Risk-Averse Set-Based Design and Applications
to Shaping a Hydrofoil
J.O. Royset∗
Professor
Naval Postgraduate School
Monterey, CA 93943; [email protected]
L. Bonfiglio and G. Vernengo
Postdoctoral researchers
Mechanical Eng., MIT
Cambridge, MA 95616
S. Brizzolara
Associate Professor
Aerospace and Ocean Eng., Virginia Tech, Blacksburg, VA 24061
The paper presents a framework for set-based design under uncertainty and demonstrates its viability through designing a super-cavitating hydrofoil of an ultra-high speed
vessel. The framework achieves designs that safely meet
requirements as quantified precisely by superquantile measures of risk (s-risk) and reduces the complexity of design under uncertainty. S-risk ensures comprehensive and decisiontheoretically sound assessment of risk and permits a decoupling of parametric uncertainty and surrogate (model) uncertainty. The framework is compatible with any surrogate
building technique, but we illustrate it by developing for the
first time risk-tuned surrogates that are especially tailored to
s-risk. The numerical results demonstrate the framework in
a complex design case requiring multi-fidelity simulation.
Introduction
Set-based design was pioneered by Toyota and later
adopted by the U.S. Navy [1]. It responds to the practical
need of finding satisfactory designs or, equivalently, finding designs that can be eliminated from further consideration. Thus, a wider set of designs tends to be seriously
considered than is the case for an optimization-driven approach; see [2–4] and references therein for motivation and
examples. In this paper, we develop a novel risk-based formulation of set-based design, coined Risk-Averse Set-Based
(RASB) design, and demonstrate its viability by shaping a
hydrofoil for an ultra-high speed vessel. The consideration of
uncertainty in connection with set-based design is not new;
see for example the use of utility-theory in [5]. However, the
manner in which we account for uncertainty in quantities of
interest (QoIs) and their models has not been explored.
We rely on superquantile measures of risk (s-risk) [6],
invented under the name conditional value-at-risk in [7] and
extensively used in finance, to assess whether QoIs meet requirements. There are three main advantages of s-risk: First,
it permits a natural decoupling of parameter uncertainty (due
to loads, material properties, etc.) from surrogate uncertainty
(due to modeling approximations). This realization, not recognized earlier, results in significant simplifications. Second,
s-risk reduces in a rigorous sense the complexity of set-based
design. Third, it provides a decision-theoretically sound assessment of risk [6, 16].
RASB design is flexible, addresses uncertainty specified by probability distributions as well as those given by
intervals of uncertainty, and permits any type of surrogate
modeling and multi-fidelity scheme such as those described
in [8–10]. Although any type of surrogate building scheme
can be used, we develop for the first time a scheme that
constructs surrogates of s-risk directly without attempting to
build models of intermediate QoIs.
1
∗ Corresponding
author
There is an extensive literature on design under uncertainty; see for example the special issues [11, 12], the recent
articles [13–18], and the earlier studies [19–21]. It is beyond
the scope to provide a comprehensive review; some comparisons are included below. However, we note that much of
the literature quantifies performance relative to a threshold
by the corresponding failure probability, which increases the
complexity of design under uncertainty as discussed below.
For the first time we consider s-risk in a real-world design case that requires multi-fidelity simulation of complex
physics. We examine a new family of dual-operating mode
super-cavitating hydrofoils [22] devised to ensure high efficiency both in super-cavitating and in fully-wet conditions.
These unconventional hydrofoils are designed to serve as lifting surfaces in a new ultra-high speed small waterplane-area
twin-hull vessel reaching more than 100 knots [23].
The paper continues in Section 2 with development of
RASB design, in Section 3 with surrogate construction, and
in Sections 4 and 5 with demonstration of the design case.
2
RASB Design Framework
The framework for RASB design as developed in this
section relies on a novel formulation of set-based design under uncertainty based on s-risk and the subsequent decoupling of the treatment of parametric uncertainty and surrogate uncertainty. Set-based design centers on finding one,
several, or all designs that meet a set of requirement. These
candidate designs would typically be the output of a stage of
the design process that has as objective to reduce the number of designs (concepts) under consideration. The candidate
designs might then become input to a next stage where adjusted QoIs and requirements would be studied, and further
reduction of the design space will take place. We develop a
mathematical formulation of the precise meaning of “candidate design” in the presence of uncertainty. Our formulation
applies to any stage of a design process despite the fact that
the stages might involve different QoIs and requirements.
In addition to the ability to extensively explore a design space, set-based design has another advantage over
optimization-based approaches previously not fully recognized. The situation is best illustrated by a simple example. Ignoring uncertainty and other complications temporarily, suppose that the QoI is f (x) = ε(x − 1), given as a function of a design variable x that is permitted to vary in the
design space [0, 2]. The parameter ε is near zero. Clearly, an
optimization-based approach will be highly sensitive to the
value of ε; the design minimizing f (x) jumps from x = 0 to
x = 2 with arbitrarily small changes of ε around zero. In contrast, a set-based approach is much less sensitive to changes
in ε. In fact, the set of designs that meet the requirement
f (x) ≤ |ε| is always [0, 2], regardless of the value of ε. The
same conclusion holds for essentially all functions [24] and
provides a rigorous justification for set-based design.
2.1
Accounting for Uncertain Parameters
Suppose that there are K QoIs, with numerical values
gk (x, v), k = 1, ..., K, parameterized by a vector x containing design variables and a vector v consisting of uncertain
parameters. The uncertain parameters represent inputs that
are beyond the control of the designer and that can affect
the performance of the design such as environmental conditions and manufacturing imprecision. Under design x and
parameter v, gk (x, v) is the (actual) numerical value of the kth
QoI, which is usually analytically unavailable and/or computationally expensive to evaluate. Without loss of generality,
we assume that low values of the QoIs are preferred to higher
values and that the requirement associated with the kth QoI
is that gk (x, v) should not exceed a number rk . If a practical
situation demands high values, we simply replace gk (x, v) by
−gk (x, v) and rk by −rk . The requirement rk might be specified by regulatory requirements, given as target at the current
design stage, or is simply varied. The latter case shows that
a goal of minimizing gk (x, v) can be achieved by carrying
out a one-dimensional search over rk . Thus, RASB design
addresses also situations demanding optimization.
The goal is to identify designs x with gk (x, v) ≤ rk , but
this problem is ill-posed because the values of the uncertain
parameters v are not known. We assume that the uncertainty
in v can be quantified by a known probability distribution.
Typically, such distributions are estimated based on data, but
we omit a discussion of this subject. We permit as a special case the possibility that the distribution of a parameter is
simply given by upper and lower bounds on its support and
therefore enable modeling using interval uncertainty; additional comments about this possibility are included below.
When viewing the uncertain parameters as a random vector
with a given distribution, we denote it by V. A realization of
V is still denoted by v. Now gk (x, V) is a random variable
for each x, which allows us to bring in s-risk.
S-risk reduces a random variable to a representative
number that can be used in comparison with requirements.
Specifically, for risk-parameter α ∈ [0, 1], the s-risk of
gk (x, V), denoted by Rα (gk (x, V)), is understood as the
average of gk (x, V) in the worst (1 − α)100% outcomes.
This expression can be taken as the definition if gk (x, V) has
a continuous distribution. If the risk-parameter α = 0, then
Rα (gk (x, V)) is simply the expected value E[gk (x, V)]. If α =
1, then Rα (gk (x, V)) is the largest possible value of gk (x, V).
In the case of parameter uncertainty represented by intervals,
the latter possibility simply propagates that uncertainty and
no probability distribution is needed. A value of α between
these two extremes provides a middle ground between focusing on the average performance and the absolutely worst
outcome. If the random variable gk (x, V) is normal with
mean µ and standard deviation σ, then Rα (gk (x, V)) = µ +
σφ(Φ−1 (α))/(1 − α), where φ is the standard normal probability density function and Φ is the standard normal cumulative distribution function. Generally, Rα (gk (x, V)) =
the minimum value of c + E[max{0, gk (x, V) − c}]/(1 − α)
across all scalars c [16]. If gk (x, V) happens to be a deterministic constant ck , i.e., V is deterministic and/or the system negates all variability, then Rα (gk (x, V)) = ck . If it is
not constant, then Rα (gk (x, V)) is always greater than the
expected value of gk (x, V) unless α = 0 when it is equal to
the expected value. S-risk is actually continuous in spaces
of random variables and also with respect to α. It is therefore inherently stable with regard to misspecification of the
risk-parameter.
S-risk is fundamentally appealing as it has a simple interpretation, accounts for the tails of probability distributions, and is grounded in axiomatic foundations of decision
theory [16]. In contrast to the probability that gk (x, V) exceeds rk , which is largely insensitive to the nature of the tail
of the probability distribution of gk (x, V), s-risk accounts for
the magnitude of possible exceedance. Moreover, minimizing such failure probabilities and finding designs with sufficiently low failure probability are fundamentally difficult
as such problems are nonconvex even if gk (x, v) is convex
in x for all v. Thus, the consideration of failure probability increases the complexity of the design process as compared to a deterministic approach that ignores uncertainty
in parameters. S-risk retains the complexity of a determin-
istic approach: if gk (x, v) is convex in x, then the s-risk
Rα (gk (x, V)) will also be convex in x regardless of the distribution of V. If that distribution is discrete, then linearity of
gk (x, v) in x translates into a linear programming expression
for s-risk. Thus, at this fundamental level, finding a design
x with Rα (gk (x, V)) ≤ rk is no harder than finding an x with
gk (x, v) ≤ rk for a given v, i.e., ignoring uncertainty. RASB
design therefore reduces the complexity of design under uncertainty compared to approaches relying on failure probabilities and similar concepts that do not preserve convexity
including those that attempt to approximate the failure probability such as FORM, MVFOSM, and Moment Matching
Methods [25].
We are now in a position to formalize the first component of RASB design. Letting αk ∈ [0, 1] be the riskparameter associated with the kth QoI, the requirement that
gk (x, V) should not exceed rk is rigorously defined as having Rαk (gk (x, V)) ≤ rk . We say that gk (x, V) is safely ≤ rk
when this condition holds. This safeguarding implies that
even on average over a set of worst outcomes, the value of
the QoI will not exceed rk . The specific definition of “worst
outcomes” depends on αk . We can also interpret the condition probabilistically [6]: Rαk (gk (x, V)) ≤ rk is equivalent to
having the buffered failure probability ≤ 1−αk and it implies
that the (usual) failure probability is ≤ 1 − αk .
It is apparent that the choice of αk needs to reflect the demands of an application. In the case of interval uncertainty
for the parameters v, the choice αk = 1 might be suitable, but
it is otherwise often too conservative. The relation to the failure probability provides general guidance: If there is a need
for a reliability level corresponding to a failure probability of
10−3 , then αk = 1 − 10−3 achieves this goal. Since s-risk is
continuous in αk , small changes in αk implies small changes
in the s-risk, which partially alleviates the need to select αk
“correctly.” When relying on failure probabilities the situation is dramatically worst as small changes in the reliability
level might be associated with large jumps in the range of
requirements that is satisfactory.
Using the notation sk (x) = Rαk (gk (x, V)), we highlight a
fundamentally convenient property: the requirements on an
uncertain system is translated into the deterministic requirement sk (x) ≤ rk . The presence of uncertain parameters have
been addressed and the challenge is reduced to that of dealing with the complex, but deterministic functions sk (x).
Accounting for Uncertainty in Surrogates
The s-risk sk (x) is rarely easy to compute for physical
systems; the underlying values gk (x, v) and the corresponding probability distribution of the QoI might be unavailable
or costly to evaluate. The predicament of a complex, deterministic function is well-known and we turn to surrogates
(response surfaces) as usual. Let Sk (x) be a surrogate of
sk (x), which we view as a random field over the space of
design variables. We assume that the randomness in Sk (x)
is due to uncertainty associated with the accuracy of the
surrogate. Examples of surrogates include those based on
kriging [8, 10, 26, 27], where the surrogate error is immedi-
ately available in a probabilistic form, at least if the underlying Bayesian assumptions are accepted. However, Sk (x)
can be constructed by any surrogate building approach assuming that the resulting surrogate is associated with probabilistic estimates of accuracy for example obtained by crossvalidation methods and their enhancements [26, 28–31].
The original requirement sk (x) ≤ rk is now replaced by
one involving the surrogate. In a situation analogous to that
above we find the requirement Sk (x) ≤ rk ill-posed as Sk (x)
is a random variable for every x due to the unknown error in
the surrogate relative to the actual value sk (x). Thus, again,
we turn to s-risk. For risk-parameter βk ∈ [0, 1] we adopt
the requirement Rβk (Sk (x)) ≤ rk , which is well-posed. It
ensures that even the average value of Sk (x) over the worst
(1 − βk )100% outcomes of Sk (x) do not exceed rk . We stress
that here the average is over possible values of an uncertain
surrogate. These values are unrelated to the uncertain parameters V. Since the surrogate is imperfect, Sk (x) is random and
we account for this fact when making the comparison with
rk . As a concrete example, suppose that the surrogate Sk (x)
is Gaussian with mean µ(x) and standard deviation σ(x), for
example obtained by kriging. Then, Rβk (Sk (x)) ≤ rk translates into having µ(x) + σ(x)φ(Φ−1 (βk ))/(1 − βk ) ≤ rk . Although general recommendations are difficult, this expression provides guidance regarding the choice of βk . If βk = 0,
then we are simply asking the (posterior) mean of the Gaussian surrogate to satisfy the requirement. Clearly, the actual
value sk (x) might very well be higher than µ(x) leading to a
nonconservative prediction of the performance of the system.
As βk increases, the requirement becomes more stringent as
we account for uncertainty in the surrogate in a manner that
resembles that of quantile-based improvement strategies in
Bayesian optimization; see for example [32]. Again, since
s-risk is continuous in its risk-parameter, small changes in βk
implies small changes in the s-risk, which makes the exact
choice of βk less critical.
We are then in a position to define the set of candidate
designs of the RASB design framework as
designs x with Rβk (Sk (x)) ≤ rk for all k = 1, 2, ..., K.
Designs that meet these requirements are safeguarded in a
rigorous manner against poor performance, accounting for
both parametric uncertainty in V (using risk-parameters αk )
and surrogate imprecision (with risk-parameters βk ).
2.2
Algorithm for RASB design
1. Select risk-parameter αk ∈ [0, 1] to capture desired level
of safety against uncertain parameters v.
2. Construct surrogate Sk (x) of s-risk sk (x) including uncertainty (error) estimates.
3. Select risk-parameter βk ∈ [0, 1] to capture desired level
of safety against uncertainty in the surrogate Sk (x) relative to the actual s-risk sk (x).
4. Compute the set of candidate design.
We note that in Step 2 any surrogate building technique can
be used. Step 4 is usually trivial as after Sk (x) is built, the
s-risk of Sk (x) is easily computed. For example, when Sk (x)
is Gaussian, it only entails computing mean and standard deviation as described above.
The actual representation of a set of candidate design might vary with an application’s need. If only a single candidate design is needed, any standard optimization
algorithms can be applied to the problem of minimizing
maxk=1,...,K {Rβk (Sk (x)) − rk } over the design space and be
terminated when objective function values becomes nonpositive. If the goal is to find a few “different” designs within
the set, one can again solve a sequence of optimization problems after the notion of “different” is formulated; see for example [33]. If we seek many candidate designs, the simplest
approach might be Monte Carlo sampling of the points in the
design space paired with requirement checks, which also addresses the previous goals in a brute-force manner. However,
we note that these calculations only involve the surrogate and
usually are inexpensive.
3
Risk-Tuned Surrogates
The framework of RASB design permits any surrogate
Sk (x). However, in an attempt to explore the special structure of sk (x), here for the first time, we develop a surrogate
that is tuned to estimating s-risk. For simplicity, we omit the
subscript k as the process will be identical for each QoI.
For a QoI given by g(x, v), typically from a high-fidelity
simulation, we leverage a low-fidelity approximation h(x, v)
to construct the surrogate. There is no difficulty with including many levels of fidelity, but we omit the details. We
are given a data set {(gi , hi , xi , vi )}Ni=1 consisting of designs
xi , parameter values vi , and the corresponding simulation
output gi = g(xi , vi ) and hi = h(xi , vi ). We adopt a model
gi ≈ c0 + c⊤ b(hi , xi ), where b(h, x) is a given vector-valued
function, for example one of the usual polynomial expansions, and c is a vector of coefficients and c0 is a scalar coefficient both to be determined. Although the scalar coefficient
c0 could be incorporated into b(h, x), we make it explicit due
its special role below. The essence of the surrogate building
approach is the method by which the coefficients are fitted.
The method is tailored to the risk-parameter α ∈ (0, 1) used
in capturing parameter uncertainty. Specifically, (c0 , c) is determined by α-quantile regression using the same α-value.
That is, find coefficients (c0 , c) that minimize
(
1
(1 − α)N
N
)
∑ max{0, ei }
i=1
+
1 N
∑ ei ,
N i=1
where the fitting error ei = gi − [c0 + c⊤ b(hi , xi )]. We do
not carry out standard least-squares regression, but a regression that is tuned to the chosen risk-parameter α. We discard
the optimal c0 , but name the other optimized coefficients ĉ.
Finally, we set ĉ0 = s-risk of {gi − [c0 + c⊤ b(hi , xi )]}Ni=1 . Extending Corollary 4.2 in [34], we obtain that Rα ({gi }Ni=1 ) ≤
Rα ({ĉ0 + ĉ⊤ b(hi , xi )}Ni=1 ). Consequently, over the given data
set the model ĉ0 + ĉ⊤ b(h, x) leads to an upper approximation
of the s-risk of the QoI. This motivates the use of the model
as an approximation of the s-risk of the QoI, i.e.,
Rα (g(x, V)) ≈ Rα (ĉ0 + ĉ⊤ b(h(x, V), x)).
(1)
There are two reasons why this approximation might fall
short of an actual inequality. First, fitting of the coefficients
of the surrogate might be carried out using a data set of size
N and not the true distribution of the random vector V. Second, Eqn. (1) makes comparisons at individual designs x,
which might generally be different than considering “averages” across a variety of designs as relied on while fitting the
coefficients. The idea of such risk-tuned regression can be
traced back to [34,35], but has not previously been developed
in this form. It is central that the risk-parameter α used in the
regression matches that of interest so that the coefficients in
the model are “tuned” appropriately. In the case of α = 0, we
simply determine the coefficients by means of standard leastsquares regression, and for α = 1 we reference [34]. Eqn.
(1) gives a deterministic surrogate, but we would like to account for surrogate imprecision. There is a vast literature on
the subject and existing approaches based on cross-validation
can be adopted; see for example [26, 28–31]. Section 5 illustrates one possibility.
The above approach to developing a surrogate for sk (x)
is novel and preliminary results in Section 5 demonstrate
that reasonable accuracy can be obtained. Clearly, further
testing is necessary to establish the merit of the approach.
Here, we briefly contrast the approach with other possibilities. Conceptually, any surrogate building technique can be
applied to develop a response surface ĝk (x, v) for gk (x, v)
in the combined x-v space. Then, the s-risk sk (x) can be
estimated on a grid in the design space using the formula
minc {c + E[max{0, ĝk (x, V) − c}]/(1 − αk )}, which involves
optimization over c and numerical integration. This provides
(approximate) function values sk (x) that subsequently can be
used to fit a surface in the design space; see [10] for such
an approach. Several improvements are available, but it is
apparent that such a layered approach is complex. In contrast, the proposed scheme develops directly the surrogate
from simulation output without the need for additional optimization, numerical integration, and intermediate models of
gk (x, v) in a potentially high-dimensional space.
4
Hydrofoil Design
As a design case, we consider a surface-piercing supercavitating hydrofoil of an ultra-high speed vessel. RASB design offers possibilities of reaching higher speeds for such
vessels, avoiding the hysteresis phenomena typically affecting conventional super-cavitating hydrofoils. Traditional
super-cavitating hydrofoils are designed to operate at high
speeds only where flow conditions ensure the presence of a
stable vapor cavity enveloping the entire suction surface and
closing many chords aft a blunt trailing edge. The shape of
the unconventional hydrofoil presented in [22, 36] features a
Table 1.
Fig. 1.
Time-averaged vapor content contours for the benchmark
design; flow is left to right. Red indicates stable cavity.
pointed leading edge ensuring cavity detachment at the operating angle of attack and cavitation index in design conditions and a sharp edge on the pressure side triggering base
cavitation when working at zero angle of attack. The blunt
trailing edge, typical of conventional super-cavitating profiles, is instead tapered in a tail which is designed to be enclosed in the supercavity at high speed while producing a
good pressure recovery and higher lift at lower speeds when
cavitation disappears. This guarantees the performance of
the hydrofoil also in sub-cavitating conditions. The tail is
functionally separated by the forward body of the profile
through a sharp corner (face cavitator) on the face in order
to trigger base cavitation at intermediate speeds or at high
speeds and lower angles of attack. We aim to shape this
hydrofoil using a parametrization with composite B-Spline
curves.
We compare with a benchmark hydrofoil obtained by
means of a differential evolution optimization algorithm in
[37], without considering uncertainty, using a low-order
Boundary Element Method (BEM) to solve for the steady
potential flow with cavitation. Figure 1 presents a timeaveraged snapshot of the flow around the benchmark design; Reynolds number 4.58 · 107 , angle of attack 6 degrees,
and cavitation index 0.05. The cavity shape can be inferred
from the color scale indicating vapor content. The optimization leads to a thin cavity enveloping the hydrofoil surfaces
showing the drawback of the design: it is highly sensitive to
changes in operating conditions. In this case the existence of
a super-cavity relies on the presence of a vapor regime over
the hydrofoil suction surface. It is evident that manufacturing errors or variable conditions (especially in terms of angle
of attack) could significantly affect the flow regime and hydrofoil performances for a design with a thin cavity.
4.1
Description and Parameters
We let the design vector x consist of 15 variables representing coordinates of the control polygon of the composite B-spline curves describing the shape. Figure 2 gives a
schematic representation of the 15 variables together with a
resulting hydrofoil shape. The pressure side is described by
five control points, one of which allows to change the vertical position of the face cavitator, i.e., the sharp edge at the
beginning of the hydrofoil tail. Variations of the suction side
Benchmark design and bounds of design variables.
Variable
Benchmark
ximin
ximax
x1
5.556 · 10−2
5.556 · 10−2
1.667 · 10−1
x2
2.227 · 10−1
2.222 · 10−1
3.333 · 10−1
x3
3.900 · 10−1
3.889 · 10−1
5.000 · 10−1
x4
5.576 · 10−1
5.556 · 10−1
6.667 · 10−1
x5
5.556 · 10−3
0
5.556 · 10−3
x6
1.111 · 10−2
1.111 · 10−3
1.111 · 10−2
x7
3.456 · 10−2
1.333 · 10−2
3.756 · 10−2
x8
1.833 · 10−2
5.556 · 10−3
2.778 · 10−2
x9
5.556 · 10−3
−1.111 · 10−3
5.556 · 10−3
x10
1.979 · 10−1
1.977 · 10−1
2.417 · 10−1
x11
4.208 · 10−1
4.000 · 10−1
4.889 · 10−1
x12
5.604 · 10−1
5.500 · 10−1
6.722 · 10−1
x13
3.833 · 10−2
3.200 · 10−2
3.911 · 10−2
x14
6.711 · 10−2
5.600 · 10−2
6.844 · 10−2
x15
8.267 · 10−2
7.044 · 10−2
8.267 · 10−2
are performed by five control points; three of them move together vertically following the 15th coordinate. The leading
and trailing edge position as well as the control points of
the tail pressure side are kept fixed. The control points on
the hydrofoil face regulate lift and drag performance while
those on the back suction side control the cavity thickness
and maintain a sufficient inertia modulus.
We consider uncertainty related to manufacturing errors using a random vector V = (V1 , ...,V15 ) describing relative displacement values to the control point coordinates.
The actual shape manufactured from a (nominal) design
x = (x1 , ..., x15 ) is assumed to be x̄i = xi (1 +Vi ), i = 1, ..., 15.
We refer to the vector x̄ = (x̄1 , ..., x̄15 ) as the (random) shape
of the hydrofoil under design x. The designer selects x from
a range given by the bounds ximin ≤ xi ≤ ximax , i = 1, ..., 15;
see Table 1, which also gives the benchmark design in Figure 1. The values of the design variables are relative to the
hydrofoil chord.
The distribution of V is discrete with 898 equally
likely realizations essentially uniformly distributed in the
box [−0.06, 0.06]15 , with some adjustment to eliminate unrealistic manufacturing errors. Thus, “noise” up to ±6% is
added to xi . Although no standard guideline exists for hydrofoils, these numbers are motivated by manufacturing tolerance for ship propeller blades (ISO standard 484/1). The
framework is indifferent to the distribution of V. Here, we
select a discrete distribution to avoid errors due to sampling
and other approximations, and more easily highlight the features of RASB design. Using chord length c = 0.66m, we
define five (K = 5) QoIs: (i) Profile inertia modulus w, which
is required to exceed the minimum value wmin = 8.1 · 10−6
(X12;X15)
(X15)
(X15)
(X11;X14)
(X10;X13)
(X1;X5)
(X4;X8)
(X2;X6)
(X3;X7)
(X9)
Fig. 2. Design variables x = (x1 , ..., x15 ). Arrows indicate allowed
movement of control points; some are fixed and illustrated by green
squares. Three control points move together with x15 .
m3 . (ii) Profile thickness tP2 at 2% of the chord, which
needs to exceed the minimum value tPmin /c = 0.2%. (iii)
Drag-over-lift ratio CD /CL , which needs to meet a requirement of 0.1 with CD = D/(0.5ρcU 2 ) and CL = L/(0.5ρcU 2 )
being normalized drag D and lift L under operating speed
U = 61.667 m/s and water density ρ = 997.3 kg/m3 . (iv) Lift
CL , which has a requirement of 0.2625 at the design angle of
attack αdes = 6 degrees and at the design cavitation number
σdes = 0.05. (v) Negative lift −CL , which must meet the requirement −0.23. The risk-parameters are α1 = α2 = 0.7
(for the geometric properties), α3 = 0.9 (for drag-over-lift),
and α4 = α5 = 0 (for lift). Consequently, we seek designs
that on average in the worst 30% of the realizations has
−w ≤ −wmin and −tP2 ≤ −tPmin . Moreover, on average in the
worst 10% of the realizations we have CD /CL ≤ 0.1 and the
average of CL is between 0.23 and 0.2625.
The first two QoIs do not need surrogates as there are
explicit formula for inertia modulus and profile thickness.
For drag-over-lift and lift we develop surrogate models of
the kind given in Section 3 using β3 = 0.95 for drag-over-lift
and β4 = β5 = 0 for lift.
4.2
Simulations and Surrogates
We have to our disposal a high-fidelity solver for an unsteady viscous method for the solution of Reynolds Averaged Navier-Stokes equations (URANSE) as well as a lowfidelity potential flow solver that both return for a given
shape x̄ = (x̄1 , ...., x̄15 ) lift CL and drag CD . The URANSE
and potential flow output are referred to as high- and lowfidelity output, respectively. In the present context, each
high-fidelity solution takes about 6 hours on 4 cores, while
each low fidelity run requires 5 seconds on a single core.
Figure 1 displays high-fidelity output.
The potential flow solver is based on a BEM in which
a Laplace equation is solved for a perturbation potential and
a Bernoulli equation is solved instead of continuity and momentum equations for a steady non-viscous flow. Friction
forces on the wetted surface of the hydrofoil are included using an empirical local friction coefficient. A cavitation model
solving for the cavity thickness on the hydrofoil suction surface is included by imposing an additional set of kinematic
boundary conditions that are satisfied through singularities
distributed on the hydrofoil suction surface and wake panels.
The model predicts mid-chord cavity closure, but not bubble dynamic for cavities developing on the pressure surface;
see [37] for a detail description.
The solution of URANSE is achieved for a multi-phase
flow consisting of a fluid mixture of liquid and vapor using a
finite volume technique with collocated arrangement of variables where PDEs are solved for the fluid mixture through a
scalar quantity γ indicating the relative content of vapor with
respect to liquid within each cell [38]. A transport equation
for γ, representing the vapor content, is solved together with
RANS equations when the source term indicating the specific
net mass transferred in the cavitation process is known. Condensation and vaporization regulating the above mentioned
mass transfer are found through the cavitation model [39].
The super-cavitating flow around the hydrofoils is solved using a hybrid structured-unstructured grid; a structured grid
close to the hydrofoil surface, two nested unstructured grids
in the region of cavity, and a coarse grid elsewhere [40].
For the shapes resulting from the benchmark design with
the 898 realizations of V we run the URANSE solver and obtain CD and CL for all 898 resulting shapes. The same input to
the potential flow method results in some estimates of CD and
CL , but also approximately 50% computational failures, due
to the predicted presence of face midcord cavitation that is
not presently resolved by the method. Consequently, instead
of using the actual low-fidelity simulations for each of the
898 shapes, we compute for each shape a weighted average
over all the successful runs, with weights determined by the
distance from the current shape to the shape of those runs.
Throughout, we deal exclusively with this modified lowfidelity data set. Figure 3 illustrates the high- and low-fidelity
drag-over-lift values across the 898 shapes; Figure 4 gives
the corresponding errors in the low-fidelity estimates. These
runs generate the data from which the surrogates are fitted by
the procedure in Section 3 using c0 + c⊤ b(h(x, V), x) = c0 +
c1 x1 + ...c15 x15 + c16 h(x, V) + c17 [h(x, V)]2 , where h(x, V) is
the modified low-fidelity values. We estimate surrogate uncertainty by simply diving the data set of 898 drag-over-lift
values into nine roughly equally large groups, on which separate fitting is carried out. The division ensures a reasonably
large sample size (about 100) in each case. Many other crossvalidation approaches could have served the same purpose.
The choice of β3 = 0.95 to assess uncertainty in the resulting drag-over-lift surrogate SD/L (x) implies that the highest
of the nine fits are used in comparison with the requirement.
The choices of β4 = β5 = 0 for the lift surrogate SL (x) imply
that the average of the nine fits are employed in assessment.
5
Results and Candidate Designs
We are now in a position to generate sets of candidate
designs. For the five QoIs, we have the conditions
Rα1 (−w) ≤ −wmin
(sufficiently large inertia modulus)
Rα2 (−tP2 ) ≤ −tPmin
(sufficiently large thickness)
Rβ3 (SD/L (x)) ≤ 0.1
Rβ4 (SL (x)) ≤ 0.2625
Rβ5 (−SL (x)) ≤ −0.23
(sufficiently low drag-over-lift)
(sufficiently low lift)
(sufficiently high lift)
Benchmark Design
Drag/lift using high-fidelity solver
0.089
0.0885
Benchmark Design
0.088
Design No. 2
0.0875
0.087
Design No. 2
0.0865
Design No. 4
0.086
0.0855
0.085
0.08
Design No. 4
0.0805
0.081
0.0815
0.082
0.0825
0.083
Drag/lift using low-fidelity solver
Fig. 3.
Low- and high-fidelity estimates of drag-over-lift (meter).
Fig. 7. Time-averaged vapor content (left column) and pressure coefficients (right column) contours. From top to bottom: benchmark
design, Design 2, and Design 4. Flow conditions as in Figure 1.
80
60
40
20
0
4.5
5
5.5
6
6.5
Drag/lift error: high-fidelity minus low-fidelity
7
×10 -3
Fig. 4. Histogram of errors in low-fidelity estimates of drag-over-lift
(meter).
0.1
Benchmark
Profile #2
0.08
z
0.06
0.04
0.02
0
-0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 5.
Design 2 (blue curve, triangles for control points) and bench-
mark (black curve, inverted triangles for control points); axis scale in
meter
0.1
Benchmark
Profile #4
0.08
z
0.06
0.04
0.02
0
-0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 6. Design 4 (red curve, circles for control points) and benchmark (black curve, inverted triangles for control points); axis scale in
meter.
defining the candidate designs, where the first two requirements avoid surrogate models and thus Rβ1 and Rβ2 become
immaterial. A design x that satisfies these inequalities is
guaranteed to meet the specified requirements in a mathematically precise sense accounting for both parametric uncertainty in the manufacturing process of the hydrofoil and
uncertainty in surrogate models.
The above five conditions are trivial to evaluate for each
design x as they involve only explicitly or easy-to-compute
quantities. No high-fidelity simulations are required. Lowfidelity output is needed to evaluate the surrogates due to the
presence of h(x, V) in Eqn. (1), but they are quickly obtained. It is therefore easy to check whether a given design
satisfies these conditions and also to generate a collection
of candidate designs satisfying the conditions. For the sake
of demonstration, we simply randomly generate 500 designs
by uniform sampling within the bounds of Table 1 and for
each check the five conditions. This results in four candidate
designs that satisfy the conditions. (Many more could have
been obtained by further sampling.) In the following, we
concentrate on two of them, labelled Design 2, with (in meters) x = (0.0834, 0.2629, 0.4258, 0.5611, 0.0040, 0.0099,
0.0147, 0.0094, 0.0027, 0.1887, 0.3929, 0.5602, 0.0300,
0.0602, 0.0742), and Design 4, with x = (0.1069, 0.2724,
0.3749, 0.5962, 0.0009, 0.0072, 0.0283, 0.0084, 0.0045,
0.1877, 0.4047, 0.5058, 0.0297, 0.0578, 0.0742); see Figures 5 and 6. Designs 2 and 4 are quite different, with the
two other candidate designs resembling these.
Table 2 summarizes the performance of Designs 2 and
4 as well as the quality of the surrogates. Columns 2 and
3 give the drag-over-lift for Designs 2 and 4, respectively.
Rows 3-11 give the estimates provided by the nine surrogates obtained by partitioning the data and the overall assessment Rβ3 (SD/L (x)), which equals the highest of the nine
numbers above, given in the second-to-last row. For the sake
of validation, we compute the actual s-risk of drag-over-lift
using 898 high-fidelity simulations and obtain the results in
Table 2.
Performance of Designs 2 and 4.
Drag/Lift
Lift
Model
Design 2
Design 4
Design 2
Design 4
S(1) (x)
0.0959
0.0971
0.2354
0.2367
S(2) (x)
0.0937
0.0949
0.2346
0.2368
S(3) (x)
0.0925
0.0930
0.2346
0.2397
S(4) (x)
0.0958
0.0988
0.2341
0.2355
S(5) (x)
0.0954
0.0982
0.2352
0.2368
S(6) (x)
0.0954
0.0990
0.2323
0.2337
S(7) (x)
0.0962
0.0989
0.2345
0.2381
S(8) (x)
0.0932
0.0951
0.2345
0.2381
S(9) (x)
0.0956
0.0970
0.2363
0.2390
Rβ (S(x))
0.0962
0.0990
0.2346
0.2371
Actual
0.0956
0.0971
0.2396
0.2465
detachment position and virtually increasing the operating
angle of attack. Design 4 is characterized by a double curvature leading to lower drag-over-lift. As shown in Figure
6, the vertical position of the face cavitator does not move
compared with the benchmark shape. Hence, the increase in
lift generation is mainly due to the double curvature leading
to a double pressure peak as evidenced in Figure 7, which
shows pressure as well as vapor content for all three designs.
Design 4 experiences a second pressure peak upstream the
face cavitator, locally behaving as a stagnation point. The
presence of bump on the face in between the two pressure
peaks generates, as seen in Figure 7, a lower pressure region within the maximum curvature points. Both Designs 2
and 4 present a reduced suction surface curvature with respect to the benchmark design. This feature increases cavity
thickness close to the leading edge (see Figure 7) and ensures more robust cavitating regimes than for the benchmark
shape.
6
the last row. We observe that although some individual surrogates (for example S(3) (x)) underestimate drag-over-lift,
Rβ3 (SD/L (x)) is conservative relative to the actual drag-overlift. This highlights the importance of considering the uncertainty in surrogates. Columns 4 and 5 give similar results
for lift; again surrogates are reasonably accurate. For reference, the benchmark design has actual s-risk for drag-overlift of 0.0880 and for lift of 0.2675, the latter being excessively high.
The s-risk of the inertia modulus for the benchmark,
Design 2, and Design 4 are 7.400 · 10−6 , 8.738 · 10−6 , and
8.48489 · 10−6 , respectively. We see that in face of manufacturing errors, the benchmark design fails to meet the requirement of wmin = 8.1 · 10−6 . The s-risk for the corresponding profile thicknesses are 0.00118, 0.00135, and 0.00185.
Again, the benchmark design fails to meet the requirement
of tPmin = 0.00132.
Figures 5 and 6 shows that the suction side control points
of Designs 2 and 4 have moved to achieve a thicker cavity by
lowering the back profile surface. The pressure side control points are shifted closer to the leading edge. The control points of the pressure side have been changed in different ways to ensure the overall hydrodynamic performance:
Design 2 has a single curvature shape with a maximum at
about 0.42m horizontally from the left, while Design 4 shows
higher curvatures with a slope change at 0.28 m horizontally
from the left. Design 4 has pressure side shape similar to
that of the benchmark design, but the maximum curvature of
the concave part is shifted towards the trailing edge. Design 2 has a single curvature face and Design 4 a double
curvature face. For Design 2, the single curvature pressure
surface leads to lower lift force generation, which for this
particular shape has been compensated through the vertical
displacement of the face cavitator. Moving the sharp edge on
the pressure side has the direct effect of changing the cavity
Conclusions
The framework of RASB design generates robust candidate designs that tend to perform well for a range of values of
uncertain parameters and therefore are superior to those obtained by methods ignoring uncertainty. In Section 5, we see
that the obtained designs have only slightly lower efficiency
than a benchmark design, due to a larger cavity thickness,
but their performance are more robust to changes in geometry because of manufacturing errors.
Compared to optimization-based approaches, candidate
designs obtained by the framework are less sensitive to errors and misspecifications in the formulations due to our setbased approach as exemplified at the beginning of Section 2.
Compared to reliability-based approaches (relying on failure
probabilities), candidate designs have several favorable characteristics because of the superior properties of s-risk over
that of failure probabilities. Their performance as well as the
set itself are stable under changes in risk-parameters αk , βk ,
and other quantities and they have reduced possibility of significant under-performance relative to the requirements. The
set of candidates is always convex provided that Sk (x) is convex for all k = 1, ..., K, which offers computational and other
advantages.
The proposed framework is broad and permits the use
of any surrogate. However, the risk-tuned surrogates differ
from other surrogate-based design processes by directing the
surrogate building towards approximating s-risk of QoIs and
not the QoIs directly. The framework differs by assessing the
uncertainty (error) in any surrogate using s-risk instead of by
the variance and related quantities. The difference between
the two assessment is especially noticeable when the surrogate errors are asymmetric, in which case a large variance
might be due to a benign and even beneficial heavy lower
tail of the corresponding distribution.
As in all approaches that achieves robustness in the presence of uncertainty, there is a potential downside in terms of
bulkier, heavier, and costlier designs. The exact trade-off between cost and robustness is of course application dependent
and, in fact, the RASB design framework facilitates such exploration because the cost can be defined as one of the QoIs.
The risk-parameters αk and βk are central tools in the exploration as their values drive the level of safeguarding and thus
the level of robustness.
Acknowledgements
This work is supported by DARPA under N66001-15-24055, HR0011517798, HR0011619431, HR0011620572.
References
[1] Singer, D. J., Doerry, N., and Buckley, M. E., 2009. “What is set-based
design?”. Nav. Eng. J., 4(121), pp. 31–43.
[2] Canbaz, B., Yannou, B., and Yvars, P.-A., 2014. “Improving process
performance of distributed set-based design systems by controlling
wellbeing indicators of design actors”. ASME J. Mech. Des., 136(2),
pp. 021005–021005–10.
[3] Inoue, M., Nahm, Y.-E., Tanaka, K., and Ishikawa, H., 2013. “Collaborative engineering among designers with different preferences: Application of the preference set-based design to the design problem of
an automotive front-side frame”. Concurrent Eng. Res. A., 21(4),
pp. 252–267.
[4] Hannapel, S., and Vlahopoulos, N., 2014. “Implementation of setbased design in multidisciplinary design optimization”. Struct. Multidiscip. Optim., 50(1), pp. 101–112.
[5] Jr, R. J. M., Aughenbaugh, J. M., and Paredis, C. J. J., 2009. “Multiattribute utility analysis in set-based conceptual design”. Comput.
Aided Design, 41(3), pp. 214–227.
[6] Rockafellar, R. T., and Royset, J. O., 2010. “On buffered failure probability in design and optimization of structures”. Reliab. Eng. Syst.
Saf., 95, pp. 499–510.
[7] Rockafellar, R. T., and Uryasev, S., 2000. “Optimization of conditional Value-at-Risk”. J. Risk, 2, pp. 493–517.
[8] Forrester, A., Sobester, A., and Keane, A., 2008. Engineering Design
via Surrogate Modelling: A Practical Guide. Wiley.
[9] Freitag, S., Cao, B. T., Ninic, J., and Meschke, G., 2014. “Surrogate
modeling for mechanized tunneling simulations with uncertain data”.
In Proceedings of the 6th International Workshop on Reliable Engineering Computing (REC 2014), Chicago, pp. 44–63.
[10] Perdikaris, P., Venturi, D., Royset, J., and Karniadakis, G., 2015.
“Multi-fidelity modeling via recursive co-kriging and gaussian markov
random fields”. P. R. Soc. A, 2179(471).
[11] Papalambros, P. Y., 2012. “Special issue: Design under uncertainty”.
ASME J. Mech. Des., 134(10), pp. 100201–100201–1.
[12] Li, M., Mahadevan, S., Missoum, S., and Mourelatos, Z. P., 2016.
“Special issue: Simulation-based design under uncertainty”. ASME J.
Mech. Des., 138(11), p. 110301.
[13] Ayyub, B., Akpan, U., Koko, T., and Dunbar, T., 2015. “Reliabilitybased optimal design of steel box structures. i: Theory”. ASCE-ASME
J. Risk Uncertainty Eng. Syst. A, 1(3), p. 04015009.
[14] Akpan, U., Ayyub, B., Koko, T., and Rushton, P., 2015. “Development of reliability-based damage-tolerant optimal design of ship
structures”. ASCE-ASME J. Risk Uncertainty Eng. Syst. A, 1(4),
p. 04015013.
[15] Meng, D., Li, Y.-F., Huang, H.-Z., Wang, Z., and Liu, Y., 2015.
“Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic transmission mechanism design”. ASME J. Mech. Des., 137(5), pp. 051402–
051402–9.
[16] Rockafellar, R. T., and Royset, J. O., 2015. “Engineering decisions
under risk-averseness”. ASCE-ASME J. Risk Uncertainty Eng. Syst.
A, 1(2), p. 04015003.
[17] Dutta, S., Ghosh, S., and Inamdar, M., 2016. “Reliability-based design
optimization of frame-supported tensile membrane structures”. ASCEASME J. Risk Uncertainty Eng. Syst. A, p. G4016001.
[18] Wang, S., and Qing, X., 2016. “A mixed interval arithmetic/affine
arithmetic approach for robust design optimization with interval uncertainty”. ASME J. Mech. Des., 138(4), pp. 041403–041403–10.
[19] Antonsson, E. K., and Otto, K. N., 1995. “Imprecision in engineering
design”. ASME J. Mech. Des., 117(B), pp. 25–32.
[20] Royset, J. O., Der Kiureghian, A., and Polak, E., 2001. “Reliabilitybased optimal structural design by the decoupling approach”. Reliab.
Eng. Syst. Saf., 73(3), pp. 213–221.
[21] Royset, J. O., Der Kiureghian, A., and Polak, E., 2006. “Optimal
design with probabilistic objective and constraints”. J. Eng. Mech.,
132(1), pp. 107–118.
[22] Brizzolara, S., and Bonfiglio, L., 2015. “Comparative CFD investigation on the performance of a new family of super-cavitating hydrofoils”. J. Phys. Conf. Ser., 656(1), pp. 12147–12150.
[23] Brizzolara, S., 2014. “Watercraft device”. US Patent(US 13362298),
09.
[24] Royset, J. O., 2016. Approximations and solution estimates in optimization. In review.
[25] Haldar, A., and Mahadevan, S., 2000. Probability, reliability, and
statistical methods in engineering design. Wiley.
[26] Goel, T., Haftka, R. T., and Shyy, W., 2009. “Comparing error estimation measures for polynomial and kriging approximation of noise-free
functions”. Struct. Multidiscip. Optim., 38(5), pp. 429–442.
[27] An, D., and Choi, J. H., 2012. “Efficient reliability analysis based on
bayesian framework under input variable and metamodel uncertainties”. Struct. Multidiscip. Optim., 46(4), pp. 533–547.
[28] Meckesheimer, M., Booker, A. J., Barton, R. R., and Simpson, T. W.,
2002. “Computationally inexpensive metamodel assessment strategies”. AIAA J., 40(10), pp. 2053–2060.
[29] Viana, F. A. C., Haftka, R. T., and Steffen, V., 2009. “Multiple surrogates: How cross-validation errors can help us to obtain the best
predictor”. Struct. Multidiscip. Optim., 39(4), pp. 439–457.
[30] Zhang, J., Chowdhury, S., Mehmani, A., and Messac, A., 2014. “Characterizing uncertainty attributable to surrogate models”. ASME J.
Mech. Des., 136(3), p. 031004.
[31] Mehmani, A., Chowdhury, S., and Messac, A., 2015. “Predictive
quantification of surrogate model fidelity based on modal variations
with sample density”. Struct. Multidiscip. Optim., 52(2), pp. 353–373.
[32] Picheny, V., Ginsbourger, D., Richet, Y., and Caplin, G., 2013.
“Quantile-based optimization of noisy computer experiments with
tunable precision”. Technometrics, 55(1), pp. 2–13.
[33] Drezner, Z., and Erkut, E., 1995. “Solving the continuous p-dispersion
problem using nonlinear programming”. J. Oper. Res. Soc., 46(4),
pp. 516–520.
[34] Rockafellar, R. T., and Royset, J. O., 2015. “Measures of residual risk
with connections to regression, risk tracking, surrogate models, and
ambiguity”. SIAM J. Optimiz., 25(2), pp. 1179–1208.
[35] Rockafellar, R. T., Uryasev, S., and Zabarankin, M., 2008. “Risk
tuning with generalized linear regression”. Math. Oper. Res., 33(3),
pp. 712–729.
[36] Brizzolara, S., 2015. “A new family of dual-mode super-cavitating
hydrofoils.”. In Proceedings of the Fourth International Symposium
on Marine Propulsors, smp15, Austin, Texas, USA, June.
[37] Vernengo, G., Bonfiglio, L., Gaggero, S., and Brizzolara, S., 2016.
“Physics-based design by optimization of unconventional supercavitating hydrofoils”. J. Ship Res., 60(4), pp. 187–202.
[38] Hirt, C. W., and Nichols, B. D., 1981. “Volume of fluid (vof)
method for the dynamics of free boundaries”. J. Comput. Phys., 39(1),
pp. 201–225.
[39] Kunz, R. F., Boger, D. A., Stinebring, D. R., Chyczewski, T. S., Lindau, J. W., Gibeling, H. J., Venkateswaran, S., and Govindan, T., 2000.
“A preconditioned navier–stokes method for two-phase flows with application to cavitation prediction”. Comput. Fluids, 29(8), pp. 849–
875.
[40] Bonfiglio, L., and Brizzolara, S., 2016. “A multiphase ranse-based
computational tool for the analysis of super-cavitating hydrofoils”.
Nav. Eng. J., 128(1), pp. 67–85.