Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
Table 3.1: Uncertainty parameters for SCI development model. Name Description p0 E1 Young’s Modulus of truss 1 & 2 72 GPa E2 Young’s Modulus of truss 3 & 4 72 GPa m1 00 11 000 111 00 11 d1 d2 000 111 d2 00 11 000 111 00 11 000 111 00 11 000 111 E2 m 2 E 1 E 1 E 2 Y d1 000 111 000 111 000 111 000 111 000 111 Carlo propagation, depending on the application. The Monte Carlo propagation serves as a baseline comparison to verify the assumption of convexity. In the following section these techniques are demonstrated through an uncertainty analysis of the PT SCI design. 3.1.3 Example: SCI Development Model In general, material properties represent a significant source of parametric uncer- tainty. Properties such as Young’s modulus can be difficult to measure or predict accurately and often vary from sample to sample of the same material. Therefore, in the SCI sample problem the Young’s modulus of the four truss segments, taken in pairs, are chosen as the uncertainty parameters, as listed in Table 3.1. The param- eter, E1 is the Young’s modulus of the truss segments with negative-x coordinates, and E2 corresponds to the truss segments with positive-x coordinates, as shown in the accompanying figure. The nominal values of 72 GPa are based on the material properties of aluminum [17]. The Young’s Modulus of the truss segments are con- sidered in pairs to reduce the computation required for the development model. In application, any number of uncertainty parameters can be considered, but the com- putation time required for uncertainty analyses and robust optimizations increases significantly with the number of parameters. The addition of uncertainty parameters to the model results in a performance met- 76 X
ic, σz(�x, �p), that is dependent on both the tailoring, �x and uncertainty parameters, �p. Like the tailoring parameters, the uncertainty parameters affect the performance through the finite element matrices. In this particular case, only the stiffness matrix is affected, since Young’s modulus does not appearinthemassmatrixatall(Equa- tions 2.2 and 2.3). The uncertainty model is bounded and uniformly distributed about the nominal parameter value over a ranged defined by ∆i: � 1 − ∆i � � pi0 ≤ pi ≤ 1+ 100 ∆i � pi0 100 (3.1) where pi is one of the uncertainty parameters in Table 3.1. ∆i is a percent of the nominal parameter value and is referred to throughout as the uncertainty level. The uncertainty model is propagated through the PT design with both a vertex search method and Monte Carlo analysis. In the vertex search, only the vertices, or corners, of the uncertainty space are considered. The number of performance evaluations necessary for this propagation method, npv, grows exponentially with the number of uncertainty parameters, np: npv =2 np (3.2) As discussed previously, the worst-case performance is at a vertex if the uncertainty space is convex. To check this assumption, a Monte Carlo analysis is run in which values for the uncertainty parameters are chosen randomly from their distributions. The performance is then evaluated at each uncertainty combination. If the convexity assumption holds, all of the performance values from the Monte Carlo should be at or below the worst-case value from the vertex search. The results of the uncertainty analyses on the PT design are plotted in Figure 3-1. The uncertainty level for all parameters is 10%, and the Monte Carlo analysis is run with 500 random uncertainty values. The Monte Carlo results are shown in a his- togram plotted against RMS performance. It is clear that even a small amount of uncertainty in the Young’s Modulus values results in a large spread on the perfor- mance prediction. The dotted line to the right of the plot indicates the worst-case 77
- Page 26 and 27: Table 1.1: Effect of simulation res
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- Page 30 and 31: has been found that structural desi
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- Page 41 and 42: ometer (SCI). In the following sect
- Page 43 and 44: The equations of motion of the unda
- Page 45 and 46: The frequency response functions fr
- Page 47 and 48: the output covariance matrix, Σz,
- Page 49 and 50: where the subscript indicates the i
- Page 51 and 52: 2.3.3 Design Variables The choice o
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- Page 55 and 56: algorithms begin at an initial gues
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- Page 61 and 62: and the RMS OPD is computed using E
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- Page 79 and 80: Magnitude, OPD/F x [µm/N] Magnitud
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- Page 83 and 84: metric to the cost function. Note,
- Page 85 and 86: tion: ∂hi (z,�x, �pi) ∂�x
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- Page 89 and 90: Table 3.3: Algorithm performance: a
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- Page 97 and 98: Norm. Cum. Var. [µm 2 ] PSD [µm 2
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- Page 103 and 104: RMS performance, [µm] 400 350 300
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- Page 109 and 110: on a physical truss. Since tailorin
- Page 111 and 112: Table 4.1: Tuning parameters for SC
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- Page 119 and 120: Norm. Cum. Var. [µm 2 ] PSD [µm 2
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ic, σz(�x, �p), that is dependent on both the tailoring, �x and uncertainty parameters,<br />
�p. Like the tailoring parameters, the uncertainty parameters affect the performance<br />
through the finite element matrices. In this particular case, only the stiffness matrix<br />
is affected, since Young’s modulus does not appearinthemassmatrixatall(Equa-<br />
tions 2.2 and 2.3). The uncertainty model is bounded and uniformly distributed<br />
about the nominal parameter value over a ranged defined by ∆i:<br />
�<br />
1 − ∆i<br />
� �<br />
pi0 ≤ pi ≤ 1+<br />
100<br />
∆i<br />
�<br />
pi0<br />
100<br />
(3.1)<br />
where pi is one of the uncertainty parameters in Table 3.1. ∆i is a percent of the<br />
nominal parameter value and is referred to throughout as the uncertainty level.<br />
The uncertainty model is propagated through the PT design <strong>with</strong> both a vertex<br />
search method and Monte Carlo analysis. In the vertex search, only the vertices,<br />
or corners, of the uncertainty space are considered. The number of performance<br />
evaluations necessary for this propagation method, npv, grows exponentially <strong>with</strong> the<br />
number of uncertainty parameters, np:<br />
npv =2 np (3.2)<br />
As discussed previously, the worst-case performance is at a vertex if the uncertainty<br />
space is convex. To check this assumption, a Monte Carlo analysis is run in which<br />
values for the uncertainty parameters are chosen randomly from their distributions.<br />
The performance is then evaluated at each uncertainty combination. If the convexity<br />
assumption holds, all of the performance values from the Monte Carlo should be at<br />
or below the worst-case value from the vertex search.<br />
The results of the uncertainty analyses on the PT design are plotted in Figure 3-1.<br />
The uncertainty level for all parameters is 10%, and the Monte Carlo analysis is run<br />
<strong>with</strong> 500 random uncertainty values. The Monte Carlo results are shown in a his-<br />
togram plotted against RMS performance. It is clear that even a small amount of<br />
uncertainty in the Young’s Modulus values results in a large spread on the perfor-<br />
mance prediction. The dotted line to the right of the plot indicates the worst-case<br />
77
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