Mathematical Problems in Engineering

Volume 2019, Article ID 7498526, 19 pages

https://doi.org/10.1155/2019/7498526

## Sensitivity Analysis Based on Polynomial Chaos Expansions and Its Application in Ship Uncertainty-Based Design Optimization

^{1}Key Laboratory of High Performance Ship Technology, Wuhan University of Technology, Ministry of Education, Wuhan, China^{2}School of Transportation, Wuhan University of Technology, China

Correspondence should be addressed to Haichao Chang; moc.361@374961163

Received 23 October 2018; Accepted 2 January 2019; Published 23 January 2019

Academic Editor: Krzysztof Puszynski

Copyright © 2019 Xiao Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2], including robust design optimization (RDO) and reliability-based design optimization (RBDO). Papanikolaou et al. [3] considered uncertainties related to ship’s seakeeping responses and wave induced loads and presented their recent advances in modelling the combined hydrodynamic responses of ship structures using cross-spectral combination methods and using uncertainty models for the development of modern decision support systems as guidance to ship’s masters. Recently, Plessas and Papanikolaou [4] developed the procedure which may be used as a Decision Support Tool (DST) for interested ship investors including a Life Cycle Assessment (LCA) model that takes into consideration uncertainties of the most dominant economic parameters in tanker ship operation to demonstrate the optimization of alternative hull/engine/propeller setups for a defined operational scenario of a tanker that includes ship’s operational profile in calm seas and representative weather conditions, as well as all relevant safety and efficiency regulatory and technical constraints. Hou et al. [5] took Autonomous Underwater Vehicle (AUV) as the research object, the resistance as the optimization objective, and the characteristic parameters of AUV as the uncertainty and then evaluated the performance of AUV with the 6 sigma design criterion to ensure the operation reliability of AUV under the environmental interference. After that, 4 different forms of uncertainty and their applications in the RDO were considered [6], where the standard Wigley ship model was taken as the research object and EEOI of the whole route was used as the optimization objective to verify feasibility and superiority of this method. The same method was also applied in the hull form optimization of a bulk carrier [7].

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 [8] and proposed a two-stage ship design optimization method considering the uncertainty [9]. Then they systematically demonstrated RDO, introduced an integral form uncertainty quantification method, and compared various robust optimization forms to verify the superiority of the method through a robust optimization example of a bulk carrier [10]. For the epistemic uncertainty, a Bayesian method was proposed to quantify this kind of uncertainty while the probability density function was still used to model the aleatory uncertainty [11]. Subsequently, he proposed a variable-accuracy metamodel-based architecture to improve the efficiency of multidisciplinary robust optimization design and applied it to the hydrostatic resistance optimization of DTMB5415 model [12]. In order to accurately establish the probability distribution of uncertain variables, Diez and his team obtained a large amount of operational data through a two-month data collection of five sister ships and derived a probability distribution of displacement and speed [13], where some modifications of a small portion of the hull were proposed in order to increase significantly the performances of the hull and decrease the operative cost of the ship.

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 [14], which belongs to the broader class of variance-based methods. These methods do not assume any kind of linearity or monotonicity of the model and rely upon the decomposition of the response variance as a sum of contributions of each input factor or combinations thereof, that is, partial variance. Various methods have been investigated for computing Sobol’ indices; among them, Monte Carlo (MC) method is still the most commonly used method to sample and estimate the integral in indices; however, a prominent drawback of this classical method is that it often requires thousands of model evaluations, which requires substantial computational resources and makes it impractical in terms of time consumption. Therefore, it is obviously infeasible to use MC method directly. In other words, a method as accurate as the MC method but with lower computational cost is desired.

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 [15] and applied to replace the traditional Sobol’ MC method to calculate Sobol’ indices. In the field of the stochastic modelling of subsurface flow and mass transport, PCE has been widely used and proved to be able to perform SA comprehensively and reliably at low computational cost. Fajraoui et al. [16] and Younes et al. [17] applied a PCE-based global SA to flow and mass transport in a heterogeneous porous medium and established the transient effect of uncertain flow boundary conditions, hydraulic conductivities, and dispersivities; in the radionuclide transport simulation in aquifers, Ciriello et al. [18] analyzed the statistical moments of the peak solute concentration measured at a specific location; Sochala and Le Maitre [19] considered uncertain effects of soil parameters upon 3 different physical models of subsurface unsaturated flow; Formaggia et al. [20] applied PCE-based sensitivity indices to evaluate the influence of the uncertainty in hydrogeological variables on a basin-scale sedimentation process; Deman G. et al. [21] combined the computation of Sobol’ indices with sparse PCE method to effectively analyze the influence of input parameter uncertainty of the ground water over the life cycle on model outputs.

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. [22] discussed the use of PCM to solve elliptic stochastic partial differential equations; Foo et al. [23] used PCM to study the three-dimensional problem with random loads and the spatial variability material; Li et al. [24] applied PCM to the simulation of the ground water seepage and the solute transport; Isukappali [25, 26] thought that the number of collocation points should be twice as many as the number of undetermined coefficients and then used the regression method to calculate the coefficients. Unfortunately, Jiang et al. [27] found that sometimes increasing the number of collocation points to more than 8 times the number of the undetermined coefficients still cannot maintain the calculation accuracy. In order to deal with the unbalance between the number of sample points and the calculation accuracy, an improved probabilistic collocation method (IPCM) based on the linear independence principle is adopted to 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 for solving polynomial coefficients of PCE, thereby improving the efficiency of the uncertainty-based optimization.

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 [28]. In contrast to the previous work, this paper uses Sobol’ indices based on PCE for dimension reduction, reduces the sample points by using an improved probability collocation method, and solves failure probability via maximum entropy method (MEM) and is organized as follows. In Section 2, a brief outline of uncertainty-based design optimization and its formulation are presented. Section 3 introduces PCE theory and summarizes the process of solving the polynomial; moreover, it provides concepts of SA with Sobol’ indices and the computation of those indices using PCE. In Section 4, the basic theory of PCM is introduced and its improvement based on the linear independence principle is given with numerical examples to verify its feasibility. Finally, the proposed method is applied to the uncertainty-based optimization of the bulk carrier preliminary design in Section 5, whereas final conclusions are shown in Section 6.

#### 2. Uncertainty-Based Design Optimization

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.

##### 2.1. Robust Design Optimization (RDO)

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 1, point 1 is the optimal point of deterministic optimization (DO) while point 2 is the optimal point of RDO. If the influence of uncertain factors is not considered, the objective function value of point 1 is minimum, which should be the optimal solution of objective function. However, when the design variable perturbs in the range of , the perturbation of objective function at point 1 violates constraints, resulting in failure of the product. The perturbation is obviously less than , and all the objective outputs are within the constraint. Therefore, the robust optimal solution is a relatively stable optimal solution.