An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures

When carrying out robust design optimization for complex engineering structures, they are computed by finite element software and are always computation-intensive. Aim at this problem, the paper proposes an efficient integrated framework of Reliability-based Robust Design Optimization (RBRDO). Firstly, the conventional RBRDO problem is changed as percentile form, that is, the improved percentile formulation of computing the objective robustness and probabilistic constraints is presented by resorting to the employment of Performance Measure Approach (PMA). Secondly, the above improved RBRDO problem is simplified by a series of new approximation methods due to the need of reducing computation. An efficient approximation method is proposed to estimate PMA functions of the RBRDO formulation. Based on it, the above improved RBRDO problem can be transformed into An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 3 a sequence of approximate deterministic sub-optimization problems, whose objective function and constraints are expressed as the approximate explicit form only in relation to the design variables. Furthermore, use the trust region (TR) method to solve the above sequence of sub-optimization. Lastly, several examples are used to demonstrate the effectiveness and efficiency of the proposed method. PACS(optional, as per journal): 75.40.-s; 71.20.LP


Introduction
Reliability-based Robust Design Optimization (RBRDO) as the advanced method is wide-spread used in different fields of engineering applications (Fran et al., 2020;Gang et al., 2016;Hao et al., 2019;Panzeri et al., 2018;Papadimitriou & Papadimitriou, 2016;Tsao & Thanh, 2020). In RBRDO, the objective function or constraint function usually includes probability factors to consider the influence of uncertainty. In the probabilistic framework, the typical RBRDO problem is formulated as follows: is the vector of controllable arbitrary design variables, whose mean value  X is controllable.
is the vector of wild arbitrary design parameters or noise factors, whose mean value is uncontrollable. n is the number of the constraints. Prob(·) refers to the probability of the jth constraint. j F p alludes to the failure of probability of the jth constraint. Superscripts "L" and "R" denote lower and upper limits, respectively.
Generally, the past literature on RBRDO can be classified into three different formulation approaches (Rathod et al., 2013). The first is the moment-based RBRDO formulation (Javed et al., 2019). It describes the robust design quality by using the statistical moments of a performance function (e.g. mean and variance). The second is the percentile difference-based RBRDO formulation (Castaldo et al., 2018). It obtains the percentile performance differences by comparing the right and left tails of the performance distribution and use it to approximately evaluate performance variation.
An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 5 Then it brings such performance variation into the objective function to describe the robustness. The third is hybrid quality-loss-functions-based RBRDO formulation (Ali et al., 2016). It endeavors to obtain the trade-off solutions that satisfy several constraints (objectives) to the extent possible. For all of the above methods, the uncertainties are quantified by conducting experimental design or Taylor series expansion or Monte Carlo simulation (MCS) and its variation-reduced methods. For most practical engineering problems, large-scale MCS and its variation-reduced methods are wildly used since they are simple and universal. However, MCS and its variation-reduced methods require to compute time-consuming functions for many times. Hence, serious computational obstacles are harmful to real-life engineering applications. Eq.
(1) shows that RBRDO is an double-loop optimization. The outer loop optimizes the design vector d , while the inner loop determines the uncertainty of the objective function and constraints. In most engineering applications, each performance function always should be computed by large engineering software, e.g. the finite element software, and each time of computation is time-consuming. Therefore the repeated computation of performance functions required by MCS and its variation-reduced methods in the inner loop is more computation-intensive, let alone such repeated job of the inner loop due to the loop of the outer optimization. Aiming at this problem, scholars have put forward relevant research methods. Gu and Lu(Gu & Lu, 2014) performed reliability-based robust design for occupant restraint system. They built the Support Vector Regression (SVR) surrogate by using the Optimal Latin Hypercube Sampling Technique (OLHST) and used the surrogate model to replace actual simulation model during the optimization process. Sun et al. (Sun et al., 2011) carried out crash worthiness design of the vehicle by using robust optimization. They used the OLHST and the dual An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 6 response surface method to obtain the surrogate model, which were also used for robust optimization. Zhu (Zhu et al., 2009)  fuel (Doh et al., 2018). Gao et al. established the explicit mapping relationship between the casting process parameters and the casting quality by using the double-layer Kriging method and used the explicit model for relieving the computation when solving the robust optimal process parameters (Gao et al., 2020 Hence, the above two problems can be solved to a large extent.
The paper is organized as follows: Section 2 introduces the basic theory of RBRDO and presents an improved RBRDO formulation. Section 3 presents a series of approximation methods for RBRDO. In Section 4 analyzes a benchmark test and two practical engineering problems to show the effectiveness and efficiency of the presented method. Section 5 concludes the paper.

The formulation of the RBRDO
An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 8

The general formulation of the RBRDO
Generally, Eq. (1) is usually written in the form of the Weighted Sum (WS): where F  and F  denotes the mean and standard deviation of the performance function ( , , ) F d X P , respectively.
[0,1]   is the weighting factor.  and  are normalization In order to apply the presented method, the deterministic design vector d is assumed as normal random one with its variations limited in very narrow boundaries. σ σ σ σ , respectively. For the sake of focusing on stating the method introduced in this paper, all random variables in U are assumed as independent and normal. As for correlated arbitrary random variables , they can be transformed into An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures normal independent ones by Rosenblatt transformation (Rosenblatt, 1952 In U-space (the independent and standardized normal space), U denotes the standardized normal vector u and the relationship between the components in U and those in u are as follows: ( ) / 1,..., , 1,..., , 1,..., Then Eq. (3) can be further changed to: large. In the following paper, an efficient integrated framework of RBRDO is presented.
It attaches importance to greatly reduce the repeated job of building the surrogate models by the sampling simulation.

Improvement of the RBRDO formulation
For the convenience of applying the approximate methods for RBRDO, PMA is used to improve the RBRDO formulation. PMA can be used to describe the probabilistic uncertainty of the performance function. It solves the following optimization problem (Hao et al., 2019)(see Eq. (6)) to judge whether a design vector meets the probabilistic constraint corresponding to a given reliability index j  : An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures min ( ; )  By resorting to PMA, fp1 and fp2 can be represented as: s.t: For the sake of convenient statement in the following paper, fp1, fp2 and * ( ; ) j j g u V are called PMA function, since they are obtained by PMA.
There are several advantages of using the PDM (i.e. fp2-fp1) to replace the general standard deviation f  in Eq. (3) for robustness assessment (Du et al., 2004), but it's not applicable to the case where the performance function is not monotonic (Lee et al., 2008). For example, if F(u) = u 2 [u ~ N(0, 1)] and f  is 3, then two MPPs become 1.732 and -1.732. Thus, the percentile performance difference (i.e. fp2-fp1) is 0. To assessing the robustness of the performance function f, the following improvement of PDM takes into account three characteristic values, i.e. fp0, fp1 and fp2, as shown in Eq. (10).
Through over steps, the RBRDO form of Eq. (5) can be changed as: An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures * ( ; ) Using penalty function method, Eq. (11) can be further formulated to: where  is a penalty factor.
By slightly modifying the procedure of Fig

Approximate estimation of the PMA function
In Eq. (12), it necessitates to work out the MPP to obtain the response value of the PMA functions, i.e. fp1, fp2 and * ( ; ) j j g u V . As for the arbitrary PMA function K , the MPP on the disappointment surface in U-space is computed as follows (Ang & Tang, 1984;Yang et al., 2020): An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures where * i u is the chid element of * u , and 2 * * According to Eq. (4) , / i K u   can be rewritten as : Combining Eq. (4) and Eq. (14), Eq. (13) can be transformed as Eq. (15) within the original design space (Ang & Tang, 1984): (12) and Eq. (15), the realization value of the design vector V is corresponding to a specific value of the MPP * U . Hence, * U is changed with the updated V during the optimization process. To obtain the MPP * U , an iterative strategy (Rackwitz & Flessler, 1978) is required since it is within the both sides of Eq. (15). But by doing so, it increases the computation times of the PMA function.
Instead of iteratively computing the MPP * U , the paper approximates it by using the mean point: In order to compute / An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures , c is a small value and the smaller the better.
Resorting to Eq. (17), / i U K    can be expressed in the finite difference form as: It is worth noticing that when the PMA function * ( ; ) K U V is constant, monotonously increasing, or monotonously increasing, the above method can precisely approximate its real value. However, if the PMA function is non-monotonical, it is difficult to search the MPP near the saddle points and the approximate value of the PMA function * ( ; ) K U V has a larger error. This disadvantage is ubiquitous for all MPP-based methods (Shan & Wang, 2008). As a result, when using the above method, it should be given full attention for the non-monotonical cases. An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures where the components of / K  V is derived as follows: With the help of Eqs. (17)- (20), the PMA functions in Eq. (12) are expressed as the explicit function only in relation to the design vector V . By using such explicit function, the computation times for the performance function can be greatly decreased.
To this end, the sequential approximate optimization for the RBRDO is presented.

Sequential approximate RBRDO
In order to ease the frequent update of the design vector during the optimization process, a new sequence of approximate optimizations is suggested to replace the original optimization and the TR method is used to ensure the solutions of the new sequence of optimizations to converge to the original ones.
In the optimization process, a sequence of approximate optimizations is constructed. At the kth iterative step, the sequence optimization for Eq. (11) can be written as: An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures where r f  , k δ and k V are the mean values of f , the TR radius vector and the current design vector at the kth iterative step, respectively. k δ changes with the iterative step. If k δ is a small enough interval, the updated V is in the vicinity of k V . Then it is possible to use the local explicit function suggested in Section 3.1 to approximate the PMA values in Eq. (21). The following approximate optimization model for Eq. (21) is changed as: where ~( ) p H V (termed as "approximate penalty function") is computed also based on the above approximate PMA functions. And particle swarm optimization (PSO) is An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures used to solve Eq. (23) in this paper, since it possesses a fine global convergence performance and has been widely used in an amount of literature (Dimopoulos, 2007;Praveen & Duvigneau, 2009).
PSO updates the current position of each particle in the swarm by adjusting the velocity vector:  (Wu et al., 2011) for more information about the PSO. Since Eq. (23) is the deterministic optimization and the actual model simulation is avoided, the computational effort required by PSO is acceptable.
In order to ensure the approximate accuracy between Eq. (12) and Eq. (23), the side constraints for the design vector V in Eq. (23) should be modified at each optimization cycle according to certain strategy. In this paper, the TR method as the strategy is used.
At the kth iterative step, it adjusts k δ by the criterion, that is current optimization in Eq.
(23) has good approximation to the original optimization problem.
Such good approximation is quantified by the approximate degree of the penalty function in this paper. Hence, the trust index containing the information of the penalty function is defined as follow: An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures where *k V is the optimum for the kth approximation optimization problem in Eq. (23).
In Eq. (26), the approximate penalty function ~p H can be obtained by solving Eq. (23).
And it also necessitates the computation of p H , which denotes the actual value of the penalty function (termed as "actual penalty function") and is always unknown and implicit for most of the complex engineering problems.
In general, there are many TR methods with respect to the update of the TR radius for sequential approximate optimization (Conn et al., 1988). They are widely used in engineering optimization due to their good convergence. In this paper, the updating strategy shown in Eq. (27) is used (Jiang et al., 2008;Rodriguez et al., 1998): Once the inner loop in Eq. (23)   It is worth mentioning that the computed value of the penalty function is approximate rather than real in a rigorous sense. In spite of the above difference, this approximation is precise enough. Hence, it is still called "actual penalty function"，distinguishing from the "approximate penalty function". A design problem of beam is shown in Fig. 6 (Wang, 2003). The random vector is The design problem is to minimize the vertical deflection on the premise of satisfying the cross-section area and stress constraints: where min ( , ) F d X represents the minimization the vertical deflection, 1 ( , ) G d X denotes that the cross-section area is no more than 300 cm 2 , 2 ( , ) G d X is the stress constraint.
Correspondingly, the maximum generations and the population size for PSO used in the general method are intended to be 100 and 30.

Convergence performance
Next job is to test cases with different starting design points and different TR radii for analyzing the convergence ability of the proposed method. First of all, analyze four cases with different starting design points and the same TR radius. In Table 2, the TR radius of the four cases is specified as one 16th of the design space, i.e. (4.375, 2.5).
The iterative optimization process is shown is  In the following section, four cases with different starting design points and the same TR radius are also studied. The corresponding radii is one half, one 4th, one 8th and one 16th of the design interval, respectively. The starting design point is specified as (10, 10). Table 3 lists the optimization results. Fig. 8 shows the iterative trajectory of the solutions. It can be found that the convergence trajectories of these four cases quickly converge to the same optimum P0, i.e. (80.00, 50.00), though the TR radii are different. In conclusion, the optimization results converge quickly with high speed, though the starting conditions are quite different. And it also shows that the convergence ability of the proposed method is robust.

Optimization efficiency
In this test, it is supposed that the objective function and the constraint functions in For the outer PSO, the total generated number of the particles is 3000, since 30 particles with 100 iterations are carried out. For each particle, being a trial design vector, it needs to called the inner PSO fourth to calculate four PMA functions, and each particle also requires 3000 times of calculating the actual simulation models. Hence, the total computation times of the actual simulation model is 3000×3000×4 = 3. times of the PMA function is 480 for 20 iterative cycles. Hence, the ratio of the calculation scale for the above two methods is 3.6×10 7 to 480. In a word, the proposed method is quite efficiency by greatly reducing the computation times of the actual simulation model, and it indicates that such advantage contributes to widespread engineering application.

Analysis of the uncertainty level
In the following paper, influences of the uncertainty level are examined. In the following paper, the RBRDO with different uncertainty levels of the design variables and random variables are carried out, respectively.
An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures

Analysis of the uncertainty level of the design variable
As mentioned above, the design variables are treated as random ones in RBRDO.
According to Ref. (Yu et al., 2013), the uncertainty levels of the design variables should be small enough, or the optimization accuracy can't be satisfied. In Table 4, cases with nine different quantitative levels for variation range of the design variables are examined. Meanwhile, the starting design point and the TR radius of the above cases are specified as (10, 10) and (4.38, 2.5), respectively. As showed in Table 4, the optimum results are inaccurate, when the variation range ξ is not small enough, i.e.

Analysis of the uncertainty level of the random variable
In Table 1, the ratio between the variation range of random variables and their mean values is ±10%. Besides, the other cases with the ratio being ±20%~±90% are also investigated. The starting design points and the trust radius vectors of the above cases are set as (10, 10) and (4.375, 2.5), respectively. Obviously, the larger the variation An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures range is, the smaller the possibility of the existing optimum that satisfies the constraints will be. In order to make sure that there are optimums for all of the cases, the constraints * 1 1 ( ; ) g u V and * 2 2 ( ; ) g u V in Eq. (29) are relaxed by trial and errors, and they are equivalently corresponding to the first term and the second term in Eq. (31), respectively.
For comparisons, Table 5 shows the optimization results computed by both the proposed method and the general method. Apparently, when the uncertainty level is less than ±60%, the deviations from the precise optimum are small and acceptable.
However, with increases of the uncertainty level, the deviations are unreliable. Table 5 Optimization results under different uncertainty levels of the random variables. 4.2Application 1 Fig. 9 A 72-bar truss.
The robust optimization design of a 72-bar truss is considered in Fig. 9. In this structure, a concentrated load being 50kN is acting on the point A along the xcoordinate. The maximum allowable stress and displacement for all bars should not exceed 150MPa and 0.003m, respectively.  The design problem is the minimization of the structural weight with the limitation of the stress constraint, displacement constraint and fundamental frequency constraint.
That is, The structure is modeled by using the FEM, and the solution of the stress and the vertical displacement constraints in Eq. (32) must call FEM. Here the computation times of the FEM is concerned. The factors  , , ,  , f  , j  and c are specified as 0.5, 1, 1, 10 5 , 3 ,1.28 and 0.001, respectively. Maximum generations and the population size for PSO are specified as 100 and 30. The starting design point is specified as (15, An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 15, 5) and the maximum iterative number is specified as 20 for the sequential optimization cycle.   Fig. 10 Iteration history of the 72-bar truss.

Application 2
This example considers an optimization of the piezoresistive pressure sensor. As for this sensor shown in Fig. 11, a silicon membrane is bonded on a glass support. And a piezoresistor (PZT) is pasted on the silicon membrane for converting the mechanical stress into a voltage. Then the pressure acted on the silicon membrane can be obtained by measuring the voltage of PZT.

Iterative step
Penalty function Fig. 11. Schema of pressure sensor. Fig.12 The main sizes of pressure sensor.  Table 8.
The device is also modeled by using an axisymmetric finite element model (FEM) and its maximum stress, the maximum displacement and the fundamental frequency can be simultaneously obtained. Hence, the evaluation number of the FEM is concerned.  Table 9 lists the optimization results. Fig. 13 shows the iterative optimization history. Seen from Table 9 and Fig. 13 Fig. 13 Iteration history of the pressure sensor design.

Conclusion
Aiming at the uncertain optimization design of the computation-intensive structures, the paper proposes an efficient integrated framework of RBRDO. Through PMA, the RBRDO problem is transformed into an improved percentile form. In the improved formulation above, a series of new approximation methods are proposed to accelerate the computation. Firstly, PMA functions are estimated by an efficient approximation method. Secondly, the improved form above to RBRDO problem is

Availability of data and materials
All data generated or analysed during this study are included in this published article.

Competing interests
The authors declare that they have no competing financial interests.

Funding
The funding is :National Natural Science Foundation of China (Grant No. 51175194) and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No.ZQN-PY504).

Authors' contributions
Laixiongming: proposes the main thinking efficient integrated framework of reliability-based robust design optimization and guides the paper writing.
Huangju: proposes the approximate estimation of the PMA function, carries out example validation and writes the paper.
Wangcheng: finishes algorithm implementation and participates in paper discussion and modification.
Zhangyong: proposes approximate calculation of the actual penalty function and participates in paper discussion and paper modification. Table 1 Input parameters for the beam design problem. Table 2 Optimization results under with different starting design points. Table 3 Optimization results under with different TR radii. Table 4 Optimization results under different uncertainty levels of the design variables. Table 5 Optimization results under different uncertainty levels of the random variables. Table 6 Input parameters for the 72-bar truss design problem. Table 7 Optimization results of the 72-bar truss (m). Table 8 Input parameters for the pressure sensor design problem. Table 9 Optimization results of the pressure sensor (m -3 ).            An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures Table   Table 1 Input parameters for the beam design problem.   Table 2 Optimization results under with different starting design points.  Table 3 Optimization results under with different TR radii. 1 2 3 4 Table 4 Optimization results under different uncertainty levels of the design variables.    Table 9 An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures  An efficient integrated framework for reliability-based robust design optimization of computation-intensive structures 54 Fig. 2 The distribution of the performance function f.     Fig. 11. Schema of piezoresistive pressure sensor. Fig.12 The main sizes of pressure sensor. Fig. 13 Iteration history of the pressure sensor design.

Iterative step
Penalty function