Modeling Features With Shape Algebras and Formal Logic

Objective: Develop a feature-based geometric reasoning system, with algebraic representations used in shape grammars and logic specifications of features.

Need(s) Addressed: Research in the development of design modeling systems has identified the need for evolutionary models which support dynamic schema modification. However, the development of current design systems does not easily support such a goal. They tend to be constructed in a bottom-up manner, with the design of low level data structures and operations first. This can be seen as a 'kit-of-parts' approach, and is often done in order to develop efficient operations for object manipulation. What this generally does is force the designer/user into a specific manner of representing and manipulating objects. Thus, the structure of a model must be decided at the start. Essentially this is akin to the philosophy of reductionism, which considers the universe to be composed of separate parts which, in various combinations, make up the whole.

The decision to classify and structure up front may preclude the possibility of other desirable forms and structures in the future. It is extremely difficult, if not impossible, to anticipate all possible ways in which one might wish to view or classify parts of a model. This often requires an unmanageable amount of information. The problems with this approach were among the causes of the failure of early CAD building modeling systems in the 1970s and early '80s, which often required the predetermination of all types of information of interest, and for this information to be stored in a single model.

On the other hand, the philosophy of holism considers the universe to be a whole rather than the sum of its parts. A system which forces no preconceived structure upon the user, but rather, allows one to find all sorts of emergent features and properties from within the whole, would be extremely desirable. This might enable an easier development path in a top-down fashion, from the abstract to the specific.

Technical Approach: The algebras of shape, as used in shape grammars, can support both holistic and reductionist views. By considering shapes as finite sets of elements which can carry fixed properties, a reductionist view is supported. The real power of such algebras, however, lies in the fact that the elements of a shape and their properties may be defined in such a manner as to enable the emergence of features which are not apparent in the initial formulation of a shape. In addition, the generality of their representations, their reliance upon a minimum of structure, and their use in combination can provide the semantic richness needed for design generation and analysis.

Representing shapes and features in first order predicate logic provides an easy way to develop complete computer systems for reasoning about designs. The use of logic provides a natural, intuitive method of generating precise definitions of parametric shapes and high level spatial relations. In addition, logic formulation serves as a powerful specification language for computer programs: classes of logic specifications can be easily transformed in working programs, with ongoing research seeking to improve these techniques.

The proposed work builds upon doctoral research recently completed, in which a model of shape, spatial relations and non-spatial properties was developed, constructed from first principles of geometry, topology and logic. A major effort here will consist of computer implementation of the modeling system described above. This involves the construction of a feature recognition system, with the features encoded from the dependency network of shape relations previously specified as logical expressions. The likely implementation platform will be a deductive database, which provides a simple mechanism for encoding facts and rules.

To date, the application domains tested have been architectural design and geographic information systems. Here, we intend to examine the applicability of the approach to the domain of mechanical engineered parts. This would involve the development of libraries of feature sets and a comparison to other work in feature-based modeling and reasoning.

A further look at non-geometric properties of shapes is also proposed. The existing model permits computations with labels (set-based properties) and weights (properties that can combine), which can provide an additional richness for defining designs. An important question is, can properties such as functionality be captured using these constructs? In the future, an additional research goal is an investigation of how current STEP standards for product modeling can be extended to incorporate such emergent feature based representations.