NESC9588: 3DGEOELE, 3-D Nonlinear Least Square Fit

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NESC9588 3DGEOELE. (Abstract last modified 09-MAY-1989)

3DGEOELE, 3-D Nonlinear Least Square Fit

Abstract Table of Contents:

1. NAME OR DESIGNATION OF PROGRAM - 3DGEOELE.

2. COMPUTER FOR WHICH PROGRAM IS DESIGNED AND OTHER MACHINE VERSION PACKAGES AVAILABLE -

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Explanation of the status codes.

Program-name Package-ID Status 3DGEOELE NESC9588/01 Tested

Machines used: Package-ID Orig.Computer Test Computer NESC9588/01 DEC PDP-11 DEC VAX 8810

3. DESCRIPTION OF PROGRAM OR FUNCTION - 3DGEOELE is a collection of six programs - LINE, PLANE, CYL, CONE, RING, and TOROID - for nonlinear least squares fitting to a line, plane, circular cylinder, circular cone, ring, and toroid, respectively, in three-dimensional space of a set of measured x, y, z coordinates. The measurements are made by multi-axis coordinate measuring machines. The analysis is based on the residuals being calculated in a direction perpendicular to the surface or line. All six types involve nonlinear relationships, but in the case of the straight line or plane, nonlinear equations are avoided by using linear equations iteratively in repeated two- dimensional fits. In the case of the cylinder, cone, ring, and toroid, the equation of the geometrical shape (element) in the general position is found by taking the standard equation of the element relative to a coordinate system where the principal axis of the element is the z axis of that coordinate system and then by means of transformation equations, converting the equation to be relative to the original coordinate system. These transformation equations are based on three shift constants (in the x, y, z direction) and on two angles of rotation (first counter-clockwise about x, and then counter-clockwise about y). There are two modes of execution: experimental and production. The experimental mode is for checking each program to ensure proper operation and sufficient accuracy in the final results. The production mode is used with measured data where the answers are unknown in advance.

4. METHOD OF SOLUTION - The least squares solution used minimizes the sum of the squares of the perpendicular deviations or residuals from the fitted line to the data points. The nonlinear normal equations from the least squares analysis are solved using Brown's iteration method, which is Newton-like and based on Gaussian elimination but converges faster than Newton's method. If the residuals are in the realm of hundreds of microinches or less, convergence will occur in less than 20 iterations to 12 significant figures.

5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM - Maximum of -
30 iterations

6. TYPICAL RUNNING TIME -

NESC9588/01:

NEA-DB ran the test cases included in this package on a VAX 8810 computer in less than 8 second sof CPU time.

7. UNUSUAL FEATURES OF THE PROGRAM -

8. RELATED AND AUXILIARY PROGRAMS -

9. STATUS
NESC9588/01: 09-MAY-1989 Tested at NEADB

10. REFERENCES  -
- 3DGEOELE, NESC No. 9588 Tape Description,
  National Energy Software Center Note 87-69, May 21, 1987.
- K.M. Brown
A Quadratically Convergent Newton-like Method Based Upon Gaussian Elimination,
  SIAM Journal of Numerical Analysis, Vol. 6, No. 4, pp. 560-569,
  December 1969.
NESC9588/01:
- S.R. Drake:
  Computer Programs for Nonlinear Least Squares Analysis of Various
  Three-Dimensional Geometrical Elements
  BDX-613-3741  (April 1987).

11. MACHINE REQUIREMENTS -

NESC9588/01:

The different modules run on a VAX 8810 computer in 22K bytes (43 pages) of main storage.

12. PROGRAMMING LANGUAGE(S) USED -
NESC9588/01: FORTRAN-IV

13. OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED - RSX-11M.

NESC9588/01:

V5.0-1 (VAX 8810).

14. OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS - The measured coordinates must be available in a stored data file prior to execution; all additional information required is supplied manually by the user and is menu driven.

15. NAME AND ESTABLISHMENT OF AUTHORS -
          S.R. Drake
          Bendix Kansas City Division
          Allied Corporation

17. CATEGORIES -

- P. General Mathematical and Computing System Routines

Keywords: COMPUTER-AIDED DESIGN, COORDINATES, LEAST SQUARE FIT, NONLINEAR PROBLEMS, X-Y-Z