12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 10 - 12 September 2008, Victoria, British Columbia Canada

AIAA 2008-5928

Probabilistic Optimization of Integrated Thermal Protection System Sunil Kumar, Diane Villanueva, Bhavani V. Sankar and Raphael T. Haftka University of Florida, Gainesville, Florida, 32611, USA Abstract The paper considers probabilistic optimization of integrated thermal protection system (ITPS) that combines the thermal protection function with the structural load carrying function. For ITPS design, structural and thermal requirements usually conflict. Increased structural thickness helps carry loads but increases heat conduction. Designers need to allocate risk between structural and thermal failure modes. In deterministic designs, this risk allocation is implicit in the choice of safety factors. Probabilistic design allocates risk explicitly. This paper uses a simple case to illustrate the difference between deterministic and probabilistic risk allocation for ITPS design.. For this example the deterministic design allocates risk about equally between thermal and structural failure, while the probabilistic design allocates most of the failure to thermal failure. Keywords: Integrated Thermal Protection System Design, Probabilistic Optimization, risk allocation Nomenclature θ = angle of corrugations ρ = density ds = Saffil foam thickness k = thermal conductivity p = half the length a unit cell of the corrugated-core sandwich panel tB ,tT ,tW = thickness of top face sheet, bottom face sheet and web β, γ = non dimensional parameters to predict maximum bottom face sheet temperature C = specific heat Capacity of ITPS panel CB = specific heat Capacity of Bottom Face Sheet C1 = thermal capacity of ITPS, i.e. Allowable Maximum Bottom face sheet temperature C2 = structural capacity of ITPS panel R1 = thermal response of ITPS, i.e. maximum Bottom face sheet temperature reached R2 = structural response of ITPS wt = mass per unit area of ITPS panel SM = Safety Margin SF = Safety Factor Φ-1 = inverse cumulative density function of standard normal distribution βallow= minimum allowed reliability index of ITPS panel g1, g2 = constraint functions for thermal and structural constraints

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

1.

INTRODUCTION

Thermal protection systems for protecting space vehicles during atmospheric reentry traditionally do not perform structural functions. In fact, as in the case of the space shuttle they are fragile and susceptible to damage. Therefore, for many years, there has been interest in integrated thermal protection systems (ITPS) that combine thermal and structural functions [1]. The design of such systems presents an inherent conflict between structural requirements that require robust structural members and the thermal requirements that seek to minimize heat conduction through these members. When conflicting requirements lead to design compromises, it can be expected that deterministic approaches will lead to different compromise than probabilistic designs. The reason is that deterministic designs are usually based on mandated safety factors, while probabilistic designs adjust the safety factors to the magnitudes of the uncertainties and the cost of providing safety against competing failure modes. In other words, the probabilistic optimization will in general result in different risk allocation between the different modes of failure. The objective of this paper is to explore the difference in terms of risk allocation between deterministic and probabilistic design of an ITPS for spacecraft reentry based on a corrugated core sandwich panel concepts [3]. Structural failure constraints include limits on stresses and buckling loads and thermal failure is defined by a temperature limit. As a first step in exploring the difference between the probabilistic and deterministic design compromise we consider a simplified version of the ITPS design problem. A response surface approximation of the maximum bottom face sheet temperature was developed in [1]. They used mild simplifying assumptions allowed to reduce the number of variables in the approximation to two non-dimensional variables. We add a greatly simplified buckling constraint, and a response surface for the stresses. in the work in [3] suggested an ITPS panel based on alumino-silicate/Nextel 720 composites for top face sheet and web and beryllium for bottom face sheet. We seek the optimal dimensions for this material choice. .Probabilistic structural optimization is expensive because repeated analyses are required for calculating probability as the structure is being re-designed. Several methods have been proposed to alleviate the computational burden [1]. In particular, when Monte Carlo simulation is used for obtaining probabilities of failure, cost can be reduced considerably for applications where the failure criterion can be separated into a capacity and a response, with the response depending on one set of random variables and the capcity depending on another set, uncorrelated to the first set. This separable Monte Carlo technique [5] is used here to reduce computational cost. In the paper, we first present problem description in section 2.1 and then do formulation of optimization in section 2.2. Later on in section 2.3 we explore reliability issues. Section 3 deals with results. There we first present deterministic optimization in section 3.1 and then results for probabilistic optimization in section 3.2. Finally, we conclude in section 4. Acknowledgements and references are also presented at the end of the paper.

2.

INTEGRATED THERMAL PROTECTION SYSTEM

2.1 Problem Description The thermal protection system (TPS) of space vehicles needs to satisfy a wide range of requirements [1-4]. During ascent and reentry, TPS has to withstand high temperatures and must also be light weight in order to reduce the overall weight of the vehicle. The Integrated Thermal Protection System (ITPS) analyzed here is an extension of the ARMOR TPS design [4], using ia corrugated-core sandwich panel (see Figure 1).

Figure 1. A unit-cell of the simplified ITPS design

It is expected that by suitably designing the corrugated-core sandwich structure a robust, operable, weight-efficient, load-bearing TPS can be developed. The design process for the corrugated-core ITPS was started by simplifying the geometry of the panel so as to include a minimum number of geometric (design) variables. These variables include the top face thickness (tT), bottom face thickness (tB), thickness of the foam (ds), web thickness (tw), corrugation angle (θ), and length of unit cell (2p). An optimum deterministic design consisting of alumino-silicate/Nextel 720 composites for top face sheet and web and beryllium for bottom face with dimensions given in Table 1 was proposed by Gogu et al. [1]. For the current ITPS design problem, a quadratic response surface approximation of maximum bottom face sheet temperature was adopted to solve the optimization problem. The response surface approximations are functions of design variables. For this preliminary design optimization, the maximum bottom face sheet temperature is chosen. Figure 3 shows the heat flux profile at top of ITPS that was used to generate response surface shown in figure 2 to predict maximum bottom face sheet temperature. Parameters tT tB ds

Values 2.1 mm 3.1 mm 117.3 mm

tw θ p

5.3 mm 870 117 mm

Table 1: Value of the preliminary Geometric Design parameters

To explore the different risk allocations of the deterministic and probabilistic approaches, we first analyze a simplified problem where the design constraints are the maximum bottom face sheet temperature and a simplified buckling constraint. Using two non dimensional parameters defined in Eq. (6)

(6) and response surface shown in figure [2], we can obtain the maximum bottom face sheet temperature for any design in the vicinity of design suggested by Gogu et al. [1].

Figure 2: Response surface approximation of bottom face sheet temperature using Non Dimensional parameters

Figure 3: Heating profile used for preliminary design of ITPS

2.2 Optimization Formulation For the deterministic design the temperature constraint is given in terms of a safety margin S on the calculated temperature Tcal Tcal (β, γ ) + S ≤ Tallow

(7)

The buckling constraint uses a simplified approach assuming that the web is the weakest link (because thermal considerations push it to be thin) and that its buckling is overall Euler buckling, so that the buckling load Pbuckling is proportional to the cube of the thickness and inversely proportional to its length. The second assumption is that the compressive force P in the web is proportional to the temperature differential between the upper and lower face sheets t w3 Pbuckling = c1 2 ds

P = c2 ∆T

(8)

The buckling constraint is then expressed as the condition that the ratio between the load and the buckling load is the same as for the deterministic design

P / Pbuckling ≤ ( P / Pbuckling )

deterministic

(9)

which yields 2 2    ∆Tds  ≤  ∆Tds   t 3   t 3   w  deterministic  w cal

(10)

The objective function is the weight of panel per unit depth given by Eq. (9).

ρW tW d s (11) sin(θ ) Only three of the six geometric design parameters are used to illustrate the different risk allocation. These are the web thickness, bottom face sheet thickness and panel width. Other parameters were fixed to the design suggested by Gogu et al. [1]. For temperature constraint a safety margin of 48.5 K used which corresponds to the allowable maximum bottom face sheet temperature of 433.9 k. The critical value of the buckling criterion is chosen to give us a safety factor of 1.38 for deterministic design and this value also leads to about 1% probability of failure of the system. These values of initial safety margins and safety factors are chosen to prevent both failures almost equally. wt = ρT tT p + ρ B t B p +

2.3 Uncertainty Model

Due to modeling and computational limitations there is error involved in calculating maximum bottom face sheet temperature. This error, eT is modeled here as an uniformly

distributed random variable over a 10% range of the calculated temperature from response surface, see Eq. (12). This introduces a corresponding error in the compressive force in the web. In addition, we assume that there is a similarly distributed error es in the calculation of the buckling load as shown in Eq. (13). Ttrue = Tcal (1 + eT )  ∆Td 2  (1 + e ) P s  T =   3 Pbuckling  tw cal . (1 + eS )

(12) (13)

The constraints can be modeled by Eq. (12) and (15).

g1 =

g2 =

Tallowable − Tcal ≡ C 1 − R1 1 + eT

(1 + eS )t 3

w ,initial

(1 + eT )d 2

s ,initial

∆Tinitial

 t 3  −  w  ≡ C 2 − R2 2  ∆Tds cal .

(14)

(15)

In the thermal constraint, the error has been moved to the capacity side in order to allow us to use the more efficient separable Monte Carlo sampling. As written in Eq. (14), the uncertainty in the capacity depends only on the error, and the uncertainty in the response (calculated temperature) depends only on the variability in material properties and dimensions. The buckling constraint has been similarly re-formulated for the same purpose. The uncertainties in material properties of plates were modeled as uniform random variable varying between the ranges given in Table 2 below. The heat capacity and conductivity of saffil foam is very temperature dependent property. They were modeled as uniform random variables varying between a ranges obtained by changing representative temperatures from 583k to 683k. The geometric parameters were assumed to have 5% range about the mean values and they were also modeled as uniformly varying between those extremes. The time of heat influx was assumed to have 10% range about the mean value. Material Property Density Thermal Conductivity Heat Capacity

Nextel 720 Range 2450 2600 2.52 2.93 950 1100

Beryllium Range 1840 1860 190 216 1820 1930

Saffil Foam Range 24 24 0.0896 0.1208 1099 1142

Table 2: Material Properties of the Materials used in ITPS Panel

Units 3 kg/m W/mK J/Kg/K

3.

RESULTS

3.1 Deterministic Optimization First deterministic optimization procedure was implemented through ‘fmincon’ function of MATLAB which uses sequential quadratic programming. Results are presented in the Table 3 which are very close to those obtained by Gogu et al. [1]. The total probability of failure of this optimum design is 2% according Ditlevsen’s first-order upper bound [15] given by Eq (16). Pf1 and Pf2 are probabilities of two constraints being violated. They are both 1%. det (16) PFS (= Pf 1 + Pf 2 ) ≤ PFS Design variables

Initial design

Final design

Top thickness, tw

5.0 mm

3.1 mm

Bottom face sheet thickness, tB

6.0 mm

5.1 mm

Foam thickness, ds

115 mm

120 mm

Weight per unit depth

3.66 kg/m

2.68 kg/m

Table 3: Deterministic optimum results based on three design parameters

3.2 Probabilistic Design Optimization In this part of the design process we fix the probability of failure of the system to an acceptable level comparable to Deterministic design. Next, we perform probabilistic optimization which redistributes risk between maximum temperature constraint and buckling criteria. The problem formulation is given by Eq. (17). The constraint on the probability of failure is replaced by a constraint on the reliability Index, β. The reliability Index is obtained from the probability of failure using the inverse cumulative density function of standard normal distribution as shown in Eq. (18). min wt w ,t

det s.t. PFS (= Pf 1 + Pf 2 ) ≤ PFS FS βallow = −Φ−1(Pdet. )

(17)

(18)

The probabilistic optimization problem was not handled well by fmincon. Therefore, we solve the problem graphically by constructing the Pareto frontier between weight and reliability for designs near deterministic optimum. A Pareto frontier consists of all Pareto

optimal Points, which are points that cannot be improved in all objectives (here weight and reliability) simultaneously. The weight and reliability index of a large number of candidate designs and the Pareto front are shown in Figure 4. By choosing an acceptable level of reliability, we can predict the minimum weight and other design parameters using some interpolation technique between two close designs on Pareto Frontier (see Figure 5). The result linear interpolation of design parameters is shown in Table 3. Figure 6 confirms that we got a good probabilistic optimum. We can see that weight contours and reliability index contours touch at the predicted optimum. The risk allocation favored by the probabilistic formulation may be explained by the cubic dependence of the buckling load on the web thickness. The increase in temperature due to increased in web thickness is only linear. In addition, reduction in the bottom face sheet thickness increases the temperature, but it reduces the weight and reduces the compression in the web, so it was selected as the primary mechanism to reduce the weight. The deterministic design had safety factors such that probabilities of failure for both thermal and buckling constraint were almost equal. In contrast, the probabilistic optimum allocates less risk to buckling failure and compromises a little on the thermal margin to reduce weight for the same expected reliability of the design. This can be seen in the Tables 4 and 5. Contours of percentage contribution to system probability of failure by thermal constraint in Figure 6 show that as we go from the deterministic optimum to probabilistic optimum percentage contribution of thermal failure increases from about 50% to 90%, thereby eliminating most of buckling contribution to system probability of failure. Design points when compared for reliability and weight

Reliability Index of sum of two probabilities

5

4

3

2

1

0

-1

-2 2.2

2.3

2.4

2.5 2.6 2.7 Weight in Kg/m

2.8

2.9

3

Figure 4: Weight and reliability for some Designs close to deterministic optimum

Pareto Optimal Design points

Reliability Index of sum of two probabilities

2.7 2.6 2.5 Optimal Design

2.4

Initial Design Point

2.3 2.2 2.1 2 1.9 2.63

2.64

2.65

2.66 2.67 2.68 Weight in Kg/m

2.69

2.7

2.71

2.72

Figure 5: Pareto Front between Reliability Index and weight

Web thickness, tw (mm)

Deterministic Design 3.15

Probabilistic Design 3.2

Bottom face sheet thickness, tB mm)

5.12

4.9

Saffil Foam thickness , ds

120.0

120.2

PF(Temp)

0.0038

0.0097

PF(buckling) PTotal

0.0057

0.0006

0.0095

0.0104

Reliability

2.34

2.34

Weight (kg/m)

2.68

2.66

Optimum

Table 4: Comparison of Deterministic and Probabilistic optimum results

Safety Margins and Safety Factors Thermal Safety Margin

Deterministic Design 48.5

Probabilistic Design 45.8

Buckling Safety Factor

1.38

1.44

Table 5: Changes in Safety factors and safety margins indicated by Probabilistic optimization

Figure 6: Optimality of Probabilistic Design

4.

CONCLUSION

The paper presented a study intended to illustrate the difference in risk allocation between structural and thermal failure in an integrated thermal protection system (ITPS). A simplified model of buckling response was used to represent structural considerations. It was found that while the deterministic design allocated risk about equally between structural failure and thermal failure, the probabilistic design allocated almost all of the risk to the thermal failure. This was explained due to the cubic dependence of the buckling load on the thickness of the ITPS web as compared to linear dependence of the bottom face sheet temperature. ACKNOWLEDGEMENT The material is based upon work supported by NASA under award No. NNX08AB40A. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Aeronautics and Space Administration. REFERENCES 1. Gogu, C., Bapanapalli, S.K., Haftka, R.T., Sankar, B.V., “Analysis and Design of Corrugated-Core Sandwich Panels” 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, May 2006, Newport, Rhode Island

2. Du, X., and Chen, W., “Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” ASME Journal of Mechanical Design, Vol. 126, No. 2, 2004, pp. 225-233. 3. Bapanapalli, S.K., Martinez O.M., Gogu, C., Haftka, R.T., Sankar, B.V., “Comparison of Materials for Integrated Thermal Protection Systems for Spacecraft Reentry” 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 2007, Honolulu, Hawaii 4. Poteet, C. C., Abu-Khajeel, H., Hsu, S-Y, “Preliminary thermal-mechanical sizing of a metallic thermal protection system,” Journal of Spacecraft and Rockets, Vol. 41, No. 2, Mar – Apr 2004, pp. 173-182. 5. Smarslok, B.P., Haftka, R.T., and Kim, N.H., “Taking Advantage of Conditional Limit States in Sampling Procedures,” AIAA Paper 2006-1632, 47th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, May 2006 6. Lee, T.W and Kwak, B.M., “A Reliability-based Optimal Design Using Advanced First Order Second Moment Method,” Mechanics of Structures and Machines, Vol. 15, No. 4, 1987, pp. 523-542. 7. Ba-abbad, M.A., Nikolaidis, E., and Kapania, R.K., “New Approach for System Reliability-Based Design Optimization,” AIAA Journal, Vol. 44, No. 5, May 2006, pp. 1087-1096. 8. .Kreyzig E., Advanced Engineering Mathematics, Wiley, New York, pp. 848. 9. Oberkampf, W.L., DeLand, S.M., Rutherford, B.M., Diegert, K.V. and Alvin, K.F., “Estimation of Total Uncertainty in Modeling and Simulation”, Sandia National Laboratory Report, SAND2000-0824, Albuquerque, NM, April 2000. 10. Oberkampf, W.L., Deland, S.M., Rutherford, B.M., Diegert, K.V., and Alvin, K.F., “Error and Uncertainty in Modeling and Simulation,” Reliability Engineering and System Safety, Vol. 75, 2002, pp. 333-357. 11. Acar, E., Kale, A. and Haftka, R.T., “Effects of Error, Variability, Testing and Safety Factors on Aircraft Safety,” Proceedings of the NSF Workshop on Reliable Engineering Computing, 2004, pp. 103-118. 12. Acar, E., Kale, A., and Haftka, R.T., "Comparing Effectiveness of Measures that Improve Aircraft Structural Safety," submitted, ASCE Journal of Aerospace Engineering, 2006. 13. Haftka, R.T., and Gurdal, Z., “Elements of Structural Optimization,” Kluwer Academic Publishers, 3rd edition, 1992. 14. Elishakoff, I., Haftka, R.T., and Fang, J., “Structural Design Under Bounded Uncertainty—Optimization with Anti-optimization,” Computers and Structures, Vol. 53, No. 6, 1994, pp. 1401-1405.

15. Rober E. Melchers, Structural Reliability Analysis and Prediction, Wiley 2002, pp 157-158.

Probabilistic Optimization of Integrated Thermal Protection System

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