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THESIS

Illlllllllllllll‘lillll ill Willill‘llllllil

3129301688

This is to certify that the

thesis entitlcd

SOAR Telescope Primary Mirror, Surface Distortion

Simplification and Minimization,
Using Finite Element Approach

presented by

Szu-Han Hu

has been accepted towards fulfillment
of the requirements for

Master's Mechanics

degree in

 

 

WKW

Major professor

Date 7/2 FI/7y

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

LIBRARY

Michigan State
University

 

 

 

 

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v a. .-—._— *‘O W"

PLACE 1N RETURN Box
to remove this checkout from your record.
TO AVOID FINE-3 return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/98 WWW-p.14

 

SOAR TELESCOPE PRIMARY MIRROR
SURFACE DISTORTION SIMPLIFICATION AND MINIMIZATION
USING FINITE ELEMENT APPROACH
By

Szu-Han Hu

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Materials Science and Mechanics

1998

35m.

ABSTRACT

SOAR TELESCOPE PRIMARY MIRROR
SURFACE DISTORTION SIMPLIFICATION AND MINIMIZATION
USING FINITE ELEMENT APPROACH
By

Szu-Han Hu

The surface distortions of the primary mirror of the SOAR telescope are a major con-
cern. Due to its nonhomogeneous coefficient of thermal expansion, the distortions after a
temperature rise have to be minimized. In 1992, Mr. Larry Barr formed a study about the
primary mirror using a coarse mesh and obtained a satisfactory result. The goals of this
research project are: first, to verify Barr’s assumption about the simplification of the coef-
ficient of thermal expansion; second, to improve the finite element model which will allow
a further detailed study about interfacial mechanics; and to obtain a near-circular form of
surface distortions (i.e. minimize global warping).

By repeating Barr’s procedures, it was found that the axial coefficient of thermal
expansion cannot be simplified. Accordingly, a more sophisticated finite element model
was required to analyze the distortion problem. The validation for adopting the finite ele-
ment software, MARC, has been verified by analytical solutions and supported by another
software, ANSYS. A FORTRAN code was developed to assist the software to capture the
variation of coefficient of thermal expansion within each boule. Axisymmetric analyses

were adopted to study the interfacial mechanics and predeterrnine suitable boule

placement combinations. After the studies and trials, a satisfactory semi—circular form of
surface distortion was achieved. 10 cm thick mirror was proved to be an appropriate

design. The maximum principal stresses were far less than the material ultimate strength.

To my parents and sisters

iv

ACKNOWLEDGEMENTS

I would like to thank the Department of Physics and Astronomy at Michigan State
University for granting me the Opportunity to get involved with the SOAR project and the
financial support. I would also like to thank Mr. Larry D. Barr for his generosity in allow-
ing me to follow his previous work and for providing me with substantial information. I
must also thank my CMRG colleagues for their helpful discussions. Special thanks
should give to Dr. Venkateshwar R. Aitharaju, Mr. Xinjian Fan and Miss Tammy Cum-
mings for their tremendous help. I also greatly appreciate Dr. Dahsin Liu, Dr. Edwin D.
Loh, and Dr. Hungyu Tsai for making efforts to serve my thesis committee.

Thank you, loving Tananya, for being so supportive. Your encouragements have com-
forted and carried me through out the entire project. I gratefully thank my beloved parents
and sisters for the years of unconditional support.

Finally, I would like to give special gratitude to Dr. Ronald C. Averill, my research
advisor. There were times of uncontrollable severe loss during this period of time. His
academic, as well as personal guidance throughout the duration Of this project are grate-
fully acknowledged.

Without them, all of this would not be possible.

Szu-Han Hu

July, 1998
E. Lansing, Michigan

 

2,3

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ............................................................................................................. x
CHAPTER I INTRODUCTION ...................................................................................... l
1.1 Preliminary Information ....................................................................................................... l

1.2 Objective and Approach of Present Study ........................................................................... 5

1.3 Organization of the Thesis ................................................................................................... 6
CHAPTER II LITERATURE REVIEW .......................................................................... 8
2.1 Thermodynamics of Solids .................................................................................................. 8

2.2 Mechanics ............................................................................................................................ 9

2.2.1 Kinetics .................................................................................................................. 9

2.2.2 Kinematics ........................................................................................................... 11

2.2.3 Failure .................................................................................................................. 12

2.3 Methods of Solution ........................................................................................................... 13

2.3.1 Comparison of Numerical Method ...................................................................... 13

2.3.2 Computational Cost and Interpolation Function of FEM .................................... 14

2.3.3 Error Associated with FEM ................................................................................. 15

CHAPTER III VERIFICATION OF PREVIOUS WORK ............................................ 19
3.1 Mr. Barr’s Work from 1992 ............................................................................................... 19

3.2 Verification of Barr’s Work ................................................................................................ 26

3.2.1 Beam Models ....................................................................................................... 27

3.2.2 Three Dimensional Flat Mirror Models .............................................................. 31

3.3 Summary ............................................................................................................................ 34

CHAPTER IV IMPLEMENTATION OF NONHOMOGENEOUS CTE AND

VALIDATION OF FEM RESULT ...................................................................... 36
4.1 Assumptions ....................................................................................................................... 36
4.2 CTE and Temperature Relation ......................................................................................... 37
4.3 The Effects of the Mirror Curvature .................................................................................. 40

vi

 

 

 

CHKPT.

4.4 Local CTE Determination .................................................................................................. 44
4.5 FORTRAN Code Development ......................................................................................... 45
4.6 Using Exact Solutions to Insure the Correctness of MARC .............................................. 48
4.6.1 A Thin Isotropic Circular Plate with Arbitrary Radial Temperature Variations..49

4.6.2 A Hollow Isotropic Sphere with Arbitrary Centrifugal Temperature Variations 55

4.7 Numerical Comparison Between MARC and ANSYS ..................................................... 61
4.8 Single Boule Analysis: Convergence ................................................................................. 62
4.9 Summary ............................................................................................................................ 64
CHAPTER V MODEL SIMULATION AND INITIAL SOLUTION .......................... 66
5.1 Mirror Thickness ................................................................................................................ 66
5.1.1 The Axisymmetric Analysis for the Different Thicknesses ................................ 67

5.2 Initial Analysis of Full Scale Model with an Introduction to the Biaxial CTE Variation..72
5.3 Summary ............................................................................................................................ 80

CHAPTER VI SURFACE DISTORTION SIMPLIFICATION AND MINIMIZATION .

................................................................................................................................ 81

6.1 Axisymmetric Analysis and Interfacial Mechanics ........................................................... 81

6.2 Coarse Full Scale 3-D Model Simulation .......................................................................... 84

6.2.1 Initial Attempts for the 3-D Simulation ............................................................... 84

6.2.2 Final Boule Placement Patterns for the Coarse 3-D Model Simulation .............. 86

6.3 Surface Distortions by Refined 3-D FE Model (the final result) ....................................... 93

6.3.1 On-axis ................................................................................................................ 93

6.3.2 Off-axis ................................................................................................................ 99

6.4 Summary .......................................................................................................................... 104
CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS ............................. 105
7.1 Conclusions ...................................................................................................................... 105

7.2 Recommendations ............................................................................................................ 106
REFERENCES ................................................................................................................ 109
APPENDIX A PROPERTIES OF ULETM BOULE AND CTE VARIATION ............ 114
APPENDIX B FE CONTOUR PLOT OF BEAM MODELS ..................................... 140
APPENDIX C FORTRAN CODE, CTE.F .................................................................. 147

vii

APPEN]

APPEXl

APPENDIX D BOULE PLACEMENT AND SURFACE DISPLACEMENT
OF ON-AXIS MIRROR ...................................................................................... 160

APPENDIX E BOULE PLACEMENT AND SURFACE DISPLACEMENT
OF OFF-AXIS MIRROR ..................................................................................... 178

viii

Table 3. 1

Table 3. 2

Table 3. 3

Table 4.1

Table 6. 1

Table A1.

LIST OF TABLES

The temperature distribution of Barr’s FE beam models (°C) .................... 21
Mean CTE values and mechanical properties used for Figure 3.3. ........... 25
Result comparison of beam analysis. ......................................................... 28
Total number of nodes in each model and displacement error ................... 63
Results of different boule combination for axisymmetric analysis ............ 81
ULETM properties. .................................................................................... 115

ix

Figure 1.1

Figure 1.2

Figure 1.3
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7
Figure 3.8
Figure 3.9
Figure 4.1
Figure 4.2
Figure 4.3

Figure 4.4

LIST OF FIGURES

Geometry of the primary mirror: on-axis design with a one-meter diameter
center hole .................................................................................................... 3

Geometry of the primary mirror: Off-axis design without the hole (same

dimensions as Figure 1.1) ............................................................................ 4
Contour plot of an idea form Of surface distortions. .................................... 5
Barr’s FE beam models. ............................................................................. 20
Barr’s FE mirror model (brick element). ................................................... 22
Barr’s final boule pair placement [2]. ........................................................ 24
FE models for present study, Quad 4, Type 11 [20]. ................................. 27
Y-displacement caused by a linear temperature distribution, model 8-layer
A (see Table 3.1) ........................................................................................ 29
Y-displacement caused by a nonlinear temperature distribution, model 8-
layer C (see Table 3.1) ............................................................................... 30
2-layer, 20 cm flat mirror FE model, Hex 8, Type 7 elements [20]. .......... 31
Contour plot, z-displacement of the 20 cm flat mirror. .............................. 32
Contour plot, resultant displacement of the 20 cm fiat mirror. .................. 33
Enthalpy change with respect to temperature, for SiOz. ............................ 39
Enthalpy change with respect to temperature, for Ti02. ........................... 39
A circular plate without curves, Quad 4, Type 3, 1500 elements [20] ....... 41

A circular plate with curvature (16 m), Quad 4, Type 72, 1536 elements [20].
.................................................................................................................... 42

 

 

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Figure 4.7

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Figure 4.9

Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Figure 4.19
Figure 4.20
Figure 4.21

Figure 5.1

Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5

Figure 5.6

Data measurement of boule CT E. .............................................................. 44

The flow chart of the FORTRAN code. ..................................................... 47
The temperature distribution of Equation 4.10. ......................................... 48
Exact solutions of equations 4.11~4.13. .................................................... 50
3-D FE model for the solid thin circular plate, Hex 20, Type 21. [20] ...... 51
Radial displacements of the thin circular plate. ......................................... 52
Radial stresses of the thin circular plate. ................................................... 53
Tangential stresses of the thin circular plate. ............................................. 54
Exact solutions of equations 4.14~4.16. .................................................... 56
1/8 of the 3-D sphere model, Hax 20, Type 21[20]. .................................. 57
Radial displacements of the 1/8 sphere ...................................................... 58
Radial stresses of the 1/8 sphere. ............................................................... 59
Tangential stresses of the 1/8 sphere. ......................................................... 60
Z—displacement contours, by AN SYS. ....................................................... 61
Z-displacement contours, by MARC. ........................................................ 62
Convergence of the z-displacement of the center node. ............................ 63
Convergence of the z-displacement of a vertex node. ............................... 64

Graphical representation of the maximum-normal-stress theory (Juvinall

[17], pp. 214) .............................................................................................. 67
FE axisymmetric model. ............................................................................ 68
Principal stress maximum of the axisymmetric 10 cm model. .................. 69
Principal stress maximum of the axisymmetric 20 cm model. .................. 70
Surface displacement of the axisymmetric model ..................................... 71
Gradient of surface shear Of the axisymmetric model. .............................. 72

xi

 

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Figure 5.1:

”final

Figure 5.7

Figure 5.8
Figure 5.9
Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10
Figure 6.11

Figure 6.12

On-axis full scale FE model, Hex 20, Type 21 [20]. (the reflecting surface

is faced the negative z-direction) ............................................................... 73
Off-axis full scale FE model, Hex 20, Type 21 [20]. ................................. 74
Constrains of off-axis FE model. ............................................................... 74
The initial boule placement for 10 cm thick mirror. .................................. 75

On-axis principal Cauchy stress maximum for Figure 5.10 boule
placement, on-axis ..................................................................................... 76

off-axis principal Cauchy stress maximum for Figure 5.10 boule
placement, off-axis ..................................................................................... 77

Mirror surface distortions, z-displacement, for Figure 5.10 boule
placement, on-axis ..................................................................................... 78

Mirror surface distortions, z-displacement, for Figure 5.10 boule
placement, Off-axis ..................................................................................... 79

Gradients of shear strains of the models in Table 6.1. ............................... 83

Off-axis coarse mesh FE model, Hex 20 and Type 21 [20], 4 elements

through the thickness ................................................................................. 85
On-axis coarse mesh FE model, Hex 20 and Type 21 [20], 4 elements

through the thickness ................................................................................. 85
The final boule placement pattern for on-axis. .......................................... 87
Surface displacement contours of Figure 6.4, 3-D coarse model. ............. 88
Principal stress maximum contours of Figure 6.4, 3-D coarse model. ...... 89
The final boule placement pattern for off-axis. .......................................... 90
Surface displacement contours of Figure 6.7, 3-D coarse model. ............. 91
Principal stress maximum contours of Figure 6.7, 3-D coarse model. ...... 92
Final result of surface displacement, on-axis ............................................. 94
The final result of principal stress maximum, on—axis. .............................. 95
The final result of inplane shear Cauchy stress 12, on-axis. ...................... 96

xii

 

 

 

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Figure A1.
Figure A2.
Figure A3.
Figure A4.
Figure A5.
Figure A6.
Figure A7.
Figure A8.

Figure A9.

Figure A10.
Figure All.
Figure A12.
Figure A13.
Figure A14.

Figure A15.

The final result of transverse shear Chuchy stress 31, on-axis. ................. 97
The final result of transverse shear Cauchy stress 23, on-axis. ................. 98
Final result of surface displacement, off-axis. ........................................... 99
The final result of principal stress maximum, off-axis. ........................... 100
The final result of inplane Shear Chuchy stress 12, off-axis. ................... 101
The final result of transverse shear Chuchy stress 31, off-axis. ............... 102
The final result of transverse shear Cauchy stress 23, Off-axis. ............... 103
Flow chart of iiGA for this particular mirror analysis. ............................ 108
CTE variation of boule A. ........................................................................ 116
CTE variation of boule B. ........................................................................ 117
CTE variation of boule C. ........................................................................ 118
CTE variation of boule D. ........................................................................ 119
CTE variation of boule E. ........................................................................ 120
CTE variation of boule F. ......................................................................... 121
CTE variation of boule G. ........................................................................ 122
CTE variation of boule H ......................................................................... 123
CTE variation of boule I. ......................................................................... 124
CTE variation of boule J. ......................................................................... 125
CTE variation of boule K. ........................................................................ 126
CTE variation of boule L. ........................................................................ 127
CTE variation of boule M. ....................................................................... 128
CTE variation of boule N ......................................................................... 129
CTE variation of boule O. ........................................................................ 130

xiii

 

 

 

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Figure A
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Figure A3.

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Figure 84.
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Figure B8,

Figure B9,

Figure A16.
Figure A17.
Figure A18.
Figure A19.
Figure A20.
Figure A21.
Figure A22.
Figure A23.

Figure A24.

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Figure B3.
Figure B4.
Figure B5.
Figure B6.
Figure B7.
Figure B8.
Figure B9.
Figure B10.
Figure B11.
Figure B12.
Figure D1.

Figure D2.

CTE variation of boule P. ......................................................................... 131

CTE variation of boule Q. ........................................................................ 132
CTE variation Of boule R. ........................................................................ 133
CTE variation of boule S. ........................................................................ 134
CTE variation of boule T. ........................................................................ 135
CTE variation of boule U. ........................................................................ 136
CTE variation of boule V. ........................................................................ 137
CTE variation of boule W. ....................................................................... 138
CTE variation of boule X. ........................................................................ 139
X—displacement, 2-layer beam (A) ........................................................... 141
Y-displacement, 2-layer beam (A). .......................................................... 141
X-displacement, 4-layer beam (A) ........................................................... 142
Y-displacement, 4-layer beam (A). .......................................................... 142
X-displacement, 4-layer beam (B). .......................................................... 143
Y-displacement, 4-layer beam (B). .......................................................... 143
X-displacement, 8-layer beam (A) ........................................................... 144
Y—displacement, 8-layer beam (A). .......................................................... 144
X-displacement, 8-layer beam (B). .......................................................... 145
Y-displaeement, 8-layer beam (B). .......................................................... 145
X-displacement, 8-1ayer beam (C). .......................................................... 146
Y—displacement, 8-layer beam (C). .......................................................... 146
Trial #1, on-axis. ...................................................................................... 161
Trial #2, on-axis. ...................................................................................... 162

xiv

 

 

 

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Figure D16.

Figure D17.

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Figure E2.
Figure E3.
Figure E4.
Figure E5.
Figure E6.

Figure E7.

Trial #3, on-axis. ...................................................................................... 163

Trial #4, on-axis. ...................................................................................... 164
Trial #5, on-axis. ...................................................................................... 165
Trial #6, on-axis. ...................................................................................... 166
Trial #7, on-axis. ...................................................................................... 167
Trial #8, on-axis. ...................................................................................... 168
Trial #9, on-axis. ...................................................................................... 169
Trial #10, on-axis. .................................................................................... 170
Trial #11, on-axis. .................................................................................... 171
Trial #12, on-axis. .................................................................................... 172
Trial #13, on-axis. .................................................................................... 173
Trial #14, on-axis. .................................................................................... 174
Trial #15, on-axis. .................................................................................... 175
Trial #16, on-axis. .................................................................................... 176
Trial #17, on-axis. .................................................................................... 177
Trial #1, off-axis ....................................................................................... 179
Trial #2, Off-axis ....................................................................................... 180
Trial #3, Off-axis ....................................................................................... 181
Trial #4, Off-axis ....................................................................................... 182
Trial #5, off-axis ....................................................................................... 183
Trial #6, off-axis ....................................................................................... 184
Trial #7, off-axis ....................................................................................... 185

XV

 

 

SOAR
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CHAPTER I
INTRODUCTION

SOAR project is a new-generation telescope. It will provide Opportunities for people
to engage in the major scientific issues of modern astronomy, such as the formation of the
chemical elements in the Big Bang, the growth and the fate of the universal structure, the

formation of galaxies, and the formation of planetary systems. [1]

1.1 Preliminary Information

SOAR project is a 28 million dollar joint project between Brazil, Chile, Michigan
State University (MSU), US National Optical Astronomy Observatory (NOAO), and the
University of North Carolina at Chapel Hill (UNC). The SOAR design employs active
control of Optics and tip/tilt wavefront stabilization to attain median visible-band images
of 0.4 arcsec across at least a five-arcmin field of view [1]. This telescope will be the first
high-resolution telescope in the southern Hemisphere that can be remotely observed by
the educational institutions and scientific organizations in the United States.

The primary mirror of SOAR telescope is designed with a four meter diameter and a
sixteen meter radius of curvature. This primary mirror will be composed of pieces of

ULE“ boules. ULE stands for ultra low expansion. It is a titanium Silicate glass

 

 

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manufactured by Corning Incorporated. These boules will be cut into full and partial hex-
agonal elements and assembled in a mosaic, and then fused in a furnace to join the ele-
ment interfaces to form a four meter diameter mirror. The geometry of the primary mirror
is shown in Figure 1.1 and 1.2.

The primary mirror will be formed and polished at room temperatures in Arizona,
US, and then shipped to Cerro Pachon, Chile, where is the permanent site of SOAR tele-
scope. The average temperature at night in Cerro Pachon is about 0° C. Due to the consid-
erable variations in the coefficient of thermal expansion (CTE) in the manufacture of
ULE” boules, significant mirror warping (due to a 25°C temperature difference) is antici-
pated and must be minimized. Unfortunately, the CTE variation was not able to be con-
trolled precisely during the ULE” boule manufacture process.

The process for making ULETM boules involves flame vapor deposition of $02
(92.5%) and TiOz (7.5%) onto a rotating turntable. The condensation Of the materials is
formed by an overhead gaseous flame that jets onto the turntable in a furnace. Due to
some uncontrollable changes of furnace condition during the formation, the thickness of
these boules is not uniform. The thicknesses of these boules range from 10.16 to 16.51
cm. The CTE values of these boules do not vary linearly in either the axial (z-axis) or
radial (xy-plane) directions. The mean CTE values of ULE” boules range from -3.3 to
6.5 ppb (10").1 Nevertheless, due to the deposition process on the rotating turntable and
despite the nonuniforrnity, the CTE variation Still retains circular symmetry about the

rotating axis (i.e. central axis of each boule) [51].

 

1. See Appendix A. for the detailed CTE variation.

 

 

 

 

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Figure 1.1 Geometry of the primary mirror: on-axis design
with a one-meter diameter center hole.

 

 

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Figure 1.2 Geometry of the primary mirror: Off-axis design
without the hole (same dimensions as Figure 1.1).

 

 

 

Since the mirror is used for high precision optics, the mirror surface distortions are of
an extreme concern. The magnitude of the surface distortions are desired to be less than
0.02 pm (after corrections). Since the CTE variation was uncontrollable during the boule
formation, the mirror distortion due to the temperature change has to be minimized by
arranging the boules’ neighboring patterns. A semi-circular distribution of surface distor-
tions with respect to the mirror of the central axis is desired (i.e. global warping has to be
minimized). Figure 1.3 shows ideal surface distortion contours without global warping.
By obtaining the simple form of distortions, external hydraulic forces which are controlled
by a figure-sensing optical system can then be easily applied to correct local surface dis-

tortions.

 

 

 

 

 

Figure 1.3 Contour plot of an idea form of surface distortions.

1.2 Objective and Approach of Present Study

In 1992, an initial finite element analysis of this primary mirror was preformed by Mr.
Barr for the University of North Carolina. Accordingly, the tasks of this research project
are to verify the work done by Mr. Barr in 1992 [2], and to create reasonable finite element
models that allow more detailed study of surface distortions and interfacial mechanics.
This would entail examining how different boule neighboring patterns, thickness, and
designs affect these mechanisms. Presently, there are 24 boules in inventory. Due to some
serious internal flaws, only 22 out of these 24 boules are usable. This quantity is only

enough to form 11 boule pairs to make a 20 cm thick mirror. In order to save the material,

 

 

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reduce the cost, and to have faster thermal response, a 10 cm thick mirror is also proposed
to be studied. The thermal distortions and interfacial stresses and strains of both on-axis
and off-axis designs as well as both thicknesses, 10 cm and 20 cm, will be studied and
compared.

Due to the geometry complexity and nonhomogeneous CTE values, numerical solu-
tion seems to be the only approach to analyze such a problem. Using finite element
method seems to be a good alternative since it has many diverse commercial packages and
allows one to input material properties that are dissimilar. All numerical results will be
carried out by a computer SUN1 Ultra 1 that has 128 MB of RAM and 2 GB hard drive.

Insights gained from these studies will help in identifying optimum boule neighboring
patterns in order to obtain a simple form of global surface distortion and minimize the

interfacial stresses.

1.3 Organization of the Thesis

The goal of this research project is to minimize mirror warping. Chapter One gives
some brief background of the SOAR project, and preliminary information about the mirror
material. Some substantial papers and materials have been reviewed and summarized in
Chapter Two that help one to understand the material behaviors and to determine the final
results. Barr’s work in 1992 is presented and verified in Chapter Three. Chapter Four
deals with the verification of correctness of the finite element method and adopted soft-

ware. The FORTRAN code, a critical tool which was used to assist the FE models in

 

1. SUN Microsystems
http://www.sun.com

 

 

order to ca;
forming an

cussed in Cl
forms surfac
model. The
rors are also

darions are 9:

order to capture the biaxial CT E variation, is developed in Chapter Four as well. By
forming an axisymmetric analysis, the decision of which mirror thickness to use is dis-
cussed in Chapter Five, along with an initial full scale mirror analysis. Chapter Six pre-
forms surface distortion simplification and minimization analyses using the axisymmetric
model. The final results of the global surface distortion of both on-axis and Off-axis mir-
rors are also presented by full scale fine mesh models. Final conclusions and recommen-

dations are given in Chapter Seven.

 

 

 

The re”:
numerical m1
egorized in ii:
cles that ha
composites {u
material beha‘

nonhomogenen

l&}ered compo

 

CHAPTER II
LITERATURE REVIEW

The review addresses material thermal expansion properties, mechanics behavior, and
numerical method suitabilities. Accordingly, previous contributions are reviewed and cat-
egorized in thermodynamics of solids, solid mechanics, and methods of solution. The arti-
cles that have been reviewed are fundamental mechanics theories, multilayered
composites (which are treated as layer-wise isotropic), and orthotropic and homogeneous
material behaviors. Although no article was found which directly related to analyzing a
nonhomogeneous material structure, the concepts and knowledge gained by understanding

layered composites and orthotropic materials are indeed essential for this research project.

2.1 Thermodynamics of Solids

CTE is a thermodynamic property that provides a measure of density change in
response to a change of temperature with constant pressure. The isobaric CTE is essen-
tially a function of material’s enthalpy. The enthalpy is defined as the heat required to
increase the temperature of one mole of solid from T1 to T2 at constant pressure. The var-
iation of this thermodynamic property is also associated with temperatures. For a majority

of materials, enthalpies are relatively insensitive to the pressure in the range of 0 to 1 atm.

Since 1h:
tion can

or :1 din.
510,. an;

enthaipy

 

of mirror

Stnitd [he

23 Merl

One of
due to a n“.

0“ W1. 1h.

Since the SOAR telescope is Operated on the earth surface, the atmosphere pressure varia-
tion can be neglected [3]. The CTE of ULETM should be regarded as either a static value
or a dynamic value that would depend upon the enthalpies of the majority composition,
SiOz, and the range of temperature variation. Regarding the primary rrrirror analysis, the
enthalpy variation of Si02 within the 25° C range has to be determined prior to a decision
being made of whether or not to use a static or a dynamic value. The design consideration
of mirror material was carried out by Gulati [54]. Edwards, Bullock, and Morton pre-

sented the absolute thermal expansion measurement for ULEWl glass in 1996 [51].

2.2 Mechanics

One of the causes of stress in a body is nonuniform heating. When a body expands
due to a rise in temperature, this expansion generally cannot proceed freely in a continu-

ous body, then stresses due to the heating take place.

2.2.1 Kinetics

In this primary mirror case, even though we assume a uniform temperature increase
through out the body, the nonuniform CTE value will cause the same kind of stresses as
those caused by temperature gradients. Since linear thermal expansion for homogeneous
isotropic materials is the product of CTE and changing temperatures, the expansion
depends upon the CTE change and/or the temperature change. If we simulate the nonho-
mogeneous CTE boule as a multi-layer composite, it will have in-plane stresses (on, Oyy,

Oxy) acting in the layers themselves, along with transverse Shearing (on, 6,2), and

 

 

level and
stress is I
With an it
transverse

stresses d.

matron the.
thickness-{C

IO

transverse normal stresses (on).

Suhir [4] studied and proved that for multilayered elastic films the in-plane stresses are
responsible for the strength of the layers and the transverse stresses are responsible for
peeling failure. These interfacial stresses increase with an increase in the in-plane stress
level and in the thicknesses of the layers. It should be noted that the maximum interfacial
stress is located on the “free ends” of the structure. The interfacial stresses also increase
with an increase in the interfacial stiffness. For a thick-section composite, even though the
transverse stresses are generally smaller than other stress components, the transverse
stresses do have a significant interactive effect on material failures under in-plane stress,
especially when this stress is compressive.

Reddy [16] states the shortcomings of classical plate theory. Kirchhoff-Love’s
hypotheses neglects the transverse shear deformation, known as the Love’s first-approxi-
mation theories (Love, 1888). The theories yield sufficiently accurate results when a
thickness-to-diameter ratio is small. The thickness-tO-diameter ratio of this primary mir-
ror is small ( s 1%). The accuracy of these theories, however, is only held in the case if
material anisotropy is not severe. Since the anisotropy caused by the nonhomogeneous
CTE is present in the ULETM glass, we conclude that the transverse shear stresses will def-
initely draw our attentions toward this research.

By studying a bonded dissimilar material, Chou [5] stated that stresses are stresses no
matter how they originate. Stresses due to externally applied loads, internally residual
stresses, phase transition-induced stress, thermal-mismatch stresses, and various combina-
tions of stresses all cause materials to fail. The combined Stresses always exceed a certain

fracture limit and occur where the temperature gradient is maximum between two bonded

 

 

materials
mostly be
obtain the
are impor
neighborin
Exact s
direction a:
are found 11
Iheiranajj 5,
effective CT
tions Were i
LObode [316]
mUOdUCed eq
50iUthns of a

hfipful fOr Ver

2.22 K“Nina

Chou [5] Cc

face p r0Fifties.

”Wis CTE ShOu]

nalg [Or “in
I‘ ‘1 ‘

Edie,” in p»

 

11

materials. In this mirror case, the interfaces between these hexagonal boule elements will
mostly be the locations that failure takes place, if fracture occurs at all. While trying to
obtain the simple form of surface distortions, the interfacial stresses and strain gradients
are important and will provide essential information to help arrange the desired boule
neighboring patterns.

Exact solutions of a thin circular disk with arbitrary temperature variation in its radial
direction and a hollow sphere with arbitrarily centrifugal symmetry temperature variation
are found in both Timoshenko [12] and Boley [25]. Wu, Tarn, and Yang [31] presented
their analyses in therrnoelastic laminated Shells and Lutz [33] studied thermal stresses and
effective CTE of a functionally gradient sphere.l Some orthotropic therrnoelastic solu-
tions were presented by Tauchert [35], Kalam and Tauchert [55] (cylinder problem),
Loboda [36] and Baker [37] (slab problem). Khoma [38], Shvets and Flyachok [39]
introduced equations and theorems for anisotropic Shells, and Ma [40] presented analytical
solutions of anisotropic bimaterial elastic wedges. These studies and solutions can be

helpful for verifying numerical solutions.

2.2.2 Kinematics

Chou [5] concluded that thermal expansion and the resultant strains and stresses are
strongly affected by the gradient of thermal-mismatch, geometries, dimensions, and sur-
face properties. Therefore the curvature of the mirror, the thickness, and the nonhomoge-

neous CTE should all be evaluated in the final results. Neither one of these properties can

 

1. Materials for which their microstructure is controlled during fabrication in order to create a desired spa-
tial gradient in properties such as elastic moduli and CTE.

 

be neglc

The
equal to
a surfac
L'gural [
from the
about the
loading [1
mid} [his
merits, T1

655315 Site

12

be neglected or analysed separately.

The mirror is regarded as a thin shell since its thickness to diameter ratio is less than or
equal to 1/20. For homogenous materials, the plate and shell bending theory indicates that
a surface displacement due to external forces varies inversely with thickness cubed.
Ugural [11] also stated that due to the shell curvature, any arc length located a distance
from the mid-surface is a function of the radii of principal curves. A similar statement
about the curvature effect was also shown on a problem of a Spherical shell under thermal
loading by Timoshenko and Goodier [12]. These analytical solutions supported Chou’s
study that the mirror thickness and curvature result in Significantly different displace-
ments. Therefore, the analysis cannot be carried out without considering the geometrical

effects such as different mirror designs and thicknesses.

2.2.3 Failure

Distortion, or plastic strain, is associated with Shear stresses and involves slip along
natural Slip planes of materials. Failure is defined as the point at which the plastic defor-
mation reaches an arbitrary limit. The maximum-distortion-energy theory postulates that
any elastically stressed body has a capacity limit to absorb energy of distortion. This
energy tends to change material’s shape but not the Size. The energy attempts to subject
the material to larger amounts of distortion energy which causes it to yield. This is known
as von Mises yield criteria. When the equivalent stress exceeds the material’s yield stress,
the material failure that occurs is often governed by Tresca criterion or maximum-Shear-
stress theory. This theory correlates reasonably well with the yielding of ductile materi-

als, and the distortion energy theory agrees very well with the maximum-shear-stress

 

 

theolF JU‘
uith 155‘ d
maximum"
mechaniCS.
principal str

pa] planes.

2.3 Metho

The gox'ei
lyrically when
the primary m
glass. the com
unsoli'ahle ana.

only choice.

13.1 Compari

L' .
Sing numc.

There are three

 

Ihfi‘erence Mei-

 

l3

theory. Juvinall [17] stated that maximum-normal-stress theory correlates reasonably well
with test data for brittle fractures. Because of the brittleness of the UL T" glass, the
maximum-normal-stress criterion will be adopted for failure analysis. For interfacial
mechanics, we will study components of Cauchy stress (based on deformed body) and
principal stress of certain locations since maximum normal stress at a point acts on princi-

pal planes.

2.3 Methods of Solution

The governing partial differential equations of plates and shells cannot be solved ana-
lytically when complex geometries, boundary conditions, and loadings are present. Since
the primary mirror is dome-Shaped and is composed of many pieces of hexagonal UL T”
glass, the complex thermal expansion properties and the geometries make the problem
unsolvable analytically. Therefore, adoption of a numerical approach that seems to be the

only choice.

2.3.1 Comparison of Numerical Method

Using numerical methods facilitate the solution of these partial differential equations.
There are three numerical methods that have been substantially studied; they are Finite
Difference Method (FDM), Boundary Element Method (BEM), and Finite Element
Method (FEM). Although FDM has advantages of simplicity, ease of use, and minimal
computer storage required, it lacks geometric flexibility in fitting awkward geometries.

BEM substantially reduces computer storage and requires smaller amount of data to define

 

the georn
gross inh«
interest.

defined ox
at internal
tions of di:

this primar

133. Corr

14

the geometries, but it is more difficult to program for efficient computer execution and
gross inhomogeneities associated with distinct changes of material within the region Of
interest. Fenner [30] compared these three methods and concluded that for equations
defined over arbitrary domains and that have a substantial number of values of unknowns
at internal points, FEM is the most effective method. In addition, FEM allows connec-
tions of dissimilar materials in distinct subregions which iS the most important concern of

this primary mirror analysis.

2.3.2 Computational Cost and Interpolation Function of FEM

In order to reduce the cost of FE analysis, generally, we take advantage of symmetry of
one or more axes of the model. This reduction is based upon symmetric boundary condi-
tions, symmetric geometries, and symmetric meshes (if possible) [19]. Quite Often in FE
analysis, the determination of symmetry can dramatically reduce the efforts of modelling
and the costs of computation. Therefore for FE analysis, the determination of symmetry is
a primary concern when constructing a FE model. Choosing an appropriate interpolation
function (i.e. element type and class) can reduce costs as well. The choice depends upon
the nature of the problem. Certainly these two tactics can be properly combined to pro-
mote the savings.

Based on the total potential energy principle, the finite-element model for the classical
plate theory requires third or higher-order interpolation functions in the approximation of
the transverse deflection. Another facilitated approach, the total potential energy principle
of the first-order shear deformation theory allows the finite element models to use linear or

higher-order interpolation functions for all displacements. Thus the classical theory of

 

plates iS 31.91
theory helps
from the 50‘s"
by using red.
methods caut:

According
mesh over the
thcless. intern
Although usir.
elements. the
through the th;

is constructed.

2.33 Error A

There are ,
These errors aft
0' numerical m.

“Ch approxtm

 

DlScrethau-OH [A

 

15

plates is algebraically complex and computationally expensive. The shear deformation
theory helps to reduce the computational cost, but the Shear deformable elements suffer
from the so-called locking phenomenon [16]. Although the locking can be circumvented
by using reduced integration or through other techniques, one should'adopt numerical
methods cautiously to prevent an unreliable result.

According to the geometry of the primary mirror, shell elements seem to be suitable to
mesh over the curved domain in case a three-dimensional (3-D) model is needed. Never-
theless, interior detailed mechanics information is desired when a 3-D analysis is required.
Although using shell elements to capture the curved domain would be better than brick
elements, the former does not present transverse stresses (i.e. stresses vary linearly
through the thickness) [20]. Therefore brick elements will be used when a 3-D FE model

is constructed, and a fine mesh Should compensate its shortcomings.

2.3.3 Error Associated with FEM

There are sources Of error that may cause inaccuracy of finite element solutions.
These errors are associated with the “approximation” which is most common type of error
of numerical methods. This inaccuracy can be greatly improved by cautiously adopting

each “approximation” step.

Discretization error
FEM discretizes a whole domain into a specific finite number of subregions called ele-
ments. Accordingly, the geometrically complex domain is then approximated by these

simple subregions. This leads to a result which is then approximated in an element-wise

 

 

manner.
of the o

ensured

Formula
lnexa
error. In
from the ;
successful

and its sen

Locking
Another

PTOdUCes res

aCCOUm in [it

fit .

«at “ere Chc

mm“ anttcn

5011130118“) lo

PittnounCed in

In the Past,

ii
If shear S

Erin.

Ciiied

16

manner. Therefore, an inappropriate element mesh which lacks a detailed representation
of the original domain can lead an inaccurate result. This approximate result may be

ensured by stages of mesh refinement [21].

Formulation error

Inexact evaluation of shape functions and boundary conditions can cause formulation
error. In practical FEM analyses, the boundary conditions are often difficult to abstract
from the physical situation. Simulation of reasonable boundary conditions is the key for a
successful FE analysis. A lack of knowledge of the real structure, its working function,

and its serving environment can cause a FEM result to fail [22].

Locking

Another severe but implicit error is caused by the locking phenomenon. Such problem
produces results that are only a fraction of a percent of the correct results at any practical
level Of discretization. The cause of locking is due to consistency not being taken into
account in the interpolation function. The consistency requires the interpolation functions
that were chosen to initiate the discretization process to also ensure that any Special con-
straints anticipated must be allowed for in a consistent way. Failure of consistency causes
solutions to lock to erroneous answers. The difficulties caused by inconsistency are most
pronounced in the lowest-order elements based on linear interpolation functions [23].

In the past, beam, plate, and shell elements formed poorly because they could not meet
the shear strain-free condition. These elements were excessively stiff and exhibited SO-

called shear locking. Later, many “tricks” (reduced integration, energy balancing, etc.)

 

were trier
ments [22
plate theo
pler in 19‘
neither ret
ertheless.
interpolatj.
as it has be
refinement
“hen c
Smitture is
131“ 31 plate
amOdel )’ie
1011! of Cm.“
”Wane 1.

[1118 locking l

pOLSSOD's rat

When a fi

17

were tried out on the displacement formulations and resulted in acceptably accurate ele-
ments [23]. A simplest rectangular 4-node element for thin plate using Reissner-Mindlin
plate theory that is free of shear locking was presented by Oguamanam, Hansen, and Hep-
pler in 1998 [43]. The element was numerically proved free of shear locking and required
neither reduced integration nor transverse shear strain interpolation or manipulation. Nev-
ertheless, this new element is not available yet in MARC. MARC uses assumed strain
interpolation formulation to improve the bending performance for class 4 elements. Thus,
as it has been mentioned earlier, high-order type of elements, reduced integration, or mesh
refinement can be adopted to avoid shear locking.

When dealing with thin flexible structures, an efficient way to construct a thin flexible
structure is using a shell. If the shell surface is replaced with flat triangular and/or quadri-
lateral plate elements, a membrane stiffness is superimposed on a bending stiffness. Such
a model yields inaccurate results with coarse meshes [23]. Namely, the very poor behav-
iour of curved elements cause looking as they become thin. This phenomenon is known as
membrane locking. A consistent representation of membrane strains is desired to avoid

this locking [45].

Poisson’s ratio stiffening effect

When a flexural action of a beam, plate, or shell model is meshed by a single four-
node plane stress element or eight-node brick element through the model depth (thick-
ness), a phenomenon called parasitic shear takes place. This effect can be removed by
using a reduced integration for the shear strain energy which makes the element field con-

sistent. However, even with such field-consistent element, it is found that results are

 

stiffer by
mentionet
mesh refit

the depth

18

stiffer by about 10%, depending upon the Poisson’s ratio of materials [23]. This effect, as
mentioned previously, only occurs in the case with one element through the depth. A
mesh refinement in the depth should lead to a better result Since more elements through

the depth can correctly sense the change of Sign of strains (81) through the depth.

 

Mr. Barr
Squires data

be less than

CHAPTER III
VERIFICATION OF PREVIOUS WORK

Mr. Barr presented that the variation of boule CTE values could be simplified by least
squares data fitting to approximate the actual CTE values. The overall error was proved to
be less than 10% by FE beam models. Accordingly the analysis was Simplified upon the
assumption of using a constant CTE through the radial direction (on xy-plane), and the
CTE variation was only counted in the axial direction (through the thickness, z-direction).
A 3-D FE rrrirror model was modelled as a flat circular plate without curvatures. By using
mean CTE values and a flat coarse mesh finite element model (Figure 3.2), a simple form

of global surface distortion was achieved.

3.1 Mr. Barr’s Work from 1992

Barr [2] stated that in order to simplify the FE model for the primary mirror analysis, it
was desirable to limit the number of element layers. He suggested that the least squares
linear fitting technique would be a good approach to simplify the CTE variations. Three
mulit-layered rectangular FE beam models, with dimensions of 30 in x 8 in x 8 in, were
created to verify the CTE simplification analysis. The beam model may be treated as a

thin radial Slice of a circular boule. The beams’ thermal distortion would approximate the

19

 

behaviors of
with difl'eren
The bear
quadrilateral
fired at the 1.
late the free .
Ppb throught
Simulate the

Mid.

20

behaviors of a circular multi-layered disk having symmetric properties about its center, but
with different absolute distortion amplitudes.

The beam models were constructed with two, four, and eight layers of linear four-node
quadrilateral elements. Figure 3.1 illustrates Barr’s FE beam models. All beams were
fixed at the left end to simulate the center of the boule and freed at the other end to simu-
late the free outer rim. The beam models were imposed with a constant CTE value of 15
ppb throughout the entire beam. Different temperatures were assigned to each layer to
simulate the variations of CTE. Table 3.1 lists the temperature distributions of each beam

model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2-layer beam

M‘UN—

 

4-layer beam

 

Ofl~IOM§UN~

Y I 8-layer beam

Figure 3.1 Barr’s FE beam models.

 

Ta

The temp
along its leng
direction),l r
for these ihrer
Same linear \-

then [he “116$

21

Table 3. l The temperature distribution of Barr’s FE beam models (°C).

MWWW

(A) (A) (B) (A) (B) (C)
1 45 45 4O 45 40 45
2 25 35 40 40 40 35
3 5 25 25 35 40 40
4 15 10 30 25 55
5 5 10 25 25 -25
6 20 25 45
7 15 10 35
8 10 10 -10
9 5 10 5

The temperatures were made constant across the width (z-direction) of the beam and
along its length (x-direction) so that the critical variable would be the axial variations (y-
direction).1 The objective was to compare the distortion contour patterns and amplitudes
for these three beam models to see if different temperature distributions would result in the
same linear variation of distortions through the depth of the beams (y-direction). If so,
then the linear fit simplification would be applicable to reduce the model layer.

Barr’s FE beam models displayed a deflection contour line which showed that there
was hardly any difference between these models in both x and y-directions. The differ-
ences between each contour pattern were difficult to detect visually. The results Showed
that the maximum difference of deflection between these models was 9%. It was con-
cluded that “one could use least squares fits to CTE data without introducing large errors
in the finite element results and that fewer layers would still produce acceptably accurate

results” [2].

1. See Figure 3.1

A three

the entire m

elements 1Fi

 

 

22

  

\Dn‘ ill!!!“
owed. .. in?
flfliioooooo b‘l‘oflifllflo

§
~fl¢¢¢fiflflfl B... O‘fi’

a... $3» an... .. :45:

i... ll l‘ o 9 iii. -
.....x.§.. eavnfiss
as. «nun» u flue...“
3:0"47‘4 fioflfliizf
u. o oiflfflo o o o flflilo -
- ooflflufihu I’oo’o oolliflf
9...... ”fiooovliiuflfi
ooillvofiliu
oi.§§§\\\§

      
  
    
   
      

  

                 

A three dimensional FE model was constructed to simulate the surface distortions of

the entire mirror. By using a FE program, GIFTS], the model was meshed with 960 brick

elements (Figure 3.2).2

Figure 3.2 Barr’s FE mirror model (brick element).

Instead of assigning different material properties to the FE model, Barr derived an arti-

ficial temperature formula to calculate local temperatures to simulate the CTE change.

The temperature difference to be experienced by the nrirror was assumed to be constant,

however, the thermal expansion/contraction of an isotropic elastic solid is just the product

 

1. Casa GIFTS, Inc., Tucson, AZ

2. The element class and type were not specified in Barr’s report [2]

 

 

23

Of the change in temperature and CTE. Therefore by taking the temperature difference to
simulate the CTE variation in the material was a clever idea. The derivation of the artifi-

cial temperature is as follows:

[(Tz-T1)+AT]a=(T2—Tr) (01+Aor) (3.1)

AT = A01 (T2 - T1)/0t (3.2)

let the artificial temperature be

T’ = (T2 - T1) + AT (3.3)

therefore

T’ = (T2 - T1) (1 + Atx / a) (3.4)

Where

(T2 - T1) = 25 is the predetermined temperature change

AT is the temperature increment

or is the average (71' E

Ad is the difi'erence between the local CT E and the boule mean CT E
T’ is the artificial temperature that simulates the local CT E variation

With the implementation of the artificial temperatures and some adjustments (trial and
error), the FE model shown in Figure 3.2 gave satisfactory surface distortions in a simply

circular form with a maximum amplitude of 0.45 pm [2]. Figure 3.3 shows the final boule

pair placement.

b-im

Fm A _.

24

 

mum

    
  

b-inv/u

\nv/u

- inv/u

      

e-inv/d

 

b-inv b-inv/u 2L

X

1

 

 

 

Figure 3.3 Barr’s final boule pair placement [2].1

 

l. Boule pairs are made by fusing two boule together to form a 20 cm thick structure.
The “inv” stands for inverse, wherein the boule is flipped up side down.

Tab}
Boule ID
mm
e-im‘fd
f/c

a/m

25

Table 3. 2 Mean CTE values and mechanical properties used for Figure 3.3.

Boule ID Mean CTE (ppb) Young’s modulus (Nlcmz) Poisson’s ratio

w/l-inv -0.88 6.76E7 0.17
e-inv/d 2.33 6.76E7 0.17
f/c 3.02 6.76E7 0. 17
a/m 2.70 6.76E7 0.17
p/t 4.12 6.76E7 0.17
g/i 4.97 6.76E7 0.17
s/r 3.37 6.76E7 0.17
h/j 3.08 6.76E7 0.17
x-inv/u 0.98 6.76E7 0.17
q/v 4.78 6.76E7 0.17

b-inv/u 1 .96 6.76E7 0. l7

26

3.2 Verification of Barr’s Work

The following assumptions and results were presented by Barr’s study in 1992. The

first step of this study was to verify these results.

The CTE variation was simplified by a least square linear fit through the data provided
by Coming, and three FE beam models were employed to support this simplification.
The maximum difference in tip deflection between these beam results was less than
10%.

A simply circular form of surface distortion was achieved and the maximum amplitude
was 0.45 pm

In order to verify Barr’s results, the same boule pair selections, dimensions, and boule

placement pattern as Barr’s will be adopted. FE beam model and a two-layer brick ele-

ment mirror FE model were constructed for this verification. A finite element computa-

tional software named MARCl was employed through out this present study for all

computations. The knowledge gained through this verification will build up a basis which

spans further study for interfacial mechanics.

 

l. MARC Analysis Research Corporation
260 Sheridan Avenue, Suite 309, Palo Alto, CA 94306, USA
http://www.marc.com

3.2.1 Beam Models

 

 

27

 

 

 

Figure 3.4 FE models for present study, Quad 4, Type 11 [20], 8-layer.

 

28

The beam models have dimensions of 76.2 cm x 20.32 cm x 20.32 cm which are fixed
at the left end (u = v = 0) and analyzed using plane stress analysis. FE beam models
shown in Figure 3.4 are meshed with Quad 4, Type 11 elements [20] and are subjected to
the temperature distributions from Table 3.1. The results show that the magnitudes of x
and y-displacement are not in an agreement with Barr’s results. The maximum error is
12.5% (Table 3.3). The node at the right top comer of the beams are selected for displace-
ment comparison. The contour patterns are similar between these beam models (Appen-
dix B). Nevertheless, the eight-layer model C (nonlinear temperature distribution) shows
distinct contours in the y-direction. And thus, the number of layers has to be increased to
offset the severity of nonlinearity. The y-direction is an analogy of the surface distortions
of the primary mirror. Figures 3.5 and 3.6 show contour-line plots of deflections in the y-

direction for linear and nonlinear temperature distribution.1

Table 3. 3 Result comparison of beam analysis.

Beam model X-Deflection (um) X-Error compare to Ref. Y-Deflection (um) Y-Error compare to Ref.

2-Layer (A) 0.5260 Ref. -0.8492 Ref.

4-Layer (A) 0.5185 -1.4% -0.8196 -3.5%
(B) 0.5215 -0.9% -0.8408 -1 %

8-Layer (A) 0.5164 -1.8% -0.8115 -4.4%
(B) 0.5919 12.5% -0.9104 7.2%
(C) 0.5290 0.6% -0.8691 2.3%

 

1. All displacement legends of contour plots through out this thesis are scaled in cm,
and all stress legends of contour plots through out this thesis are sealed in N/cmz.

29

 

“it.“ '(
Til- 2 1.0000000 w U

   

'l_

z x

Dual-cm u

 

 

 

unit: cm

Figure 3.5 Y—displacement caused by a linear temperature distribution,
model 8-layer A (see Table 3.1).

30

 

If: 3 E M A r‘
n- : 1.0000000 WM’J’“

   

L

z x

Duel-cm u

 

 

 

unit: cm

Figure 3.6 Y—displacement caused by a nonlinear temperature distribution,
model 8-layer C (see Table 3.1).

31

3.2.2 Three Dimensional Flat Mirror Models

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.7 2-layer, 20 cm flat mirrorFE model,
Hex 8, Type 7 elements [20].

A two-layer (two elements through the thickness) 20 cm thick flat mirror FE model
meshed with 1596 Hex 8, Type 7 elements (Figure 3.7) was generated using MENTAT].
The model was fixed at the central hole where all nodes on the circumference were con-
strained with u = v = w = 0. Since Barr [2] did not specify the artificial temperature distri-
butions on his FE model, the present FE model for this verification was given the average
CI'E values to each hexagonal boule (see Figure 3.3 and Table 3.2). Since homogeneous
expansion/contraction over a given span is the product of changes in temperature and

CTE, the FE model experiences a temperature rise statically. Because heat transfer is not

 

l. MENTAT is the pre and post processor of MARC.

32

a concern for this analysis, mechanical-thermal coupled analysis is not necessary. An iso-
thermal statical mechanical analysis was adopted for the FE analysis. The CTE for each
boule was given a reference temperature of 0° C, then the model experienced a tempera-

ture rise of 25° C. The results of the z-displacement and the resultant displacement are

shown in the following figures.

 

In: x 1 .h

 

 

 

 

 

unit: cm

Figure 3.8 Contour plot, z-displacement of the 20 cm flat mirror.

33

 

xMAnc

 

 

Job!

 

 

DllDlacOI-nt

 

unit: cm

Figure 3.9 Contour plot, resultant displacement of the 20 cm flat mirror.

Figure 3.8 shows the contour plot of the z-displacement. The contours do not give a
simply circular form of distortion in the z-direction, like those shown in Figure 1.3. The
maximum peak-to-valley amplitude is 0.01369 pm. Referring to Figure 3.3 and Table 3.2,
each boule was assigned an average CTE value. After experiencing a temperature rise, the
flat mirror shows z—displacements in a “localized fashion” (Figure 3.8). Due to the ther-
mal mismatch, the interface regions of these hexagonal boules appear to have slope dis-

continuities. Figure 3.8 distinctly shows the individual boule expansion and the effects of

34

thermal mismatch in the z-direction. Figure 13 of Barr [2] showed that the “w-displace-
ment” in circular form has a peak-to-valley amplitude of approximate 0.45 pm. Some-
how, a discrepancy exists between Barr’s “w-displacement” and the “resultant displac-
ement” shown here in Figure 3.9. Apparently the Figure 13 in Barr [2] may be compara-
ble to Figure 3.9 on the previous page. Even though there are similarities in the form of
distortions, the maximum peak-to-valley amplitude in Figure 3.9 is 0.1799 um that is 60%

less than Barr’s result.

3.3 Summary

Even though these four-node plane elements are stiff for a beam bending analysis, the
through thickness (y-direction) refinement would ensure a satisfactory result (refer to
Chapter Two, Section 2.3.2). Therefore, the MARC result of 12.5% in Table 3.3, does not
agree with “fewer layers would still produce acceptably accurate results” [2]. Even
though the deflection contours are not significantly influenced by the through thickness
refinement under linear temperature distributions, this conclusion is not applicable to non-
linear temperature distributions. When nonlinear temperature variations act on a FE mod-
el, a finer mesh is able to capture better temperature variation details. Thus, nonlinear
temperature distributions do draw our attention towards the primary mirror analysis.

The flat mirror FE model gives a maximum peak-to-valley amplitude of 0.1799 um
which is 60% less than Barr’s result of 0.45 pm. The error may be due to having a finer
mesh and a different element interpolation function. It could also be caused from not
imposing the same artificial temperatures to the present FE model that Barr mentioned.

Without considering the discrepancy of Barr’s w-displacement” and the “resultant

35

displacement” in the present result, the distortion forms are very similar to one another.
However, the two abrupt changes on the right side of the contours shown in Figure 3.9
requests a further adjustment of the boule neighboring.

In conclusion, the work thus far verifies previous assumption and sets a path of the
development for this research. The biaxial CTE variations (artificial temperatures) have to
be imposed to FE models. A fine mesh, especially through the thickness, is needed for
obtaining a good result. Also an actual radius of curvature of the mirror will be consid-

ered.

CHAPTER IV
IMPLEMENTATION OF N ONHOMOGENEOUS CTE
AND VALIDATION OF FEM RESULT

4.1 Assumptions

Isothermal expansion; the temperature change is a constant, 25° C.

The temperature is determined independently of the deformations of the mirror.
. The deformations of the mirror are small.
. The material deforms elastically.

All material properties are independent of temperatures.

. No internal residual stresses in these boules present after the formation and cooling
process.

The boules contain no flaws and inclusions.

The first and second of the above assumptions allow the omission of transient effects
(heat transfer) and mechanical coupling terms disappear. In such cases, the displacements
are subsequently calculated after determining the temperature field in the body. Therefore
the distribution of some boundary traction loading would not influence the temperature
distribution in the body. The third assumption implies that the displacements are suffi-
ciently small so that no distinction is needed between the coordinates of a particle before
and after deformation, and that the displacement gradients are sufficiently small so that

their product terms may be neglected (when an analytical solution is adopted for

36

37

verifications). The fourth avoids a large temperature change that causes large stresses.
Since the temperature change and the range are essentially small so the ULE1M mechanical
properties have no significant change, the fifth assumption facilitates the analysis. The
CTE value should be treated as either a static value or dynamic value which will be dis-
cussed in a later section. The sixth indicates that the ULETM boules are initially in a stress
free state. The last assumption permits us to describe the macroscopic motion and defor-

mation of a solid continuum and to define stress and strain at a point.

4.2 CTE and Temperature Relation

CTE is a thermal property and for most materials, it is a function of temperature.
Whether the CTE is significantly changed by 25° C temperature rise (under atmospheric
pressure) or not has to be determined by looking into the UL TM’s compositions. The
ULE” is composed of Si02 (92.5%) and Ti02 (7.5%). By referring to Chapter Two, Sec-
tion 2.1, the isobaric CTE is essentially a function of a material’s enthalpy. The enthalpy
is defined as the heat required to increase the temperature of one mole of solid from T1 to

T2 at constant pressure. The CTE and enthalpy have the following relations [3]:

a = $.(3_;1)p (4.1)
(3%)7 = —T-a- V+V (4.2)

[H]

38

H = J'deT (4.4)

where

H is enthalpy

V is volume

P is pressure

T is temperature

0t is CT E

Cp is heat capacity

the subscript denotes a constant condition

and the heat capacity functions for SiOz and TiOz are as follows:
[CP(T)]Si02 = 43.93 + 0.03883 T - 9.96E5 T '2 (4.5)

[Cp(T)]1102 = 73.35 + 0.00305 T - 17.03135 T '2 (4.6)

For a majority of materials, enthalpies are relatively insensitive to the pressure in the
range of 0 to 1 atm. Since the SOAR telescope is operated on the earth surface, the effect
of the atmosphere pressure variation can be neglected [3]. The CTE of ULE” should be
regarded as either a static value or a dynamic value that depends on enthalpies of the
majority composition, SiOz, and the range of temperature variation. The following figures
show the enthalpy variation with respect to temperature for SiOz and Ti02 (plots of Equa-

tion 4.4 with substitutions of Equations 4.5 and 4.6).

39

 

 

 

 

 

 

Enthalpy (Si02)
4E+4
9 .
E 3E+4
5 234.4 ,
::
CEO I l l l I l r
298 373 448 523 598 673 748 823 898
Temperature (K)

 

 

Figure 4.1 Enthalpy change with respect to temperature, for Si02.

 

 

 

 

 

Enthalpy (Ti02)
1.5E+6
Q IE+5 «
E
3 5E+4 ‘
:1:
0E0 T l I T T
298 598 898 1198 1498 1798
Temperature (K)

 

 

 

Figure 4.2 Enthalpy change with respect to temperature, for Ti02.

According to these two plots and Equations 4.1 and 4.4, the CTE change of the ULETM

is insignificant within the ambient tempreature range. Therefore, the CTE value can be

treated as a static value for the mirror analysis.

40

4.3 The Effects of the Mirror Curvature

Barr’s FE mirror was not modelled with curves. Chou [5] stated that thermal expan-
sion and the resultant strains and stresses are strongly affected by the gradient of thermal-
mismatch, geometries, dimensions, and surface properties. Ugural [11] stated that due to
shell curvature, any arc length located a distance from the mid-surface is a function of the
radii of principal curvature. The following analytical solution for a homogeneous hollow
sphere with T = T(r) from Timoshenko [12] (pp. 453) and two FE numerical results (Fig-

ures 4.3 and 4.4) show the significance of the curvature effect.

H

+ l 2 62
u:1_v.a.;3-J:(T-r)drarcl-r7 (4.7)

<

 

41

 

 

1.111.“

0.101.“

 

 

 

Figure 4.3 A circular plate without curves,
Quad 4, Type 3, 1500 elements [20].

 

 

 

 

42

 

 

w“

 

 

 

Figure 4.4 Acircular plate with curvature(l6m),
Quad 4, Type 72, 1536 elements [20].

 

The C1 31
boundm' C0
ture spatial \
rm, This re
of principal '
4311500<fle
kammflm
rrmst EVE:
ndmeekm
sgmficant

lnconclu
88h curves.
Curved plate .
construct an 8

Variations.

43

The c: and C2 in Equation 4.7 are constants of integration to be determined by specific
boundary conditions and the a is the inner radius of the sphere. By ignoring the tempera-
ture spatial variation in Equation 4.7 (take T = constant) then Equation 4.7 reduces to u =
rTa. This result shows that the radial displacement, u, is strongly influenced by the radius
of principal curvature, r, with or without temperature variations. The contours in Figure
4.3 (1500 elements) show the maximum displacement of a flat circular plate is about 50%
less than the plate with 16 m of radius of curvature that shown in Figure 4.4 (1536 ele-
ments). Even though the numerical results of these two models depend upon the meshes
and the element types, the displacement differences between these two models are still
significant.

In conclusion, the displacement of a flat plate would result in only a fraction of the one
with curves. This, however, depends on how big the radii of principal curvature of the
curved plate is. The following sections will develop some techniques that enable one to
construct an axisymmetric and a 3-D full scale mirror model with the artificial temperature

variations.

4,4 L002

The (113
range beta

mam“ th.

cm Q in) i

{0'25 in) in
{at}. is to;

ACCOrdinQ

mined 55' ti

4,4 Local CTE Determination

rm A a:

 

 

 

 

 

 

\ °:::";._
II+

::::..+az

,9 eeeeeee TTT~+~I
\ g/iafi—

 

Figure 4.5 Data measurement of boule CT E.

The diameter of the ULETM raw boules is 152.4 cm (60 in) and the thicknesses are in a
range between 10.16 to 16.51 cm (4 ~ 6.5 in.) Corning [51] used an ultrasonic method to
measure the average radial CTE values of each boule through the entire thickness at 5.08
cm (2 in) intervals starting from the boule center. The axial CT E values were obtained
from samples cut from the outer rim of each boule through the full thickness, at 0.635 cm
(0.25 in) intervals beginning from the top surface (Figure 4.5). The measurement accu-
racy is :t0.25 ppb. Here we designate the average radial CT E as on, and axial CT E as ocz.
According to the previous statements, the local CTE values within each boule are deter-

mined by the following equations.

The 10c;
tions are est
""l- Therefc
Equation 4.9

Since the
Of this mirror
"ariathn

into MARC.

for a nonlinea

mt allow a Sir

is unavoidable

Such Circumst;

 

 

’,(r)
ar(r) - arm“) (4-3)
(10. z) = a,(r) - (12(2) (49)

where
6t,(r) is the through thickness average radial CTE at a distance from the center
(7. (r) is the through thickness average radial CT E at the edge of each boule

r

(XX?) is designated as the radial CT E, it ’s a function of radius of a boule
(12(2) is designated as the axial CTE, it ’s a function of the thickness of a boule
a (r; z) is an actual local CT E, it ’s a function of both radius and thickness
The local CTE is taken as a product of the axial and radial CI'E function. These func-
tions are estimated from the measured data by using curve fitting techniques (Appendix
A). Therefore Equation 4.9 has biaxial variations. The local CI'E values calculated by

Equation 4.9 are then substituted into the Equation 3.4 as A0 to calculate artificial tem-

peratures for each individual boule.

4.5 FORTRAN Code Development

Since the mirror elements are cut from 11 different ULETM boules, the CTE properties
of this mirror will vary considerably from point to point. In order to closely represent the
CTE variation in the FE model, Equation 3.4 (artificial temperature) has to be introduced
into MARC. Although MARC allows users to input material and temperature functions
for a nonlinear process, the geometry and material complexities of the primary mirror do
not allow a simple function input. Using manual input is impractical because human error
is unavoidable when a fine mesh is used. Thus a computer program is proposed under

such circumstances. A computer program can precisely calculate the local CTE values for

each boule
smfimm
can be copi
A FOR”
8% progra
boule from
isfined “it
moved. and
ousresultir
generated a
eated until ;
MWCmm

join m‘0 or

46

each boule according to each appropriate boule CI'E function and nodal coordinates. A
specifically formatted output file can be written by the program, and then the output file
can be copied and pasted into the MARC data file for a process without human error.

A FORTRAN code name cte.f was developed for the above purpose. In brief, the code
was programmed to read nodal coordinates, set ID’s and node numbers of each mirror
boule from the MARC data file. According to the set ID and nodal coordinates, each node
is fitted with particular functions. During the process, the nodes’ location is converted,
moved, and projected to an appropriate position for the curve fitting. Substitute the previ-
ous result into] Equations 4.9 and 3.4, the artificial temperature (nodal temperature) is then
generated and stored before it is written to a formatted output file. This process is rep-
eated until all nodes are finished. The formatted output file can then be pasted into the
MARC data file for numerical processing. Some nodal temperatures, for those nodes that
join two or more boules, are calculated separately and averaged before the final output.
This FORTRAN code is also converted into different versions for different model uses.
Namely, the code is model dependent, but it is mesh independent. The flow of the FOR-
TRAN program is illustrated on the next page (Figure 4.6) and an example code is listed in

Appendix C.

47

read data from file
and store in arrays

 

 

)4
[ read data set from array /

 

 

 

> V
node projection

match
boule ID

calculate local
nodal CTE

1

counter

 

 

 

 

 

 

 

 

 

 

 

 

NO last node YES

 

 

 

in the set?

finish all sets

 

nodal CTE
counter

print MARC
data file

Figure 4.6 The flow chart of the FORTRAN code.

 

 

 

 

 

4.6 ['sir

Sincel
numerical
of the FE l
interpolati
verif) its ;
sections. t'
ante and c
0nd is an

llimitation .

48

4.6 Using Exact Solutions to Insure the Correctness of MARC

Since FEM discretizes a whole domain into a specific finite number of subregions, the
numerical solution given by FEM is only an approximation. The accuracy and correctness
of the FE result, besides boundary conditions, is mainly influenced by meshes and element
interpolation functions (element types). Before adopting a FE software, it is necessary to
verify its performance and correctness by some simplified problems. In the following two
sections, two examples from Timoshenko [12] are employed to verify MARC perform-
ance and correctness. The first is a thin isotropic homogeneous circular plate and the sec-
ond is an isotropic homogeneous hollow sphere. A third order temperature function

(Equation 4.10) is imposed on both cases.

 

T = -l.796732 + 0.8693409 r - 0.06908982 r2 + 000173349 r3 (4.10)
T 6 '_
I
L
4 L
2 L
l

 

Figure 4.7 The temperature distribution of Equation 4.10.

4.6.1 A T
Variation

The tet
radial tem;

placements.

Wllé’re

EiS YOll);
V l5 PM;
b is Ill? 0
a is the I".
r i" an ar.

results of [hes

 

49

4.6.1 A Thin Isotropic Circular Plate with Arbitrary Radial Temperature
Variations

The temperature variation in the radial direction of this thin plate is an analogy to the
radial temperature variation of the primary mirror. The analytical solution of radial dis-

placements (u), radial stresses (0', ), and tangential stresses (6,) are as follows [12]:

1 0t
u=(1+v)-0t-;-J:Trdr+ro(l—v)-;—2-]:Trdr (4.11)
l J-b 1 )
o =0t-E- —- Td -—- Td 4.12
r (b2 0 rr r2 J; rr ( )
co:a.E.(—T+l-frrdr+i-J'Trdr) (4.13)
b2 0 r2 0

where

E is Young 's modulus

v is Poisson ’s ratio

b is the outer radius

a is the inner radius ( a=0 for a solid disk)

r is an arbitrary radius from the center of the disk

This case is made up with E = 6.76E6, v = 0.17, b = 28, a = 0, and 0t = 4.03 ppb. The

results of these analytical solutions are plotted on the next page in Figure 4.8.

5()

radial displaomnt

2.5x10"»
2x10"
1.5x10"
1x10"i

5x10"l

 

 

 

s ‘10‘ ‘ 15‘ ‘20‘ “’ 25 r
-Graphics-
radial stnass

0.06»

0.05

0.02

0.01

 

 

 

~8raphics-

tangential strass

 

 

. "25’ ” r
-o.ozs

'0.05;
-0.075l
-o.1;

 

oGraphics-

Figure 4.8 Exact solutions of equations 4.11~4.13.

Since I
adopted ul
8 3-D mic
Therefore.
stress circr
The follou
The fine m:
410). The
cte.f. The

‘BSUmption

51

Since this case is specified as a thin plate, a plane stress analysis is supposed to be

adopted when doing the FE analysis. Yet later on a 20-node brick element will be used for

This verification also conducts a test for element performance.

a 3-D mirror analysis.

Therefore, the FE model simplification is not a concern for now. Otherwise this plane

stress circular disk analysis could be performed by adopting an axisymmetric analysis.

, Type 21 elements [20].

The following FE model is constructed by MENTAT using Hex 20

The fine mesh is used to ensure a good representation of temperature variations (Equation

4.10). The temperature variations are imposed to the model by using the FORTRAN code,

cte.f. The model has a thickness to diameter ratio of 0.1228 that fulfills the plane stress

assumption (Figure 4.9).

 

 

 

.22....
i I t
I (it!

     

nu.p

l
are

It

.5.

1 :11: p
a: n I .1 p I .
bun...” "I. t a

    

   

a
tawny-10
in n5 :2.

 

 

Figure 4.9 3-D FE model for the solid thin circular plate,

Hex 20, Type 21. [20]

52

The results given by the FE model in Figure 4.9 almost agree perfectly with the exact
solutions in Figure 4.8. The error is less than 1%. The MARC solutions are plotted and

shown in the following figures.

 

CCMARC

 

 

 

4
\x

 

 

 

 

/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Length (x10)

 

Figure 4.10 Radial displacements of the thin circular plate.

53

 

 

5.915

 

 

 

 

 

 

 

 

 

\

 

 

 

 

 

 

 

 

 

 

 

 

 

Length (1:10)

 

Figure 4.11 Radial stresses of the thin circular plate.

 

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.591
\ 4 J,—
0 \MI
\
4.053 0 2.8
Length (3:10)

 

Figure 4.12 Tangential stresses of the thin circular plate.

 

4.6.2 A E
Variation

The tei
0g) t0 the

EqUaliOn
in this Cm ‘.
inner radlUs ‘
mined by Spe
30d OUICr 5111']

he results at

 

55

4.6.2 A Hollow Isotropic Sphere with Arbitrary Centrifugal Temperature
Variations

The temperature variation in the centrifugal direction of this hollow sphere is an anal-
ogy to the axial temperature variation of the primary mirror. The analytical solution for

radial displacements and stresses are as follows:

 

 

 

_1+v l_ . . C_2
u-j°a';2 LT rdr+c1 r+r2 (4.14)
2.“.5 1 2 E-cl 2-E-c2 1
,_— 1_v r3 LT rdr+1_2v— 1” r3 (4.15)
_a-E 1 2 E'cl 5'62 1 a-E-T
9- l—v 3 LT rdr+l—2v+1+v _3— l—v (4°16)

r

Equation 4.14 is the same as Equation 4.7. Notations, properties, and parameters used
in this case are the same as those used for the flat circular plate on page 49, except the
inner radius a is equal to 24 in this case. c, and c; are constants of integration to be deter-
mined by specific boundary conditions. For the hollow sphere, radial stresses on the inner
and outer surface are zero, and the constants of integration or and c; are solved accordingly.

The results are plotted and shown on the following page.

56

radialdfiaplacenant

5.2x10'7'

25 26 27 ’28 r

4.8x10'7’

 

4.6x10"

-Graphic3-

radidlstress

0.004.
0.003I

0.0022

0.001I

 

25 26 i ‘20 ‘ ‘ ‘ 28 r
-Graphics-

temgentidlstress

0.04.
0.021

MM;

25 ‘ ‘20 ‘ 27 ‘ ‘ ‘ ‘28 r

 

-0.02.

-0.04Z

 

-Graphics-

Figure 4.13 Exact solutions of equations 4. 14~4.16.

The 2!
Three eler
non. By:
B)" using ‘

are easier

 

57

The 20—node brick element is also used here to construct this hollow sphere model.
Three elements through its thickness gives seven nodes to represent the temperature varia-
tion. By simplifying the model, the hollow sphere is only modelled as US of the whole.
By using this 1/8 hollow sphere model, the analysis is facilitated. Boundary conditions

are easier to assign to the 1/8 model (Figure 4.14).

 

 

Y
KL
X

 

 

 

Figure 4.14 1/8 of the 3-D sphere model, Hax 20, Type 21[20].

The results given by this 1/8 model are also in close agreement with the exact solu-
tions. The error is less than 3%. The MARC results are plotted in Figures 4.15 to 4.17.

The comparison of the accuracy can be made by referring to Figure 4.13.

58

 

 

5.317

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.551 /

 

 

 

Length

 

Figure 4.15 Radial displacements of the 1/8 sphere.

 

59

 

 

4.202

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ac/i/

 

Length

 

Figure 4.16 Radial stresses of the 1/8 sphere.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Length

 

Figure 4.17 Tangential stresses of the 1/8 sphere.

 

47 hhur

The cc
single min
boundary
processing
Terence in

codes is le

’ Omputatit
CA&SL2(

61

4.7 Numerical Comparison Between MARC and ANSYS

The comparison of numerical results is also made between MARC and ANSYS.l A
single mirror boule with the realistic curve, biaxial temperature variations, and symmetric
boundary conditions were constructed and submitted to both MARC and ANSYS for
processing. The z-displacement contours are presented in Figures 4.18 and 4.19. The dif-
ference in predicted maximum z-displacement (absolute value) between the two software

codes is less than 2%.

 

ANSYS 5. 3

 

 

 

Boula_I

 

Figure 4.18 Z-displacement contours, by AN SYS.

 

1. Computational Applications and System Integration Inc.
CA&SI, 2004 S. Wright Street, Urbana, IL 61821

the analyll

4-8 Sing

One m.
lfifinemEm.

In order to

62

 

1%

fl?

‘fiflllfiw’
Vmfifiwr

9M”
1M0

aflll
2::E
8%;-
mg

858‘“!
8%..-

 

 

 

Figure 4.19 Z-displacement contours, by MARC.

Biaxial temperature variation is imposed to this model. This case is a supplement for

the analytical solutions in the previous section that lack the biaxial temperature variation.

4.8 Single Boule Analysis: Convergence

One more concern about the numerical solution is convergence. Because of the inho-
mogeneity of local CTE, the FE models will not give a smooth convergence after stages of
refinement. Also due to the computer hardware capacity, the mesh refinement is limited.

In order to show the tendency of convergence, a single mirror boule I (same as Figure

4.19) “'3

and radiz

node an:

.1 . .0... ..i .4 f..4. .r . ._ a.
f. . x ‘1‘ - .11 d . u
2. .2. «.1... .mrv a. .1 . M941. 2.. :44
at .v ...1 u r. 4 flit. 4.1 .
A . 4 E .4 2.1.4
at- I . Gil 0|. T. 0
Q h I h h

«EU» EOEIDIEQB

63

4.19) was refined in stages to show the convergence. The model was refined in both axial
and radial directions (Table 4.1). The MARC results of surface displacement of the center

node and a vertex node were selected for the convergence plot (Figures 4.20 and 4.21).

Table 4.1 Total number of nodes in each model and displacement error.

Total number of nodes Model Z-dlaplaeomont attire cantor (om) _’ Error compare to model 6

287‘ 1 4.1812150? , 19.82% 7

V 3451 2. 455342507... . ._ __ , ....-12-_23%...
10321 3 . -1.61557E-07 , 7 . -9.66%_
22905 4 -151917507 5.12%
35469 5 -147319507 ref.

 

 

Tot-Inumboromdu Model Hum-cementumHort-Hem) Errorcompmtomodols

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,, 28722,, ,1 ,,,,,, _ 3587640506 WW_,’_1'16.00%’_W ______
_ , 3461. _.2 4931850505 . 13.87%
.. 10321 3 -2.195610E-05_ 2.11% ,_ ..
229075 4 2243630505 ‘ ’ . _ 0.04%
35469 5 -2.242840E-05 ref.
Convergence of a Single Boule (at the center)
-100000507 . . . .
410000507 1 2 3 4 5
287 nodes
E 420000507 ‘\
1; 450000507
E \
° - .4 - 7
3 1 000050 \ 35469
‘3 -150000507
5 \ 10321 22905
.1 .60000507
470000507
model

 

Figure 4.20 Convergence of the z-displacement of the center node.

dlaplacamont (cm)

FlgUTE:

temperatur.

Convergence of a Single Boule (at a vortex)

 

 

 

 

 

 

 

 

 

-1000000507 . . . .
1 2 3 4 5
.. -5.100000E-06 1131mm:
2 \
2.
E -1010000505
. \ l
3 4510000505 ‘
5 \\
-2010000505 3461 “3‘ 22965 35469
2510000505

 

 

Figure 4.21 Convergence of the z-displacement of a vertex node.

Figures 4.20 and 4.21 show that as the number of nodes approaches “infinity”, the

temperature variation can then be perfectly captured.

4.9 Summary

The CTE change is insignificant within the ambient temperature range. CTE can be
treated as a static property for this primary mirror analysis. The confidence of using
MARC for this primary mirror analysis is reinforced by conducting two simple model
analyses. The fine meshes ensure the model can closely capture the temperature varia-
tions. The margin of error between the analytical solution and FE solution is essentially
small. Two numerical solutions, MARC vs. AN SYS, were conducted to support the solu-

tions of 3-D FE models with biaxial temperature variation. The numerical solutions

between tl
vergent te
4JOtends
values. A

tifies and s

65

between these two software codes closely agree with one another. The numerically con-
vergent tendency of the FE model was shown by analyzing a single mirror boule. Figure
4.20 tends to have mesh dependency that is caused by the inhomogeneity of local CTE
values. Although the convergence curve is not smooth, the tendency of the curve still jus-

tifies and supports the correctness of this FE model.

9?

5.1 Mir

Thech
generated
ing to the
pTCSSlVe 51
is498E4

Whmhthe

CHAPTER V
MODEL SIMULATION AND INITIAL SOLUTION

5.1 Mirror Thickness

The decision of whether to adopt the 10 cm or 20 cm design depends upon the stresses
generated by thermal expansion (excluding the consideration of mirror support). Accord-
ing to the maximum-normal-stress theory, failure occurs when the greatest tensile or com-
pressive stress exceeds the uniaxial tensile strength. The ultimate tensile stress of ULETM
is 4.98E4 N/cmz. Figure 5.1 illustrates that failure is predicted for any state of stress in

which the principal Mohr circle extends beyond either of the dotted vertical boundaries.

66

An axi
StresgeS am
full Scale 3
bOUles, all .

[hmugh thic

5"" The A

The Ems;

Quad 4. ripe

67

 

uniaxial compression uniaxial tension

Sue Sut

 

 

principal Mohr circle must
lie Within these bounds .
to avord failure

 

 

 

Figure 5. 1 Graphical representation of the
maximum-normal-stress theory (Juvinall [17], pp. 214).

An axisymmetric analysis is a simple and good approach to examine these thermal
stresses and thus help us to obtain useful information for decision making without using a
full scale 3-D FE model. A FE axisymmetric model containing three different ULE”
boules, all of which have severe thermal mismatch and nonlinear temperature distribution

through thickness, is constructed for this purpose.

5.1.1 The Axisymmetric Analysis for the Different Thicknesses

The axisymmetric model contains three boules that have ID and mean CTE value of J:
1.49 ppb, X: 6 ppb, and N: -3.3 ppb respectively (Figure 5.2). The model is meshed with

Quad 4, Type 10 elements [20] (8 elements through the thickness) that has a temperature

distributi:
stresses, :

faces for

SlllCe U
mum provn

mnclpal Str

68

distribution identical to the case in Table 3.1, eight-layer model C. Maximum principal
stresses, surface displacements, and gradients of shear stresses at the locations of inter-

faces for 10 cm and 20 cm thick designs are compared in the following.

 

N: -3.3 ppb —> “

 

J: 1.49 ppb +1,

 

 

 

Figure 5.2 FE axisymmetric model.

Since the normal stress maximum occurs at principal planes, the principal stress maxi-
mum provides evidence for the failure analysis. Figures 5.3 and 5.4 show the maximum

principal stress distribution for this model for the 10 cm and 20 cm design respectively.

Fig

69

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

acme
3.262 K
>—
7 l
i
i
l.
r
t 159 j 1
0 1.999
Length (.100)

 

 

unit: N/cm2

Figure 5.3 Principal stress maximum of the axisymmetric 10 cm model.

F1 gl

70

 

M -
1 WMARC

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lor‘th (x100)

 

 

 

. 2
unit: N/cm

Figure 5.4 Principal stress maximum of the axisymmetric 20 cm model.

We c:
knmm
and 5.6 s
thinner rr
combinat
also can t
a. Figure
lower that
bility for 1
severe CT
(8 Perfect 1

This matcl

SignlfiCamd‘

We can see that the maximum principal stress of both 10 cm and 20 cm models are far
less than the ultimate tensile strength of ULE”, 4.98E4 N/cmz. Furthermore, Figure 5.5
and 5.6 show that when both designs have the same intensity of thermal mismatch, the
thinner mirror has a lower stress gradient but a higher surface distortion (some other CTE
combinations have been tried as well, and they exhibit the same pattern). These results
also can be justified by Equations 4.14 to 4.16, by setting r equal to a constant and varying
a. Figures 5.3 and 5.4 show that the maximum principal stress of 10 cm model is 57%
lower than the 20 cm model.1 This fact promotes the mirror’s long term mechanical dura-
bility for the thinner thickness design. The high displacement (Figure 5.5) is cause by the
severe CI'E mismatch which is a made-up combination that was used for this studied case

(a perfect bond was assumed, therefore displacement has no discontinuity at the interface).

71

This match will not be used in the real mirror boule placement.

 

0.000000E+00

-5.000000E-05

-1 .000000E-04

-1 .500000E-04

doplacement (cm)

4000000504

-2.500000E—04

 

Surface Dlsplacement, 10 cm vs. 20 cm Deslgn

(outer Interface)

 

123450709101112

 

 

 

 

 

 

 

 

+10cm
M. +200.“
W
nodepdh

 

Figure 5.5 Surface displacement of the axisymmetric model

 

1. Since the mirror displacements are essentially small, engineering stresses and Cauchy stresses have no

significant difference.

 

 

72

 

Gradlent of Shear on Mirror Surface,
10 cm vs. 20 cm Doslgn (outer Interface)
5.00501 "1

4.00501 ‘
3.006-01
2005-01
E 1.00E-01
E 000500
-1.00E-01
-2.00E-O1
000501 E
4.00501 ~51
node pdh

    

 

 

+100m
+20cm

 

 

 

 

 

 

Figure 5.6 Gradient of surface shear of the axisymmetric model.

From a fracture mechanics and an economic point of view, the 10 cm design appears to
be safe from failure. It also saves material as well as transportation costs. Even though it
has higher surface displacement, it can be corrected by using different kinds of mirror sup-
port and external hydraulic forces, as long as the maximum stress does not exceed the
material uniaxial tensile strength, and it provides a faster thermal response. Accordingly,

10 cm will be an acceptable thickness for this primary mirror.

5.2 Initial Analysis of Full Scale Model with an Introduction to the
Biaxial CTE Variation

Two 10 cm thick 3-D full scale FE models were constructed for an initial run. There
are four 20-node brick elements through the thickness (Figures 5.7 and 5.8). The models

were made with the actual curvature of 16 m. They also included the biaxial CTE

variations
fixed aror
ditions. '.
fixed (u =
the thickr

=Ol.as sl

1\

. fills is app

73

0. The off-axis model has loose boundary con-

0) through

 

 

 

Type 21 mi.

V=W

ad!” 44.... 6% ~96

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t .4. .oa a .
a o. .6 ’6 2.. .8 a .
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(the reflecting surface is faced the negative z-direction).

variations along with the boule placement shown in Figure 5.10. The on-axis model is
ditions. The central boule is loosely constrained by a cross, the vertical line of the cross is

fixed around the centeral hole, 11

0) through the thickness, the horizontal line of the cross is fixed (v

fixed (u

the thickness, and only one node on the mirror back surface at the cross center is fixed (w

0), as shown in Figure 5.9.

 

 

 

Figure 5.7 On-axis full scale FE mode], Hex 20,

1. This is applied to all of 3-D mirror FE models.

 

74

 

 

 

 

 

 

 

 

Figure 5.8 Off-axis full scale FE model, Hex 20, Type 21 [20].

 

 

 

 

 

Figure 5.9 Constrains of off-axis FE model.

The res
the milXimt
higher than
tensile mes
axis mOde]
Global wall
densed at S]

a

adJuStmtint.

“Con Bar.

75

 

 

 

 

 

Figure 5.10 The initial boule placement for 10 cm thick mirror.1

The results are shown in Figures 5.11 to 5.14. By comparing the results we see that
the maximum principal Cauchy stress of on-axis model is 9.433 N/cm2 that is about 2%
higher than off-axis model’s 9.23 N/cmz. They are both far less than the material ultimate
tensile stress. Nevertheless, the maximum peak-to-valley surface displacement for the on-
axis model is 0.51059 ttm that is about 40% less than the off-axis model’s 0.71238 ttm.
Global warping, however still exists on both models and stress concentrations are con-
densed at small areas (Figures 5.11 and 5.12). The boule placement requires further

adjustment.

 

1. Base on Barr’s report, November 1996.

76

 

!“ : I
1v n
u— ; 1.000.460 "' " C

 

 

 

soon

Principal Cam Stra- In:

 

unit: N/cm2

Figure 5.1 1 On-axis principal Cauchy stress maximum
for Figure 5.10 boule placement, on-axis.

 

77

 

 

5"

  

 

x_l

Jam

2

Principal cam Sena- In

 

unit: Nlcm2

Figure 5.12 off-axis principal Cauchy stress maximum
for Figure 5.10 boule placement, off-axis.

 

78

 

 

 

 

D‘Ifllm! x r

 

unit: cm

Figure 5. 13 Mirror surface distortions,
z-displacement, for Figure 5.10 boule placement, on-axis.

 

 

79

 

 

F... .,. .
in: r | M .
u- : Loco-00 *M’ch

 

 

mulxa-M l l

 

Figure 5.14 Mirror surface distortions,
z-displacement, for Figure 5.10 boule placement, off-axis.

 

5.3 Summ:

10 cm ll'lll
less than ulti
response cont
following aria

Further b(
the surface di
that having a
40% less pea}.
has 2% highe
from this poii
However, the
SCAR commi
prOlCCl integr;
efore the anal

anMYSTS. Bot}

80

5.3 Summary

10 cm thickness will be very cost-effective since its maximum principal stress is much
less than ultimate strength of the material. In 1996, due to economical and thermal
response concerns, some project partners also favored the 10 cm design. Accordingly, the
following analysis will only focus on the 10 cm design for the proceeding optimization.

Further boule adjustment is needed since stress concentration occurs at small areas and
the surface distortions are not in a circular form. Figures 5.13 and 5.14 provide evidence
that having a center hole weakens the global expansion. In this case, on-axis model has
40% less peak-to-valley surface displacement than off-axis model, but on-axis model only
has 2% higher principal stress than off-axis model. The on-axis design is a better design
from this point of view. This fact helps us for setting up further optimization analyses.
However, the final decision of adopting either on-axis or off-axis will be made by the
SOAR committees. This decision depends not only on thermal mechanics, but also on the
project integration of housing structure, mirror support, and optics-image concerns. Ther-
efore the analysis for both on-axis and off-axis will still be carried out for the optimum

analysis. Both results will be provided to the SOAR committee for the final decision.

6.1 Axisyi

An axisy
[onion simpl
metry, the a,
ively. The 111

1116 ”Mel, f0

Table

CHAPTER VI
SURFACE DISTORTION
SINIPLIFICATION AND MINIMIZATION

6.1 Axisymmetric Analysis and Interfacial Mechanics

An axisymmetric FE model as shown in Figure 5.2 is again used here for surface dis-
tortion simplification/nunimization analysis. Although the mirror has no real axial sym-
metry, the axisymmetric model still helps to gather useful initial information inexpens-
ively. The model contains three boules. A different boule combination is imposed upon

the model, for each run, as shown in Table 6.1 below.

Table 6. 1 Results of different boule combination for axisymmetric analysis.

 

 

 

 

 

 

lit-161-0119 ,.'_I!é'di|".1-nti l[111699179511.fffli'litélfél’l’flicflffi.
Boule combinatlon ‘_' A354 . G: 6.52 7 0.1.73 . p _ 10:17.3 7
ID&CTE(ppb)_ :.. .. . P:4.977 94.97 . 8:246 _ (8:246,
6: 6.52 A: 3.54 O: 4.10 J: 1.46

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.low 4.6.7.5507. 4964507 4.439508...,.4-4305‘08..

. ' sou-prawn? jhig'a {52.491594} [#196616~ f 44“11187245164.410245"
(rt-m). 3'0" . 341600.506 .,-,5.-000507. ,..2-5OOEO?-. 1550506...

 

81

 

Each bOI
to Table 6.1
lowest (5101
between bor
ferences. ab
difference 0
low CTE at
Also the Me
tive CTE at
Strain singu]
that gradien

“TD and OE

82

Each boule combination has relatively low, middle, and high mean CTE values. Refer
to Table 6.1, the boule placement of each run is arranged by CI'E values from highest to
lowest (Model GPA), lowest to highest (Model APG) with the highest CTE difference
between boules being about 2 ppb. Model OBD and JBD have relatively lower CTE dif-
ferences, about 1 ppb between boules. This arrangement tends to find out if a mean CTE
difference of 2 ppb is a critical condition for interfacial stresses, and to find out whether a
low CTE at the center or a high CTE at the center gives a smaller global displacement.
Also the Model OBD has a negative mean CTE at the center boule to determine if a nega-
tive CTE at the center lowers the global displacement. An illustration of interfacial shear
strain singularity of these axisymmetric models is given in Figure 6.1. The figure shows
that gradient curve of Model APG is almost coincident with Model GPA and so as Model

JBD and OBD. The gradients in the figure were determined using finite difference.

 

 

 

Fig

83

 

.. <50 1------

 

A83 .3009?

 

, QmO

am: a.

.mnN

 

mafia. _

 

 

 

 

 

..:\

89:35

_+mc.n

 

 

 

1.614649606046666

afloa-

. amoe-

. 3...?

mime."-

Bod

mum—ON

wine-v

wfiod

named

ruotperg

 

 

Figure 6.1 Gradients of shear strains of the models in Table 6.1.

The result
ence of 2 ppl
strength of thc
facial strain g
depends on th
provides evid

lower global l

62 Coarse

6-2-1 Initial

B)” exam
With the Stud
are establish-
a Centrifugal
ues on One S
Each boule i
magnitudes E
interfacial St]

Referring
Can Sm] me-
full Scale 3~[

The mOdels C

84

The results in Table 6.1 indicate that the interfacial stresses caused by a CT E differ-
ence of 2 ppb is still not severe. These stresses are only about 1/360 of of ultimate
strength of the material. The CTE value difference of 2 ppb causes six times higher inter-
facial strain gradient than a difference of 1 ppb (Figure 6.1). However, this contrast also
depends on the CTE variation patterns at the adjacent zone between two boules. Table 6.1
provides evidence that placing a relatively lower CTE boule at the mirror center gives

lower global displacements and lowers stresses.

6-2 Coarse Full Scale 3-D Model Simulation

6.2- 1 Initial Attempts for the 3-D Simulation

By examining the results and trends from the previous axisymmetric analyses along
with the study about the CTE variations (refer to Appendix A), boule placement patterns
are established. Due to an insufficient number of boules to arrange the CTE distribution in
a Centrifugal pattern, the presumed placement patterns arrange the highest mean CTE val-
UCS on one side and lower mean CTE values on the opposite side (Appendix D and B).
Each boule is joined with the boules that have relatively similar CTE variations and close
magnitudes around their boundaries. This boule placement pattern tends to minimize the
interfacial stresses and the gradients of surface displacement.

Referring to Section 4.7, even though the accuracy is not good enough, a coarse mesh
Can still provide adequately accurate distortion patterns. Accordingly, coarse meshes on

full Scale 3-D FE models are adopted to simulate all of the presumed placement patterns.

The models contain about 1,470 20-node brick elements, it takes about 80 minutes to run.

85

 

 

 

M
M
a
a
.

 

2 Off-axis coarse mesh FE model,
Hex 20 and Type 21 [20], 4 elements through the thickness.

6

Figure

 

 

 

 

 

h FE model,

18 coarse mes

3 On-ax

Figure 6.
Hex 20 and Type 21 [20], 4 elements through the thickness.

Figures
terns have I
are provide
By referrin
share the s.-
on the cen
these 3-D
effectively
made earli-
deerenes ;

coarse 3-1]

6.2.2 Fin.

Oil-axis
Figure
displacem
tl\'ely_ Ev-
Iration int:
is 2-8885‘,

86

Figures 6.2 and 6.3 show the coarse mesh FE models. Numerous boule placement pat-
terns have been studied. The boule placement patterns and surface displacement contours
are provided in Appendix D and Appendix E for on-axis and off—axis mirrors, respectively.
By referring to these results, we notice that on-axis and off-axis models do not necessarily
share the same behavior for identical boule placement patterns. Certainly the CTE effect
on the center boule is weakened by the hole for the on-axis design. Furthermore, from
these 3-D analyses we also learn that placing a higher CTE boule at the mirror center
effectively dominates the mirror global distortion. This is in contrast to the statement
made earlier at the end of Section 6.1, that a relatively low CTE boule at the mirror center
decreases global displacement. The best boule placement patterns among the numerous

coarse 3-D simulation for both on-axis and off-axis are presented in the following.

6.2.2 Final Boule Placement Patterns for the Coarse 3-D Model Simulation

On-axis

Figure 6.4 shows the final boule placement pattern for the on-axis model. The surface
displacements and maximum principal stress are shown in Figures 6.5 and 6.6 respec-
tively. Even though there are stress concentrations on this boule arrangement, the concen-
tration intensity is still not a concern. The maximum peak-to-valley surface displacement
is 2.88857 pm. The maximum principal stress is 1.451E2 N/cm2 which is only 1/343 of

the ultimate strength of ULE“.

 

 

7

"1V" Slam

87

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.4 The final boule placement pattern for on-axis.1

 

1. “inv” stands for inverse, obtained by flipping the boule up side down.

Figure

88

 

Inc: 1 M

Yin. : 1 .0000900 MARC

'7'"

._ 15250-04
._ 2. 7290-04
_. 2.632I-04
... 2.55I-04
24400-04
2. 3430‘04
2 . 2470-04
2. 1510-04
2.055l-04
l .m-O‘

 

 

Jobl x Z

Dispute-en! z

 

 

 

unit: cm

Figure 6.5 Surface displacement contours of Figure 6.4, 3-D coarse model.

89

 

 

lrI: 1 leARC

 

 

J Y
Jdll x

PMr-crpal tundra 32m. Mu

 

unit: N/cm2

Figure 6.6 Principal stress maximum contours of
Figure 6.4, 3-D coarse model.

 

Off-axis
Figure (
momm-
principal st
occur in thi
maximum

stress is l.(

Off-axis

Figure 6.7 shows the final boule placement pattern for an off-axis model which appe-
ars to be the same pattern as the on-axis model. The surface displacements and maximum
principal stress are shown in Figures 6.8 and 6.9 respectively. Stress concentrations also
occur in this boule arrangement, and the concentration intensity is still not a concern. The
maximum peak-to-valley surface displacement is 3.932 pm. The maximum principal

stress is 1.091E2 N/cm2 which is only 1/456 of the ultimate strength of ULE”.

 

 

 

 

 

Figure 6.7 The final boule placement pattern for off-axis.

 

91

 

 

 

M!

nuance-um. l

 

 

unit: cm

 

Figure 6.8 Surface " ' ‘ contours of

I

Figure 6.7, 3-D coarse model.

92

 

 

Inc : I M
rm 1 Looooooo wMARC

 

 

_|Y
4°01 X 2

Principal may SIN” Hal 1

 

unit: N/cm2

Figure 6.9 Principal stress maximum contours of
Figure 6.7, 3-D coarse model.

 

63 Su

The '
The refit
nodes) v

model cc

through ‘

6.3.1 0

The f
6.4. The
and 6.1 1
(Figures
The max
of ULE“

ment arO

93

6.3 Surface Distortions by Refined 3-D FE Model (the final result)

The previous 3-D models were refined in this section in order to obtain better results.
The refined on-axis FE model contained 6,144 20-node brick elements (total of 30,528
nodes) with four elements through the thickness (Figure 5.7). The refined off-axis FE
mode] contained 5,888 20-node brick elements (total of 28,985 nodes) with four elements

through the thickness as well (Figure 5.8). The results are presented in the following.

6.3.1 On-axis

The final boule placement pattern for the refined on-axis model was the same as Figure
6.4. The surface displacements and maximum principal stress are shown in Figures 6.10
and 6.11 respectively. Components of Cauchy shear stresses are also listed for reference
(Figures 6.12 to 6.14). The maximum peak-to-valley surface displacement was 3.035 pm.
The maximum principal stress was 6.261E2 N/cm2 which was 1/75 of the ultimate strength
of ULE”. The large increase of principal stresses may have been caused by the refine-

ment around the center hole which was fixed 11 = v = w = 0.

94

 

F
inc: 1

 

“no : LOOQOOO

 

 

unit: cm

Figure 6.10 Final result of surface displacement, on-axis.

 

95

 

 

(Sr-mo

 

 

—' Y
Jml. x

Principal County Stun flux 1

 

unit: N/cm2

Figure 6.11 The final result of principal stress maximum, on-axis.

 

96

 

 

I 1 [Mil

 

 

le "—f 2

Coup 12 of Cluchu Stmo 1

unit: N/cmz

Figure 6.12 The final result of inplane shear Cauchy stress 12, on-axis.

 

97

 

 

Inc x 1 M
rm 1 Low-.00 ”~th

 

 

Y
Jab: l

Coco 31 of Cauchy Stmo l

 

unit: N/cm2

Figure 6.13 The final result of transverse shear Chuchy stress 31, on-axis.

 

 

98

 

In: x 1 M A ‘
n- : hm *M’J‘C

 

 

J Y
Jobl x z

Cor-p 23 or Cauchy Stmu 1

 

 

 

unit: N/cm2

Figure 6.14 The final result of transverse shear Cauchy stress 23, on-axis.

99

6.3.2 Off-axis

The final boule placement pattern for the off-axis model was the same as in Figure 6.7.

The surface displacements and maximum principal stress are shown in Figure 6.15 and
6.16 respectively. The maximum peak-to-valley surface displacement was 3.964 pm. The
maximum principal stress was 1.425E2 N/cmz which was 1/349 of the ultimate strength of

ULE”. Components of Cauchy shear stresses are also listed (Figures 6.17 to 6.19).

 

 

Inc : I M ‘
Tm : 1.0mm» .. J WMAF‘C

 

. ' Y
Jam J

Ill-plum: z 1

 

 

 

unit: cm

Figure 6.15 Final result of surface displacement, off-axis.

 

 

Inc : I
xMAHc

' 7f
s

 

 

—] Y
Jml

Principal Car-icky Stun flax

 

3
unit: N/cmz

Figure 6.16 The final result of principal stress maximum, off-axis.

 

101

 

 

In: : 1
xMAnc

 

 

J0“ x Z

Soap 12 of touch; atm.

 

 

unit: N/cm2

Figure 6.17 The final result of inplane shear Chuchy stress 12, off-axis.

102

 

 

-l . 327.601
-1 . 023.601

 

 

Jam

Can 31 of Candy Stret-

xiv-mo

 

unit: N/cm2

Figure 6.18 The final result of transverse shear Chuchy stress 31, off-axis.

 

103

 

 

Int: : 1
I'm- : Lab-000

3 1.338.002
_. 1.247.002
.. I. 150001
-— LWI
9.7230000
Em“

 

Jobl-

Cw 23 of Cm Sem-

X

xMARc

 

_JY

2

 

unit: N/cm2

Figure 6.19 The final result of transverse shear Cauchy stress 23, off-axis.

 

104

6.4 Summary

Satisfactory surface distortions for both on-axis and off-axis mirror designs were
achieved. Flipping the boules up side down provided more choices for matching the CTE
variations around boule boundaries. By doing so, there were 48! probabilities of boule
placement pattern combinations. The axisymmetric analysis successfully provided good
information for initial boule arrangements and the coarse 3-D FE model gave good results
without trying through all 48! boule combinations. Although the axisymmetric analyses
suggest that placing a relatively low CTE boule at the center of the mirror reduces global
distortions, the coarse 3-D model provides evidence that a relatively high CTE boule at the
center efficiently dominates global distortions (compare the final boule patterns with Fig—
ures E16 and E17). In order to correct the surface distortions after the mirror experiences
a temperature change, the distortion form is the main concern. Since all of the principal
stresses generated by the high CTE boule at the mirror center are far below the failure cri-
terion, they are no longer a concern. Therefore, adopting a high CTE boule at the mirror
center to dominate the global distortions is rather favorable. The maximum surface dis-
placement of the on-axis mirror was 3.035 pm which was 30% less than the off-axis mir-
ror, 3.946 um, while the maximum principal stress of the off-axis mirror (1.425E2 N/cm’)

was only 23% of the on-axis mirror (6.261E2 N/cm’).

CHAPTER VII
CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

In the case when surface distortion is a major concern, the axial CTE variation
becomes more critical than the radial CTE variation. The beam and axisymmetric models
have shown that a fine mesh through the thickness is necessary to ensure the FE models to
closely capture the axial CTE variations, especially when the through thickness CTE var-
iations are severe. Furthermore, the significance of curvature effect mentioned in Section
4.2 claims that the real mirror curvature should be included in the models. The magnitude
of global displacements are significantly increased by the geometric curvature, a fact
which should not be ignored.

Due to the material property and geometric complexities, it takes about 10 to 15 hours
to run a full scale fine mesh FE model. It is still desired to simplify FE models in order to
save the cost of the analyses, but the simplified FE models should not lose their represen-
tation of the actual problem. The axisymmetric FE model is adopted under such consider-
ations. As a result, the axisymmetric FE model gives good approximations about the
patterns of surface distortion and the singularity of interfacial stresses prior to conducting
the full scale model simulation. In contrast, the axisymmetric model only takes two min-

utes to obtain results.

105

106

When committing to a numerical software, it is desired to verify the correctness and
determine the accuracy of the software. Two simple cases of analytical solution and
another FE software have been employed to verify MARC’s performance. The results
were in very close agreement with one another. The verification of MARC has been done,
thus the confidence for adopting MARC is gained.

A critical tool, FORTRAN code (cte.f), successfully assists MARC to closely capture
the biaxial CI'E variations within the mirror. The code was modified into different forms
for axisymmetric models and 3-D models. A sample code is listed in Appendix C.

Based upon the information obtained from the numerous axisymmetric analyses and a
close study of individual boule CTE variations, a satisfactory boule placement pattern was
found by trial and error. The surface distortion patterns are in simple and semi-circular
form as we desire. Through out these trials, we also learned that placing a high CTE boule
at the mirror center can effectively dominate global distortions. Although the global dis-
tortions are enhanced by this arrangement, the circular form of surface distortions can be
corrected relatively easier by applying external forces and local heating/cooling devices.
The final boule placement pattern give maximum peak-to-valley surface displacement of
3.035 pm for the on-axis model and 3.946 pm for the off-axis model. The maximum prin-
cipal stresses for the on-axis (6.261E2 N/cm’) and the off-axis model (1.425E2 N/cm’) are
both far less than the ULETM ultimate strength (4.98E4 N/cm’). There is thus no concern

for failure upon the assumptions of this mirror analysis.

7 .2 Recommendations

The mirror support (housing) plays an important role in the surface distortions. In

107

order to obtain more realistic results, the support type and the contact locations have to be
simulated with the FE models. A good result, however, relies on correct boundary condi-
tions. Details of the mirror support should be provided prior the mirror analysis. Upon
the determination of actual boundary conditions, gravity effects can be imposed upon the
FE models, which will give a more realistic simulation. By then, the final boule placement
pattern provided here should be rerun with the new boundary conditions to ensure its
validity. External forces can also be applied to the 3-D model to correct those local sur-
face distortions. This external force simulation will also help in setting up force actuators
to correct surface distortions when the mirror is in service.1

In terms of the thermal distortion control (when the telesc0pe is in service), instead of
or in addition to using external force actuators the concept of the FORTRAN code can be
further extended to control some local heating/cooling devices to correct local distortions.
In brief, a computer constantly senses the ambient temperature and calculates local tem-
peratures needed (based on local CTE variations) to maintain a constant mirror shape. It
sends out these local temperature signals to individually control the local heating/cooling
devices and local temperature sensors to respond appropriately. This method works well
if the thermal conductivities of ULE” glass does not vary with the service temperature
range of SOAR telescope.

In order to save computational costs “Injection Island Genetic Algorithm” (iiGA) can
be adopted to perform the simplification/nfinimization process of this mirror analysis [56].
Basically the method used in this thesis has employed the concept of iiGA but performed

it manually. iiGA can be automatically coded and processed for this particular analysis as

 

1. http://www.gemini.edu/optics/optics.html

108

shown in the following flow scheme.

        

islands
.— ’ \
I " ’ ‘ r
axisymmetric axisymmetric 0 axisymmetric
analysis 1 analysis 2 analysis It

     

 

 

 

 

coarse 3-D analy9 0 Game 3D analysia

fine mesh 3-D model

 

 

 

 

 

Final Result

Figure 7.1 Flow chart of iiGA for this particular mirror analysis.

iiGA does crossover, mutation and inversion between the islands and injects good
results to next “higher resolution” islands, etc. until it reaches the final best result. GA is
an evolutionary, combinatorial technique that is partial to problems with discrete solution
spaces.

The tactics and techniques performed in this thesis can be employed for other similar
projects such as the international Gemini eight-meter telescope projectlas well as other

nonhomogeneous structures.

 

l. http://www.gemini.edu/

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mam. AThesis, Michigan State University, 1997.

APPENDIX A
PROPERTIES OF ULETM BOULE
AND CTE VARIATION

114

115

1

Table A1. ULETM propenies.

 

 

 

 

 

 

 

 

:0 0NN000 00.000... -:N0.00N0 00.0 ...... x:
-- ,N...0-- 2----..NN000N.-1 ---1-00000001---- ---.-.N0-.00.N0 a-.. 0.0.------.-.-,-3 .......

N0.0- . ..NN000 -- 0000-00-0 _ N0.00N. 0...-- -. 00.0... ,. _ N. .-
-.1-2.0 ..:NN00 0 0.00.000... 0.00Nm- -- 1-00.0-1..- f3.-
N ..0 ...NN00 0 , --.-..0..000 0 ---N0.00N 0 - . 0.0.0, N:
-- N...0---- -. ...... .-..---0-0.N.00.0. --1-0.0.0.000; Now-0.2.0-.-. 0..-.0-- 2.0
N00 .NN00..0--- m - -00....000. 0. ..N0.00N 0 . 0N0. ..... 5m
N00. -- - .j.N-N.00. 0..--- 1--...0-0-00-00 -N0..00-N0 ., ..-1-1...”.-- . .0:
N00 , -..NN000 ., .. 00.00.00 -. N0.00N0 _- w-N0.0. . .0 .
N00. .NN000 00.000.0- ...... N0.00N0 - - 0......- io
N. ..0, 0NN050 0 00.000. 0 N0.00N .0- .-,00.0...-. 5 z ,
N..0 - 0-.NN00 0-- .-.--.00.00N00 30.: 0 0.0..N-E 0..

N ..0 ..NN000 00.0000 N0.00N 0- 0N...- 3
N... . .28... , 3.00.3 .. 3.00.... 0.. BN- 2.. ..
.N..0,, .. .NN000. . - 00.000.0- - H..H-..N0..00.N...0.-- . - .00....- , 0.
.N...0 . .NN000 - . 00.000.0- ; . -. N0.00N0 : 00.0.,- _
N00 .NN00.0 .-.00.0000 . . N0.00N0 - 00.....- 0..,
N...-0- ,..NN000- 00.0000 . . _. N0.00N0 ...N0-0 ........ o
.N..,0 - ..NN00 0,- .. 00.000. 0 N0.00N0 . ---.,.N0..0..m-- ,... ..
N00- - .0.NN00..0_ -. 00.0000 ; -..N0..0.0N0- -0.-N..N. - _ 0

N00 . ...-NN00..0 00.0000 -: N0.00N0 ,,,,, 0.0.-.. ..... 0

N ..0 0NN000 00.000. 0 3.0.2. -..0 .- .00.,N - :0
N00. .NN00.0. 00-.000. 0.... N0.00N0- - - .00.N,, -. ...0. .,
N00 ‘ ,.NN00.0-.,_. 00.0000 N0.00N0 :000‘ -. .0;

2.0.. 0.000060. 8080.90 .5000... 000.0. 9080sz 0:00:00 0.09.0.5 @080sz 03.3001 0.00380 308 who :00... .o._ 0.300

 

l. Barr’s report, November 1996.

116

AC=6.692332—0.518469*W+O.5145772*W**2-0.08778082*W**3

 

CTE (99b)

 

 

 

on ---- - ‘00
00 0‘ 10 15 202.530 35494550 530.0015 70

Add (In)

 

RC=3.1 12074+1.972035*R-O.5966239*R**2+0.05764301*R"3-0.002299781*R"4—I>0.00003283*R**5

ID

 

 

03

00 o 0
~03

 

 

 

.1.o ----- -----
oacounnnnnnnaaan

MM

Figure A1.CTE variation of boule A.

117

AC=O.5396559-2.045625*W+1.477424‘W“2-O.2464461*W**3

CTE (PM!)

 

en - - - 1 1
3.6
an i
u i
20 i o
1:
111 i
on i
on
on
.111
11.11 '
1211 i
-u i
an i
1:11 i
49
45
on '
41.11 [

~00 ‘ e 1
00 0.6 1D 1.6 20 2.6 an 35 Q0 4.6 60 65 00

MCI (In)

 

 

 

 

RC=-1.796732+0.8693409"'R-0.06908982"‘R"I I"2+0.00173349"'R“""3

CT! ["5]

 

 

 

 

 

l
.4 o - . 1 1 0 - - L N 1 - -
0 2 4 I O ‘0 12 14 fl 1. D D 24 a a so

Figure A2.CTE variation of boule B.

118

AC=-0.1291866-7.l61156*W+8.047375*W**2-3.449349‘W**3+0.6424932‘W“4-0.04376068*W‘"‘5

"5 W)

on .. - - -
02’
on’
on
42
45'
an
cni
.1. i
an
43' .
oz
45
an
4.1
4.‘
41

an A A A A - 1. A A A A A A
on 05 1o 15 29 25 an as an 45 on as on

ms (111)

 

 

 

 

 

RC=-0.3803922+0.9290383*R-0.08789862*R"2+0.00234l *R“ *3

31! fluid

 

 

 

 

 

O 2 4 O I 10121401020222120310

Figure A3.CI'E variation of boule C.

119

AC=5.545455-0.01897233‘W

 

 

CTE [will
1‘. 3

 

 

 

00 06 10 15 20 2‘ 30 35 40 48 60 65 0.0

Axlallh]

RC=—2.800654+1.27496*R-0.1 159412*R**2+0.003061 1 1*R**3

 

CI! ["51

bbbbh500uugoo~aoo

 

 

 

 

I 10 12 14 0. 1. 20 22 24 211 2. E

R1“ M

O
N
b
0

Figure A4.CTE variation of boule D.

120

AC=4.49029-5.790166*W+1 2.2021 3"W”2-9. l 46586*W**3+3.232425*W**-O.5425023*W"‘*5+0.03456165*W"*6

 

CTE Mb]
3 3

 

05

on A A A A A A A A A
on on w u an u u u 4.0 45 on u on

W00)

 

 

 

RC=-1.636601+1.033141*R-0.08246023"'R"2+0.0020451*R**3

 

 

 

 

 

24.01012MDGOIORN820N

M [it]

Figure A5.CTE variation of boule E.

121

AC=9.77436-22.71638*W+29.42734*W**2-16.97739*W“3+4.95331*W“4—0.7147381*W“5+0.04027234“W**6

 

 

 

 

 

)
0.0 05 1.0 15 20 2.5 OD 3.5 4D 4.8 GD .3 0.0

RC=O.7029412+1.05916*R-0.09579024*R**2+0.00238028*R**3

 

 

 

 

.wA AAA A
ozaoonunnnmnzaaaan

 

Figure A6.CTE variation of boule F.

122

AC=3.276l45-0.5414188*W+0.7995721*W“ ’2-0. 1424589*W**3

 

CTE [PPM

 

 

 

 

00 0.5 1.0 1.6 2D 2.6 3.0 3.8 4.0 4.5 5.0 6.6 00

mum

RC=8.592484»0.1587197*R-0.02607646*R**2+0.001 16003*R“3

 

CT! ["5]

 

3.0 o

20

 

 

 

on A AAA AA
ozaouwunnnnnuaan

Figure A7.CTE variation of boule G.

123

AC=8.853064—12.62087*W+8.694S4*W"2-2.085492*W**3+O.1612028*W**4

 

WW]
8::

 

 

 

 

0.0 0.0 1.0 1.5 2.0 2.0 3.0 0.5 4.0 4.0 6.0 0.6 00

RC=5.028431+0.5253537‘R-0.07985887*R**2+0.002341*R**3

0.0 v a

 

C15 ["5]

 

1.0
on
onAAAAAAAAAAAAAAA
02400101214101.2022241031)

Minn]

 

 

 

Figure A8.CTE variation of boule H.

124

AC=4.752985-19.74305*W+22. l2489*W"2-8.993916*W**3+1.582345*W**4—0.1038646‘W**5

 

 

 

 

 

RC=O.2584967+1.64624*R-0.1520842*R**2+0.00359268*R“3

 

c113 [pm

 

 

 

24001012141010208243203

MM

 

Figure A9.CTE variation of boule I.

125

AC=O.8815086+3.756955*W-3.18592*W*"2+1.464537*W**3-0.3060963*W**4+0.02181849*W**S

 

WW]
3:

 

 

 

 

'oo oo 10 15 an 25 an an 40 45 so as on

Axhllku

RC=1.429739+O.3197691‘R-0.03735501*R**2+0.00088508*R“

 

CT! (".1

 

 

 

 

ozaaowuunuzozzxnaao

Rllhllku

Figure A10.CTE variation of boule J.

126

AC=3.943191-3.506161*W+1.618764*W**2-0.2306513‘W**3

 

an - - e - - - a - -
55’
an’
43’
4n 9
u
an
25
an
15
1o
05’
on’
05’
.19
45'
.20>
.25’
49'
on
40’ A A A A A A A A
on 0.5 1.0 u an as an as an 45 so u 0.0

WW1

 

 

 

 

RC=-6.715359+1.539639*R-O. 1247688*R**2+0.00275473*R* *3

 

c1: [ppb]

 

 

 

 

02400101214101.2022242020”

ms on:

Figure A11.CTE variation of boule K.

127

AC=4.082707-2.4905 37*W+1 .046854*W**2-0.1395742*W“3

 

00
00’
00 O
40
C0
3.6b
3.0>
20
20

c" ("'1

10
10
05
00
00
40

 

4.0

 

 

 

b
.10 A A A A A A A A A
00 05 10 15 20 25 30 35 00 CA 00 08 00
mal [II]

RC=-3.24085+l.401635*R-0.1352865*R* *2+0.00312395*R“3

 

c1: [pm

 

 

 

 

ooAAAAAAAAAAAAAAA
ozaoaaoizunuzozazazozozo

Mal [In]

Figure A12.CTE variation of boule L.

128

AC=5.91749-3.336218*W+2.582592*W**2-0.6817157*W"3+0.05270092*W**4

 

 

 

 

-2o AAAAAAAAAAA
on 0.5 10 u 20 u so ”was on 55 on u 7.0 n 0.0

mm

 

RC=3.558738-1.261 102*R+0.1876794*R**2-0.0106886*R**3+0.00020274*R“4

 

CT! [”5]

 

 

 

 

02400101214101.322242020”

MM

Figure A13.CTE variation of boule M.

129

AC=0.1230349-244144‘W+3.439277*W**2-1.108452‘W**3+0.1004006*W"4

 

mo»:
38

 

 

 

 

0.0 0.9 10 1.9 20 25 M 3‘ 40 4.9 on 6.6 on
Will]

RC=-2.43S415-0.8794024‘R+0.1306292*R**2-0.00756096*R* I"3+0.00014769"‘R""“4

 

CT! ["51

 

 

 

 

oaaoowiznsosozozazozoaao

RI“ [In]

Figure A14.CTE variation of boule N.

130

AC=9.523582-2.769093*W+0.3791138*W"2-0.02929062*W**3

100 v v - v v v f - v v

 

m ("'1
8

 

 

 

 

00 0.0 10 10 20 25 00 30 0.0 4.0 00 00 00

RC=-1.5941 18+0.1782213*R-0.01375646*R"2+0.00040064*R**3

 

CT! 1".)

 

 

 

 

20°

~25

4.0Lk‘ AA
02400101214101.202224302030

Figure A15.CTE variation of boule O.

131

AC=14.34699-9.504163*W+7.500762*W**2-1.962608*W**3+0.1571725*W**4

 

on W1
.8
h b

 

 

 

 

00 00 10 15 20 25 30 35 10 10 00 05 OD
mum]

RC=5.587375-0.6626495*R+0. 1073 28‘R“2-0.00695281 I'R‘” *3

00 v v v 4a . v . AA
D
00

 

CT! ["5]

 

20 0

 

 

 

15’
10’
05’
F
one #2 a o‘ekso‘cz‘MAsoAao m‘nAn ”A2030

Figure A16.CI'E variation of boule P.

132

AC=7.444272-4.748372*W+5.448423‘W"*2-2.024053*W“3+0.2128269*W**4

on
0.5 1
on ' o
u ’
u
on
on
on
so ’
u
on
u
so
u
an
1 5
10
on
co
on
.10
.15

 

m on)

 

 

 

 

.2 0
00 0.6 1.0 1.6 2.0 2.6 an 3.6 4.0 4.6 60 6.6 60
mm [In]

RC=8.37401 1-2.006773 *R+0.2988679*R“2-0.01635692‘R‘*3+0.00029569*R**4

10.0 v v v 1 v v y

 

CT! ["51

 

2.0
1.6

19
05'
MA A +1 A AA
ozooowauuuzonzaaoaao

mm

 

 

 

Figure A17.CTE variation of boule Q.

133

AC=2.149351+1.838947"W-2.100118*W“*2+0.6090008*W“3-0.0490396*W**4

CT! ["51

o‘ak'4-‘104aunuugouooowuooo

 

 

 

 

 

RC=4.923719-0.5392579*R+0.09301115*R“*2-0.005764~45*R”3+0.0001 1561*R"4

cm [ppb]

 

 

 

 

 

4

010121416103222‘2620fl
Radialtln]

Figure A18.CTE variation of boule R.

134

AC=2.409357-1 . 10605*W+0.9503337*W**2-0. 1851621 *W**3

 

an - 3
u ’
on
u
‘D O
as
an
25
an
15
1o ’
on ’
on ’
on
.10 1
.
.15
.20
.25
an
.35 ’

40
0.0 0.6 10 16 20 2.6 3.0 3.6 6.0 4.6 60 6.6 60

Axial [In]

CT! ["51

 

 

 

 

RC=3.76634+0.2251924*R—0.04252406*R"2+0.00129095*R**3

6.0 - . 1

 

01‘ ["51

 

 

 

 

1n 0 9
r
as
on g
o 2 4 o 0 1o 12 14 so 10 no 22 24 an a an

Figure A19.CTE variation of boule S.

135

AC=1.027348-9.032187*W+8.24326‘W**2-2.283174*W**3+0.1910286*W**4

 

 

 

 

 

00 0.6 10 1.6 2.0 20 3.0 36 40 4.6 60 6.6 0.0

Axum

Rfi-Z.181373+1.974646‘R-0.1479409*R**2+0.003103*R**3

9.0 ' I ' I V r v T v v V
D
80-

 

6.0 >
5.0
4.0 -
3.0 l
2.0
1.0 l
0.0 -
-10 '-

CW [PPM

 

 

 

-20 4 1 A 1 A 4 4k 1 A

0 5 10 15 20 25 A 30
Rldlll

 

Figure A20.CTE variation of boule T.

136

AC=5.014202+0.6735526*W-0.08286857*W**2-0.055228 l *W"3

 

09. .00.

CT! ["51

 

 

 

 

'on as 1n 1.: 2.0 u an as on u on u on

RC=-6.253922+1 .8%267*R-0. 1321981 *R**2+0.00267094*R* *3

 

CW ("'1

 

 

 

02660101214161.20222426203

 

Figure A21.CTE variation of boule U.

CTE W]

137

AC=2.836815+1.292151‘W-0.05330322*W* *2

 

 

 

 

 

Asia: [In]

RC=1.557843+1.429415*R-0.1 141995*R**2+0.00252954*R*‘3

01'! [ppb]

 

0.0 v . . v T v 4 v v
0.6

D
0.0 o

 

1.0

0.11

0.0 - - A A
o

 

 

 

240010121410102132224202330

Figure A22.CTE variation of boule V.

138

AC=7.220021-12.66894*W+13.02921*W**2-5.894468*W“3+1.198597*W**4-0.09060564*W“5

 

CT! ["51
3

 

 

 

 

'oo 05 1o 15 20 u 30 3.1.1 an 45 so as oo

Arid [h]

RC=2.795666-0.9847553*R+0.1479296*R* I"2-0.00851709"‘R"“"3-1-0.00015882“'R"‘ *4

6.0

 

 

 

 

 

ozooownumnannauun

Figure A23.CTE variation of boule W.

139

AC=6.52905-2.873149*W+0.7502868*W**2-0.09785729*W"‘*3

 

01! [ppb]
’3

 

 

 

 

o A 4* A A A A A A A A
0.0 0.6 10 1.6 20 2.6 3.0 3.6 40 4.6 6.0 6.6 0.0

Minn]

RC=-1.662109+].2727]8*R-0.2053296*R**2+0.01050944*R**3-0.00016331*R**4

 

CTE ["51

 

 

 

 

on-A- A--AA-4A-
02400101214101.202224202030

Rudd [In]

Figure A24.CTE variation of boule X.

APPENDIX B
FE CONTOUR PLOT OF BEAM MODELS

140

141

 

Figure B 1 .X-displacement, 2-1ayer beam (A).

 

Figure BZ.Y—displacement, 2-1ayer beam (A).

 

 

142

 

Figure B3.X-disp]acement, 4-layer beam (A).

 

Figure B4.Y-displacement, 4-layer beam (A).

 

 

143

 

 

 

 

Figure B5.X-displacement, 4-layer beam (B).

 

 

 

 

 

 

Figure B6.Y-disp1acement, 4-1ayer beam (B).

144

 

 

Figure B7.X-displacement, 8-1ayer beam (A).

 

 

Figure B8.Y-displacement, 8-1ayer beam (A).

145

 

 

 

1
E
t
1
1
I
I
!

 

 

 

 

Figure B9.X-displacement, 8-layer beam (B).

 

 

 

 

 

 

 

 

Figure B10.Y-displacement, 8-layer beam (B).

146

 

 

Figure B11.X-displacement, 8-layer beam (C).

 

 

Figure BlZ.Y-displacement, 8-1ayer beam (C).

APPENDIX C
FORTRAN CODE, CTE.F

147

148

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
PROGRAM: CI'E

PURPOSE: INTERFACIAL COEFFICIENT OF THERMAL EXPANSION SIMULATION
FOR SOAR TELESCOPE PROJECT

DATE: FALL 1997

00000000

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

#********#******#**#**#*itittt*¢*********¢##*#*#*##**********#*t********#t************#**#*i**#t

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

 

C !! DATA INPUT !!

C

C > TITLE (A30)

C >MATERIALID (llAl)capita1 1!!
C > INVERSEPARAMETER (l 1A1) “Y” or “N”
C > BOULE POSITION (1113)

C > NUMBER OF NODE ON EACH BOULE (1116)

C > RADIAL CTE AT BOUNDARY (1113)

C > MEAN CTE OF EACH BOULE (1 1F6.2)

C > NODEL NUMBER AND COORDINATES (IS,3FlO.5)
C > NODEL NUMBER ON EACH BOULE (918)

C

C

C

C " VARIABLES "

C

C TITLE: Title of the run

C NONB: Node number

C BOUNB: Total of boule used

C SETNO: Total sets stored

C SETID: Sets’ ID used

C BLOC: Boule’s location on the mirror

C NOBO: Number of boule used in the model

C X' Coordinates in x-direction

C Y Coordinates in y-direction

C Z: Coordinates in z-direction

C U: Sub-X

C V: Sub-Y

C W Sub-Z

C R: Distance for localization

C AC: Axial CTE value

C RC: Radial CI'E value

C ZARC: Z-coor. of the arc (has to convert to plane)

C NBNO: Total number of node (change, according to model)
C SETNOD: Number of nodes in each set

C NODESE: Nodes’ number in each set (give the maximum)
C RCB: Radial cm at boundary

C MCI'E: Mean CI‘E valur for each boule used

C LCTE: Local CTE

C ART: Artifical temperature

C CUNTER: Counter to avoitectede ART overlap

C DART: Data of ART

C NODE: Array of node number (global)

C INV: Inverse parameter
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

149

IMPLICIT NONE

INTEGER NBNO, NONB, SETNO, SETNOD, BOUNB. I, J, K, L,
RCB, NODESE, BLOC, CUNTER, NODE, M. IUNIT.
OUNI'T], OUNIT2

PARAMETER (IUNI'T=10, OUNIT1=12, OUNIT2=13, NBNO=29626.

SETNO=11)
!Change this number accounding to the model

REAL X(NBNO), Y(NBNO). Z(NBNO), MCIE(SETNO),
U, V, W, R

DIMENSION NONB(NBNO), SETNOD(SETNO), RCB(SETNO),
BLOC(SETNO), NODESE(5000), lMay change
CUNTER(NBNO), NODE(NBNO)

DOUBLE PRECISION ZARC, LCTE, ART, AC, RC, AT, DART(NBNO)

CHARACTER‘30 TITLE

CHARACTER” SE'TID(SETNO), INV(SETNO)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C MAIN PROGRAM
C

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

OPEN(UNIT=10,FILE=’cte.data’, STATUS=‘old’)
OPEN(UNIT=12,FILE=’data.out’ , STATUS=’ unknown’)
OPEN(UNIT=13,FILE=’cte.out’, STATUS=’unknown')

CALL INPUTDATA (TITLE, BOUNB. BLOC, SETID, SETNOD, RCB,
I, J ,K, L, M, MCTE, IUNIT, OUNIT], INV)

CALL READDATA (NONB, X, Y, Z, ZARC, SETNOD, NODESE. I, J, K,
DART, CUN'IER, AC, RC, LCTE, AT, NODE, U, V, W,
BLOC, SETID, MCIE, RCB, R, IUNIT, OUNITIJNV)

CALL PRINTMARC (NBNO, DART, CUNTER, ART, NODE, L, OUNIT2)
END

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

C Subroutine Input Data

C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

SUBROUTINE INPUTDATA (TITLE, BOUNB. BLOC, SE'I'ID, SETNOD,
RCB, I, .I, K, L, M, MCTE, IUNIT,
OUNIT] , INV)

IMPLICIT NONE

INTEGER NBNO, BOUNB. BLOC, SETNO. SETNOD, RCB, I,
J. K. L, M, IUNIT, OUNIT]

150

PARAMETER (NBNO = 29626, SETNO = 11)

DIMENSION SETNOD(SETNO), RCB(SE'TNO), BLOC(SETNO)
REAL MCIE(SETNO)

CHARACTER‘30 TITLE

CHARACTER’] SETID(SETNO), 1NV(SETNO)

 

WRITE(OUNIT1,*) 'cte.f written by Szu-Han Hu’
READ(IUNIT,*) TITLE
WRITE(*,"‘) ’TITLE OF THIS EXECUTION IS:’, TITLE
WRITE(OUNIT],*) "ITTLE: ’, TITLE
WRITE(OUNIT1,‘)

WRI'IE(*,*) ’THE MAXIMUM NODE NUMBER OF THIS MODEL ’, NBNO
WRITE(OUNIT] ,2) NBNO
WRI'IE(OUNIT1,*)

WRITE(*,*) ’HOW MANY BOULE USED?’
READ(*,"') BOUNB
WRITE(OUNIT1,3) BOUNB
WRITE(OUNI'T1,"‘)

WRITE(*,*) SETNO, ’ SET STORED’
WRITE(OUNIT1,4) SETNO
WRITE(OUNIT1,*)

* ------------- input boule IDs that corespond to each set ------------------

READ(IUNIT,7) (SETID(I), I=1, SETNO)
WRITE(OUN1T1,*) ’BOULE USED'
WRITE(OUNIT1,9) (SETID(I), 1:1, SETNO)
WRITE(OUNIT1,*)

" ----- input inverse parameter, "Y" to invert the boule, "N" for not to -----

READ(IUNITJ) (INV(I), I=1, SETNO)
WRITE(OUNIT1,*) 'INVERSE PARAMETER'
WRITE(OUNIT1,9) (INV(I), I=1, SETNO)
WRITE(OUNIT1,*)

* ................. inupt boule’s location on the mirror

 

READ(IUNIT,5) (BLOC(I), I=l, SETNO)
WRITE(OUNIT1,*) ’BOULE LOCATION’
WRI'IE(OUNI'T1,5) (BLOC(I), I=1, SETNO)
WRITE(OUNI'T1,*)

*---------- input the total number of node corespond to each set ------------

READ(IUNIT ,6) (SE'TNODO), J=], SETNO)
WRITE(OUNIT1,*) "TOTAL NUMBER OF NODE IN EACH SET’
WRI'IE(OUNIT 1,6) (SETNODO), J=1, SETNO)
WRITE(OUNIT1,*)

151

* ------- input the extreme boundary radial CTE for each boule coresponding to
*------- the above ID

READ(IUNIT,5) RCB(K), K=1, SETNO)
WRITE(OUNIT1,") ’(all CTE values are in ppb)’
WRITE(OUNIT1,*) ’EXTREME BOUNDARY RADIAL CTE'
WRITE(OUNIT],5) (RCB(K), K=l, SETNO)
WRITE(OUNIT1,*)

‘-------- mean CTE value for each boule coresponding to the above ID --------
READ(IUNIT,8) (MCTE(L). L=l, SETNO)
WRITE(OUNIT1,") ’MEAN CTE VALUE FOR THE BOULE USED’
WRITE(OUNIT1,8) (MCI'E(L), I;l , SETNO)
WRITE(OUNIT1,"')

2 FORMAT (’ TOTAL NODE OF THIS MODEL: ’, I6)
3 FORMAT (I3, ’ BOULE USE’)

4 FORMAT (I3, ’ SETS STORED’)
5 FORMAT (1 113)

!depend on mesh and the element used
6 FORMAT (IX, 1116)

7 FORMAT (HAD

8 FORMAT (1 1F6.2)
9 FORMAT (1 1A3)
RETURN
END
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C Subroutine Read Data
C

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

SUBROUTINE READDATA (NONB, X, Y, Z, ZARC, SETNOD, NODESE. I,
J, K, DART, CUNTER, AC, RC, LCTE, AT,
NODE, U, V, W, BLOC, SETID, MCIE, RCB,
R, IUNIT, OUNITI, INV)

IMPLICIT NONE

INTEGER NBNO, SETNO, RCB, IUNIT, OUNITI
!Change this number accounding to the model

PARAMETER (NBNO = 29626, SETNO = 11)

INTEGER NONB, SETNOD(SETNO), NODESE. I, J, K,

CUNTER(NBNO), BLOC(SETNO), NODE(NBNO)

DIMENSION NONB(NBNO), NODESE(5000), !May change
. RCB(SETNO)

152

REAL X(NBNO), Y(NBNO), Z(NBNO), U, V, W,
MCI'E(SETNO), R
DOUBLE PRECISION ZARC, AC, RC, LCI'E, AT, DART (NBNO)
CHARACTER‘ 1 SETID(SETNO), 1NV(SEI'NO)
* ------------ -- read node number and x, y, z-coordinate ---------------

WRITE(OUNIT1,*) 'NODE NUMBER AND COORDINATE’
WRITE(OUNIT1,*) ’(X, Y, and Z in centimeter)’

DO 15 1:1, NBNO
READ(IUNIT,18) NONB(I), X(I), Y(I), Z(I)

* --------- localize z-coordinate in order to apply the CTE functions --------
!Compensate the mirror curve to be plane

ZARC=(10.00680-0.00054485*SQRT(X(I)**2+Y(I)**2)
.+0.00032137*(X(I)**2+Y(I)*‘2)-0.00000002*SQRT(X(I)“2+Y(I)“2)**3)

Z(I)=ABS(ABS(Z(I))-ABS(ZARC))
WRITE(OUNIT],18) NONB(I), X(I), Y(I), 2(1)
15 CONTINUE

WRITE(OUNIT1,“)

!this format is depend on MARC’s data file
18 FORMAT (IS, 3F10.5)

 

* initiation

 

DO K: 1, NBNO

DART(K) = O !artifical temperature(final)
CUNTER(K) = 1 !counter
NODE(K) = 0 ! global node number

END DO
U = 0
V = O
W = O
R = 0
AC = O !axial CTE
RC = 0 !radial CTE
LCI'E = 0 !local CTE
AT = 0 !artifical temperature (before counter)

CALL CALCULATECTE (I, J, K, SETID, SETNO, NODESE. NBNO,
SETNOD, BLOC, MCIE, RCB, CUNTER,
DART, NODE, RC, AC, AT, U, V, W.
R, X, Y, Z. IUNIT, OUNIT 1,1NV)

24 CONTINUE

153

RETURN

END
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C Subroutine Calaulate CTE Values
C

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
SUBROUTINE CALCULATECIE (I, I, K, SETID, SETNO, NODESE.
NBNO, SETNOD, BLOC, MCTE. RCB,
CUNTER, DART, NODE, RC, AC, AT.
U, V, W, R, X, Y, Z, IUNIT, OUNI'Tl, INV)

IMPLICIT NONE

INTEGER SETNO, NODESE(5000), NBNO,
SETNOD(SETNO), I, J, K,
RCB(SETNO), CUNTER(NBNO),
NODE(NBNO), BLOC(SETNO), IUNIT,
OUNITl

REAL X(NBNO), Y(NBNO), Z(NBNO),
MCI'E(SETNO), U, V, W, R

DOUBLE PRECISION LCTE. AC. RC. AT. DART (NBNO)
CHARACTER’] SETID(SETNO), INV(SETNO)

* read sets and node numbers

 

 

WRITE(OUNIT1,"')
WRITE(OUNIT1,") ’MATERIAL SET ID and # of NODE’

40 DO 850 J=1. SETNO
WRITE(OUNIT1,")

WRITE(OUNI'T1,42) SETID(.I), MCIEU), SETNODU)
READ(IUNI’TAS) (NODESE(I), I=l , SETNOD(J))
WRITE(OUNIT1,*) ’NODE #’
WRITE(OUNIT 1,45) (NODESE(I),I=1, SETNOD(J))

42 FORMAT (’Corning ID: ’, A1, ’ MEAN CIE: ’, F6.2,
. ’ 'IUTAL NODE: ’, I6)
45 FORMAT (918)

WRI'IE(OUNIT1 3’)
WRITE(OUNIT1,*) ’NODE LOCAL CTE ARTIFICAL TEMP’

 

DO 700 K=l, SETNODU)

U = ABS( X(NODESE(K)) )
V = ABS( Y(NODESE(K)) )
W = ABS( Z(NODESE(K)) )/ 2.54
* ------- localize coordinates in order to apply the CTE equations ---------

IF (BLOC(J) .EQ. 1) THEN !HEXAGON H1

154

R = SQR'T( U“*2 + V*"'2)
ELSE IF (BLOC(J) .EQ. 2) THEN !HEXAGON H2, H4, H5, H7

R = ABS( SQR’T( U**2 + V**2 ) - SQRT( 6.262801E1**2
. + 1.084749E2“2 ) )
ELSE IF (BLOC(J) .EQ. 3) THEN !HEXAGON H3, H6

R = ABS( SQR'T( U“2 + V“2 ) - 1.25256E2)
ELSE IF (BLOC(J) .EQ. 4) THEN !HEXAGON H9, H10

R = ABS( SQRT( U*"2 + V”2 ) - SQRT( 1.25E2**2
+ 7.216875E1"2 ))

ELSE IF (BLOC(J) .EQ. 5) THEN !HEXAGON H8
R = ABS( SQR’T( U**2 + V**2 ) - 1.443376E2)

ELSE !HEXAGON Hl lTR, H1 lCR, H1 1BR, H1 111., H11CL, H1 lBL
!!!They are all located at the same radius (depend on the cut)

R = ABS( SQRT( U“2 + V“2 ) - 142)

END IF
Cuuuuuuuu Reserve for different kind of cut tuna-suntan”
C
C HEXAGON H1 1TR
C 417 R=ABS( SQRT( U**2+V**2) -
C
C HEXAGON HllCR
C 418 R=ABS( SQR’T( U**2+V**2) -
C
C HEXAGON H1 IBR
C 419 R=ABS( SQRT( U**2+V**2) -
c
C HEXAGON HllTL
C 420 R = ABS( SQR'T( U**2 + V**2) -
C
C HEXAGON HllCL
C 421 R = ABS( SQRT( U**2 + V**2) -
C
C HEXAGON HIIBL
C 422 R = ABS( SQR'T( U”2 + V"2 ) -
C

Ct*tit*tttttttttt*00*0*#*#¥titttttttttittttttttttfittt*t******#*

* match CTE function with boule ID
*------- calculate axial and radial CTE value and set it equal AC and RC ----
!The geometry was built in the unit of centimeter
!The CTE data was measured in the increment of inch

 

 

R = R / 2.54

IF (INV(J) .EQ. 'Y’) THEN
W=ABS(W-6.50)

END IF

IF (SETIDU) .EQ. ’A’) THEN

155

* ----- BOULE A
601 AC=6.692332-0.518469*W+0.5145772*W**2-0.08778082*W**3

RC=3.1 l2074+l .972035*R-0.5966239*R**2+0.05764301*R**3-000229978
.1 *R**4+0.00003283*R* *5

ELSE IF (SE'I'IDU) .EQ. ’8’) THEN
* ----- BOULE B

602 AC=0.5396559-2.045625*W+1.477424*W**2-O.2464461 *W**3
RC=-1.796732+O.8693409*R-0.06908982*R**2+0.00173349*R**3
ELSE IF (SETIDU) .EQ. ’C’) THEN
* ----- BOULE C

603 AC=-0.]291866-7.l61 156*W+8.047375*W**2-3.449349*W* *3+0.6424932*
. W**4-0.04376068*W**5

RC=-0.3803922+0.9290383*R-0.08789862*R* *2+0.002341*R* *3
ELSE IF (SETIDO) .EQ. ’D’) THEN
*-----BOULE D
604 AC=5.545455-0.01897233*W
RC=-2.800654+1.27496*R-0.1 1594] 2*R**2+0.003061 1 1*R**3
ELSE IF (SETIDU) .EQ. ’E’) THEN
*-----BOULE E

605 ACfl.49029-5.790166*W+12.20213*W**2-9.146586*W**3+3.232425*W**4
. -0.5425023*W**5+0.03456165*W**6

RC=-1.636601+1.033141*R-0.08246023*R**2+0.0020451*R**3

ELSE IF (sauna) .EQ. ’F') THEN
* ----- BOULE F

606 AC=9.77436-22.71638*W+29.42734*W**2-16.97739*W**3+4.953306*W**4
. -0.7147381*W**5+0.04027234*W* *6

RC=0.7029412+1.05916*R-0.09579024*R**2+0.00238028*R**3
ELSE IF (SETIDU) .EQ. ’G’) THEN
* ----- BOULE G
607 AC=3.276]45-0.5414188*W+0.7995721*W**2-0.1424589*W* *3
RC=8.592484—0.1587197*R-0.02607646*R**2+0.00116003*R**3
ELSE IF (SETID(.I) .EQ. ’H’) THEN
* ----- BOULE H

608 AC=8.853064-12.62087*W+8.69454*W* *2-2.085492*W* *3+0.1612028*W* *4

156

RC=5.028431+0.5253537*R-0.07985887*R**2+0.002341 *R* *3
ELSE IF (SETIDU) .EQ. ’1’) THEN

* ----- BOULE]
609 AC=4.752985-l9.74305*W+22.12489*W“2-8.9939l6*w**3+l.582345*W“4
. -0.1038646*W**5
RC=0.2584967+1.64624*R-0.l520842*R**2+0.00359268*R**3
ELSE IF (SETIDU) .EQ. ’1') THEN
* ..... BOULE]

610 AC=0.88]5086+3.756955*W—3. l 8592*W**2+1.464537*W**3-0.3060963*
. W**4+0.02181849*W**5

RC=1.429739+0.3197691*R-0.03735501*R**2+0.00088508*R* *3
ELSE IF (SETIDO) .EQ. ’K’) THEN
* ----- BOULE K
61] AC=3.943191-3.50616l*W+1.6l8764*W**2-0.2306513*W**3
RC=-6.715359+].539639*R-0. ] 247688*R**2+0.00275473*R* *3
ELSE IF (SETIDO) .EQ. ’L’) THEN

1 ----- BOULE L
612 AC=4.082707-2.490537*W+1.046854*W**2-O.1395742*W**3

RC=-3.24085+1.401635*R-O.l352865*R**2+0.00312395*R**3
ELSE IF (SETIDO) .EQ. ’M’) THEN
* ----- BOULE M

613 AC=5.91749-3.336218*W+2.582592*W**2-0.6817157*W**3+0.05270092*
. W**4

RC=3.558738-1.261 102*R+0. 1876794*R**2-0.0106886*R**3+0.00020274
. 0R*$4

ELSE IF (SETIDU) .EQ. ’N’) THEN
* ----- BOULE N
614 AC=0.1230349-244144*W+3.439277*W**2-1.108452*W**3+0.1004006*W**4
RC=-2.435415-0.87940?A*R+0.1306292*R**2-0.00756096*R**3
. +0.00014769*R**4
ELSE IF (SETIDO) .EQ. ’0’) THEN
*----BOULE O
615 AC=9.523582-2.769093*W+0.3791 138*W**2-0.02929062*W**3

RC=-1.594] 18101782213*R-0.01375646*R**2+0.00040064*R**3

157

ELSE IF (SETIDU) .EQ. ’P’) THEN
* ----- BOULE P
616 AC=14.34699-9.504163*W+7.500762*W**2—l .962608*W**3+O.1571725*W“4
RC=5.587375-0.6626495*R+0.107328*R**2-0.00695281*R**3
ELSE [F (SEI'ID(J) .EQ. ’Q’) THEN
* ----- BOULE Q
617 AC=7.444272-4.748372*W+5.448423*W**2-2.024053*W**3+0.2l28269*W**4

RC=8.37401 1-2.006773*R+0.2988679*R**2-0.01635692*R**3
. +0.00029569*R**4

ELSE IF (SETIDU) .EQ. ’R’) THEN
*-----BOULE R
618 AC=2.149351+1.838947*W-2.1001 l8*W**2+0.6090008*W**3-0.0490396*
. W**4

RC=4.9237 l 9-0.5392579*R+0.09301 l 15*R**2—0.00576445*R**3
. +0.00011561*R**4

ELSE IF (SETIDO) .EQ. ’8’) THEN
* ----- BOULE S
619 AC=2.409357-1.10605*W+0.9503337*W**2-0.1851621*W**3
RC=3.76634+0.2251924*R-0.04252406*R**2+0.00129095*R**3
ELSE IF (SETIDU) .EQ. ’T’) THEN
* ----- BOULE T
620 A&1 .027348-9.032187*W+8.24326*W**2-2.283174*W**3+0.1910286*W**4
RC=-2.181373+1.974646*R-0.1479409*R**2+0.003103*R**3
ELSE IF (SETID(J) .EQ. ’U') THEN
* ----- BOULE U
62] AC=5.014202+0.6735526*W-0.08286857*W**2-0.0552281*W**3
RC=-6.253922+1 .890267*R-0.1321981*R**2+0.00267094*R**3

ELSE IF (SEI'ID(J) .EQ. 'v') THEN
* ..... BOULE v

622 AC=2.836815+1.292151*W-0.05330322*W**2
RC=1.557843+1.429415*R-0.1 141995*R**2+0.00252954*R**3

ELSE IF (SETIDa) .EQ. ’W’) THEN
*-----BOULE w

158

623 AC=7.220021-12.66894*W+13.02921 *W**2-5.894468*W**3+1.198597*W**4
. -0.09060564*W* *5

RC=2.795666-0.9847553*R+0.1479296*R* *2-0.00851709*R**3
. +0.00015882*R**4

ELSE
* ----- BOULE X
624 AC=6.52905-2.873 l49*W+0.7502868*W**2-0.09785729*W**3

RC=-1.662109+1.272718*R-0.2053296*R**2+0.0]050944*R**3
. -0.00016331*R**4

ENDIF

* calculate local CTE value

 

 

LCTE=AC*(RC/RCB(J))

 

* calculate artifical temperature ----------------
AT=25*(1+(LCTE-MCTE(J))/MCTE(I))
WRITE(OUN1T1,650) NODESE(K), LCI'E, AT
650 FORMAT(I6.1, 6X, E125, 10X, E12.5)

* set a counter to avoide overlap ................

 

IF ( DART (NODESE(K)) .EQ. 0 ) THEN
DART(NODESE(K)) = AT
ELSE
CUNTER(NODESE(I()) = CUNTER(NODESE(K)) + 1
DART (NODESE(KD = DART (NODESE(K» + AT
END IF
NODE(NODESE(K)) = NODESE(K)
700 CONTINUE
850 CONTINUE

RETURN
END

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

C Subroutine Print MARC Data File

C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

159

(cte.out)

 

 

SUBROUTINE PRINTMARC (NBNO, DART, CUNTER, ART, NODE, L,

. OUNIT2)

IMPLICIT NONE

INTEGER NBNO. OUNIT2

INTEGER CUNTER(NBNO), NODE(NBNO), L
DOUBLE PRECISION DART (NBNO), ART

 

WRITE(OUNIT2,880)
WRITE(OUNIT2.*)

DO 900 L = 1, NBNO
ART: DART(L) / CUNTER(L)

WRITE(OUNIT2,860) ART
WRITE(OUNIT2,870) NODE(L)

860 FORMAT (lPEl 1.5E1)
870 FORMAT (18)

880 FORMAT (’point temp’)
900 CONTINUE

RETURN
END

APPENDIX D
BOULE PLACEMENT AND
SURFACE DISPLACEMENT OF
ON -AXIS MIRROR

160

161

 

 

 

 

 

 

 

Figure D1. Trial #1, on-axis.

162

 

 

 

 

 

 

Figure D2. Trial #2, on-axis.

163

 

 

 

 

 

, Oil-3X18.

Figure D3. Trial #3

 

 

 

 

 

 

 

Figure D4. Trial #4, on-axis.

165

 

 

 

 

 

 

 

Figure D5. Trial #5, on-axis.

166

 

 

 

 

 

 

 

 

 

Figure D6. Trial #6, on-axis.

167

 

 

 

 

 

 

Figure D7. Trial #7, on-axis.

 

168

 

 

 

 

 

 

Figure D8. Trial #8, on-axis.

169

 

Figure D9. Trial #9, on-axis.

 

 

 

 

 

 

 

170

 

 

 

 

 

 

 

 

Figure D10. Trial #10, on-axis.

171

 

 

 

 

 

 

 

Figure D11. Trial #11, on-axis.

172

 

 

 

 

 

 

Figure D12. Trial #12, on-axis.

173

 

 

 

 

 

 

Figure D13. Trial #13,on-axis.

174

 

 

 

 

 

 

 

Figure D14. Trial #14, on-axis.

175

 

 

 

 

 

 

 

Figure D15. Trial #15,on-axis.

176

 

 

 

 

 

 

 

 

 

Figure D16. Trial #16, on-axis.

177

 

 

 

 

 

Figure D17. Trial#17, on-axis.

APPENDIX E
BOULE PLACEMENT AND
SURFACE DISPLACEMENT OF
OFF-AXIS MIRROR

178

179

 

 

 

 

 

Figure E1.Tria1 #1, off-axis.

180

 

 

 

 

 

 

 

Figure E2.Trial #2, off-axis.

181

 

 

 

 

 

 

Figure E3.Tria1 #3, off-axis.

182

 

 

 

 

 

 

 

Figure E4.Tria1 #4, off-axis.

183

 

 

 

 

 

 

 

Figure E5.Tria1 #5, off-axis.

184

 

 

 

 

 

 

Figure E6.Trial #6, off-axis.

185

 

 

 

 

 

Figure E7.Tria1 #7, off-axis.

HICHIGQN STQ

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