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Research Papers

Creating Polytope Representations of Design Spaces for Visual Exploration Using Consistency Techniques

[+] Author and Article Information
Srikanth Devanathan

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

Karthik Ramani1

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907ramani@purdue.edu

“n-ary” is the generalization of unary (1-ary), binary (2-ary), ternary (3-ary) and so on. “Arity” is defined in Sec. 3.

1

Corresponding author.

J. Mech. Des 132(8), 081011 (Aug 18, 2010) (10 pages) doi:10.1115/1.4001528 History: Received May 06, 2009; Revised February 18, 2010; Published August 18, 2010; Online August 18, 2010

Understanding the limits of a design is an important aspect of the design process. When mathematical models are constructed to describe a design concept, the limits are typically expressed as constraints involving the variables of that concept. The set of values for the design variables that do not violate constraints constitute the design space of that concept. In this work, we transform a parametric design problem into a geometry problem thereby enabling computational geometry algorithms to support design exploration. A polytope-based representation is presented to geometrically approximate the design space. The design space is represented as a finite set of (at most) three-dimensional (possibly nonconvex) polytopes, i.e., points, intervals, polygons, and polyhedra. The algorithm for constructing the design space is developed by interpreting constraint-consistency algorithms as computational-geometric operations and consequently extending (3,2)-consistency algorithm for polytope representations. A simple example of a fingernail clipper design is used to illustrate the approach.

Copyright © 2010 by American Society of Mechanical Engineers
Topics: Design , Approximation , Space
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References

Figures

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Figure 1

Formal (mathematical) model maps a point in the space of design variables to a point in the performance space. Constraints form the boundary of the design space.

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Figure 2

Different representations for solution and design spaces.

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Figure 3

Procedure to construct a polyhedral label involving variables xi, xj, and xk

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Figure 4

Subdividing a facet into smaller facets

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Figure 5

Illustration of the filling procedure to construct solution space around a starting point

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Figure 6

(3,2)-consistency algorithm to prune the solution spaces

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Figure 7

Function to revise the solution space Lijk using the solution spaces Liuv, Ljuv, and Lkuv

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Figure 8

Convolution in three dimensions

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Figure 9

Geometrically, ∏uvLiuv∣xi=αi is computed by obtaining a intersecting Liuv with the plane xi=αi

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Figure 10

Example of directional convexity in planar shapes.

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Figure 11

Schematic of fingernail clipper design (adapted from Otto and Wood (1)).

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Figure 12

Initial label of ternary constraint C134≡{tm,hb,x}

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Figure 13

Initial label of constraint C346≡{hb,x,x6}

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Figure 14

Initial label of constraint C256≡{Dh,d,x6}

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Figure 15

The initial (light) and final (dark) label of constraint C134≡{tm,hb,x}

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Figure 16

The reduced label of constraint C346≡{hb,x,x6}(fmax=3 lbs)

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Figure 17

The reduced label of constraint C346≡{hb,x,x6}(fmax=2.5 lbs,  λ=0.05)

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Figure 18

The reduced label of constraint C346≡{hb,x,x6}(fmax=2.1 lbs,  λ=0.025)

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Figure 19

Erroneous approximations can be obtained for large values of λ

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