A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures

Abstract

Purpose

In order to solve the problems faced by First Order Reliability Method (FORM) and First Order Saddlepoint Approximation (FOSA) in structural reliability optimization, this paper aims to propose a new Reliability-based Design Optimization (RBDO) strategy for offshore engineering structures based on Original Probabilistic Model (OPM) decoupling strategy. The application of this innovative technique to other maritime structures has the potential to substantially improve their design process by optimizing cost and enhancing structural reliability.

Design/methodology/approach

In the strategy proposed by this paper, sequential optimization and reliability assessment method and surrogate model are used to improve the efficiency for solving RBDO. The strategy is applied to the analysis of two marine engineering structure cases of ship cargo hold structure and frame ring of underwater skirt pile gripper. The effectiveness of the method is proved by comparing the original design and the optimized results.

Findings

In this paper, the proposed new RBDO strategy is used to optimize the design of the ship cargo hold structure and the frame ring of the underwater skirt pile gripper. According to the results obtained, compared with the original design, the structure of optimization design has better reliability and stability, and reduces the risk of failure. This optimization can also better balance the relationship between performance and cost. Therefore, it is recommended for related RBDO problems in the field of marine engineering.

Originality/value

In view of the limitations of FORM and FOSA that may produce multiple MPPs for a single performance function, the new RBDO strategy proposed in this study provides valuable insights and robust methods for the optimization design of offshore engineering structures. It emphasizes the importance of combining advanced MPP search technology and integrating SORA and surrogate models to achieve more economical and reliable design.

Citation

Yang, S., Meng, D., Guo, Y., Nie, P. and Jesus, A.M.P.d. (2023), "A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures", International Journal of Structural Integrity, Vol. 14 No. 5, pp. 809-826. https://doi.org/10.1108/IJSI-06-2023-0049

1. Introduction

Design optimization method based on safety factor is widely used in the design of maritime engineering structures (Motlagh et al., 2021; Amin et al., 2021; Gong et al., 2018; Liang et al., 2022; Liu et al., 2017, 2019; Hegseth et al., 2020; Zhu et al., 2021; Bhardwaj et al., 2023). Motlagh et al. (2021) performed design optimization based on safety factor for offshore jackets using genetic algorithms. Gong et al. (2018) used ANSYS software to establish the hydraulic skirt pile gripper model and optimize it. Then, they used the fuzzy matter-element method to select the optimal design. After optimization, the maximum equivalent stress is reduced by 14.75%, and the volume is reduced by 4.8%. Hegseth et al. (2020) used design optimization method based on safety factor to carry out integrated optimization on 10 MW spar floating wind turbine.

However, this method always relies on the experience of the designer. Lowering the safety factor can reduce manufacturing costs, but this may reduce lifetime safety. The safety factor can be set larger to increase the service life of the structure (Liu et al., 2022; Ai et al., 2022a; Bagheri et al., 2021). This results in higher production costs as material needs will increase. Furthermore, the main idea of the safety factor-based approach is a deterministic design and optimization (Cherid et al., 2021; Meng et al., 2023a). The optimal solution of deterministic optimization is often located on the constraint boundary. However, design variables are often subject to uncertainty (Ai et al., 2022b; Meng et al., 2020a; Wang et al., 2022; He and Deng, 2022; Ku et al., 2022; Wakjira et al., 2022a). Therefore, due to the influence of uncertain factors, the optimal design may violate the set constraints (Li et al., 2020; Xiao, 2021; Song and Xiao, 2022; Xue and Deng, 2022; Meng et al., 2021a; Wakjira et al., 2022b). In other words, the optimal design may not be able to complete the set function within the specified time. Reliability-based Design Optimization (RBDO) is an emerging optimization strategy that can fully consider variable and parameter uncertainty information (Yang et al., 2020). Recently, RBDO has attracted more and more attention in the maritime engineering structure design fields (Clark and DuPont, 2018; Stieng and Muskulus, 2020; Young et al., 2010; Meng et al., 2020b, c, 2021b, 2022a, 2023b, c; Song et al., 2011, 2021; Radfar et al., 2022). Young et al. (2010) developed a RBDO methodology for adaptive marine structures. The results proved that the RBDO method is more suitable for the design optimization of adaptive marine structures than the deterministic optimization method. Meng et al. (2020b) proposed a RBDO method based on saddle point approximation to improve the reliability of offshore structures. Meng et al. (2020c) proposed a decoupling strategy called sequential optimization and reliability assessment together with a collaborative uncertainty design and optimization model. Three instances demonstrate the application of this hierarchical strategy in contemporary distributed maritime engineering design workflows. Song et al. (2011) used the RBDO strategy to optimize the design of a riser support installed on a floating production storage and offloading. The solution obtained through RBDO strategy presented improved design performances under various riser operating conditions. Meng et al. (2023b) proposed a novel Kriging-model-assisted reliability-based multidisciplinary design optimization strategy to optimize offshore wind turbine tower.

At present, the methods of RBDO are mainly divided into three categories as shown in Figure 1: (a) Double-loop algorithm (Meng et al., 2022b); (b) Single-loop algorithm (Jeong and Park, 2017; Yang et al., 2022); (c) Decoupling algorithm (Du and Chen, 2004; Yu and Wang, 2019). The double-loop algorithm is the most basic and direct method, but its efficiency is the lowest. The single-loop method uses the K-K-T condition to omit the reliability analysis loop. Therefore, original double-loop optimization becomes single-loop optimization (Meng et al., 2022c). The solution efficiency is greatly improved. However, this method may fail when the initial design is far from the optimal point. The decoupling method continuously utilizes the Most Probable Point (MPP) information to convert the RBDO into a deterministic optimization. This method can better balance the relationship between efficiency and accuracy.

The First Order Reliability Method (FORM) is one of the widely used models for finding MPP (Zhang et al., 2022; Zhong et al., 2020; Zhao et al., 2020; Liu et al., 2023; Zhu et al., 2022; Xiong et al., 2021; Gao et al., 2022). In the FORM, first the random variables are transformed from the original space (X space) to the standard normal space (U space). Then, the MPP is defined as the shortest distance form performance function to the origin. However, this transformation of random variables may reduce optimization accuracy. Du and Sudjianto (2004) introduced First Order Saddlepoint Approximation (FOSA) to address this challenge. In the FOSA, the MPP is defined as the maximum joint probability density function point on the performance function surface. However, FOSA may provide multiple MPPs for a performance function. Zhu et al. (2020) proposed a novel probability model (Original Probabilistic Model (OPM)) for searching MPP in structure reliability analysis. The OPM can avoid the limitation of FORM and FOSA in structural reliability analysis. In current decoupling RBDO of maritime study, there is no RBDO algorithm that can avoid the limitation of multiple MPP points. Therefore, a novel RBDO strategy for maritime engineering structures is proposed based OPM and decoupling strategy in this study. Ship cargo hold and frame ring of underwater skirt pile gripper are two important components in maritime engineering structure. Therefore, two RBDO problems of these two components are used to illustrate the effectiveness of the proposed strategy.

The aim of this research is to review the general formulation of RBDO strategy and MPP search algorithm (see Section 2). Then, in Section 3, the proposed RBDO strategy for maritime engineering structures is explained in detail. Two maritime engineering structure case studies, including Ship cargo hold and frame ring of underwater skirt pile gripper, are given to prove the effectiveness of the proposed method in Section 4. Finally, the conclusions are summarized in Section 5.

2. Review of RBDO and MPP search strategy

In this section, some preliminaries, essential definitions, the general formulation of RBDO strategy and MPP search algorithm are briefly reviewed.

2.1 The general formulation of RBDO

The mathematical formulation of the RBDO is shown as follows (Meng et al., 2023c):

(1){minDV C(d,μX)s.t. Pif[gi(di,X)≤0]≤Pt,if i=1,2,...,ndL≤d≤dU,XL≤μX≤XUDV={d,μX}

where g(·) is the performance function; g(·)≤0 represents the occurrence of failure event; Pif[·] represents the probability that the failure event to occur; n represents the number of uncertain constrain; Ptf is target failure probability; C(·) is the objective function; DV is the design variables set; d is the vector of deterministic design variables; X is the vector of random design variables; μ represents a vector of mean values of random variables; the superscripts L and U denote the lower and upper bounds of the design variables, respectively.

Sequential Optimization and Reliability Assessment (SORA) is one of the most popular decoupling strategies. SORA uses shifting vector s to move the deterministic constraint continuously toward the feasible region as shown in Figure 2.

For the kth cycle, the shift vector s(k) is calculated as follows (Du and Chen, 2004):

(2)s(k)=μx*(k−1)−xMPP,(k−1)

where μx*(k−1) is optimal solution of the (k-1)th deterministic optimization; xMPP,(k−1) is the MPP of the (k-1)th cycle. It is worth noting that when k = 1, s(k)=0.

After shifting vector is acquired, the kth deterministic optimization is shown as follows (Du and Chen, 2004):

(3){minDV C(d(k),μX(k))s.t. gi(d(k),μX(k)−s(k))≥0 i=1,2,...,ndL≤d(k)≤dU,XL≤μX(k)≤XUDV={d(k),μX(k)}

2.2 MPP search algorithm

In FORM algorithm, the MPP is defined as the shortest distance form performance function to the origin. The MPP can be found by the following optimization problem (Meng et al., 2015):

(4){Find UMPPmin L(U)=‖U‖=UTU=[(X−μxσx)T(X−μxσx)]s.t. g(U)=0

where UMPP is MPP in the U-space; U=(u1,u2,...,um) is the standard normal variable vector, which can be acquired from the original random variable, i.e. the original random variable space (X-space) is transformed into a standard normal space (U-space); L(U)=‖U‖ represents the modulus of vector U which can be calculated as follows:

(5)‖U‖=u12+u22+⋯+un2=∑i=1mui2

In FOSA algorithm, the MPP is defined as the maximum joint probability density function point on the performance function surface. The MPP can be found by the following optimization problem (Du and Sudjianto, 2004):

(6){Find XMPPmax L(X)=∏i=1mfX(xi)s.t. g(X)=0

where XMPP is MPP in the X-space found by FOSA algorithm. fX(·) represents the Probability Density Function (PDF) of the random variable.

In the standard normal space (U-space), the joint probability density function given as follows:

(7)∏i=1m12πexp(−12ui2)=12πexp(−12∑i=1mui2)

Therefore, minimizing ∑i=1mui2 is equal to maximizing ∏i=1m12πexp(−12ui2). In other words, the core ideas of FORM and FOSA are the same. However, there is a flaw in this idea. FORM and FOSA may provide multiple MPPs for a single performance function. For example, let's consider a performance function g(X)=−0.5⁡cos(x) and assuming that random variable x obeys standard normal distribution. Then, for FORM and FOSA, the MPP is acquired by Eq. (8). By calculation, the MPPs of this performance function are −π2 and π2, respectively. The location of MPPs is shown in Figure 3.

(8){Find XMPPmax L(X)=exp(−x22)2πs.t. g(X)=−0.5⁡cos(x)=0

From Figure 3 it can be seen that there are two MPPs for this one performance function, which is unreasonable. In order to address this challenge, Zhang et al. (2022) defined MPP as the point corresponding to the largest cumulative distribution function in the failure domain by theoretical derivation and experimental verification. The schematic diagram of finding MPP is shown in Figure 4. They named this method OPM. The underlying mathematical model of this method is shown in Eq. (9) (Zhu et al., 2020).

(9){Find XMPPmax L(X)=∏i=1mFX(xi)s.t. g(X)=0

where FX(·) represents the Cumulative Distribution Function (CDF) of the random variable.

The above performance function (g(X)=−0.5⁡cos(x)) and random variable information (x∼N(0,1)) are used as examples to verify the effectiveness of OPM. According to Eq. (9), the mathematical model of this problem is defined as follows:

(10){Find XMPPmax L(X)=Φ(x)s.t. g(X)=−0.5⁡cos(x)=0

By calculation, the MPP of this performance function is π2. The location of MPP is shown in Figure 5. From Figure 5 it can be seen that there is one MPP for this one performance function. Therefore, the OPM can effectively avoid the shortcomings of FORM and FOSA.

3. The proposed RBDO strategy for maritime engineering structures

In this section, the proposed RBDO strategy for maritime engineering structures is explained in detail. The flowchart of the proposed approach is shown in Figure 6. The specific calculation steps are explained as follows:

• Step1: The mathematical model of RBDO is determined for the optimization problem of engineering structure. First, determinate target failure probability, design variables and their uncertainty information. Second, the optimized objective function and performance function should also be defined. Final, according to Eq. (1), the mathematical model of RBDO is determined for the optimization problem of engineering structure.

• Step2: A surrogate model is used to approximate the performance function and the objective function, because the performance function or optimization objective function of marine engineering structures is often implicit. In iterative optimization, the output response of each design variable is often obtained through experiments or Finite Element Analysis (FEA), which is a very time-consuming process. Therefore, surrogate models are usually used to approximate actual response (Liu et al., 2021; Li et al., 2021; Chen and Deng, 2022; Luo et al., 2022a, b; Yang et al., 2023a, b; Ling et al., 2022). Currently, three popular surrogate models are the second-order polynomial response surface (Shanock et al., 2010; Wang et al., 2023), the Kriging model (Yu and Li, 2021; Teng et al., 2022), and the artificial neural network (Lazakis et al., 2018). The basic process for building a surrogate model is the same. First, Design of Experiment (DoE) is used to generate a set of initial training samples. The output responses of these samples are generated by experiments or FEA. Second, these samples and responses are used to build an initial surrogate model. Third, an update strategy for surrogate model is used to choose an update sample. This sample is the point where the accuracy of the model can be improved the most. Surrogate model is rebuilt by initial samples and a new update sample. Finally, when the model satisfies the stop criteria, the accuracy of this surrogate model is adequate for the RBDO of the maritime engineering structures. In this study, Kriging model is adopted in two example studies.

• Step3: The RBDO problem is transformed into a deterministic optimization problem. First, the MPPs of each performance function is found by the OPM. The detailed search process is shown in Section 2.2. Second, for the kth cycle, the shift vector s(k) is evaluated by Eq. (2). Third, the RBDO problem is transformed into a deterministic optimization problem by Eq. (3).

• Step4: Solve deterministic optimization problems. Deterministic optimization problems usually have two types of solutions, i.e. gradient-based optimization algorithms, and intelligent optimization algorithms, such as genetic algorithm, particle swarm algorithm, simulated annealing algorithm among others. Because the intelligent optimization algorithm has efficient global convergence, it is widely used in this RBDO field (Deb et al., 2009; Chakri et al., 2018; Meng et al., 2021c; Safaeian Hamzehkolaei et al., 2016). Therefore, this method is also used in this study. After obtaining the optimal solution of the kth cycle, the judgment of convergence is carried out. The judgment of convergence is defined as Eq. (11):

(11)|Ck(·)−Ck−1(·)|≤δC

where δC represents allowable error; Ck(·) and Ck−1(·) represent the objective function of the optimal designs for the kth and (k-1)th cycles, respectively; if judgment of convergence and constraints are not satisfied, the cycle goes back to Step 3. If judgment of convergence and constraints are satisfied, the algorithm outputs the optimal design.

4. Example studies

In this section, two RBDO of maritime engineering structures are used to illustrate the effectiveness of the proposed strategy.

4.1 Ship cargo hold structure

A schematic diagram of the ship cargo hold structure is shown in Figure 7.

In Figure 7, thickness of the plate ti(i=1∼16) represents 16 design variables. They all obey a normal distribution N∼(μti,0.02μti). In this optimization problem, the total weight of the cargo compartment is the optimization target. The performance function is the maximum equivalent stress of each plate σmax⁡,j and the yield strength of the corresponding part σs,j. The calculation condition about maximum equivalent stress is that the ship leaves the port with full load. Therefore, the RBDO model of ship cargo hold structure is shown as follows:

(12){minDV m(μti)s.t. P1f[g1=σmax,1(μti)−σs,1≤0]≤Pt,1fP2f[g2=σmax,2(μti)−σs,2≤0]≤Pt,2fP3f[g3=σmax,3(μti)−σs,3≤0]≤Pt,3fP4f[g4=σmax,4(μti)−σs,4≤0]≤Pt,4fP5f[g5=σmax,5(μti)−σs,5≤0]≤Pt,5fP6f[g6=σmax,6(μti)−σs,6≤0]≤Pt,6fti>0 (i=1∼16)DV={μti}

The optimization results of the proposed method are shown in Table 1. It can be seen from Table 1 that the optimized volume is 2.63×108mm3. Compared with the original design, the volume is reduced by 3.38%.

4.2 A frame ring of underwater skirt pile gripper

The underwater skirt pile gripper is a very important equipment in the construction of the offshore jacket platform. Frame ring is an important part of skirt pile gripper. Their actual schematic is shown in Figure 8. The schematic diagram of the geometric structure of the frame ring is shown in Figure 9.

In Figure 9, D1 represents small diameter of annular hole; D2 represents large diameter of annular hole; D3 represents the inner diameter of frame ring; D4 represents the outer diameter of frame ring; H represents the height of frame ring (Yang et al., 2018).

When the underwater skirt pile gripper is working normally, the reaction force of the pile to the gripper acts on the annular surface of the opening of the frame ring. In other words, the circumferential direction of the opening of the frame ring is subjected to the pressure Ps of the pipe pile as shown in Figure 10. The original frame geometry and finite element model are shown in Figure 11.

In this study, the volume of frame ring V is used to define the objective function. The material of the frame ring is high strength shipbuilding steel DH36. Its yield strength σs is 355 MPa. The performance function is the difference between the maximum equivalent stress experienced by the frame ring and the yield strength. The target failure probability Ptf is 10−4. The design variables are the five geometric parameters of the frame (D1,D2,D3,D4,H). Their uncertainty information is shown in Table 2.

The RBDO model for a frame ring of underwater skirt pile gripper is shown as follows:

(13){minDV V(μD1,μD2,μD3,μD4,μH)s.t. Pf[σmax(μD1,μD2,μD3,μD4,μH)−σs≤0]≤PtfD2>D1,D4>D3D1,D2,D3,D4,H>0DV={μD1,μD2,μD3,μD4,μH}

The optimization results of the proposed method are shown in Table 3. It can be seen from Table 3 that the optimized volume is 2.63×108mm3. Compared with the original design, the volume is reduced by 5.05%. The failure probability of the optimization design is greatly reduced. Therefore, the reliability of the frame ring is improved. When using traditional FORM method (Du and Chen, 2004) to solve MPP, the optimal solution is oscillating. Therefore, the RBDO method based on the traditional FORM cannot get the optimal design scheme. The stress and strain distribution cloud diagram of the optimal design is shown in Figure 12. It can be seen from the figure that the maximum equivalent stress of the frame ring is 273.87 MPa, which meets the strength requirement.

This optimal solution has significant value for two reasons. First of all, the design of the frame circle has fully considered the influence of geometric uncertainty factors. Such a design will improve the reliability and stability of the pile gripper and reduce the risk of failure, thereby reducing maintenance costs and potential safety hazards. Secondly, the proposed RBDO method balances the relationship between performance and cost during the optimization process, which makes the design of the frame ring more economical while meeting the engineering requirements. This optimization result helps enterprises to reduce production costs and improve market competitiveness.

5. Conclusion

In this study, a novel RBDO strategy for maritime engineering structures has been proposed, incorporating a new probabilistic model for searching the MPP. This approach overcomes the limitations of FORM and FOSA that may yield multiple MPPs for a single performance function. By employing SORA and surrogate models, the efficiency of solving RBDO problems is significantly enhanced.

The application of this innovative technique to other maritime structures has the potential to substantially improve their design process by optimizing cost and enhancing structural reliability. The case studies presented demonstrate the effectiveness and practical implications of the proposed strategy for maritime engineering structures. Key findings from this research that stand out and have a significant impact on future research include the successful implementation of the novel MPP searching method, which addresses the challenges faced by FORM and FOSA.

The proposed RBDO strategy provides a systematic and scientific design approach for marine engineering structures. Although this paper only illustrates the application of the proposed method to the ship cargo hold structure and frame ring of underwater skirt pile gripper, this method can be extended to the design and optimization of other similar structures. This will contribute to the development and technological progress of the entire industry.

In conclusion, the novel RBDO strategy presented in this study offers valuable insights and a robust approach for optimizing the design of maritime engineering structures. It highlights the importance of incorporating advanced MPP searching techniques and integrating SORA and surrogate models to achieve more cost-effective and reliable designs. Future research should focus on extending the proposed method to account for variable-dependent effects, as the current study is limited to uncorrelated random variables. This extension will further enhance the applicability and impact of the proposed strategy in the field of maritime engineering.

Figures

Figure 1

Conventional RBDO solution method

Figure 2

The diagram of the constrained movement

Figure 3

MPP location

Figure 4

The schematic diagram of finding MPP by OPM

Figure 5

MPP location

Figure 6

The flowchart of the proposed RBDO strategy for maritime engineering structure

Figure 7

A schematic diagram of the ship cargo hold structure

Figure 8

The actual schematic of underwater skirt pile gripper

Figure. 9

The schematic diagram of the geometric structure of the frame ring

Figure 10

Frame circle force diagram

Figure 11

The original frame geometry and finite element model

Figure 12

The stress and strain distribution cloud diagram of the optimal design

Table 1

The optimization results of the frame ring

Design variable	Original design	Optimization design	Design variable	Original design	Optimization design
t1 (mm)	18	18.51	t10 (mm)	15	13.07
t2 (mm)	18	20.17	t11 (mm)	15	13.34
t3 (mm)	16	17.63	t12 (mm)	12	10.13
t4 (mm)	14	15.07	t13 (mm)	12	10.41
t5 (mm)	9	8.11	t14 (mm)	12	11.44
t6 (mm)	12	9.73	t15 (mm)	10	8.47
t7 (mm)	12	10.39	t16 (mm)	7	6.83
t8 (mm)	12	10.64	m (t)	487.413	470.934
t9 (mm)	15	13.99

Source(s): Author's own creation/work

Table 2

Uncertainty information of design variables

Design variable	Distribution type	Mean	Standard deviation
D1	Normal	μD1	0.01 μD1
D2	Normal	μD2	0.01 μD2
D3	Normal	μD3	0.01 μD3
D4	Normal	μD4	0.01 μD4
H	Normal	μH	0.01 μH

Source(s): Author's own creation/work

Table 3

The optimization results of the frame ring

Design variable	Original design	*a Du and Chen (2004)	*b
D1 (mm)	305	–	301.11
D2 (mm)	350	–	337.99
D3 (mm)	2190	–	2227.43
D4 (mm)	2330	–	2349.64
H (mm)	700	–	683.27
V (mm3)	2.77×108	Not converge	2.63×108
Pf	0.005	–	6×10−5

Note(s): *a: The optimal design by FORM method

*b: The optimal design by the proposed method

Source(s): Author's own creation/work