Probabilistic Optimization of Integrated Thermal
Protection System
Sunil Kumar, Bhavani V. Sankar and Raphael T. Haftka
University of Florida, Gainesville, Florida, 32611, USA
1.
INTRODUCTION
Probabilistic structural optimization is expensive because repeated analyses
(typically finite element analyses) are required for calculating probability as the structure
is being re-designed. Several methods have been proposed to alleviate the computational
burden [1]. These methods interleave reliability analysis and deterministic optimization.
The ECARD (exact capacity approximate response distribution, [2]) method goes one
step further by carrying full probabilistic optimization but with approximation of the
computationally expensive response distribution.
In many applications, 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. In a strength constraint, for
example, the failure stress is the capacity, while the structural stress is the response. The
failure occurs when the response exceeds the capacity. The response calculation is
typically much more expensive (e.g., requiring finite element analysis). ECARD
approximates only the expensive (i.e. response) distribution. In addition, with Monte
Carlo sampling (MCS) for estimating failure probability, the division into two sets of
uncorrelated variables permits the use of separable MCS.
We apply ECARD to the design of an integrated thermal protection system (ITPS)
for spacecraft reentry based on a corrugated core sandwich panel concept fulfilling both
thermal and structural functions is optimized for minimal mass [3]. Failure constraints
include limits on stresses and buckling loads and thermal failure is defined by a
temperature limit. A response surface approximation of the maximum bottom face sheet
temperature was developed in [4]. They used mild simplifying assumptions allowed to
reduce the number of variables in the approximation to two non-dimensional variables. In
combination with a material database, and on basis of graphical comparison of materials
for the different sections of the ITPS panel they have 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.
With two failure modes it may be expected that deterministic design based on
safety factor will lead to a different risk allocation between the two modes than
probabilistic design. We seek to compare the risk allocation for the deterministic and
probabilistic designs having the same system probability of failure. In that case, the
advantage of the probabilistic design will be measured by the reduction in mass.
2.
EXACT CAPACITY APPAROXIMATE RESPONSE DISTRIBUTION
(ECARD)
In probabilistic optimization, the system constraint is often given in terms of
failure probability of a performance function. We consider a specific form of
performance function, given as
g( X ) = f ( X r , X c ) ,
(1)
where x c and x r are random variables associated with capacity and response,
respectively. Both the capacity and response are functions of random variables, x . The
system is considered to be failed when the response exceeds the capacity. We assume that
the probabilistic distribution of x c is inexpensive to calculate, while that of x r requires
expensive analyses.
Since the performance function depends on two uncorrelated random functions,
x c and x r , the probability of failure can be calculated from the conditional probability as
Pf = P r[g(x; u) £ 0] =
¥
ò- ¥ FC (x) fR (x)d x .
(2)
In the above equation, FC ( x) is the cumulative distribution function (CDF) of capacity,
and fR ( x ) is the probability density function (PDF) of response. The above integral can
be evaluated using analytical integration, Monte Carlo simulation (MCS), or first/second-order reliability method (FORM/SORM), among others. Smarslok et al. [15]
presented a form of conditional MCS, called separable MCS, which is much more
accurate than the traditional MCS when the performance can be expressed in the form of
Eq. (1).
When design variables are changed during optimization, it is possible that the
distributions of both capacity and response can be changed. We model exactly the
distribution of the inexpensive capacity. However, we approximate the distribution of the
response by assuming that it is translated as a rigid body with design variable changes.
We calculate the value of the translation for the mean value of the random variables (see
Figure 1).
Figure 1: Distributions of response before and after redesign. Redesign only changes the
mean value. Distribution of capacity is also shown.
The change in the response distribution will change the probability of failure.
Then, the probability of failure at the new design truly given by
N
1
new
Pf
=
FC (ri new )
(3)
N å
i= 1
For Conditional MCS, we generate N samples of response ri = r ( xi ; u), i = 1, K , N , at
the current design. In view of Eq. (2), the probabilities of failure at the current and
perturbed design can be calculated from
N
1
Pf =
FC (ri )
(4)
N å
i= 1
N
1
=
(5)
å F (r + D )
N i= 1 C i
In ECARD, the reliability constraint is imposed such that the approximate failure
probability in Eq. (5) should be less than or equal to the failure probability at the
deterministic design. Even if Eqs. (4) and (5) are two different MCS, they can be
combined into one because the same sample, ri , will be used.
Pfapprox
3.
APPLICATION TO INTEGRATED THERMAL PROTECTION SYSTEM
The 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. Efforts are on to develop
Integral Thermal Protection System (ITPS) that not only performs the function of thermal
protection, but also withstand loads to a large extent. In a sense, it is an extension of the
ARMOR TPS design [5]. One candidate structure suitable for this purpose is a
corrugated-core sandwich panel, Figure below.
Figure 2. 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 may be taken as top face thickness (tT), bottom face thickness
(tB), thickness of the foam (ds), web thickness (tw), corrugation angle (θ), length of unit
cell (2p). A feasible and optimum design consisted of the alumino-silicate/Nextel 720
composites for top face sheet and web and beryllium for bottom face with dimensions
mentioned in table [1] was proposed by Gogu et al.[4]. 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 peak design temperature is chosen as in figure [2].
Parameters
tT
tB
ds
tw
θ
p
Values
2.1 mm
3.1 mm
117.3 mm
5.3 mm
870
117 mm
Table 1: Value of the preliminary Geometric Design parameters
To demonstrate the procedure we first apply it to a simplified problem where the design
constraints are the maximum bottom face sheet temperature and a simplified buckling
constraint. Using two non dimensional parameters, Eq. (6) and response surface shown
in figure [3], we can obtain maximum bottom face sheet temperature for any design in the
vicinity of design suggested by Gogu et al. [4]. This implies constraint on maximum
bottom face sheet temperature is given by Eq [7].
Figure 3: Response using Non Dimensional parameters
Figure 4: Heating profile used for preliminary design of ITPS
(6)
T cal . ( b , g ) £ T allow . ´ SF
(7)
The buckling constraint uses a simplified approach indicated by inequality (8).
t 3w,initial
t w3

 SF
d s 2 d s2,initial
(8)
The objective function for this part is weight of panel per unit depth given by Eq. (9).
weight  T tT p   BtB p 
W tW d s
sin( )
(9)
Only three of the six design parameters are used to illustrate the procedure. These are the
web thickness, bottom face sheet thickness and panel width. Other parameters were fixed
to the design suggested by Gogu et al. [4]. For temperature constraint a safety factor of
1.04 used which corresponds to 1% probability of failure. The allowable maximum
bottom face sheet temperature is 401.7 k. The critical value of the buckling criteria is
0.0027 mm. This gives us a safety factor of 1.24 for achieving same 1% probability of
failure. 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. This gives rise to distribution of capacity of the temperature
constraint, which is independent of randomness associated with design parameters and
other random variables. Similarly the simple model we employ will have error introduced
not just by error in temperatures (eT) evaluated but also structural buckling prediction
error, es. The constraints can be modeled by Eq. (10) and (11).
T true = T cal (1 + eT )
g1 =
T allowable
- T cal º C 1 - R1
1 + eT
ét 3 ù
T rueBucklingConditon = êê w ú
(1 + eT )(1 + eS )
2 ú
úcal .
ëêds û
t3
ét 3 ù
w ,initial
g2 =
- êê w ú
º C 2 - R2
2 ú
(1 + eT )(1 + eS )d 2
êëds ú
ûcal .
s ,initial
4.
(10)
(11)
RESULTS
First deterministic optimization procedure was implemented through fmincon function of
MATLAB produces an optimum weight. Results are presented in the table [2] which are
very close to what Gogu et al. [4] have obtained. The total probability of failure of this
optimum design is 2% according Ditlevsen’s first-order upper bound [16] given by Eq
(12). Pf1 and Pf2 are probabilities of two constraints being violated. They are both 1%.
det
PFS (= Pf 1 + Pf 2 ) £ PFS
(12)
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 2: Deterministic optimum results based on three design parameters
4.1 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. (13). But this process is very
costly if performed for full scale model involving all relevant constraints and design
parameters. Therefore we suggest approximate procedure called ECARD for the problem
in the next section. We plan to explore more elaborate model using FEM and then it will
be really difficult to perform this full probabilistic optimization.
min weight = rT tT p + r B t B p +
w ,t
s.t . PFS (= Pf 1 + Pf 2 ) £
rW tW ds
sin(q)
(13)
det
PFS
4.2 Approximate Optimization using ECARD
The proposed model of ITPS has separable capacity and response components. In this we
shift the response distribution in small steps and evaluate the direction of better
probabilistic design. We avoid performing full reliability analysis until at the end of
ECARD iterations. We expect to reduce the expensive reliability assessment by an order
of magnitude. The approximate probabilistic optimization problem is formulated based
on Eq. (14) by replacing the probability of failure with approximate one, as
min weight = rT tT p + r B t B p +
w ,t
rW tW ds
sin(q)
(14)
approx
det
s.t . PFS
(= Pfapprox
+ Pfapprox
) £ PFS
1
2
Pfapprox
and Pfapprox
are, respectively, approximations
1
2
where
of Pf 1 and Pf 2 using the
method described in Section 2. The approximate probabilities of failure are calculated as
Pfapprox
1
Pfapprox
2
N
1
=
N
å
1
=
N
å
FC 1 ( X 1i r + D 1 )
i= 1
N
(15)
FC 2 ( X 2i r + D 2 )
i= 1
where FC 1 and FC 2 are CDFs of the two capacities. The optimization in Eq. (14) is
solved iteratively until convergence criterion. The importance of the proposed method
increases with difficulty in performing reliability assessments. We will explore for more
elaborate model of ITPS which can be easily optimized by above approach.
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