models

2000 ◽  
Author(s):  
Jack Chang ◽  
Mark Ganter ◽  
Duane Storti
Keyword(s):  
Computer Aided Design ◽  
Boundary Representation ◽  
Free Form ◽  
Solid Models ◽  
Cad Cam ◽  
Implicit Modeling ◽  
Manufacturing Applications ◽  
Determine Surface Area ◽  
Implicit Solid ◽  
Aided Design

Abstract Computer-aided design/manufacturing (CAD/CAM) systems intended to support automated design and manufacturing applications such as shape generation and solid free-form fabrication (SFF) must provide not only methods for creating and editing models of objects to be manufactured, but also methods for interrogating the models. Interrogation refers to any process that derives information from the model. Typical interrogation tasks include determine surface area, volume or inertial properties, computing surface points and normals for rendering, and computing slice descriptions for SFF. While currently available commercial modeling systems generally employ a boundary representation (B-rep) implementation of solid modeling, research efforts have considered implicit modeling schemes as a potential source of improved robustness. Implicit implementations are available for a broad range of modeling operations, but interrogation operations have been widely considered too costly for many applications. This paper describes a method based on interval analysis for interrogating implicit solid models that aims at achieving both robustness and efficiency.

INTERVAL METHODS FOR RIGOROUS INVESTIGATIONS OF PERIODIC ORBITS

2001 ◽  
Vol 11 (09) ◽  
pp. 2427-2450 ◽  
Author(s):  
ZBIGNIEW GALIAS
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Discrete Time ◽  
Interval Arithmetic ◽  
Basin Of Attraction ◽  
Interval Methods ◽  
Henon Map ◽  
Invariant Part ◽  
The Basin Of Attraction ◽  
Discrete Time Dynamical Systems

In this paper, we investigate the possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical systems with special emphasis on chaotic systems. We show that methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n. We compare several interval methods for finding periodic orbits. We consider the interval Newton method and methods based on the Krawczyk operator and the Hansen–Sengupta operator. We also test the global versions of these three methods. We propose algorithms for computation of the invariant part and nonwandering part of a given set and for computation of the basin of attraction of stable periodic orbits, which allow reducing greatly the search space for periodic orbits. As examples we consider two-dimensional chaotic discrete-time dynamical systems, defined by the Hénon map and the Ikeda map, with the "standard" parameter values for which the chaotic behavior is observed. For both maps using the algorithms presented in this paper, we find very good approximation of the invariant part and the nonwandering part of the region enclosing the chaotic attractor observed numerically. For the Hénon map we find all cycles with period n ≤ 30 belonging to the trapping region. For the Ikeda map we find the basin of attraction of the stable fixed point and all periodic orbits with period n ≤ 15. For both systems using the number of short cycles, we estimate its topological entropy.

scholarly journals Method and Code for the Visualization of Multivariate Solids

2000 ◽  
Author(s):  
Karim Abdel-Malek ◽  
Jingzhou Yang
Keyword(s):  
Computational Complexity ◽  
Implicit Function Theorem ◽  
Computer Code ◽  
Boundary Representation ◽  
Implicit Function ◽  
Computer Aided Geometric Design ◽  
Exact Boundary ◽  
Computer Aided ◽  
Geometric Entity ◽  
Swept Volume

Abstract This paper is devoted to a method and computer code for the automatic visualization of multivariate solids. Example of a multivariate solids arise in computer aided geometric design when a geometric entity is swept in space, where the totality of points touched by the entity is called the swept volume and is characterized by an equation of many parameters. The method and code are presented in an integrated manner and are aimed at providing the reader with a replicable computer algorithm. The formulation for is based on the implicit function theorem; is applicable to the visualization of solids of any number of parameters; and produces the exact boundary representation. Considering the solid as a manifold (possibly with boundaries), it is shown that further stratification of the various submanifolds yields varieties that can be depicted in R3. A measure of the computational complexity is presented to give the reader a sense of robustness of the method. The code is developed using a symbolic manipulator and is presented with a number of examples.

Applications of Non-Manifold Topology

1995 ◽  
Author(s):  
William W. Charlesworth ◽  
David C. Anderson
Keyword(s):  
Automatic Generation ◽  
Boundary Representation ◽  
Surface Topology ◽  
Solid Model ◽  
Finite Element Analyses ◽  
Topological Constraints ◽  
Tool Paths ◽  
Solid Models ◽  
New Applications ◽  
Complicated Surface

Abstract It is widely recognized that a solid model based on a non-manifold boundary representation can have a more complicated surface topology than one based on a manifold boundary representation, but non-manifold topology has other capabilities that may be more valuable to the application developer. Non-manifold topology can be put to use in existing application areas in ways that differ significantly from the techniques developed for manifold modeling and it can be put to use in new applications that have not been satisfactorily solved by manifold topology. Several applications of non-manifold topology that would be difficult or impossible to implement using a purely manifold geometric modeler are illustrated: automatic formulation of finite element analyses from solid models, automatic generation of machining tool paths for 2½-dimensional pockets, and construction of geometric models using topological constraints. These applications demonstrate how a non-manifold model partitions the entire space in which an object is embedded, preserves elements of the model that would be discarded by conventional schemes, and permits the implementation of a common merge operation. All three applications have been implemented using a two dimensional non-manifold (non-1-manifold) geometric modeler.

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