Statement of Work Deliverables Action: Update Ansys and Matlab code to enhance model confidence. Bugs identified and resolved. Matlab data fed back to Ansys. Model effects of temperature profile for face sheet and truss. Capability for uniform and gradient temperature loads. Evaluate the effects of actuator uncertainties. Two different uncertainty models investigated. Document model and transfer technology to UF. Documented Matlab and Ansys GT code integrated in by UF. Result: All deliverables successfully completed.

Overview Statement of Work Deliverables Update of Previous Work – Correction of Matched CTE Cases Ansys Verification of Matlab Results Matlab Actuator Displacements Fed Back to Ansys (closed the loop) Thermal Aberration Reproduction vs. Correction Actuator Glitch Evaluation Random Glitch Discrete Glitch Actuator Forces Evaluation Concluding Remarks

Parameters Facesheet: Diameter (tip to tip) 2m F-number 1.5 Thickness m Modulus of Elasticity 95 GPa Poisson’s ratio 0.35 CTE 13E-6 /°C Substrate data: Beam diameter 1.16 mm Modulus of elasticity 68.3 Gpa Poisson’s ratio 0.33 CTE 0 /°C Actuator stiffness 2N/m Thermal loads: 1) 10°C uniform 2) 1°C/m gradient Actuators cases: 1) 15 2) 159 3) 1563

Update of Previous Work – Correction of Matched CTE Cases Problem: Similar displacements for matched and unmatched CTEs of the facesheet and truss. 10°C uniform load – 159 actuators Facesheet only Matched CTEs (incorrect) Mismatched CTEs Solution: Bug fixed in Ansys code so correct CTEs assigned in the matched case.

Update of Previous Work – Correction of Matched CTE Cases Problem: Similar displacements for matched and unmatched CTEs of the facesheet and truss. 10°C uniform load – 159 actuators Facesheet only Matched CTEs (incorrect) Mismatched CTEs Solution: Bug fixed in Ansys code so correct CTEs assigned in the matched case.

Correction of Matched CTE Cases - 10ºC Uniform, 159 Actuators Previous Facesheet only Updated Matched CTEs Previous Matched CTEs Matched CTE case is now similar to the facesheet only case.

Correction of Matched CTE Cases - 1ºC/m Gradient, 159 Actuators Previous Facesheet only Updated Matched CTEs Previous Matched CTEs Matched CTE case is now similar to the facesheet only case.

ANSYS Verification of Matlab Results Previously, Ansys provided influence coefficients and the response to thermal loads. These were imported into Matlab to calculate actuator displacements to reproduce the aberration. Currently, the Matlab-calculated actuator displacements are fed back into Ansys to either: Correct the thermally loaded system. Reproduce the thermal aberrations by actuation from an initially unloaded system (similar to previous Matlab work). Results: Correct and Reproduce yield similar results.

Summary: Reproduction vs. Correction 15 159 1563 Reproduction uniform x-grad Absolute RMS (nm) 2549 147 2083 93 341 40 RMS (%) 0.926% 0.570% 1.232% 0.992% 0.127% 1.203% 15 159 1563 Correction uniform x-grad Absolute RMS (nm) 2551 147 2084 93 342 40 RMS (%) 0.927% 0.570% 1.232% 0.995% 0.127% 1.204% Results: The correction is produced with essentially the same accuracy as the reproduction.

Ansys Plots: Reproduction vs. Correction Reproduction vs. Correction Plots: 15 actuators, 10ºC Uniform 15 actuators, 1ºC/m X-Gradient 159 actuators, 10ºC Uniform 159 actuators, 1ºC/m X-Gradient 1563 actuators, 10ºC Uniform 1563 actuators, 1ºC/m X-Gradient

10ºC Uniform, 15 Actuators Correction Reproduction Absolute RMS 2549 nm Normalized RMS 0.926 % Absolute RMS 2551 nm Normalized RMS 0.927 %

1ºC/m X-Gradient, 15 Actuators Correction Reproduction Absolute RMS 147 nm Normalized RMS 0.570 % Absolute RMS 147 nm Normalized RMS 0.570 %

10ºC Uniform, 159 Actuators Correction Reproduction 2083 1.232 2084 Absolute RMS 2083 nm Normalized RMS 1.232 % Absolute RMS 2084 nm Normalized RMS 1.232 %

1ºC/m X-Gradient, 159 Actuators Correction Reproduction Absolute RMS 93 nm Normalized RMS 0.992 % Absolute RMS 93 nm Normalized RMS 0.995 %

10ºC Uniform, 1563 Actuators Correction Reproduction 341 0.127 342 Absolute RMS 341 nm Normalized RMS 0.127 % Absolute RMS 342 nm Normalized RMS 0.127 %

1ºC/m X-Gradient, 1563 Actuators Correction Reproduction Absolute RMS 40 nm Normalized RMS 1.203 % Absolute RMS 40 nm Normalized RMS 1.204 %

Actuator Glitch Three cases of actuator glitch are compared: No Glitch – actuators have infinite resolution. Discrete Glitch – actuator displacements are multiples of 50 nm. Random Glitch – a random glitch between +/- 50 nm is added to each actuator displacement. Results: The RMS errors are not sensitive to small glitches. (Small is relative to maximum actuator strokes).

Glitch Summary No Glitch 15 159 1563   uniform x-grad Absolute RMS Error (nm) 2551 147 2084 93 342 40 RMS Error (%) 0.926% 0.570% 1.232% 0.995% 0.127% 1.204% Glitch/Max Stroke (%) 0.07% 1.47% 0.15% 2.00% 0.24% 1.79% Discrete Glitch 2556 468 2089 216 347 72 0.929% 1.813% 1.235% 2.316% 0.129% 2.146% Additional RMS Error (%) 0.00% 1.24% 1.32% 0.94% Random Glitch 2593 353 268 346 88 0.942% 1.366% 1.236% 2.871% 2.629% 0.02% 0.80% 1.88% 1.42% Result: Glitch Additional RMS Error %  Glitch/Maximum Actuator Stroke %.

Ansys Plots: Actuator Glitch No Glitch vs. Discrete Glitch vs. Random Glitch: 15 actuators, 10ºC Uniform 159 actuators, 10ºC Uniform 1563 actuators, 10ºC Uniform 15 actuators, 1ºC/m X-Gradient 159 actuators, 1ºC/m X-Gradient 1563 actuators, 1ºC/m X-Gradient

10ºC Uniform, 15 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 2551 nm % RMS 0.927 % Abs RMS 2556 nm % RMS 0.929 % Abs RMS 2593 nm % RMS 0.942 %

10ºC Uniform, 159 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 2084 nm % RMS 1.232 % Abs RMS 2089 nm % RMS 1.235 % Abs RMS 2089 nm % RMS 1.236 %

10ºC Uniform, 1563 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 342 nm % RMS 0.127 % Abs RMS 347 nm % RMS 0.129 % Abs RMS 346 nm % RMS 0.129 %

1ºC/m X-Gradient, 15 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 147 nm % RMS 0.570% Abs RMS 468 nm % RMS 1.813 % Abs RMS 353 nm % RMS 1.366 %

1ºC/m X-Gradient, 159 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 93 % RMS 0.995% Abs RMS 216 nm % RMS 2.316 % Abs RMS 268 nm % RMS 2.817 %

1ºC/m X-Gradient, 1563 Actuators No Glitch Discrete Glitch Random Glitch Abs RMS 40 % RMS 1.204% Abs RMS 72 nm % RMS 1.146 % Abs RMS 88 nm % RMS 2.629 %

Actuator Forces Force Evaluation Increasing numbers of actuators, increases required actuator forces. 15 actuators case met force specifications (< 0.1N) for both 10ºC Uniform and 1ºC/m X-Gradient loads. 159 and 1563 actuators cases exceed force specifications for both loadings. (Note: will add 45 and 93 actuator cases.)

Ansys Plots: Actuator Forces Force Plots 15 actuators, 10ºC Uniform 45 actuators, 10°C Uniform 93 actuators, 10°C Uniform 159 actuators, 10ºC Uniform 1563 actuators, 10ºC Uniform 15 actuators, 1ºC/m X-Gradient 45 actuators, 1°C/m X-Gradient 93 actuators, 1°C/m X-Gradient 159 actuators, 1ºC/m X-Gradient 1563 actuators, 1ºC/m X-Gradient

10ºC Uniform, 15 Actuators P2V 1419 m Absolute RMS 2550 nm Normalized RMS 0.926 % Maximum Force 0.026 N Maximum Stroke 68.4 actuator forces aberration Maximum actuator force within limit (< 0.1N).

10ºC Uniform, 45 Actuators P2V 592 m Absolute RMS 2,482 nm Normalized RMS 4.72 % Maximum Force 3.09 N Maximum Stroke 49.3 actuator forces aberration Maximum actuator force is exceeded (> 0.1N).

10ºC Uniform, 93 Actuators P2V 651 m Absolute RMS 2,329 nm Normalized RMS 2.03 % Maximum Force 6.92 N Maximum Stroke 38.8 actuator forces aberration Maximum actuator force is exceeded (> 0.1N).

10ºC Uniform, 159 Actuators P2V 661 m Absolute RMS 2085 nm Normalized RMS 1.23 % Maximum Force 10.0 N Maximum Stroke 32.9 actuator forces aberration Maximum actuator force is exceeded (> 0.1N)

10ºC Uniform, 1563 Actuators P2V 654 m Absolute RMS 342 nm Normalized RMS 0.127 % Maximum Force 23.0 N Maximum Stroke 20.8 actuator forces aberration Maximum actuator force is exceeded (> 0.1N)

1ºC/m X-Gradient, 15 Actuators P2V 254 m Absolute RMS 147 nm Normalized RMS 0.570 % Maximum Force 0.0006 N Maximum Stroke 3.4 actuator forces aberration Maximum actuator force is within limit (< 0.1N).

1ºC/m X-Gradient, 45 Actuators P2V 283 m Absolute RMS 121 nm Normalized RMS 0.635 % Maximum Force 0.347 N Maximum Stroke 3.45 actuator forces aberration Maximum actuator force is exceeded (> 0.1N).

1ºC/m X-Gradient, 93 Actuators P2V 170.7 m Absolute RMS 106 nm Normalized RMS 0.763 % Maximum Force 0.469 N Maximum Stroke 2.98 actuator forces aberration Maximum actuator force is exceeded (> 0.1N).

1ºC/m X-Gradient, 159 Actuators P2V 88.1 m Absolute RMS 93 nm Normalized RMS 0.995 % Maximum Force 0.58 N Maximum Stroke 2.5 actuator forces aberration Maximum actuator force is exceeded (> 0.1N)

1ºC/m X-Gradient, 1563 Actuators P2V 16.1 m Absolute RMS 40 nm Normalized RMS 1.204 % Maximum Force 4.9 N Maximum Stroke 2.8 actuator forces aberration Maximum actuator force is exceeded (> 0.1N)

Concluding Remarks Only the 15 actuator case satisfied all specified criteria RMS errors for the two thermal loads were low, less than 1%. Maximum force levels required only 25% of the holding force. Maximum stroke was about 50% of maximum. Large P2V amplitudes corrected (1419 m and 254 m). Increasing actuators increased actuator forces to unacceptable levels.