NAME OR DESIGNATION OF PROGRAM, COMPUTER, DESCRIPTION OF PROGRAM OR FUNCTION, METHOD OF SOLUTION, RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM, TYPICAL RUNNING TIME, UNUSUAL FEATURES OF THE PROGRAM, RELATED AND AUXILIARY PROGRAMS, STATUS, REFERENCES, MACHINE REQUIREMENTS, LANGUAGE, OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED, OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS, NAME AND ESTABLISHMENT OF AUTHORS, MATERIAL, CATEGORIES

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Program name | Package id | Status | Status date |
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3DGEOELE | NESC9588/01 | Tested | 09-MAY-1989 |

Machines used:

Package ID | Orig. computer | Test computer |
---|---|---|

NESC9588/01 | DEC PDP-11 | DEC VAX 8810 |

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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.

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.

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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.

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.

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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.[ top ]

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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.

- 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, included references:

- S.R. Drake:Computer Programs for Nonlinear Least Squares Analysis of Various

Three-Dimensional Geometrical Elements

BDX-613-3741 (April 1987).

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NESC9588/01

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

NESC9588/01

V5.0-1 (VAX 8810).[ top ]

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NESC9588/01

File name | File description | Records |
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NESC9588_01.001 | Information file | 148 |

NESC9588_01.002 | FORTRAN-77 source program of LINE | 388 |

NESC9588_01.003 | FORTRAN-77 source program of PLANE | 396 |

NESC9588_01.004 | FORTRAN-77 source program of CYL | 693 |

NESC9588_01.005 | FORTRAN-77 source program of CONE | 722 |

NESC9588_01.006 | FORTRAN-77 source program of RING | 695 |

NESC9588_01.007 | FORTRAN-77 source program of TOROID | 746 |

NESC9588_01.008 | Sample data of experimental mode of LINE | 5 |

NESC9588_01.009 | Sample data of experimental mode of PLANE | 5 |

NESC9588_01.010 | Sample data of experimental mode of CYL | 9 |

NESC9588_01.011 | Sample data of experimental mode of CONE | 9 |

NESC9588_01.012 | Sample data of experimental mode of RING | 5 |

NESC9588_01.013 | Sample data of experimental mode of TOROID | 17 |

NESC9588_01.014 | Sample data of production mode of LINE | 7 |

NESC9588_01.015 | Sample data of production mode of PLANE | 8 |

NESC9588_01.016 | Sample data of production mode of CYL | 9 |

NESC9588_01.017 | Sample data of production mode of CONE | 17 |

NESC9588_01.018 | Sample data of production mode of RING | 17 |

NESC9588_01.019 | Sample data of production mode of TOROID | 33 |

NESC9588_01.020 | Output of experimental mode of LINE | 75 |

NESC9588_01.021 | Output of experimental mode of PLANE | 59 |

NESC9588_01.022 | Output of experimental mode of CYL | 99 |

NESC9588_01.023 | Output of experimental mode of CONE | 97 |

NESC9588_01.024 | Output of experimental mode of RING | 90 |

NESC9588_01.025 | Output of experimental mode of TOROID | 141 |

NESC9588_01.026 | Output of production mode of LINE | 32 |

NESC9588_01.027 | Output of production mode of PLANE | 34 |

NESC9588_01.028 | Output of production mode of CYL | 54 |

NESC9588_01.029 | Output of production mode of CONE | 67 |

NESC9588_01.030 | Output of production mode of RING | 65 |

NESC9588_01.031 | Output of production mode of TOROID | 100 |

NESC9588_01.032 | Original PDP-11 source, not tested, LINE | 366 |

NESC9588_01.033 | Original PDP-11 source, not tested, PLANE | 374 |

NESC9588_01.034 | Original PDP-11 source, not tested, CYL | 673 |

NESC9588_01.035 | Original PDP-11 source, not tested, CONE | 702 |

NESC9588_01.036 | Original PDP-11 source, not tested, RING | 675 |

NESC9588_01.037 | Original PDP-11 source, not tested, TOROID | 720 |

Keywords: computer-aided design, coordinates, least square fit, nonlinear problems, x-y-z.