An Approach to Estimate the Detailed Deviation Zone in Coordinate Metrology Using Harmonic Function Properties

2014 ◽  
Author(s):  
Saeed Jamiolahmadi ◽  
Ahmad Barari
Keyword(s):  
Harmonic Functions ◽  
Closed Loop ◽  
Mean Value ◽  
Manufacturing Processes ◽  
Mean Value Property ◽  
Detailed Understanding ◽  
Manufacturing Errors ◽  
Coordinate Metrology ◽  
Geometric Deviations ◽  
Deviation Zone

Understanding the exact details of deviation zone related to a manufactured surface needs measurement of infinite number of points. Coordinate metrology provides deviation of the limited number of discrete points on a measured surface, but typically it is not capable to explore any information of the surface regions between these measured points. An approach to estimate the Distribution of Geometric Deviations (DGD) on the entire inspected surface is presented in this paper. The methodology is developed based on estimation of mean value property of the harmonic functions and Laplace equation. The resulting DGD model can be employed to estimate the deviation values at any unmeasured point of the inspected surface when a detailed understanding of the surface geometric deviations is required. Implementation of the developed methodology is described and case studies for typical industrial parts are presented. This methodology can be used for closed-loop of inspection and manufacturing processes when a compensation scheme is available to compensate the manufacturing errors based on the DGD model.

Evaluation of Deviation Zone Based on Maximum Conformance to Tolerances

2011 ◽  
Author(s):  
Ahmad Barari
Keyword(s):  
Closed Loop ◽  
Evaluation Process ◽  
Accurate Estimation ◽  
Tolerance Zone ◽  
Coordinate Metrology ◽  
Inspection Process ◽  
Geometric Deviations ◽  
The Given ◽  
Best Fit ◽  
Deviation Zone

The accurate estimation of the geometric deviations is not possible only by manipulating the Euclidian distances of the discrete measured points from substitute geometry. The real geometric deviations of a measured surface need to be calculated based on the desired tolerance zone of the surface. This fact is usually neglected in common practices in the coordinate metrology of surfaces. The importance of considering the desired tolerance zone in estimation of the optimum deviation zone is demonstrated in this paper. Then a best fit method is presented which complies with the tolerance requirements of the designed surface. The developed fitting methodology constructs a substitute geometry to minimizes the residual deviations corresponding to the given tolerance zone and the needs of down-stream operations that use the results of the inspection process. It is shown how the developed objective function can be adopted for a case of closed-loop manufacturing process, when the under-cut residual deviations of the manufactured part can be corrected by a down-stream operation. In order to validate the proposed methodology, experiments are conducted. The results show a significant reduction of uncertainties in coordinate metrology of geometric surfaces. Implementation of this method directly results in increasing the accuracy of the entire tolerance evaluation process, and less uncertainty in quality control of the manufactured parts.

scholarly journals Convergence of a finite difference approach for detailed deviation zone estimation in coordinate metrology

ACTA IMEKO ◽  
2015 ◽  
Vol 4 (4) ◽  
pp. 20 ◽  
Author(s):  
Ahmad Barari ◽  
Saeed Jamiolahmadi
Keyword(s):  
Measurement Techniques ◽  
Mean Value ◽  
Sufficient Information ◽  
Mathematical Functions ◽  
Actual Case ◽  
Novel Approach ◽  
Coordinate Metrology ◽  
Geometric Deviations ◽  
Surface Deviation ◽  
Deviation Zone

<p class="Abstract">In order to comprehend an entire surface's deviation zone, infinite measured points are required. Using the common measurement techniques through coordinate metrology, a limited number of surface actual points can be acquired. However, the obtained points would not provide sufficient information to examine the geometry thoroughly. A novel approach to predict surface behaviour via Distribution of Geometric Deviations (DGD) is examined in this paper. The methodology governs the mean value property of the harmonic functions to solve Laplace equation around each measured point. This DGD model can be used to reconstruct surface deviation values at any unmeasured point of the inspected surface based on a limited number of measured points. The convergence of the introduced approach is studied in this paper. A complete approach to implement the developed methodology is described, and the validation process is studied using actual case studies and mathematical functions. This methodology is practical in closed-loop inspection and manufacturing processes to form a scheme for compensating the surface errors during manufacturing process based on the DGD model.</p>

A Finite Element Approach to Estimate the Detailed Deviation Zone in Coordinate Metrology

2015 ◽  
Author(s):  
Amirali Lalehpour ◽  
Ahmad Barari ◽  
Saeed Jamiolahmadi
Keyword(s):  
Finite Element ◽  
Detailed Knowledge ◽  
Finite Element Approach ◽  
Manufacturing Errors ◽  
Coordinate Metrology ◽  
Geometric Deviations ◽  
Element Approach ◽  
Deviation Zone ◽  
Solution Estimates

The exact detailed knowledge of deviation zone in a manufactured surface needs measurement of infinite number of points when the coordinate metrology is utilized. The coordinate metrology process provides deviation of the limited number of discrete points on a measured surface, but typically the process is not capable to explore any information of the surface regions between these measured points. A Finite Element approach for Deviation Zone Evaluation (DZE) on the entire inspected surface is presented in this paper. The developed DZE solution estimates the deviation values at any unmeasured point of the inspected surface when a detailed understanding of the surface geometric deviations is required. Implementation of the developed methodology is described and case study for typical industrial parts is presented. This methodology can be used for closed-loop of inspection and manufacturing processes when a compensation scheme is available to compensate the manufacturing errors based on the DZE data.

scholarly journals On the mean value property of superharmonic and subharmonic functions

2006 ◽  
Vol 2006 ◽  
pp. 1-3
Author(s):  
Robert Dalmasso
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Subharmonic Functions ◽  
Mean Value Property ◽  
The Mean

We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.

On the Inverse mean value Property of Harmonic Functions on Strips

1992 ◽  
Vol 24 (6) ◽  
pp. 559-564 ◽  
Author(s):  
M. Goldstein ◽  
W. Haussmann ◽  
L. Rogge
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property

scholarly journals On the mean value property of fractional harmonic functions

2020 ◽  
Vol 201 ◽  
pp. 112112
Author(s):  
Claudia Bucur ◽  
Serena Dipierro ◽  
Enrico Valdinoci
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean

On the mean-value property of harmonic functions

1965 ◽  
Vol 14 (1) ◽  
pp. 109-111 ◽  
Author(s):  
Bernard Epstein ◽  
M. M. Schiffer
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean
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