Trivariate Simplex Splines for Inhomogeneous Solid Modeling in Engineering Design

[+] Author and Article Information
Jing Hua

Computer Science, Wayne State University

Ying He, Hong Qin

Computer Science, Stony Brook University

J. Comput. Inf. Sci. Eng 5(2), 149-157 (Jan 21, 2005) (9 pages) doi:10.1115/1.1881352 History: Received September 03, 2004; Revised January 21, 2005

This paper presents a new inhomogeneous solid modeling paradigm for engineering design. The proposed paradigm can represent, model, and render multidimensional physical attributes across any volumetric objects of complicated geometry and topology. A modeled object is formulated with a trivariate simplex spline defined over a tetrahedral decomposition of its three-dimensional domain. Heterogeneous material attributes associated with solid geometry can be easily modeled and edited by manipulating the control vectors and∕or associated knots of trivariate simplex splines. We also develop a feature-sensitive fitting algorithm that can reconstruct a compact, continuous trivariate simplex spline from measured, structured, or unstructured volumetric grids of real-world inhomogeneous objects. In addition, we propose a fast direct rendering algorithm for interactive data analysis and visualization of the simplex spline volumes. Our experiments demonstrate that the proposed paradigm augments the current engineering design techniques with new and unique advantages.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Volume rendering of 10 quadratic DMS-spline basis functions

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Figure 2

(a) A cubic trivariate DMS-spline solid corresponding to a single tetrahedral domain with 20 control points and (b) the tetrahedra of the designed solid object are scaled to show the interior of the solid

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Figure 3

A DMS-spline solid object corresponding to the tetrahedral domain: (a) a domain tetrahedralization and (b) volume rendering of the resulting solid model, which is genus-20

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Figure 4

(a) The piecewise linear boundary constraints that the user specifies; (b) the multiresolution tetrahedralization conforming to the piecewise linear boundary constraints; (c) the color map of material distribution of the designed object; and (d) volume rendering of the designed object, where we can see that all the geometric shape features are preserved.

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Figure 5

(a) The tetrahedral domain of a geometrically smooth object and (b) volume rendering of the designed object, where we can see the density discontinuities shown in different colors

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Figure 6

(a) The point view of the spx dataset, where the color indicates the density difference, (b) occupancy map of the point set, and (c) the final tetrahedralization after removing the outside tetrahedra

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Figure 7

(a) Geometric features of the spx dataset and (b) the finally constructed initial tetrahedralization

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Figure 8

The fitting results for the spx dataset: (a) and (b) fitting with control vectors only (front view and side view) and (c) and (d) fitting with both control vectors and knots (front view and side view, respectively)

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Figure 9

Simplex spline-based fitting examples: (a) the original dataset, router, in point view, where the color indicates the attribute value; (b) fitting with both control vectors and knots. The final result is rendered using our marching tetrahedra algorithm. (c) the original dataset, crosscube, in point view, where the color indicates the attribute value; and (d) fitting with both control vectors and knots. The final result is rendered using the direct volume rendering algorithm.



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