Solving topological constraints for declarative families of objects
Introduction
Parametric and feature-based modelling systems are used to create object models with a number of parameters. The set of objects that can be obtained by varying the parameters of a model, is often referred to as a family of objects, or family for short. Virtually all systems store a history of modelling operations on a B-rep. We refer to these systems as history-based modellers and to their models as history-based models. Typically, the procedure for modelling a family of objects, is to initially model a single object, called the prototype or generic object. The history of modelling operations, used to construct the prototype object, is reevaluated with different parameter values to generate other members of the family.
History-based models, even though they are the de-facto standard for commercial modelling systems, are not ideal for representing families of objects. Bidarra and Bronsvoort [1] identify six major problems with history-based models, of which the most relevant, in the context of families of objects, are the persistent naming problem, the feature ordering problem, and the inability to maintain feature semantics properly. The persistent naming problem basically is the problem of identifying topologically equivalent entities in different members of a family. This is a prerequisite for maintaining family semantics. Previous research on families of objects has focused mainly on the persistent naming problem, see, for example, [2], [3], [4].
The feature ordering problem and maintenance of feature semantics have not received much attention. The order in which features are added to history-based models affects the resulting family of objects. This makes it difficult to design and edit family models. Also, the history-based modelling scheme does not have mechanisms for adequate specification and maintenance of semantics of features, and thus the resulting families have unclear semantics. In particular, topological properties cannot be adequately specified and maintained. These problems are discussed in more detail in Section 2.
We present a new model for families of objects, the Declarative Family of Objects Model (DFOM). In this model, both geometry and topology are specified declaratively. Declarative specifications state properties of objects, typically by means of constraints, but not how to construct those objects. Procedural specifications, on the other hand, specify how to construct objects, but there is no guarantee that any property generically holds for those objects. For modelling families of objects, however, the ability to specify generic properties is essential.
Declarative specification of geometry using constraints is common practice in current modelling systems, but topology is practically always specified in a procedural way [5]. Features in history-based systems correspond to set operations, or other operations that procedurally manipulate entities in a B-rep. These operations may change topological properties of the B-rep, even when this is not desired. In our new model, topological properties are declaratively specified by topological constraints that must hold for all members of a family. These constraints are imposed on topological entities that are either explicitly defined by features in the model, or implicitly by the interaction of features in the model. A topological constraint may specify, for example, that a face of a feature must be on the boundary of the model, or that a feature may not be split into disjoint volumes. To find an explicit topology, the system of topological constraints is solved.
The complete system of constraints to be solved thus consist of geometric and topological constraints. Although geometry and topology are closely tied, in our approach they can actually be treated separately. Geometrical constraint solving is a well-developed field and is not further addressed here. For an overview of recent work we refer to [6]. Solving topological constraints on feature models, to the best of our knowledge, has not been addressed before.
In Section 2 we discuss previous work on families of objects. Section 3 is dedicated to the Semantic Feature Model, which is the basis for the DFOM. The DFOM itself is presented in Section 4. Before solving, topological constraints are mapped to Boolean constraints, which is elaborated in Section 5. Section 6 discusses the technique used for solving Boolean constraints, and its performance. The DFOM has been implemented in a prototype feature modelling system; some aspects of this are presented in Section 7. Finally, we draw some conclusions in Section 8.
Section snippets
Previous work
Parametric and feature-based models can be thought of as dual–representation schemes [7], consisting of a parametric representation, e.g. a CSG representation, and a geometrical representation, e.g. a B-rep. This view has led to considering two types of families: the parameter-space family and the representation-space family. The parameter-space family is the set of all parameter vectors that correspond to valid models. The representation-space family corresponds to the set of all objects that
The semantic feature model
The Semantic Feature Model (SFM), introduced by Bidarra and Bronsvoort [1], is a declarative model, which allows feature semantics to be adequately specified and maintained. A SFM consists of a set of features and additional constraints between features. The shape and position of all features is determined by solving the constraints specified in the features (feature constraints), and the additional constraints between features (model constraints).
Each feature is instantiated from a feature
The declarative family of objects model
The Declarative Family of Objects Model (DFOM) is a generalization of the Semantic Model Family. The DFOM is similarly defined by a set of features, including feature constraints, and model constraints. The geometrical representation is also the cellular model. However, whether the cells of the CM contain material is not determined by feature precedence, but by solving topological constraints. Another difference with the Semantic Model Family is that the DFOM can represent both families of
Mapping topological constraints
A realization of a DFOM is a set of assignments, specifying for each cell in the combined CM whether the cell contains material, such that all topological constraints are satisfied. To find the realizations of a model, topological constraints are mapped to a system of Boolean constraints on the cells in the CM, and this system is then solved. The solutions of the system of Boolean constraints yield the realizations of the model.
All the cells in the combined CM are mapped to Boolean variables.
Boolean constraint solving
The Boolean constraint problem to be solved is the following: given a set of Boolean variables and a set of topological constraints , find an assignment , where for every , such that every constraint is satisfied. This problem is known as the Boolean satisfiability problem, or SAT problem, which is an NP-hard problem [14]. The SAT problem has been well studied, and search algorithms exist that can find solutions efficiently for many
Implementation
Our implementation of the DFOM is based on Spiff, a feature modelling system developed at Delft University of Technology. Spiff originally implemented the semantic feature modelling approach presented in [1]. Because the DFOM is based on concepts of semantic feature modelling, many parts of the modelling system could be re-used. In particular, the DFOM implementation uses the same feature class definitions and canonical shape models as the semantic feature model.
The most important modification
Conclusions
The Declarative Family of Objects Model (DFOM), presented here, allows families of objects to be specified using constraints on geometry and topology. It does not have the main problems associated with history-based models. In particular, it correctly maintains feature semantics, and is not affected by the feature ordering problem. Feature and constraint definitions of the Semantic Feature Model have been reimplemented in the DFOM. The ambiguity of feature dependency analysis is overcome by
Acknowledgements
H.A. van der Meiden’s work is supported by the Netherlands Organization for Scientific Research (NWO).
Hilderick A. van der Meiden is a Ph.D. student at the Faculty of Electrical Engineering, Mathematics and Computer Science of Delft University of Technology, The Netherlands. He graduated in computer science at the same university in 2004. His research interests include feature modelling, constraint solving and semantics of families of objects.
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Hilderick A. van der Meiden is a Ph.D. student at the Faculty of Electrical Engineering, Mathematics and Computer Science of Delft University of Technology, The Netherlands. He graduated in computer science at the same university in 2004. His research interests include feature modelling, constraint solving and semantics of families of objects.
Willem F. Bronsvoortis associate professor CAD/CAM at the Faculty of Electrical Engineering, Mathematics and Computer Science of Delft University of Technology, The Netherlands. He received his M.Sc. degree in computer science from the University of Groningen in 1978, and his Ph.D. degree from Delft University of Technology in 1990. His main research area is feature modelling, in particular semantic feature modelling, multiple-view feature modelling, freeform feature modelling, and mesh generation from feature models. He has published numerous papers in international journals, books and conference proceedings, is on the editorial board of several journals and has served as co-chair and member of many programme committees of conferences.