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Research Papers: Design Theory and Methodology

Design Manifolds Capture the Intrinsic Complexity and Dimension of Design Spaces

[+] Author and Article Information
Wei Chen

Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: wchen459@umd.edu

Mark Fuge

Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: fuge@umd.edu

Jonah Chazan

Department of Computer Science,
University of Maryland,
College Park, MD 20742
e-mail: jchazan@umd.edu

1Corresponding author.

Contributed by the Design Theory and Methodology Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 16, 2016; final manuscript received February 7, 2017; published online March 23, 2017. Assoc. Editor: Harrison M. Kim.

J. Mech. Des 139(5), 051102 (Mar 23, 2017) (10 pages) Paper No: MD-16-1647; doi: 10.1115/1.4036134 History: Received September 16, 2016; Revised February 07, 2017

This paper shows how to measure the intrinsic complexity and dimensionality of a design space. It assumes that high-dimensional design parameters actually lie in a much lower-dimensional space that represents semantic attributes—a design manifold. Past work has shown how to embed designs using techniques like autoencoders; in contrast, the method proposed in this paper first captures the inherent properties of a design space and then chooses appropriate embeddings based on the captured properties. We demonstrate this with both synthetic shapes of controllable complexity (using a generalization of the ellipse called the superformula) and real-world designs (glassware and airfoils). We evaluate multiple embeddings by measuring shape reconstruction error, pairwise distance preservation, and captured semantic attributes. By generating fundamental knowledge about the inherent complexity of a design space and how designs differ from one another, our approach allows us to improve design optimization, consumer preference learning, geometric modeling, and other design applications that rely on navigating complex design spaces. Ultimately, this deepens our understanding of design complexity in general.

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Figures

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Fig. 1

Three-dimensional visualization of high-dimensional design space showing that design parameters actually lie on a two-dimensional manifold

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Fig. 2

Examples of superformula shapes

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Fig. 3

Three-dimensional visualization of the superformula design space created by a linear mapping from the high-dimensional design space X to a three-dimensional space, solely for visualization. Each point represents a design. (a) Lineardesign space (d = 1) varying s, (b) nonlinear space (d = 2) varying s and n3, and (c) design space with multiple shape categories.

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Fig. 4

Set boundary of the feasible semantic space: (a) create a convex hull of the training set in the semantic space, (b) copy the boundary of the convex hull to the grid of new designs generated from the semantic space, and (c) remove designs outside the boundary

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Fig. 5

Illustration of pairwise distance preservation. Similar designs (A and B) have similar shape representations in X, thus are closer in X than dissimilar designs (A and C). We want such relation of pairwise distances to be preserved in F (i.e., dAB < dAC) such that shapes will vary in the same manner as they do in X.

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Fig. 6

An example comparing shapes generated from the semantic space F versus the superformula parameter space P. If the embedding precisely captures the principal attributes, shapes from F should look like those from P—with neither extra unexpected shape variation nor missing diversity. (a) Shapes in the superformula parameter space and (b) generated shapes in the semantic space.

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Fig. 7

Multiple superformula categories with intersection. Our approach correctly separates the three submanifolds, even though they all connect via a common seam. (a) Shapes in the superformula parameter space, (b) result of manifold clustering (as in Fig. 3, the design space X is visualized in three dimensions), and (c) generated shapes in semantic spaces. Since there are three categories, we have three separated semantic spaces.

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Fig. 8

Multiple superformula categories with intersection and different intrinsic dimensions. Our intrinsic dimension estimator automatically detects the appropriate dimensionality of the semantic space for each design category (c). (a) Shapes in the superformula parameter space, (b) result of manifold clustering (as in Fig. 3, the design space X is visualized in three dimensions), and (c) generated shapes in semantic spaces.

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Fig. 9

Synthesized glassware shapes in a 3D semantic space. The embedding captured three shape attributes—the rim diameter, the stem diameter, and the curvature.

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Fig. 10

Synthesized airfoil shapes in a 3D semantic space. The embedding captured three shape attributes—the upper and lower surface protrusion and the trailing edge direction.

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Fig. 11

Point A has high sample density and thus higher confidence that synthesized shapes will look similar to nearby real-world samples. In contrast, point B has low sample density and thus lower confidence but higher chance of generating an unusual or creative shape. Shade darkness correlates with higher local density. (a) Arrangement of training samples in the semantic space (for simplicity, this is a 2D projection of the 3D semantic space) and (b) synthesized shapes.

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Fig. 12

Comparison of different embedding methods. The abnormal shapes generated by PCA and SdA are due to high reconstruction error and high GDI, respectively. (a) Shapes in the superformula parameter space, (b) reconstruction error and geodesic distance inconsistency, (c) embedding and shape synthesis result by PCA, (d) embedding and shape synthesis result by kernel PCA, and (e) embedding and shape synthesis result by a stacked denoising autoencoder.

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