# Presentation: 111060

## Application of NASTRAN Sensitivity Coefficients To the Design of High-Performance Optical Systems

- Mark McGinnis – Swales Aerospace
- Dr. Joe Howard – NASA/GSFC
- Marcel Bluth – Swales Aerospace

**Notes:**
Thank emcee, admire what they have done with FEMCI

## JWST Overview

- The James Webb Space Telescope (JWST) is an orbiting infrared observatory that will take the place of the HST.
- Mass:   5400 kg (12,000 lbs)
- Diameter of primary mirror:   ~6.5 m (21.3 ft)
- Optical resolution:   ~0.1 arc-seconds
- Wavelength coverage:   0.6 - 28 microns
- Orbit:   L2 halo orbit (1.5 million km from Earth)
- Operating Temperature:   <50 K (-370 °F)
- Major primary mirror innovations
- Lightweight optics
- Folding segmented mirror
- Cryogenic actuators & mirror control
- Above information courtesy of JWST project web page (http://ngst.gsfc.nasa.gov)

**Notes:**
These optical requirements create mechanical requirements to control the PM surface to 10s of nanometers. This is comparable to trying to control the roof of the LA Superdome to 3 thousands of  a millimeter.  

## NASTRAN Model

- Model logically consists of 6 sections
- Spacecraft
- Deployable Tower
- PM Backplane
- PM Segments
- Center section
- Two deployed wings
- Secondary Mirror
- Aft Optics Assembly

- ISIM Enclosure

- Aft Optics Assembly
- Secondary Mirror

- PM & PM Reaction Structure
- PMRS Backing Structure
- Deployable Tower
- Spacecraft

## Analysis Technique Objective

- Observatory designed to allow required level of performance for very long exposures
- Low-expansion materials
- Minimize thermal gradients
- Wavefront sensing and control
- Traditional STOP analysis examines basic design
- Sought a fast integrated modeling procedure
- Achieve sufficiently accurate error estimates without either NASTRAN statics or ray-tracing runs
- Allow performance of Monte Carlo simulations or consideration of large trade spaces

**Notes:**
Exposure durations on order of thousands of seconds
	Simply hopeless without extreme measures to control variations in energy flux
		Composites in metering structure tuned for low CTE at cryo
			 Composites can achieve impressively low CTE, but only over limited temp ranges
			Limited knowledge at 50K
		-----
		L2 orbit and sunshade minimize magnitude of transient heat sources. 
			Never directly see the Earth, moon, or sun. 
			Multiple layer sunshade presents very stable and cool background to telescope
			gradients are tens of milliKelvins
	Mission is equally hopeless without wavefront sensing and control
		Initial error correction eliminates most deployment-related errors
			mechanisms are expected to deliver PM segments to  within a mm of the parent optical surface
		Periodically plan to correct for very-low frequency perturbations
			exact period is still TBD, corrects for long term drifting (moisture desorption)
		Thermal analysis centered around “mid-frequency” errors
	STOP = Structural Thermal Optical, powerful integrated modeling effort 
		requires full runs of three high-end codes for each case
Many types of analyses could not be done with a traditional STOP analysis
	Tight mission requirements require reasonably accurate tool
	Sought to look at many more cases than traditional STOP could

## Sensitivity Coefficients

- MSC/NASTRAN Optimization approximates response quantities by a truncated Taylor Series expansion
- Sensitivity coefficient is the constant of the Taylor series linear term
- Sensitivity is therefore the first step in optimization
- Finds partials of response quantities, *r**i*, wrt design variable, *x**i* by means of chain rule,
- First term is unity since we are studying displacements
- Second term comes from static equilibrium equation,

- 1: Equations, example from NASTRAN optimization documentation

**Notes:**
Sensitivity Coefficients falls out of NASTRAN method of approximating responses for Optimization

As with all NASTRAN information, all response info relates back to displacements
	Obviously these optical analyses are only interested in displs, so [TRANSITION]
	Second (and only) term depends on load case (P), and stiffness matrix K

## Sensitivity Analysis Example

* *Volume
-  V	= B H L
-  V	  B0H0L **+ H****0****L** • ΔB **+ B****0****L** • ΔH **+** **0** • ΔA
-  Bending stress
-  bend 	= Mc/I = 6PL/BH2
- bend 	 6PL/B0H02 **- 6PL/(B****0****H****0****)****2** • ΔB **- 12 PL/B****0****H****0****3** • ΔH **+** **0** • ΔA
-  Limit on thermal deflection
-  δTherm 	= AL(T2-T1)
-  δTherm	  A0L(T2-T1) **+ 0** • ΔB **+ 0** • ΔH + **L****(T****2****-T****1****)** • ΔA
-  Can not make CTE(T) a design variables

- A

- Design Variables

**Notes:**
Simple example problem
	Cantilevered bar loaded normal to axis
		Height H, Width B, expansion coefficient A. Length L is neither variable nor constraint
		Loaded at tip by force P

	Examples of sensitivity coefficient calculation

	Volume is design constraint
		H0L is sensitivity coefficient for partial of volume V wrt width B
		Similarly B0L for height H wrt V
		Not surprisingly the CTE does not have any effect on volume

	Bending Stress as design constraint
		load times length divided by original area squared is coefficient for stress wrt width
		load times length divided by base-height-cubed for height

	Thermal deflection as design constraint
		Independent of everything except A (as expected)
			Does depend on beam length and applied thermal load
			Exact solution for all delta-A

## Linear Optical Modeling

- Joe Howard creates optical sensitivity array
- Introduces a small motion for each optical DOF
- Performs a ray trace
- Recovers the OPD at the exit pupil
- 3-D array of size 100 x 100 x 6n
- This work considered only rigid optics
- Exported to MATLAB
- Disturbed optical performance calculated
- Motion of each optical DOF generated in some way
- Array elements multiplied by calculated motion
- Scaled arrays summed
- Generate a single 100x100 OPD map
- Mean despace and centroid-drift removed

**Notes:**
Joe works in OSLO
	Size of perturbation should be comparable to expected motions (10 nm and nanoradians)
		One ray trace required for each optical DOF
		Neglecting ISIM-effects for now although they could be considered

	N is the number of optical degrees of freedom. For this work, N is 40 (36, SM, TM, FSM, exit pupil (CHECK THIS WITH JOE))
		100 x 100 is the ray-trace ray bundle size
		Could work in any similar tool, i.e. nothing MATLAB specific

Source and method of disturbance calculation is irrelevant, MATLAB is obviously preferred for efficiency
	The motion of each optical DOF must be scaled to OSLO-applied perturbation
	Each array element is multiplied by this scalar, and the results summed
	Routinely referred to as best-fit plane
		Estimate centroid by calculating RMS
		modify RMS by perturbing FSM (Should be whole telescope)

## Material CTE Analysis

- Monte Carlo analysis used to establish probabilistic criteria for metering structure CTE
- Determine critical parts of the metering structure
- Create design variables for each selected area
- Create relationships to the selected properties
- Create design constraints of optical DOF
- Sensitivity coefficients requested as output
- Read coefficients into MATLAB
- Calculate WFE using linear optics
- Repeated these calculations thousands of times
- Calculate mean and standard deviations of optical performance figure of merit (OPD)

**Notes:**
Evaluate design options based on wavefront error; achieved by means of linear optical modeling
	Create separate property records for each area of interest
	DESVAR records, efficient method is to create a large number of these in one run
	DVPREL records, could obviously have been DVMREL, DVGREL
	DSCON records, here we are looking at optical motions
		Be sure to create boundaries on these lest they be declared inactive
		Generally these runs set up to minimize model weight but that is irrelevant
	Case Control record controlling frequency of output short-circuits the optimizer

Create MATLAB script to read in the coefficients from an unformatted output
	Will illustrate meaning of coefficients next
	Allowed me to predict optical motion

## Analytical Judgments

- Used contractor-delivered structural math models
- Constrained kinematically
- Applied uniform 0.050K delta-T to entire model
- Magnitude comparable to the slew temperature changes
- Gradients from actual temperature sets would have complicated results interpretation
- Highest coefficients would have come from locations experiencing the greatest temperature changes during the slew
- Would have been slew-case dependent

**Notes:**
Will be free-free, decided that a normal kinematic constraint was adequate

The applied load was not any expected temperature set.
	comparable magnitude
	Allows analyst to spot those parts which most dramatically affect optical performance

## Structural Details

**Notes:**
total of 12 design variables introduced, related to 12 set of material properties

## Random CTE Generation

- Generated Gaussian random number
- μ = 0, σ = 1
- Multiplied by the 1 - σ value of CTE for particular trial
- Representing variations in manufacturing process control
- Filtered using a trial-specific bandpass
- Repeat process if value exceeds limits
- Add in nominal CTE value
- Representing variations in OTE temp. from nominal T.
- Generated random number between –1 and 1
- Uniform distribution
- Multiplied by a test uncertainty factor
- Added to the value from previous step

**Notes:**
Assuming normalized Gaussian distribution 
	MATLAB has a function to generate such a number 
	Used engineering judgment to establish likely one-sigma variations in material CTE 
		creates random CTE values with desired deviation from a mean (although mean is still zero) 
	We assume that we will be coupon testing all parts
		This is not a trivial exercise given the need for high-precision cryogenic CTE measurement
		Tantamount to rejecting the part; cuts off the tails of the distribution 
	Add the nominal value to the above number
		Operating temperature dependent in composites

	Add an additional level of uncertainty to model the unknowable test errors
		Assume they are uniform
		Make assumptions based on test-procedure error budgets as to likely range of errors
		Blurs the sharp cutoffs (some bad parts get through, some good parts are rejected)

## Analysis Cases

- Cases should be considered as four pairs to be compared to the baseline

-  	1 Stnd. 	Bandpass	Test**	**Reject	Mean  	Deviation	Filter	Uncertainty	Rate	CTE
- _Case __	__(ppb/K)__	__(ppb/K) __	__(ppb/K) __	__(%)__	__(ppb/K)__	__Comments_
- 1		± 	± 	20	 	Baseline Case
- 2	 .78 	± 	±  	10	  	Narrow CTE Spread
- 3	 1.9 	 ± 	±  	50	  	Wide CTE Spread
- 4	  	 ± .40 	±  	61	  	Tighter CTE control
- 5	  	 ± 2.0	±  	1.4	  	Looser CTE control 6 	  	 ±  	± .50 	20	  	Better CTE testing
- 7	  	 ±  	± 2.0 	20 	  	Worse CTE testing
- 8	  	 ±  	±  	20	 1.15 	35K Nominal Temp 9	  	 ±  	±  	20	 .805 	45K Nominal Temp

**Notes:**
One-sigma variation in CTE: process control (shape of distribution). 
Bandpass filter: Amount CTE is allowed to vary from mean and still be used
Test Uncertainty factor: Amount of blurring that occurs at edge of filter
Reject Rate: Used to determine the 1-sigma number, this column was selected then the first calculated to achieve it
Mean CTE: Taken from material curves in provided NASTRAN model

## Mean CTE Convergence(All Trials)

- Mean CTE Convergence(All Trials)

## Mean Displacement Convergence(All Trials)

## Design Variable Error Contributions(Nominal Case)

## Conclusions

- Detailed multi-disciplinary analyses can be augmented with sensitivity coefficient-based analysis
- NASTRAN models can generate linear estimations of the effect of changes in structural parameters
- Optical analysis tools can create similar linear models of wavefront error
- Eliminates thousands of NASTRAN model runs and equal numbers of ray-tracing runs