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Research Papers

# PDE-Constrained Gaussian Process Model on Material Removal Rate of Wire Saw Slicing Process

[+] Author and Article Information
Hongxu Zhao, Su Wu

Department of Industrial Engineering, Tsinghua University, Beijing, China

Ran Jin

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA

Jianjun Shi1

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GAjianjun.shi@isye.gatech.edu

1

Corresponding author.

J. Manuf. Sci. Eng 133(2), 021012 (Mar 23, 2011) (9 pages) doi:10.1115/1.4003617 History: Received March 09, 2010; Revised January 27, 2011; Published March 23, 2011; Online March 23, 2011

## Abstract

Thickness uniformity of wafers is a critical quality measure in a wire saw slicing process. Nonuniformity occurs when the material removal rate (MRR) changes over time during a slicing process, and it poses a significant problem for the downstream processes such as lapping and polishing. Therefore, the MRR should be modeled and controlled to maintain the thickness uniformity. In this paper, a PDE-constrained Gaussian process model is developed based on the global Galerkin discretization of the governing partial differential equations (PDEs). Three features are incorporated into the statistical model: (1) the PDEs governing the wire saw slicing process, which are obtained from engineering knowledge, (2) the systematic errors of the manufacturing process, and (3) the random errors, including both random manufacturing errors and measurement noises. Real experiments are conducted to provide data for the validation of the PDE-constrained Gaussian process model by estimating the model coefficients and further using the model to predict the overall MRR profile. The results of cross-validation indicate that the prediction performance of the PDE-constrained Gaussian process model is better than the widely used universal Kriging model with a mean of second order polynomial functions.

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## Figures

Figure 1

Contour diagram of the thickness profile of a sliced wafer (unit: m)

Figure 2

Schematic of the wire saw slicing process

Figure 3

Schematic of the slurry film in the contact zone of the wire saw and the ingot

Figure 4

Cross section of the wire saw, ingot, and slurry film in the contact zone

Figure 5

Schematic of the wire saw slicing process

Figure 6

Neighborhood of contact span y=Y

Figure 7

Schematic of the structure of the PDE-constrained GP model

Figure 8

Procedure of the modeling approach

Figure 9

Profile plot of the experimental ln(MRR) profile over all sample points

Figure 10

Profile plot of the predicted ln(MRR) using the estimated PDE-constrained GP model and the difference

Figure 11

Histogram of the differences D

Figure 12

The absolute values of the estimated parameters {βk}k=1NP when NP=30

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