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Research Papers

# $N$-Dimensional Nonuniform Rational B-Splines for Metamodeling

[+] Author and Article Information
Cameron J. Turner

Division of Engineering, Colorado School of Mines, 1500 Illinois Street, Golden, CO 80401cturner@mines.edu

Richard H. Crawford

Department of Mechanical Engineering, University of Texas at Austin, 1 University Station, C2200, Austin, TX 78712rhc@mail.utexas.edu

J. Comput. Inf. Sci. Eng 9(3), 031002 (Aug 19, 2009) (13 pages) doi:10.1115/1.3184599 History: Received November 05, 2007; Revised November 25, 2008; Published August 19, 2009

## Abstract

Nonuniform rational B-splines (NURBs) have unique properties that make them attractive for engineering metamodeling applications. NURBs are known to accurately model many different continuous curve and surface topologies in one- and two-variate spaces. However, engineering metamodels of the design space often require hypervariate representations of multidimensional outputs. In essence, design space metamodels are hyperdimensional constructs with a dimensionality determined by their input and output variables. To use NURBs as the basis for a metamodel in a hyperdimensional space, traditional geometric fitting techniques must be adapted to hypervariate and hyperdimensional spaces composed of both continuous and discontinuous variable types. In this paper, we describe the necessary adaptations for the development of a NURBs-based metamodel called a hyperdimensional performance model or HyPerModel. HyPerModels are capable of accurately and reliably modeling nonlinear hyperdimensional objects defined by both continuous and discontinuous variables of a wide variety of topologies, such as those that define typical engineering design spaces. We demonstrate this ability by successfully generating accurate HyPerModels of ten trial functions laying the foundation for future work with $N$-dimensional NURBs in design space applications.

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## Figures

Figure 1

The iterative control point (CP) addition scheme for a planar 2D-input problem. The approach is readily extensible to N-D inputs.

Figure 2

The evolution of the knot vector from two control points (first row), to three control points (second row), to four control points (third row), and five control points (fourth row) along with the corresponding set of NURBs curves and the knot vector. As control points are added, the knot vector localizes the influence of individual control points.

Figure 3

The basic fitting algorithm used in HyPerMaps to define a HyPerModel iteratively adds control points to the control net to reduce the maximum error in the metamodel. The model is refined until a stopping criterion is achieved.

Figure 4

The actual function (top) is accurately represented by the metamodel (bottom), including the slight differences in the two minima values

Figure 5

The actual function (top) and the metamodel representation (bottom). The metamodel accurately represents the high variability for small values of X, and the low variability associated with larger values of X

Figure 6

Rosenbrock’s Banana function is a classical optimization problem, characterized by a flat banana-shaped region surrounded by steep slopes, as shown in plots of the actual function (top). Both characteristics are well represented in the HyPerModel (middle). The control point network (bottom) and knot vectors are also given.

Figure 7

Plots of the six-hump camel back function (top) include four distinct optimal solutions. Each of these optima is captured in the HyPerModel (middle). The control point network (bottom) and knot vectors also are defined.

Figure 8

Plots of the actual Hansen function (top) and the HyPerModel (bottom) fit to the data set are virtually identical

Figure 9

The actual output functions (top) compare favorably with the HyPerModel (bottom). Individual correlations to each output dimension range from 97.6% to nearly 100%, with an overall correlation of 99.98%.

Figure 10

The HyPerModel simply interpolates across the void defined in the Sasena sinusoidal function (top). If the simulation can detect that a void exists, a feasibility dimension (bottom) also can be defined and modeled.

Figure 11

The actual function (top) and the HyPerModel (bottom). Some residual variability still exists in the HyPerModel.

Figure 12

The building was defined as an infeasible position in this problem by using a feasibility constraint to define crane positions within the building as infeasible. The above plot is for a specific value of the boom length so as to produce a 2D plot.

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