This paper defines a number of general operations that accept arbitrary sets of values for two variables and general relations among three variables and generate a variety of third sets that are useful in design. Although the operations are defined without respect to mathematical or engineering domain, computing these operations depends on the specific mathematical domain, and algorithms are available for only a few domains. Appropriate software could make this complexity transparent to the designer, allowing the same conceptual operations to be used in many contexts. The paper proves a number of useful characteristics of the operations and offers examples of their potential use in design.
The Labeled Interval Calculus (LIC) is a formalism for reasoning about sets of design possibilities. Examples include toleranced objects, abstract descriptions involving many possible instantiations, and varying operating conditions. It has been successful in a “mechanical design compiler”, which accepts schematics and specifications and returns catalog numbers for optimal implementations. The LIC at present operates on monotonic algebraic equations and intervals of real values, but it now appears possible to generalize it to address arbitrary types of mathematical sets and relationships. The resulting family of formalisms is expected to be useful in design by feature and other design programs.
The “labeled interval calculus” is a formal system that performs quantitative inferences about sets of artifacts under sets of operating conditions. It refines and extends the idea of interval constraint propagation, and has been used as the basis of a program called a “mechanical design compiler,” which provides the user with a “high level language” in which design problems for systems to be built of cataloged components can be quickly and easily formulated. The compiler then selects optimal combinations of catalog numbers. Previous work has tested the calculus empirically, but only parts of the calculus have been proven mathematically. This paper presents a new version of the calculus and shows how to extend the earlier proofs to prove the entire system. It formalizes the effects of toleranced manufacturing processes through the concept of a “selectable subset” of the artifacts under consideration. It demonstrates the utility of distinguishing between statements which are true for all artifacts under consideration, and statements which are merely true for some artifact in each selectable subset.
This paper defines, for use in design, rules for propagating "distribution constraints" through relationships such as algebraic or vector equations. Distribution constraints are predicate logic statements about the values that physical system parameters may assume. The propagation rules take into account "variation source causality": information about when and how the values are assigned during the design, manufacturing, and operation of the system.
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