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Research Papers

Spatial Interpolation for Periodic Surfaces in Manufacturing Using a Bessel Additive Variogram Model

[+] Author and Article Information
Yuhang Yang

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: yang221@illinois.edu

Chenhui Shao

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: chshao@illinois.edu

1Corresponding author.

Manuscript received November 1, 2017; final manuscript received January 26, 2018; published online March 7, 2018. Assoc. Editor: Xiaoping Qian.

J. Manuf. Sci. Eng 140(6), 061001 (Mar 07, 2018) (10 pages) Paper No: MANU-17-1679; doi: 10.1115/1.4039199 History: Received November 01, 2017; Revised January 26, 2018

High-resolution spatial data are essential for characterizing and monitoring surface quality in manufacturing. However, the measurement of high-resolution spatial data is generally expensive and time-consuming. Interpolation based on spatial models is a typical approach to cost-effectively acquire high-resolution data. Conventional modeling methods fail to adequately model the spatial correlation induced by periodicity, and thus their interpolation precision is limited. In this paper, we propose using a Bessel additive periodic variogram model to capture such spatial correlation. When combined with kriging, a geostatistical interpolation method, accurate interpolation performance can be achieved for common periodic surfaces. In addition, parameters of the proposed model provide valuable insights for the characterization and monitoring of spatial processes in manufacturing. Both simulated and real-world case studies are presented to demonstrate the effectiveness of the proposed method.

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Figures

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Fig. 1

A periodic anvil surface from ultrasonic metal welding: (a) optical image and (b) 3D view

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Fig. 2

The procedure of interpolation using kriging

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Fig. 3

A simulation of worn anvil surface with hemisphere-shaped knurls

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Fig. 4

Plots of variogram models for a simulated worn anvil surface. “Actual” represents the actual values of empirical variogram: (a) matern class variogram models, (b) periodic variogram models alone, and (c) periodic variogram models combined with the Gaussian model.

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Fig. 5

Plots of four fitted variogram models. The subscript numbers represent the setting sets in Sec. 3.1.2: (a) Bessel additive model, (b) Gaussian model, (c) waving additive model, and (d) SSE additive model.

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Fig. 6

Six measurements of an anvil surface in ultrasonic metal welding

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Fig. 7

Comparisons of Bessel additive model, waving additive model, and SSE additive model at each sampling time

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Fig. 8

Temporal trend of fitted parameters in the Bessel additive periodic variogram model: (a) Nugget, (b) ω, (c) a, (d) p, and (e) sill

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