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Topological structures for modeling engineering design processes

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Abstract

In this paper, a mathematical framework for describing a variety of complex and practical design processes is developed. We demonstrate that our model has the desirable quality of representing several, seemingly distinct, approaches as instances of the same framework. In addition, General Design Theory is shown to be a special case of the proposed framework. Using simple examples throughout the paper, we also hint at the potential for the framework to serve as a basis for a descriptive study of design. Various design phenomena such as design failure, identification of design knowledge bottlenecks, and benefits of collaborative design could be understood using the proposed model.

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Notes

  1. For instance, when the designer reaches a specification f n that cannot be refined due to limited knowledge, or when sufficient information is gathered that enables the “synthesis” mapping to take place.

  2. This natural type of representation has been utilized, explicitly or implicitly, as a means for describing the design process by other researchers (e.g., Pahl and Beitz 1984; Gero 1990; Takeda et al 1990; Dasgupta 1994; Simon 1996; Suh 2001). Here, we use it as a basis for modeling the underlying topological spaces.

  3. Some definitions may resemble GDT terminology (Yoshikawa 1981); nevertheless, as already discussed, GDT is only a special case of our framework.

  4. Note that we use the term “f is a generator of F” although f may only generate part of F.

  5. It could be prohibitively expensive to generate this set in real design.

  6. It may be impossible to verify this property in real design; however, engineering intuition, experience, or lack of knowledge might imply it.

  7. Note that our use of the term “implication” is not necessarily identical to “logical implication”. In a logical framework, the implication relation is associated with deduction. Consequently, by modus ponens, every description in the implication chain is implied by the initial candidate solution d 0.

  8. <fn>Recall that “d i+1is generated by d i” or “d i generates d i+1” if d i∈U F (d i+1).

  9. We use this source rather than the original papers since it summarizes GDT with simple intuitive examples.

  10. Here, in any formula of the form A→B, A is referred to as the antecedent and B as the consequent (Russell and Norvig 1995).

  11. By applying a specific knowledge base.

  12. Constraints in discrete domains can be expressed as compatibility relations between attributes, stating that certain combinations are allowed or not.

  13. Dependency-directed backtracking provides a way of taking into account the information about which pieces of knowledge contribute to the failure. This information is used in the decision of how far to backtrack.

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Authors and Affiliations

  1. Department of Industrial Engineering, Ben-Gurion University, PO Box 653, 84105, Beer-Sheva, Israel

    Dan Braha

  2. Department of Solid Mechanics, Materials and Systems, Tel Aviv University, 69978, Ramat Aviv , Israel

    Yoram Reich

Authors
  1. Dan Braha
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  2. Yoram Reich
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Corresponding author

Correspondence to Dan Braha.

Appendices

Appendix 1

Cluster point, interior, and neighborhood

Definition A.1 (Cluster point of a set)

A cluster point of a set F in a closure function space is a point f belonging to the closure of F−{f} The set of all cluster points of a set F is denoted by F′ and called the derivative of F in the closure function space. Clearly, the closure of a set F is the union of the set F with its set of cluster points.

Definition A.2 (Interior of a set)

With any closure U F for a set F there is associated the interior operation int U F , denoted briefly by int. The operation int is a function that maps elements of 2F into 2F, such that for each F⊂F, int(F)=F− U F (F− F). The set int(F) is called the interior of F in <F, U F >.

Definition A.3 (Neighborhood)

A neighborhood of a subset F of a closure function space N(F) is any subset G of F containing F in its interior. By a neighborhood, N(f), of a functional description f of F, we mean a neighborhood of the one-point set {f}. The neighborhood system of a set F⊆F (a point f∈F) in the space <F, U F > is the collection of all neighborhoods of the set F (the point f).

Appendix 2

production rules

A small sample of the domain-specific knowledge relevant to the car design domain is expressed in terms of the production rules presented in Table 3.

Table 3. A sample of rules for the car example
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Braha, D., Reich, Y. Topological structures for modeling engineering design processes. Res Eng Design 14, 185–199 (2003). https://doi.org/10.1007/s00163-003-0035-3

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  • DOI: https://doi.org/10.1007/s00163-003-0035-3

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