In order to truly reflect the ship performance under the influence of uncertainties, uncertainty-based design optimization (UDO) for ships that fully considers various uncertainties in the early stage of design has gradually received more and more attention. Meanwhile, it also brings high dimensionality problems, which may result in inefficient and impractical optimization. Sensitivity analysis (SA) is a feasible way to alleviate this problem, which can qualitatively or quantitatively evaluate the influence of the model input uncertainty on the model output, so that uninfluential uncertain variables can be determined for the descending dimension to achieve dimension reduction. In this paper, polynomial chaos expansions (PCE) with less computational cost are chosen to directly obtain Sobol' global sensitivity indices by its polynomial coefficients; that is, once the polynomial of the output variable is established, the analysis of the sensitivity index is only the postprocessing of polynomial coefficients. Besides, in order to further reduce the computational cost, for solving the polynomial coefficients of PCE, according to the properties of orthogonal polynomials, an improved probabilistic collocation method (IPCM) based on the linear independence principle is proposed to reduce sample points. Finally, the proposed method is applied to UDO of a bulk carrier preliminary design to ensure the robustness and reliability of the ship.
Ship uncertainty-based design optimization (UDO) is a method of optimizing the design space according to the requirement of the robustness and the reliability under the influence of the uncertainty [
In addition, Matteo Diez and his team have led in the ship uncertainty-based design optimization for years. They applied the PSO algorithm to the robust optimization concept design of a bulk carrier [
When the ship design process is tackled, uncertain variables of the mission profile, main dimensions, etc. need to be defined. In order to truly reflect the ship performance under the influence of uncertainties, the designer gradually considers more and more uncertainties in the ship UDO process to obtain a more robust and reliable solution. It should be acknowledged that more uncertainties are helpful to simulate the real operating environment, but the high-dimensional problems that come with it also raise considerable challenges. For UDO, it is necessary to quantify the uncertainty for every case; therefore, it is obvious that adding another uncertain parameter will definitely increase the calculation burden, resulting in low optimization efficiency. Sensitivity analysis (SA), as a dimension reduction technology, is a feasible way and has been adopted to alleviate this problem. SA can qualitatively or quantitatively evaluate the influence of the model input uncertainty on the model output, so that uninfluential uncertain variables can be determined for the descending dimension to achieve dimension reduction. SA methods can be divided into local sensitivity analysis (LSA) method and global sensitivity analysis (GSA) method. The former investigates effects of variations of input factors in the vicinity of nominal values, whereas the latter aims to quantify the output uncertainty due to variations of the input factors in their entire domain. GSA does not require the model to be a linear system like LSA and it also includes investigation of interactions between model parameters, which is currently more widely applied.
For GSA methods, there are Fourier amplitude sensitivity test (FAST), extended FAST (EFAST), random balance design (RBD), and Sobol' global sensitivity method based on variance decomposition, etc. Among several GSA methods, GSA with Sobol’ sensitivity indices is herein of interest [
In recent years, PCE has gradually been introduced in this field. Using PCE method to perform SA in UDO, the coefficients of the polynomial can not only obtain the stochastic property of outputs required (mean, standard deviation, skewness, and kurtosis) in UDO, but also directly obtain Sobol' sensitivity indices. In other words, once a PCE representation is available, calculating Sobol’ indices is only the postprocessing of coefficients of PCE; therefore, indices can be calculated analytically at almost no additional computational cost. It was originally introduced by Sudret [
Compared with MC method, PCE method can greatly reduce the computational cost; however, it still has the potential to further reduce the amount of calculation. In the process of solving the polynomial coefficient of PCE, probabilistic collocation method (PCM) is used instead of the common statistical methods, such as Latin hypercube sampling method, which select a large number of sample points to maintain the calculation accuracy. Generally, the number of sample points should be greater than the number of undetermined coefficients. However, different from statistical methods, due to the orthogonal property of the polynomial, input sample points of PCM are not randomly selected, but according to certain rules, which may lead to fewer sample points. Xiu et al. [
The paper takes the ship uncertainty design as the research object, analyzes the influence of multiple random variables on the optimization target and the constraint, and proposes a ship uncertainty design based on the multidimensional polynomial chaos expansion method. It finally applies it into the practical engineering optimization design of a bulk ship, to ensure the robustness and reliability of the ship.
The author has done some work about ship UDO [
According to different design requirements, uncertainty-based design optimization can be divided into robust design optimization, reliability-based design optimization, and reliability-based robust design optimization.
Compared with the traditional design optimization, RDO considers the interference of external factors on outputs, whose solutions are not easily perturbed by external factors.
As shown in Figure
Deterministic and robust optimal solutions.
RDO does not only improve the objective performance, but also reduce its sensitivity to the uncertainty. Therefore, considering the influence of the uncertainty, its typical measurement indexes are the mean and the standard deviation of the objective function. Its mathematical expression can be concisely expressed in
where
The core of RBDO is to deal with the influence of the uncertainty on the constraints, and its mathematical expression is as follows:
where
Changing the constraint in RDO to the probability constraint, RBRDO takes the influence of the uncertainty on the objective and the constraint into account simultaneously. Its mathematical expression is as follows:
Compared with MC method, PCE [
Constructing random output variables into a surrogate model with the randomness, for a random variable
where
In order to estimate the uncertainty of outputs, (
where the total number of PCE terms is
where
In summary, in order to do the multidimensional orthogonal polynomial chaos expansions, coefficients of polynomial chaos expansions terms
where
Higher statistical moments can be calculated similarly.
First, an input random variable is assumed as
where
The Sobol’ decomposition is unique under the condition that the integral of the expansion term for any one of independent variables is 0, as in
Equation (
The square integral of two sides of (
The left side of (
where
where
As mentioned in Section
Only when
Comparing (
It is now easy to derive sensitivity indices from the above representation. According to (
First, consider a simple polynomial function:
where the input variable
This numerical example is carried out for
In practice, the true form of the model output is unknown, thus, the order of PCE has to be chosen a priori. Thus 4 cases are considered; namely, the expansion order
Sensitivity indices of the polynomial model under different expansion orders.
Sensitivity indices | Analytical solution | Expansion order | |||||||
---|---|---|---|---|---|---|---|---|---|
| | | | ||||||
value | error | value | error | value | error | value | error | ||
| 0.2747 | 0.2752 | 0.2% | 0.2741 | 0.2% | 0.2747 | 0 | 0.2747 | 0 |
| 0.2747 | 0.2752 | 0.2% | 0.2741 | 0.2% | 0.2747 | 0 | 0.2747 | 0 |
| 0.2747 | 0.2752 | 0.2% | 0.2741 | 0.2% | 0.2747 | 0 | 0.2747 | 0 |
| 0.0549 | 0.0556 | 1.2% | 0.0552 | 0.5% | 0.0550 | 0.1% | 0.0549 | 0 |
| 0.0549 | 0.0556 | 1.2% | 0.0552 | 0.5% | 0.0550 | 0.1% | 0.0549 | 0 |
| 0.0549 | 0.0556 | 1.2% | 0.0552 | 0.5% | 0.0550 | 0.1% | 0.0549 | 0 |
| 0.0110 | 0.0091 | | 0.0109 | 1.0% | 0.0110 | 0.1% | 0.0110 | 0 |
As shown in Table
Then Sobol’ function is considered:
where the input variable
In this case,
Sensitivity indices of Sobol’ function (
Sensitivity indices | Analytical solution | PCE( |
---|---|---|
| 0.6037 | 0.6263 |
| 0.2683 | 0.2814 |
| 0.0671 | 0.0711 |
| 0.0200 | 0.0213 |
| 0.0055 | 0.0062 |
| 0.0009 | 0.0012 |
| 0.0002 | 0.0006 |
| 0.0000 | 0.0002 |
| ||
| 0.6342 | 0.6538 |
| 0.2945 | 0.3128 |
| 0.0756 | 0.0792 |
| 0.0227 | 0.0231 |
| 0.0062 | 0.0068 |
| 0.0011 | 0.0015 |
| 0.0003 | 0.0007 |
| 0.0000 | 0.0002 |
Sensitivity indices of Sobol’ function (
It can be seen that first 4 variables are the main influence on the output whether from
Sensitivity indices of simplified model of Sobol’ function with 4 variables under different expansion orders.
Sensitivity indices | Analytical solution | Expansion order | |||||
---|---|---|---|---|---|---|---|
| | | |||||
value | error | value | error | value | error | ||
| 0.6037 | 0.6334 | 4.9% | 0.6144 | 1.8% | 0.6085 | 0.8% |
| 0.2683 | 0.2778 | 3.5% | 0.2680 | 0.1% | 0.2683 | 0 |
| 0.0671 | 0.0685 | 2.1% | 0.0657 | 2.1% | 0.0666 | 0.7% |
| 0.0200 | 0.0203 | 1.5% | 0.0194 | 3.0% | 0.0198 | 1.0% |
| |||||||
| 0.6342 | 0.6334 | 1.3% | 0.6434 | 1.5% | 0.6412 | 1.0% |
| 0.2945 | 0.2778 | 5.7% | 0.2931 | 0.5% | 0.2967 | 0.7% |
| 0.0756 | 0.0685 | 9.4% | 0.0739 | 2.2% | 0.0756 | 0 |
| 0.0227 | 0.0203 | 10.6% | 0.0221 | 2.6% | 0.0225 | 0.9% |
| |||||||
Unknown coefficients | 35 | 126 | 330 |
Compared with analytical solutions, some conclusions can be drawn.
(1) When
(2) For the latter 2 cases, the model evaluations are 171 (45+126) and 375 (45+330), respectively, which is a significant reduction compared with the model of the 8th PCE with 8 variables (12870). Therefore, the simplified model can greatly reduce the computational cost of SA under the premise of ensuring the accuracy; therefore, this method can be applied into practical applications.
In conclusion, PCE-based Sobol’ indices have proven the efficiency and accuracy in SA. It is shown that the computation of Sobol’ indices after a proper expansion is precise enough while the computation cost is thus transferred to the obtention of the PCE coefficients, and the subsequent postprocessing being almost costless.
As mentioned in Section
Considering
The coefficient
Equation (
where
Equation (
Therefore, for
Roots of Hermite polynomials.
Expansion order | | Roots (collocation points) |
---|---|---|
1 | | |
2 | | |
3 | | |
4 | | |
For n-dimensional random variables, collocation points are the combination of roots of the next higher-order orthogonal polynomial. Then the number of all the available probability distribution points is
Input samples of PCM are selected according to some certain rules. Generally speaking, the number of sample points should be greater than the number of undetermined coefficients. Isukappali [
Flow chart of IPCM.
The expansion order, the distribution type, and the number of variables are determined.
The collocation points are arranged in the decreasing probability density (an ascending order according their distance from the origin). For instance, the point (
Then the sorted collocation points are selected one by one to establish the polynomial coefficient matrix row by row.
For the candidate of the (i+1)th collocation point, the (i+1)th row of the matrix should be linearly independent with the previous ith rows. Therefore, the rank of this matrix is calculated and judged whether it is equal to the row of this matrix. If it is equal, this candidate is preserved or otherwise abandoned. Then the point with next highest probability density should be tested.
The process does not end until the number of the row is equal to the number of undetermined coefficients.
When the number of random variables and the degree of PCE are both high, due to the multiple times of composing the polynomial coefficient matrix and calculating the rank of the matrix, the calculation cost of searching linearly independent points can be high as well after following the above steps. Fortunately, if the same basis polynomial chaos is adopted, for a given number of random variables and the order of PCE, collocation points for computing different outputs are the same, which means that linearly independent collocation points can be searched and saved in advance and can be used whenever you need to save the time of repeating searching the same collocation points.
To demonstrate the uncertainty quantification capability and illustrate the application of IPCM, a nonlinear nonmonotonic
where the input variable
The variation of the mean and the standard deviation of Y with the expansion order.
From Figure
Meanwhile, under the same expansion order, although the accuracy of PCM is slightly higher than that of IPCM, the number of collocation points required by IPCM is much less than that of PCM. As the order increases, for example,
As can be seen, the number of collocation points selected by PCM is greater than the number of undetermined coefficients. However, IPCM can maintain the accuracy by selecting a part of the points from collocation points generated by PCM. In order to further explain the relationship between the number of collocation points and the calculation accuracy, a part of collocation points generated by PCM, namely, 2 times, 3 times, and 4 times the number of undetermined coefficients, is selected to calculate the mean and the standard deviation by the regression method.
Tables
Comparison of the mean with the different number of collocation points.
Number of collocation points (N) | PCE | MC | |||
---|---|---|---|---|---|
| | | | (N=1000) | |
N=2P | 2.591 (25.6%) | 3.098 (11.1%) | 2.925 (16.1%) | 3.503 (0.5%) | |
N=3P | 2.614 (24.9%) | 3.112 (10.7%) | 3.441 (1.3%) | 3.478 (0.2%) | |
N=4P | 2.685 (22.9%) | 3.159 (9.4%) | 3.468 (1.3%) | 3.491 (0.2%) | 3.485 |
| |||||
Full rank | 2.912 (19.3%) | 3.279 (5.9%) | 3.472 (0.4%) | 3.489 (0.1%) | |
IPCM | 2.676 (23.2%) | 3.185 (8.6%) | 3.450 (1.0%) | 3.495 (0.2%) |
Comparison of standard deviation with different number of collocation points.
Number of collocation points (N) | PCE | MC | |||
---|---|---|---|---|---|
| | | | (N=1000) | |
N=2P | 2.786 (24.5%) | 3.352 (9.1%) | 3.016 (18.3%) | 3.743 (1.4%) | |
N=3P | 2.828 (23.4%) | 3.401 (7.8%) | 3.639 (1.4%) | 3.735 (1.1%) | |
N=4P | 2.953 (19.9%) | 3.414 (7.5%) | 3.645 (1.2%) | 3.728 (1.0%) | 3.691 |
| |||||
Full rank | 3.014 (18.3%) | 3.536 (4.2%) | 3.727 (0.9%) | 3.699 (0.2%) | |
IPCM | 2.711 (26.5%) | 3.421 (7.3%) | 3.728 (1.0%) | 3.725 (0.9%) |
In order to explain the above relationship between the calculation accuracy and the number of collocation points, Figure
Relationship between the rank of the polynomial coefficient matrix and the number of collocation points.
However, when
When using the regression method, the minimum number of collocation points that guarantee the full rank of the coefficient is 30(d=3), 84(d=5), 247(d=7), and 404(d=9). However, the number of collocation points that guarantees the full rank of the coefficient is 20(d=3), 56(d=5), 120(d=7), and 240(d=9) using IPCM. It can be seen that IPCM can reduce the number of collocation points required for the calculation greatly, thereby leading to a decline in the computational cost. The accuracy of the former is slightly higher than that of the latter; this is because the latter only uses a part of collocation points of the former. Nevertheless, compared with the exact solution, the result of IPCM can fully meet the requirement of the accuracy, and the computational efficiency is improved.
In summary, when selecting a part of collocation points generated by PCM to solve undetermined coefficients of the polynomial, the selected collocation points should satisfy the condition that the rank of the coefficient matrix should be equal to the number of the undetermined coefficient. However, for the high-dimensional high-order problems, complex uncertainty analyses, and simulations, a large amount of calculation is required. In these cases, IPCM has great advantages due to the fact that IPCM based on the linear independent principle only need the same number of collocation points as the number of undetermined coefficients, with an obvious decline in required collocation points compared with PCM.
The proposed method is applied to the ship UDO. The optimization model of the bulk carrier proposed by Hannapel S and Vlahopoulos N is considered. Details of this model can be found in [
The specific process is shown in Figure
Flow chart of ship uncertainty-based design optimization.
Preparation work: determine the optimal PCE order of original model, then perform the sensitivity analysis on uncertain variables and rank them according to their degree of influence on the objective and the constraint, fix the uncertain variable with less influence at the mean value, and reduce the dimension of the model. Redetermine the optimal PCE order for the simplified model.
Design variables and their ranges are determined.
A set of design variables, in other words, a bulk carrier scheme, is selected and transferred to the inner layer, the uncertainty analysis module.
According to the expansion order, the distribution type, and the number of uncertain variables, which are determined in advance, IPCM is used to select collocation points for uncertain variables.
PCE is used to perform the expansion of the objective and the constraint.
According to the design requirement, select one of 3 types of UDO. Then the mean and the standard deviation of the objective and the failure probability of the constraint are obtained and transferred to the outer layer, optimization module.
The optimizer judges whether the entire optimization process meets requirements of the iteration number. If the process reaches the iteration number, the optimization process ends. If not, step 3~step 6 are implemented again.
In order to give a baseline for the comparison for later results of uncertainty optimization, DO is done first. DO uses a multi-island genetic algorithm, with the subpopulation size being 10, the number of islands being 10, and the number of generations being 50. Design variables, constraints, and objective functions are presented in Table
Bulk carrier model.
Design variables | |
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Objective function | |
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constraints | |
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| |
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| |
| |
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| |
|
where
Then the uncertainty in parameters is introduced. The steel weight of the bulk carrier is given by
Equation (
Coefficient values of the objective and the constraint with different expansion orders.
It can be seen from Figure
Sensitivity indices of the objective and constraints.
| | | | |||||
---|---|---|---|---|---|---|---|---|
PCE | Sobol’ | PCE | Sobol’ | PCE | Sobol | PCE | Sobol’ | |
| 0.8692 | 0.8857 | 0.8842 | 0.8302 | 0 | 0 | 0.8507 | 0.8925 |
| 0.0615 | 0.0637 | 0.0595 | 0.0582 | 0 | 0 | 0.0649 | 0.0712 |
| 0.0002 | 0.0012 | 0.0002 | 0.0004 | 0.9999 | 0.9249 | 0.0023 | 0.0021 |
| 0.0136 | 0.0153 | 0.0128 | 0.0139 | 0 | 0 | 0.0148 | 0.151 |
| 0.0006 | 0.0005 | 0.0005 | 0.0005 | 0 | 0 | 0.0007 | 0.0006 |
| 0.9238 | 0.9159 | 0.9267 | 0.9451 | 0 | 0 | 0.9169 | 0.9237 |
| 0.1060 | 0.1155 | 0.0945 | 0.1151 | 0 | 0 | 0.1180 | 0.1244 |
| 0.0002 | 0.0003 | 0.0002 | 0.0002 | 0.9999 | 0.9249 | 0.0027 | 0.0031 |
| 0.0241 | 0.0230 | 0.0208 | 0.0314 | 0 | 0 | 0.0279 | 0.0293 |
| 0.0011 | 0.0020 | 0.0009 | 0.0012 | 0 | 0 | 0.0013 | 0.0021 |
Sensitivity indices of the objective and constraints.
According to
For this simplified model, the process of determining the optimal PCE order is repeated, and 3rd-order expansion is optimal for the model. The mean and the standard deviation of 3rd-order expansion are compared with results of MC method, as can be seen in Table
The mean and the standard deviation of the objective and the constraint.
the objective /the constraint | | | | | |
---|---|---|---|---|---|
Mean | MC (100) | 10.06 | 0.1699 | 3.408 | 0.309 |
PCE (20) | 9.92 | 0.1699 | 3.479 | 0.304 | |
| | | | | |
| |||||
Standard deviation | MC (100) | 4.21 | 0.017 | 8009 | 0.988 |
PCE (20) | 4.29 | 0.017 | 7968 | 1.021 | |
| | | | |
From results in Table
Then, according to the solved first 4 order moments of the constraints, maximum entropy method (MEM) is adopted to obtain PDF of constraints and then the failure probability is solved. Compared with MC method, the results are listed in Table
The failure probability of constraints.
Constraints | | | | |
---|---|---|---|---|
pf | MC (100) | 0% | 47% | 11.2% |
PCE (20) | 0% | 45.2% | 10.6% |
As shown in Table
Design variables, uncertain variables, constraints, and objectives of UDO.
Type of UDO | RDO | RBDO | RBRDO |
---|---|---|---|
Design variables | Design variables in Table | ||
| |||
Uncertain variables | | ||
| |||
| |||
| |||
Constraints | Constraints in Table | | |
| |||
| |||
| |||
Objectives | | | |
Optimization results.
Initial | DO | RDO | |||
---|---|---|---|---|---|
PCE(5D) | PCE(3D) | MC(3D) | |||
| 9.926 | 8.446 | 8.729 | 8.488 | 8.495 |
| - | 8.624 | 9.906 | 9.788 | 9.800 |
| - | 4.290 | 4.109 | 3.552 | 3.567 |
| 195.000 | 188.610 | 181.121 | 178.329 | 178.612 |
| 32.310 | 31.340 | 32.305 | 29.699 | 29.621 |
| 20.000 | 15.960 | 15.772 | 15.734 | 15.733 |
| 10.500 | 11.710 | 11.705 | 11.710 | 11.707 |
| 0.700 | 0.640 | 0.748 | 0.750 | 0.749 |
| 16.000 | 14.210 | 14.425 | 14.002 | 14.010 |
| 0 | 0 | 0 | ||
| 41.2% | 34.3% | 34.3% | ||
| 17.9% | 12.1% | 10.8% |
RBDO | RBRDO | |||||
---|---|---|---|---|---|---|
PCE(5D) | PCE(3D) | MC(3D) | PCE(5D) | PCE(3D) | MC(3D) | |
| 8.811 | 8.576 | 8.583 | 8.843 | 8.603 | 8.639 |
| 10.128 | 10.077 | 10.091 | 10.227 | 10.042 | 10.019 |
| 4.237 | 4.128 | 4.147 | 4.229 | 4.088 | 4.077 |
| 205.357 | 197.358 | 197.873 | 200.666 | 195.025 | 191.701 |
| 32.310 | 32.309 | 32.253 | 31.420 | 32.299 | 31.786 |
| 15.719 | 14.521 | 14.529 | 15.784 | 14.518 | 14.316 |
| 10.769 | 10.859 | 10.854 | 11.460 | 10.831 | 10.704 |
| 0.747 | 0.749 | 0.750 | 0.745 | 0.749 | 0.750 |
| 14.205 | 14.004 | 14.007 | 14.416 | 14.083 | 14.021 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0.5% | 0.4% | 0.1% |
| 0 | 0 | 0 | 0.6% | 0.3% | 0.1% |
Comparison of objective values in RDO process.
Comparison of objective values in RBDO process.
Comparison of objective values in RBRDO process.
Some conclusions can be drawn.
(1) As can be seen from Figures
(2) Table
(3) Taking the result of PCE as an example, compared with the initial design,
This paper proposed a sensitivity analysis based on polynomial chaos expansions and its application in ship UDO. Combined with the multiobjective genetic algorithm, an efficient UDO system is constructed and applied to the ship design, and some conclusions can be drawn:
(1) PCE-based Sobol’ indices are used to perform SA for the model and uncertain variables are ranked based on the indices to achieve the dimensionality reduction. Compared with MC method, the proposed method has proven the efficiency and accuracy in SA. Specifically speaking, the computation of Sobol’ indices after a proper expansion is precise enough while the computation cost is transferred to the obtention of the PCE coefficients, the subsequent postprocessing being almost costless.
(2) When using PCE, IPCM can give the optimal number of collocation points by comparing the rank of the coefficient matrix with the number of collocation points, which reduces collocation points greatly and overcomes the shortcoming, large computational complexity of traditional statistical methods.
(3) In the bulk carrier uncertainty-based design optimization, the perturbation of the objective and the constraint under the influence of multiple random variables are considered. By analyzing optimization results, first, it is necessary to take the influence of uncertainties into account where the deterministic optimal solution will violate the constraint, resulting in the failure of the solution. Meanwhile, in the optimization process, on the premise of maintaining the accuracy, the time of models after dimension reduction is significantly lower than that of original models; at the same time, the time of PCE is significantly lower than that of commonly used MC method, which shows more advantages in engineering applications.
However, the paper models all uncertain variables by using only probabilistic method and only discussed the application of the proposed method in the ship preliminary design optimization. Future work will use the probabilistic and nonprobabilistic methods to model uncertain variables according to their different characteristics and will not be limited to the ship conceptual design and will be carried out to guide the hull form design.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China [Grant nos. 51720105011, 51709213, 51609187, and 51479150] and Fundamental Research Funds for the Central Universities [WUT:2018IVB068].