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.


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
Your Session has timed out. Please sign back in to continue.



Grahic Jump Location
Figure 3

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

Grahic Jump Location
Figure 2

Different representations for solution and design spaces.

Grahic Jump Location
Figure 4

Subdividing a facet into smaller facets

Grahic Jump Location
Figure 5

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

Grahic Jump Location
Figure 6

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

Grahic Jump Location
Figure 7

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

Grahic Jump Location
Figure 8

Convolution in three dimensions

Grahic Jump Location
Figure 9

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

Grahic Jump Location
Figure 10

Example of directional convexity in planar shapes.

Grahic Jump Location
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.

Grahic Jump Location
Figure 11

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

Grahic Jump Location
Figure 12

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

Grahic Jump Location
Figure 13

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

Grahic Jump Location
Figure 14

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

Grahic Jump Location
Figure 15

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

Grahic Jump Location
Figure 16

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

Grahic Jump Location
Figure 17

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

Grahic Jump Location
Figure 18

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

Grahic Jump Location
Figure 19

Erroneous approximations can be obtained for large values of λ




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In