RELIABILITY-BASED ROBUST DESIGN OPTIMIZATION FOR SHIPS IN REAL OCEAN ENVIRONMENT M. Diez, The University of Iowa, USA; visiting from CNR-INSEAN, National Research Council - Marine Technology Research Institute, Italy X. Chen, The University of Iowa, USA; visiting from School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, P. R. China E.F. Campana, CNR-INSEAN, National Research Council - Marine Technology Research Institute, Italy F. Stern, The University of Iowa, USA. Corresponding author, email:
[email protected] SUMMARY A reliability-based robust design optimization for resistance reduction and operability increase in head waves from a real ocean environment is formulated and solved. The analysis tool consists in a URANS solver coupled with uncertainty quantification methods for variable regular wave, based on a Monte Carlo simulation with metamodels. The design optimization methods include the Karhunen-Loève expansion of the geometry modification space, coupled with design of experiments and metamodels, and a multi-objective particle swarm optimization algorithm. To reduce the computational effort, a simplified model is introduced for the effects of sea state and speed on the subsystem’s seakeeping responses. The application presented is a 100m Delft catamaran sailing in the North Pacific Ocean, with speed in the range Fr = 0.115 ÷ 0.575. An optimal design is selected, providing a 2.8% reduction for the expected value of the mean resistance and an 8.5% increase in operability, based on stochastic operations.
1.
INTRODUCTION
Sensitivity Analysis and Uncertainty Quantification to Military Vehicle Design”. The framework for UQ includes
Real-world applications in Simulation-Based Design (SBD) are permeated by different sources of uncertainty (namely operational/environmental, geometrical and numerical) and the combination of uncertainty quantification (UQ) methods with optimization algorithms is required to achieve reliability and robustness of the final product. A stochastic SBD optimization is referred to as Reliability-Based Design Optimization (RBDO) if the input uncertainties mainly affect the design constraints; whereas if the uncertainties mainly affect the design objectives then the resulting procedure is referred to as Robust Design Optimization (RDO). The more complete (and complex case) is the Reliability-Based Robust Design Optimization (RBRDO), clearly a combination of RBDO and RDO.
convergence and validation procedures using Monte Carlo methods, metamodels and quadrature formulas and has been applied to URANS studies for a NACA 0012 hydrofoil with variable Reynolds number [1], URANS and potential flow studies for the Delft catamaran in calm water with variable Froude number and geometry [2], and URANS studies for the Delft catamaran in irregular and stochastic regular wave, including variable geometry [3]. Latest research included static and adaptive metamodeling (e.g. dynamic Kriging, DKG and dynamic radial basis functions, DRBF) [4]. The background research for the Delft catamaran (deterministic) optimization is provided by [5], along with its recent extension to dimensionality reduction methods by Karhunen-Loève expansion (KLE) [6].
The stochastic SBD framework under development assesses the RBRDO of ships in real operations, in combination with accurate and expensive high-fidelity analysis tools, such as URANS solvers. The simulation of complex engineering systems requires indeed highfidelity solvers in order to include into the design cycle complex objective functions (such as the performances of the system in real operating conditions) and guarantee at the same time the accuracy of the final solution.
The objective of the present research is the RBRDO of the Delft catamaran barehull in a real ocean environment, aimed at resistance reduction and operability increase in stochastic head waves. This requires the development and the integration in the stochastic SBD of efficient UQ methods for the representation of the irregular wave statistics, including models for variable sea state and operational speed, and multi-objective optimization methods.
Background research in UQ was conducted in collaboration with NATO AVT-191 “Application of
Regular wave models (deterministic and stochastic) are investigated and used for the assessment of the expected
value of the total resistance in wave and the percent operability in the stochastic operating scenario. UQ methods use the Monte Carlo simulation with metamodels and the Importance Sampling method. The design space for barehull modifications is taken from [6], and limited to the bounds used in [6] for the initial global optimization. A multi-objective particle swarm optimization (PSO) algorithm [7] is used to define the Pareto front of non-dominated designs [8, 9]. The application presented pertains to a 100m Delft catamaran operating in the North Pacific Ocean, with Froude number in the range 0.115 ÷ 0.575. 2.
STOCHASTIC SBD FORMULATION
2.1
OCEAN ENVIRONMENT, OPERATING CONDITIONS AND OPERABILITY CONSTRAINTS
(1)
where Vd = 35 kt. For 100 m vessel, it is
2.2
"
RT f (Fr)d(Fr)
(3)
min( Fr )
where f(Fr) = 1/[max(Fr)-min(Fr)]. For the second objective function, the percent operability has to be defined and assessed. Constraints in Table 3 are assessed versus speed and sea state number k. For prove of concept, the relative heading is considered constant and equal to 0 deg (head wave). Maximum allowed speed (or speed limit) Vk* (for which all the criteria in Table 3 are met) is determined for each sea state k = 1,…,7. The percent operability is calculated from Vk* as 7
The procedure for the operability assessment is based on subsystem’s seakeeping performance criteria [12] (see Table 3) and addresses mobility (MOB), anti-submarine warfare (ASW), surface warfare (SUW) and anti-air warfare (AAW). A survey on criteria for naval missions can be found in [13]. The criteria for mobility are given in the standardization agreement NATO STANAG 4154 [14, 15]. Similarly to [12], the operational speed is assumed to span continuously the range from 20% to 100% the maximum design speed (Vd ). Specifically, a uniform distribution is assumed for the operational speed:
0.115 ! Fr ! 0.575
max( Fr )
!2 = O% =
Since the Delft catamaran (DC) is a design exercise, and a full-scale ship doesn’t exist, a conventional full-scale length of 100m is adopted. As reference, the main characteristics of the Delft catamaran are compared in Table 1 with those of the Joint High-Speed Vessel (JHSV). The design speed is taken equal to 35 kt (Fr = 0.575 for a 100 m vessel). The North Pacific Ocean is assumed as the operating environment, and its annual probability of occurrence of sea states is shown in Table 2 [10, 11].
V = U !"0.2;1.0 #$ Vd
!1 = EV ( RT ) =
MULTI-OBJECTIVE DESIGN OPTIMIZATION FOR RESISTANCE REDUCTION AND OPERABILITY INCREASE
The first objective function is the total resistance evaluated at sea state 5 (representing the average condition) and variable Froude number. The optimization objective is defined as the expected value of the mean resistance over the Froude number range. Accordingly,
(4)
where, by definition
ok =
Vk* ! 0.2 Vd 100 0.8
(5)
is the percent operability at sea state k, whereas pk is the percentage of occurrence of sea state k, as provided in Table 3. Finally, the design problem is formulated as a multiobjective RBRDO as
Minimize !1 (x1 ,..., xn );"!2 (x1 ,..., xn ) subject to lk # xk # uk , k = 1,...,n and to
g j (x) # 0,
j = 1,..., N g
(6)
where ϕ1 and ϕ2 are provided by Eqs. (3) and (4); xk are the design variables providing the barehull modifications, n is the dimension of the design space, and lk and uk are the design variables lower and upper bounds, assumed herein equal to -1 and 1, respectively. The geometrical constraints gj refer to the feasible set presented in [6] and referred to as A. 3.
(2)
1 ok pk 100 " k=1
STOCHASTIC SBD METHODS
The design optimization methods used to solve the problem of Eq. (6) include a geometry modification tool based on the KLE analysis of a high-dimensional design space [6], a multi-objective minimization algorithm [8, 9], multiple metamodels [6] coupled with novel regular wave UQ approaches for the assessment of the expected value of the mean resistance in wave and the percent operability in the stochastic operating scenario, based on subsystem’s seakeeping performance. The stochastic SBD flowchart is presented in Fig. 1.
3.1
GEOMETRY MODIFICATION
The barehull geometry modification is performed using the morphing approach used in [6]:
evaluated using all available solutions. During the swarm optimization, box and geometrical constraints are treated using a linear penalty function method. 3.3
⎛ ⎞ g ( x1 ,..., xn ) = ⎜1 − ∑ xk ⎟ g 0 + ∑ xk g k k =1 ⎝ k =1 ⎠ n
n
(7)
where g represents the modified geometry, g0 is the original geometry and gk are modified designs used as a basis for the geometry modification space. In the current study, these are given by the KLE of a free-form deformation space with 20 degrees of freedom. As shown in [6], four basis functions are required in Eq. (7) to retain the 95% of the geometric variance spanned by the original space. 3.2
MULTI- OBJECTIVE PARTICLE SWARM OPTIMIZATION
The particle swarm optimization (PSO) algorithm [7] is used herein for the global minimization. The multiobjective deterministic version of the algorithm [8, 9] is used. The resulting MOPSO iteration is expressed by
v ij = χ ⎡⎣ w v ij−1 + c1 (p j − xij−1 ) + c2 (b − xij−1 )⎤⎦
xij = xij−1 + vij
(8) (9)
where xij is the position of the j-th swarm particle at the ith iteration, pj is the closest point on the personal Pareto front of all the positions ever visited by the j-th particle and b is the closest point on the Pareto front of all the positions ever visited by all the particles; χ, w, c1 and c2 are coefficients or weights that control damping, inertia and personal/social behavior of the swarm, respectively. In the present work, multiple sets of deterministic coefficients are used to increase the variety of the swarm dynamics [6]. Specifically, the choice of coefficient values follows Eberhart and Shi [16], χ = 1.0, w = 0.729, c1 = 1.494 and c2 = 1.494; Peri and Tinti [17], χ = 1.047, w = 0.720, c1 = 2.042 and c2 = 1.150; Diez and Peri [18], χ = 0.99, w = 0.99, c1 = 0.33 and c2 = 0.67. The swarm dimension is set to 20 n (n is design space dimension). The initial swarm position, x0j, is defined using Hammersley sequence samples (HSS) [18] covering the design space provided by Eq. (7). The particles initial speed, v0j, follows the same Hammersley sequence used for the initial swarm position, as per:
v 0j =
2 ⎡ 0 (l + u ) ⎤ ⎢x j − ⎥ 2 ⎦ n⎣
(10)
where l and u represent the lower and upper bound vectors respectively, as per Eq. (6). Combining together the different coefficient sets results in 3 separate PSO procedures and the final Pareto front is
DESIGN OF EXPERIMENTS AND METAMODELS
Metamodels are trained using (i) an initial DoE based on Hammersley sequence sampling [19] (same as for PSO initialization), (ii) a DoE refinement based on MOPSO solutions. Different metamodels are applied: polyharmonic spline of first order (PHS) [20], stochastic ensemble of radial basis functions network (SRBF) with power law kernel [4], ordinary Kriging (OKG) with an exponential covariance function [21]. Once the DoE is defined, MOPSO procedures are performed for each of the metamodels above, resulting in a total of 9 optimization processes. The final optimization result is taken as the union of non-dominated solutions given by different surrogate-based MOPSOs. 3.4
IRREGULAR WAVE STUDIES
Significant single amplitude (SSA) of pitch motion, vertical acceleration and speed of bridge and flight deck respectively as per Table 3 are computed from irregular wave as
( )
( )
1 SSA J = 2 RMS J = 2 N
K
!#$ J k=1
(k )
"J% &
2
(11)
where the J(k) represent the k-th time step from irregular wave simulation and
1 J= N
K
!J
(k )
(12)
k=1
3.5
REGULAR WAVE MODELS FOR RESISTANCE IN WAVE AND OPERABILITY
3.5(a)
DETERMINISTIC MODELS
REGULAR
WAVE
Deterministic regular wave simulations are conducted at representative wave heights and periods. SSA of relevant quantities is computed following Eqs. (11) and (12). The cases shown in Table 5 are considered. Definitions of characteristic wave periods and heights follow [3] and [22]. Specifically,
H rms = H 2 =
1 2
H1/3
(13)
is the wave height that allows a single regular wave to retain the whole spectrum energy, whereas ! ! H= H rms = H 2 8 1/3
(14)
and, using the moment transport theorem
! i2 ( J ) =
K 2 8 ) +1 $ J (k ) # J & + ( J # J * " i i' i T 2 % H + , K k=1
(
(
2
is the mean wave height, assuming the Rayleigh distribution. Characteristic periods are defined as follows.
!" RMS(J i ) #$ + % J i & J T = 2m0 ' a,i 2
)
-
) +.+/ m 2
0
2
(20)
"
with m0 = # S(! )d! ; Ji(k) is the solution from the
1/4
T0 =
2! # 5 & = Tp " 0 %$ 3 ('
0
(15)
is the reciprocal of the peak frequency in energy spectrum S(ω), whereas Tp is the peak period found using S(T). #
$ S(" ) d"
T1 = 2!
0
#
$ " S(" ) d"
= 0.772T0
(16)
regular wave simulation at wave height equal to H and period equal to 2π/ωi; K is number of time steps considered for each regular wave period (here K = 100); 1 K 1 N J i = ! k=1 J i (k ) and JT = " i=1 J i fi (! ) and finally τ is a K N coefficient used as a toggle switch for the moment transport (τ = 0 indicates no moment transport, whereas when τ = 1 the moment transport is used). The mean value of the total resistance in wave is computed as
0
is the period corresponding to the true average frequency of the elemental wave components in the spectrum. #
Ta =
2!
$"
S(" ) d "
0
#
$ S(" ) d"
= 0.857T0
(17)
0
is the true average period of the elemental wave components in the spectrum. Finally, TA and HA identify the most probable condition from the joint distribution of zero-crossing period and height from Longuet-Higgins theory [23]; TB and HB correspond to their expected values. 3.5(b)
STOCHASTIC REGULAR WAVE UQ MODELS
A novel one-dimensional UQ is performed over the distribution of frequencies given by the wave energy spectrum. SSA of relevant quantities is computed using LHS and Importance Sampling (IS) as follows
( )
1 RT = N
" $1 i=1 $ K $# N
! !R
(18)
where the probability density function is based on the wave energy spectrum S as
f i (! ) =
S(! i ) m0
(19)
k=1
%
(k ) ' (( ) T ,i ' f i
(21)
'&
Two joint distributions are used for two-dimensional UQ in variable regular wave. The first, used for the first time herein, is based on the combination of the wave energy spectrum in the frequency domain and the Rayleigh distribution in the wave height domain; the second, already used in [3] is based on the Longuet-Higgins theory [23] for zero-crossing periods and heights. Specifically, for the first distribution, a two-dimensional UQ is performed extending Eq. (18) as per
( )
2
N ! SSA J $ 1 2 () ,H ) # & = '( i (J ) f i N i=1 #" 2 &%
(22)
where the probability density function is based on the wave energy spectrum S and the Rayleigh distribution fH as per
f i (! ,H ) =
2
N ! SSA J $ 1 2 () ) # & = '( i (J ) f i N i=1 #" 2 &%
K
S(! i ) f (H ) m0 H i
(23)
Equation (20) holds and Eq. (21) becomes
1 RT = N
" $1 i=1 $ K $# N
K
! ! j=1
%
RT(i), j ' f i(( ,H ) '
(24)
'&
The current probability density function is referred in the following as PDF1.
For the second distribution, a two-dimensional UQ procedure is performed using the probability density function of zero-crossing periods and heights in the wave record, as provided by [23]. This approach was used in earlier work [3]. It is
( )
2
N ! SSA J $ 1 (T ,H ) 2 # & = ' !"( i (J ) f i z z $% 2 N #" &% i=1
(25)
where K 2 +) 1 ! i2 ( J ) = * " $% J (i) # Ji & + ( Ji # J K j ' +, K j=1
(
-
) +.+/ 2
(26)
The mean resistance in wave is evaluated by
RT =
N " 1 K (i) % (T ,H ) 1 $ !RT , j ' f i z z N! '& i=1 $ # K j=1
(27)
The current probability density function is referred in the following tables and figures as PDF2. 3.6
CFD METHOD
The code CFDShip-Iowa V4.5 [24] is used for the CFD simulations. Using the code SUGGAR, the overset structured grid connectivity is obtained at running time to simulate large-amplitude motions. 6DOF capabilities are implemented using Euler angles. The blended k-ε/k-ω turbulence and single-phase level set free-surface modeling is used.. Numerical methods include advanced iterative solvers, second and higher order finite difference schemes with conservative formulations, surface-capturing (level set) method, parallelization based on a domain decomposition approach using the message-passing interface (MPI). The fluid flow equations are solved in an earth-fixed inertial reference system, while the rigid body equations are solved in the ship system. Computational girds used are provided in [3]. 4.
RESULTS
4.1
PRELIMINARY WAVE STUDIES FOR THE ORIGINAL HULL
Subsystem seakeeping performances are evaluated at sea state 6 and Fr = 0.5. Bridge and flight deck location are defined similarly to the JHSV. Accordingly, bridge (B) coordinates in the xy plane are assumed equal to (0.30L; 0.15L), with x = 0 corresponding to the forward perpendicular (FP) and x = L corresponding to the aft perpendicular (AP); z = 0 corresponds to the keel line. Flight deck (D) coordinates are taken as (0.85L; 0.10L). For the definition of the inlet boundary conditions used for the irregular wave simulation, the reader is refereed
to [3]. Results from the irregular wave simulation are taken as benchmark for deterministic regular wave and UQ methods, and are shown in Fig. 2 and Table 4. None of the criteria of Table 3 are met. Statistical convergence of the irregular wave study was shown in [3] and therefore not repeated here. Heave and pitch motions time histories from the regular wave cases are shown in Fig. 3. Figures 4 shows the convergence of the running zero-th and first harmonics, and RMS. Accordingly, 8 periods are selected for UQ studies and optimization. Figure 5 shows the response amplitude operators (RAO) of the motions from the regular wave cases, versus encounter frequency and wavelength. Comparison with irregular wave and EFD from [24] is provided. Table 6 and Fig. 6 show the comparison errors for SSA values versus the irregular wave benchmark, for each regular wave case and criterion in Table 3. Errors are quite large, ranging on average from 13 to 51%. Best prediction for pitch SSA is given by RW1 (under predicted by 6.2%); best prediction for vertical acceleration of B SSA is given by RW5 (overpredicted by 2.1%); best prediction for vertical speed of D SSA is given by RW2 (overpredicted by 0.9%). Comparison errors for total resistance are also included, showing best prediction for RW2 (0.8%). Best average absolute error is given by RW2 (12.7%). One-dimensional UQ is conducted using a surrogatebased Monte Carlo method with N = 1,000 LHS items (regular waves). Items are evaluated by metamodels (polyharmonic spline, PHS, and radial basis functions network with multiquadric kernel, RBF) and the Importance Sampling method is applied. Three training set are constructed using regular wave simulations. The first set (referred to as TS1) includes M = 17 equally spaced wave angular frequency in the same range used for irregular wave inlet boundary condition [0.3; 1.3] (rad/s); the second set (referred to as TS2) comprises M = 17 equally spaced encounter frequencies in the range where the responses are more significant [0.08; 0.4] (Hz). The third set (referred to as TS3) is based on TS2, with M = 9. Two wave heights are considered using TS1: HRMS and H . Average steepness equals respectively 1/41 and 1/46. Only H is used for TS2 and TS3. Response amplitude operators are shown in Fig. 5 using TS1, and compared to irregular wave and EFD. Comparison errors for SSA values are shown in Table 6 and Fig. 5, where M1 and M2 refer to the metamodels PHS and RBF respectively. MT refers to the moment transport. Errors are fairly small, especially if compared to the deterministic regular wave cases. Average errors range from 1.5 to 4.1%. Best results are given using H with TS2, MT and M2. TS3 with MT and M2 reveals the best compromise between computational cost and accuracy, with an average error equal to 2.32%. The two-dimensional UQ is based on a surrogate-based Monte Carlo method with N = 10,000 LHS items, predicted using PHS and RBF and using the Importance
Sampling method. Response amplitude operators from the training set items are shown in Fig. 5, and compared to irregular wave and EFD. Results from PDF1 and PDF2 are presented in Table 6 and Fig. 6. PDF1 provides an average steepness equal to 1/46, whereas PDF2 gives an average steepness equal to 1/36. Errors are reasonable, but larger than those found in the one-dimensional UQ. Average errors range from 2 to 3.4% using PDF1, whereas range from 4.6% to 5.2% with PDF2. Errors for pitch and vertical velocity of deck are quite large. Best prediction for relevant SSAs and total resistance using the two-dimensional UQ is given using PDF1, with M2 and MT (average absolute error equal to 2%). 4.2
EFFECTS OF SEA STATE AND SPEED AND SIMPLIFIED MODEL FOR NON-LINEAR EFFECTS
Effects of sea state and Froude number are studied through the one-dimensional UQ (using H , TS2, MT and M2, since most accurate as per Table 6) for each of the conditions in Table 7 (for a total of N = 272 simulations). Relevant SSAs, varying sea state and Froude number, are shown in Fig. 7. Table 7 shows the SSAs for pitch motion, vertical acceleration and speed of bridge and deck respectively. SSAs are compared to the maximum allowed values. Pitch motion criterion is violated at sea states greater than 4, regardless of the speed. Vertical acceleration of bridge and speed of deck show decreasing critical speeds as the sea state increases. Table 8 shows the operability at each sea state. Pitch motion is the most critical constraint, which is reasonable since only head waves are considered. Overall operability in head waves is close to 47%. In order to build a simplified (and less expensive) model, the operability assessment is repeated assuming that the motions response is independent of the sea state and the Froude number (assuming therefore linear response operators and transfer functions). The response at sea state 5 and Fr = 0.425 is chosen as a reference (average sea state and speed close to the constraints violation) and used to evaluate all the conditions in Table 7. Results are shown in Fig. 7 and Table 9. Some difference with the fully-nonlinear study is observed. Accordingly, a correction factor is evaluated and used to model these differences as per Fig. 8 and Table 10. This is used during the optimization process (regardless of geometry changes) to identify the speed limits and assess the overall percent operability. This procedure requires 1/16 of the computational cost of the fully-nonlinear simulation. 4.3
BAREHULL OPTIMIZATION
The barehull optimization is conducted using an initial training set (I) of different designs, Eq. (7), with size equal to 40. The first optimization is conducted with and without the simplified model for non-linearities. When the model is not applied, linear responses are assumed
without correction. Surrogate-based MOPSO results are shown in Fig. 9 (a), making use of the simplified model. The non-dominated solutions, obtained without the correction model are presented in Fig. 9 (b), whereas Fig. 9 (c) shows the non-dominated solutions using the model. The results are close, indicating that the correction does not affect significantly the operability trend in the design space. Results from different metamodels are close. A refinement (R1) of the DoE with size equal to 40 is defined, based on surrogates predictions and shown in Fig. 9. Predictions and URANS solutions are quite far, indicating that the initial DoE size is too small to represent correctly the design space. MOPSO results for the refined DoE are shown in Fig. 9 (a). Ordinary Kriging is not used since is affected by overshooting in the current problem. Metamodels give similar results, especially in the low-resistance region. These are used to define a second refinement (R2) of the DoE with size equal to 40. URANS solutions are closer to the predictions than what found in the first iteration. Associated MOPSO results are shown in Fig. 9 (a), revealing that an additional refinement is not required. Accordingly, the total number of URANS simulations in regular wave, required by the present stochastic design optimization, equals 1680. Among the solutions available from the Pareto front, the design identified by R2-P027 and shown in Figs. 9 (a) and 10 (a) is selected, providing a 2.8% reduction in expected mean resistance with an increase in operability by 8.5%. Its performances are compared to the original design in Fig. 10 (b, c, d) using RW2 at design speed and Fig. 10 (e, f, g) showing mean resistance versus Fr and motions RAOs. 5.
CONCLUSIONS AND FUTURE WORK
An RBRDO formulation for resistance reduction and operability increase in head waves from a real ocean environment has been presented and applied successfully to a 100m Delft catamaran in the North Pacific Ocean, with speed ranging from Fr = 0.115 to 0.575. The current milestone achievement includes the development of novel UQ methods based on regular waves to represent the irregular wave statistics, the extension to variable sea state and operational speed, and the stochastic design optimization. The SBD procedure is based on a URANS solver. The regular wave UQ is assessed using the Monte Carlo simulation using metamodels and the Importance Sampling method. A simplified model for the effects of sea state and speed on the subsystem’s seakeeping responses has been developed and used during the optimization. The design optimization tools include the Karhunen-Loève expansion of the geometry modification space, coupled with design of experiments and metamodels, and a multi-objective particle swarm optimization algorithm. The total computational cost of the stochastic SBD consisted in 1680 URANS simulations in regular head wave. An optimal design has been selected, providing a 2.8% reduction for the
expected value of the mean resistance and an 8.5% operability increase in the stochastic operating scenario. Future work includes the assessment of larger design spaces, namely using the design variables bounds applied in [6] to the refined DoE. A larger size for the initial DoE will be used in order to provide a more accurate approximation of the objective functions. In addition, verification of simulations at sea state 5 will be performed as well as the irregular wave simulation at sea state 5 and Fr = 0.425, for validation of UQ methods and comparison to EFD. 6.
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CHEN X., DIEZ M., KANDASAMY CAMPANA E.F., STERN F., ‘High-fidelity Global Optimization for Shape Design by Dimensionality Reduction, Metamodels and Particle Swarm,’ submitted to Engineering Optimization, 17 July 2013.
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ACKNOWLEDGEMENTS
The present research is supported by the Office of Naval Research, Grant N00014-11-1-0237 and NICOP Grant N62909-11-1-7011, under the administration of Dr. KiHan Kim, and partially by the Italian Flagship Project RITMARE, coordinated by the Italian National Research Council and funded by the Italian Ministry of Education, within the National Research Program 2011-2013. URANS computations were performed at the NAVY DoD Supercomputing Research Centre. Xi Chen is also grateful for the support from China Scholarship Council (CSC). 7.
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Survey,’ Marine Technology, Vol. 39, No. 1, 2012, pp. 29-47. 15. NATO STANAG 4154, “Common Procedures in the Ship Design Process,” Chapter 7: Seakeeping Criteria for General Application, 1997. 16. EBERHART R., SHI Y., ‘Comparing inertia weights and constriction factors in particle swarm optimization.’ Proceedings of the Congress on Evolutionary Computation, 2000. 17. PERI D., TINTI F., ‘A multistart gradient-based algorithm with surrogate model for global optimization,’ Communications in Applied and Industrial Mathematics 3(1), 2012 18. DIEZ M., PERI D., ‘Robust Optimization for ship conceptual design,’ Ocean Engineering, Vol. 37, pp. 966-977, 2010. 19. WONG T.-T., LUK W.-S., HENG, P.A., ‘Sampling with Hammersley and Halton points,’ Journal of Graphics Tools 2(2):924, 1997. 20. HARDER R.L., DESMARAIS R.N., ‘Interpolation using surface splines,’ J. Aircr. 9, 189–191, 1972. 21. PERI D., ‘Self-Learning Metamodels for Optimization,’ Ship Technology Research 56:94-108, 2009. 22. MICHEL W.H., ‘Sea spectra revisited,’ Marine Technology, Vol. 36, No. 4, Winter 1999, pp. 211-227. 23. LONGUET-HIGGINS M.S., ‘On the Joint Distribution of Wave Period and Amplitudes in a Random Wave Field,’ Proc. Roy. Soc. London, Series A, Vol. 389, 1983, pp. 241-258. 24. CARRICA P.M., WILSON R. V, AND STERN F., ‘An unsteady single-phase level set method for viscous free surface flows,’ International Journal For Numerical Methods In Fluids 53:229–256, 2007. 25. BOUSCASSE B., BROGLIA R., STERN F., ‘Experimental Investigation of a Fast Catamaran in Head Waves,’ Ocean Engineering, in press, 2013.
Table 1: Delft Catamaran full scale characteristics, compared to JHSV Parameter Propulsion Length Beam Draft Displacement Design speed
Unit m m m t kt
Delft Catamaran water jet (2x) 100 31.3 5.0 3,225 35
JHSV water jet (4x) 103 28.5 3.83 2,397 35÷43
Table 2: Characteristic annual values for sea states in North Pacific Sea State
Mean H1/3 [m]
0-1 2 3 4 5 6 7 8 >8
0.05 0.30 0.88 1.88 3.25 5.00 7.50 11.50 >14
Bales, 1982 0.00 4.10 16.90 27.80 23.50 16.30 9.10 2.20 0.10
Probability of sea state (%) Lee and Bales, 1984 Average 1.30 0.65 6.40 5.25 15.50 16.20 31.60 29.70 20.94 22.22 15.03 15.67 7.60 8.35 1.56 1.88 0.07 0.09
Exceedance 99.35 94.10 77.90 48.20 25.98 10.32 1.96 0.08 0.00
Most probable modal wave period [s] Bales, 1982 Lee and Bales, 1984 Ave. 7.50 6.30 6.9 7.50 7.50 7.5 8.80 8.80 8.8 9.70 9.70 9.7 13.80 12.40 13.1 13.80 15.00 14.4 18.00 16.40 17.2 20.00 20.00 20
Table 3: Subsystem seakeeping performance criteria (Lain et al., 1979; Kennel et al., 1985) Criterion Unit Value Roll motion (CG) deg 8* Pitch motion (CG) deg 3* Vertical acceleration bridge (B) g 0.4* Vertical velocity flight deck (D) ft/s (m/s) 6.5* (1.98*) Wetnesses 1/h 30 Slams 1/h 20 Emergences 1/h 24 (*) Values refer to significant single amplitude (SSA = 2 RMS).
Table 4: Subsystem seakeeping performance at sea state 6 and Fr = 0.5, from irregular wave simulation Criterion Unit Value* Max. allowed* Pitch motion (CG) deg 5.11 3 Vertical acceleration bridge (B) g 0.74 0.4 Vertical velocity flight deck (D) m/s 3.07 1.98 (*) Values refer to significant single amplitude (SSA = 2 RMS).
Table 5: Deterministic regular wave cases for sea state 6 Case
H Value [m]
T Definition
3.25
Most probable from f(Tz,Hz)
8.11
RWB
3.13
Expected value from f(Tz,Hz)
8.80
RW1
3.54
RW2
3.54
RW3
3.54
RW4
3.54
RW5
3.54
H/λ
Most probable from f(Tz,Hz)
1.03
1/32
Expected value from f(Tz,Hz)
1.21
1/33
2.40
1/77
1.86
1/59
1.43
1/46
Definition
RWA
Hrms (providing same energy as the whole spectrum) Hrms (providing same energy as the whole spectrum) Hrms (providing same energy as the whole spectrum) Hrms (providing same energy as the whole spectrum) Hrms (providing same energy as the whole spectrum)
λ/L
Value [s]
12.4 10.91 9.57
Tz (optimal period, corresponding to the peak frequency in the spectrum) Tp (peak period in the energy spectrum [expressed as a function of wave period]) T1 (period corresponding to the true average frequency in the spectrum)
10.63
Ta (true average period in the spectrum)
1.76
1/56
9.05
T1,e (period corresponding to the true average frequency in the encounter spectrum at Fr = 0.5)
1.28
1/41
Table 6: Comparison errors for deterministic regular wave cases and UQ at sea state 6 and Fr = 0.5; M is the number of simulations used Case
Deterministic regular wave
UQ 1D
UQ 2D
M
RWA RWB RW1 RW2 RW4 RW5 RW6 UQ1D M1 (Hrms, TS1) UQ1D M2 (Hrms. TS1) UQ1D M1 MT (Hrms. TS1) UQ1D M2 MT (Hrms, TS1) UQ1D M1 (Have, TS1) UQ1D M2 (Have. TS1) UQ1D M1 MT (Have, TS1) UQ1D M2 MT (Have, TS1) UQ1D M1 (Have, TS2) UQ1D M2 (Have. TS2) UQ1D M1 MT (Have, TS2) UQ1D M2 MT (Have, TS2) UQ1D M1 (Have, TS3) UQ1D M2 (Have. TS3) UQ1D M1 MT (Have, TS3) UQ1D M2 MT (Have, TS3) UQ2D PDF1 M1 UQ2D PDF1 M2 UQ2D PDF1 M1 MT UQ2D PDF1 M2 MT UQ2D PDF2 M1 UQ2D PDF2 M2 UQ2D PDF2 M1 MT UQ2D PDF2 M2 MT
1 1 1 1 1 1 1 17 17 17 17 17 17 17 17 17 17 17 17 9 9 9 9 129 129 129 129 129 129 129 129
Pitch motions -19.69 -12.36 -6.21 31.74 15.37 42.29 -7.69 -3.00 -3.00 -2.41 -2.41 -0.71 -0.71 -0.19 -0.19 -2.64 -2.64 -2.07 -2.07 4.12 3.97 3.73 3.59 -1.15 -0.94 0.20 0.13 -9.29 -8.91 -8.07 -7.67
Vert. acc. B 22.48 63.24 -56.03 -14.55 87.34 2.15 81.45 -1.96 -1.97 -1.96 -1.97 1.77 1.77 1.77 1.77 -0.09 -0.09 -0.09 -0.09 3.03 2.93 3.03 2.93 3.01 -4.23 3.01 -4.23 2.19 1.74 2.20 1.75
E%SSAb = (SSA-SSAb)%SSAb Vert. speed D RT mean Ave. absolute 57.14 4.91 26.05 74.85 20.70 42.79 -12.88 -7.06 20.54 0.90 0.82 12.00 44.20 39.82 46.68 1.10 5.46 12.75 87.67 29.75 51.64 8.62 2.72 4.08 8.62 2.71 4.07 8.62 2.72 3.93 8.62 2.71 3.93 8.43 0.01 2.73 8.43 0.00 2.73 8.44 0.01 2.60 8.43 0.00 2.60 3.65 -0.14 1.63 3.65 -0.15 1.63 3.65 -0.14 1.49 3.65 -0.15 1.49 2.43 0.23 2.45 2.57 0.19 2.42 2.42 0.23 2.35 2.57 0.19 2.32 9.10 -0.54 3.45 3.51 -0.22 2.22 9.10 -0.54 3.21 3.51 -0.22 2.02 9.24 0.24 5.24 8.97 0.15 4.94 9.24 0.24 4.93 8.97 0.15 4.64
Table 7: SSA for Delft Catamaran varying sea state and Fr (N=272) Sea state
3
4
5
6
7
Fr 0.425 0.500 0.575 0.425 0.500 0.575 0.350 0.425 0.500 0.575 0.350 0.425 0.500 0.350 0.425 0.500
Pitch [deg] 0.753 0.685 0.557 1.799 1.748 1.673 3.334 3.282 3.160 3.025 4.646 4.550 4.712 5.938 5.730 5.867
SSA Vert. acc. B [1/g] 0.137 0.192 0.236 0.274 0.394 0.515 0.235 0.405 0.606 0.841 0.273 0.467 0.744 0.290 0.490 0.772
SSA/SSA(max) Vert. vel. D [m/s] 0.275 0.375 0.516 0.820 1.211 1.774 1.165 1.655 2.400 3.066 1.673 2.401 3.184 2.341 3.239 4.198
Pitch
Vert. acc. B
Vert. vel. D
0.251 0.228 0.186 0.600 0.583 0.558 1.111 1.094 1.053 1.008 1.549 1.517 1.571 1.979 1.910 1.956
0.342 0.479 0.591 0.686 0.985 1.288 0.587 1.013 1.515 2.102 0.681 1.167 1.860 0.724 1.224 1.930
0.139 0.189 0.260 0.414 0.611 0.896 0.588 0.836 1.212 1.548 0.845 1.213 1.608 1.182 1.636 2.120
Table 8: Operability of Delft Catamaran in North Pacific (head waves only) Sea state 1 2 3 4 5 6 7 Operabiliy % (total)
Pitch motion Vertical acc. B Vertical vel. D Fr* Operab. % Fr* Operab. % Fr* Operab. % 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.575 100.00 0.502 84.13 0.575 100.00 0.115 0.00 0.423 66.96 0.461 75.22 0.115 0.00 0.402 62.39 0.385 58.70 0.115 0.00 0.393 60.43 0.330 46.74 51.80 76.78 81.61 (*) Maximum allowed speed
Overall Fr* Operab. % 0.575 100.00 0.575 100.00 0.575 100.00 0.502 84.13 0.115 0.00 0.115 0.00 0.115 0.00 47.09
Table 9: SSA for Delft Catamaran varying sea state and Fr (linear model, N=17) Sea state
3
4
5
6
7
Fr 0.425 0.500 0.575 0.425 0.500 0.575 0.350 0.425 0.500 0.575 0.350 0.425 0.500 0.350 0.425 0.500
Pitch [deg] 0.679 0.584 0.494 1.807 1.694 1.565 3.329 3.282 3.190 3.062 4.564 4.734 4.855 5.674 6.009 6.293
SSA Vert. acc. B [1/g] 0.096 0.117 0.133 0.237 0.317 0.399 0.268 0.405 0.564 0.737 0.316 0.501 0.735 0.353 0.571 0.853
Vert. vel. D [m/s] 0.410 0.442 0.458 0.963 1.108 1.231 1.345 1.655 1.951 2.225 1.779 2.250 2.733 2.257 2.884 3.540
Pitch [deg] 0.226 0.195 0.165 0.602 0.565 0.522 1.110 1.094 1.063 1.021 1.521 1.578 1.618 1.891 2.003 2.098
SSA/SSA(max) Vert. acc. B Vert. vel. D [1/g] [m/s] 0.239 0.207 0.292 0.223 0.333 0.231 0.592 0.486 0.793 0.560 0.997 0.622 0.671 0.679 1.013 0.836 1.409 0.985 1.843 1.123 0.790 0.899 1.253 1.136 1.838 1.380 0.884 1.140 1.427 1.456 2.133 1.788
Table 10: Correction factor for SSA of Delft Catamaran using the linear model (N=17) Sea state 3
4
5
6
7
Fr 0.425 0.500 0.575 0.425 0.500 0.575 0.350 0.425 0.500 0.575 0.350 0.425 0.500 0.350 0.425 0.500
Pitch 1.108 1.172 1.129 0.995 1.032 1.069 1.002 1.000 0.991 0.988 1.018 0.961 0.970 1.047 0.953 0.932
SSA Vert. acc. B 1.429 1.641 1.772 1.160 1.242 1.292 0.876 1.000 1.075 1.141 0.863 0.931 1.012 0.820 0.858 0.905
Vert. vel. D 0.670 0.849 1.127 0.852 1.093 1.441 0.866 1.000 1.231 1.378 0.940 1.067 1.165 1.037 1.123 1.186
Figure 1: Multi-objective stochastic simulation based design flowchart
Figure 2: Pitch angle, vertical speed (D), vertical acceleration (B) at sea state 6 and Fr = 0.5 from irregular wave simulation with associated SSAs
Figure 3: Heave and pitch motions at Fr = 0.5 from regular wave simulations
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Heave and pitch motion at Fr = 0.5: running zero-th (a, d), first (b, e) harmonics and RMS (c, f) from regular wave simulations.
(a)
(c)
(b)
(d)
Figure 5: Heave (a, b) and pitch (c, d) motion at Fr = 0.5: response amplitude operators from EFD, irregular and regular waves, including 1D and 2D UQ for sea state 6
(a)
(b) Figure 6: Comparison (a) and average errors (b) for deterministic regular wave cases and UQ at sea state 6 and Fr = 0.5
5
1.2 0.55
0.8
1
4
0.6 0.5
0.8
4
2 0.45
0.4
1
0.4
0.4 0.2
0.35
0
0 4 (1.88)
5 (3.25)
6 (5.00)
0.35
7 (7.50)
3 (0.88)
sea state (significant wave height [m])
(a)
1
6 5
3
0.4
0.6
0.45
0.4
3 (0.88)
0 4 (1.88)
5 (3.25)
6 (5.00)
sea state (significant wave height [m])
7 (7.50)
2
1.5 1
0.4 0.2
1 0.35
2 0.45
0.4
2
0.4
2.5
Fr
0.2
3
3
0.5 SSA
0.45
0.8
Fr
4
3.5
1
0.6
SSA
4
4 0.55
0.5
Fr
3
7 (7.50)
(c)
0.8
5 2
6 (5.00)
1.2 0.55
6
1
5 (3.25)
sea state (significant wave height [m])
(b) 7
0.5
-1 4 (1.88)
0.35 3 (0.88)
0 4 (1.88)
5 (3.25)
6 (5.00)
sea state (significant wave height [m])
7 (7.50)
SSA
3 (0.88)
0.55
3
1 Fr
SSA
Fr
0.6 0.45
2
0.5
0.4
0.2
3
SSA
0.55
0.5 0.35 3 (0.88)
0 4 (1.88)
5 (3.25)
6 (5.00)
7 (7.50)
sea state (significant wave height [m])
(d) (e) (f) Figure 7: SSA of pitch motion, vertical acceleration of bridge and vertical velocity of deck versus sea state and Fr, using fully non-linear responses (a, b, c; N=272) and response at sea state 5 and Fr=0.425 (d, e, f; N=17)
(a)
(b)
(c)
Figure 8: Simplified model for SSA correction: (a) pitch motion, (b) vertical acceleration of bridge and (c) vertical velocity of deck versus sea state and Fr, using response at sea state 5 and Fr=0.425
(a)
(b)
(c)
Figure 9: Multi-objective optimization results. Solutions in the objective functions space, using the simplified model for non-linearities and DoE I, R1 and R2 (a); Pareto front points (all metamodels and MOPSOs based on DoE I) without applying the model for non-linearities (b), and applying the model for non-linearities (c)
Original
Original R2-27
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 10: Comparison of original and optimized hulls; geometries (a), time histories of force (b), heave (c) and pitch (d) motions at Fr = 0.575 (design speed), mean resistance (e) using RW2 at SS 5, heave (f) and pitch (g) RAOs at SS 5 and Fr = 0.425.
