Optimization of Uncertain Structures with Interval Parameters Considering Objective and Feasibility Robustness
 Jin Cheng^{1},
 ZhenYu Liu^{2}Email author,
 JianRong Tan^{1},
 YangYan Zhang^{1, 3},
 MingYang Tang^{1} and
 GuiFang Duan^{1, 2, 3}
https://doi.org/10.1186/s1003301802443
© The Author(s) 2018
Received: 19 July 2017
Accepted: 17 April 2018
Published: 6 May 2018
Abstract
For the purpose of improving the mechanical performance indices of uncertain structures with interval parameters and ensure their robustness when fluctuating under interval parameters, a constrained interval robust optimization model is constructed with both the center and halfwidth of the most important mechanical performance index described as objective functions and the other requirements on the mechanical performance indices described as constraint functions. To locate the optimal solution of objective and feasibility robustness, a new concept of interval violation vector and its calculation formulae corresponding to different constraint functions are proposed. The mathematical formulae for calculating the feasibility and objective robustness indices and the robustnessbased preferential guidelines are proposed for directly ranking various design vectors, which is realized by an algorithm integrating Kriging and nested genetic algorithm. The validity of the proposed method and its superiority to present interval optimization approaches are demonstrated by a numerical example. The robust optimization of the upper beam in a highspeed press with interval material properties demonstrated the applicability and effectiveness of the proposed method in engineering.
Keywords
1 Introduction
The uncertainties in material properties, geometric dimensions, load conditions and so on are ubiquitous for engineering structures [1]. The optimal solutions to the optimization models of engineering structures that neglect these uncertainties may be infeasible because their mechanical performance indices will fluctuate under the uncertainties [2]. Hence these uncertainties must be considered in handling the optimization problems of uncertain engineering structures [3, 4]. Robust design optimization is a frequentlyutilized methodology to improve the robustness of structures and reducing the sensitivities of their mechanical performance indices to uncertain factors [5–8]. In the construction of various robust optimization models, the objective robustness is often achieved by simultaneously optimizing the mean of the objective mechanical performance index and minimizing its variation under uncertainties while the constraint robustness is to ensure the satisfaction of constraints when the constraint performance indices fluctuate under the influences of uncertain parameters [9].
Most researches on robust design optimization were conducted based on the assumption that the probabilistic distributions of uncertain factors were known [10, 11]. For instance, Doltsinis et al. [12] applied the perturbation technique and the incremental loading procedure for the response analysis of pathdependent nonlinear structural systems with random parameters, and evaluated the sensitivities of the mean and variance of the structural performance function by direct differentiation in the framework of stochastic finite element analysis. Tang and Périaux [13] proposed a robust optimization method capable of locating Pareto and Nash equilibrium solutions. Zhao and Wang [14] proposed an efficient approach for solving the robust topology optimization problem of structures under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Sahali et al. [15] proposed an efficient genetic algorithm (GA) for multi–objective robust optimization of machining parameters considering random uncertainties. Martínez–Frutos et al. [16] proposed a robust shape optimization approach of continuous structures via the level set method, which modeled the uncertainty in loads and material as random variables with different probability distributions as well as random fields. However, it is often difficult or computationally expensive to determine the probabilistic distributions of uncertain factors in many engineering problems [17].
In order to realize the robust optimization of uncertain structures in the absence of the probabilistic distribution information of uncertainties, several nonprobabilistic methods have been proposed in recent years to account for the uncertainties [18, 19]. Au et al. [20] proposed a robust design method based on the convex model and achieved the robustness of the objective function by minimizing the worst value of unsatisfactory degree functions of the uncertain parameters and ensured the feasibility robustness by a suboptimization conducting the worstcase analysis. Takewaki and Ben–Haim [21] represented the uncertainties in the power spectral density of load and the parameters of the structure’s vibration model by infogap models, and proposed a robust—satisficing methodology for the infogap robust design of uncertain structures. However, the above methods are very complex when the parameter numbers are large. Sun et al. [22] proposed a bilevel mathematical model with interval objective and constraint functions for robust design optimization. The single objective function was converted into two objective functions for minimizing the mean and variation while the constraint functions were reformulated with the acceptable robustness level. However, the socalled robust solution obtained by their method cannot ensure the robustness of all constraints. Karer and Skrjanc [23] proposed a robust optimization framework for PID controllers by describing the uncertain dynamics of the process as an interval model, which was firstly transformed into a deterministic model and then solved by a particle swarm optimization algorithm. Li et al. [24] proposed an actuator placement robust optimization method for active vibration control system with interval parameters. Both nominal value and radius of the performance index were considered in the interval optimization model, which was also transformed into a deterministic one by weighted processing and then solved by GA.
To sum up, present nonprobabilistic robust optimization approaches have difficulties in ensuring the robustness of all constraints and achieving the globally optimal robust solutions to real engineering problems. Moreover, the solution algorithms employed in the present robust optimization approaches based on interval models are indirect ones. That is, they firstly transformed the interval models into deterministic ones and then solved the resulting deterministic models by conventional deterministic optimization algorithms. The shortcomings of such indirect robust optimization approaches are similar to the indirect ones for solving general interval optimization models [25, 26]. Specifically, different acceptable robustness levels or satisfactory degrees of interval constraints prescribed in model transform process will lead to different optimal solutions. Additionally, the transformation of interval models into deterministic ones also deviates from the original intention of uncertainty modeling.
To avoid the limitations of indirect algorithms for solving interval optimization models, we have proposed a direct interval optimization algorithm for uncertain structures by introducing the concept of the degree of interval constraint violation (DICV) [27] based on Hu’s “center first halfwidth next” interval order relation [28]. Specifically, a design vector x has zero DICV for constraint \(g\left( {{\varvec{x}},{\varvec{U}}} \right) \le B = [b^{L} ,b^{R} ] = \left\langle {b^{C} ,b^{W} } \right\rangle\) when \(g^{C} \left( \varvec{x} \right) < b^{C}\) or \(g^{C} \left( \varvec{x} \right) = b^{C}\) and \(g^{W} \left( \varvec{x} \right) \le b^{W} ;\) otherwise, the DICV is nonzero for the constraint (where \(g\left( {{\varvec{x}},{\varvec{U}}} \right)\) is the mechanical performance index of a structure under the influence of interval parameter U, B is the given interval constant, superscripts L, R, C, W indicate the left bound, right bound, center and halfwidth of an interval). The feasibility of a design vector x is determined by the total DICV of all its interval constraints and a design vector x is regarded as feasible when its total DICV is zero. And finally the design vectors are directly sorted according to the DICV–based preferential guidelines. However, there may be \(g^{L} \left( {\varvec{x}} \right) \le b^{L}\) or/and \(g^{R} \left( {\varvec{x}} \right) \ge b^{R}\) when \(g_{{}}^{C} \left( {\varvec{x}} \right) \le b_{{}}^{C}\). That is, the constraint \(g\left( {{\varvec{x}},{\varvec{U}}} \right) \le B\) may be violated although design vector x is regarded as feasible when \(g_{{}}^{C} \left( {\varvec{x}} \right) \le b_{{}}^{C}\) according to the definition of DICV. Consequently, the DICV–based direct interval optimization algorithm cannot ensure the constraint robustness of the optimal solution.
The purpose of this paper is to put forward a direct robust optimization approach for uncertain structures with interval parameters, which can achieve the optimal solutions of objective and feasibility robustness. A novel concept of interval violation vector is proposed for describing the feasibility robustness of a design vector, which comprises two components that describe the violation degrees of the left and right bounds of an interval mechanical performance index in a constraint function. The mathematical formulae for calculating the interval violation vectors of a design vector corresponding to various constraint functions are provided. Then the objective and feasibility robustness indices of various design vectors can be calculated based on their values of total interval violation vectors. And finally, the design vectors of an uncertain structure are sorted according to the robustnessbased preferential guidelines, which is realized by integrating the Kriging technique and nested GA.
2 Robust Optimization Model of an Uncertain Structure with Interval Parameters
3 Definition of the Interval Violation Vector and its Calculation
3.1 Definition of the Interval Violation Vector for an Interval Constraint
Therefore, the following concept of interval violation vector is introduced to describe the violation degree of a design vector for an interval constraint.
Definition 1 (Interval violation vector)
The interval violation vector of an interval constraint \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) is a twodimensional vector, the components of which describe the violation degrees of the left and right bounds of the interval mechanical performance index \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right)\) of an uncertain structure under the influence of interval parameter vector U.
 (1)
There are \(0 \le v_{i}^{L} \left( {\varvec{x}} \right) \le 1\) and \(0 \le v_{i}^{R} \left( {\varvec{x}} \right) \le 1\) for any design vector x.
 (2)
There is \(v_{i} \left( {\varvec{x}} \right) = \left( {0,0} \right)\) when \(g_{{_{i} }}^{R} \left( {\varvec{x}} \right) \le b_{i}^{L}\) as shown in Figure 1(a).
 (3)
There is \(v_{i}^{L} \left( {\varvec{x}} \right) = 1\) when \(g_{{_{i} }}^{L} \left( {\varvec{x}} \right) \ge b_{i}^{L}\), see Figures 1(d)–(f).
 (4)
There is \(v_{i}^{R} \left( {\varvec{x}} \right) = 1\) when \(g_{{_{i} }}^{R} \left( {\varvec{x}} \right) \ge b_{i}^{R}\), see Figures 1(c), (e), (f).
3.2 Calculation of the Interval Violation Vector for Various Constraints
For a mechanical performance index independent of interval parameters, there is \(g_{i} \left( {\varvec{x}} \right) = g_{{_{i} }}^{L} \left( {\varvec{x}} \right) = g_{{_{i} }}^{R} \left( {\varvec{x}} \right)\), and the constraint \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) in Eq. (1) degenerates as \(g_{i} \left( {\varvec{x}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) correspondingly. Then the formula for calculating the interval violation vector of such a constraint should be adjusted as Eq. (4).
In engineering, the interval constant \(B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) in Eq. (1) may also degenerate into a real number, namely, there is \(b_{i} = b_{i}^{L} = b_{i}^{R}\). In this case, the constraint \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) in Eq. (1) degenerates as \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le b_{i}\) and the formula for calculating the interval violation vector of such a constraint should be adjusted as Eq. (5).
Furthermore, the constraint \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\)in Eq. (1) will degenerate as \(g_{i} \left( {\varvec{x}} \right) \le b_{i}\) when the mechanical performance index is independent of the interval parameters and the interval constant degenerates into a real number. And then the formula for calculating the interval violation vector for such a constraint will be simplified as Eq. (6).
As can be observed from Eqs. (3)–(6), there are \({\varvec{v}}_{i} \left( {\varvec{x}} \right) = \left( {0,0} \right)\) when \(g_{i} \left( {\varvec{x}} \right) = b_{i}^{L}\) or \(b_{i} = g_{{_{i} }}^{R} \left( {\varvec{x}} \right)\) or \(g_{i} \left( {\varvec{x}} \right) = b_{i}\), otherwise, the values of \({\varvec{v}}_{i} \left( {\varvec{x}} \right)\) can be calculated by Eq. (3). Consequently, the formula for calculating the interval violation vector of constraint \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right) \le B_{i} = [b_{i}^{L} ,b_{i}^{R} ]\) can be concluded as Eq. (7).
4 Preferential Guidelines Considering Objective and Feasibility Robustness
In order to directly solve the robust optimization model in Eq. (1), two robustness indices are introduced to evaluate the objective and feasibility robustness of a design vector. The feasibility robustness index is utilized to evaluate the acceptability of a design vector as far as the constraint functions are concerned while the objective robustness index is utilized to evaluate the superiority and robustness of a design vector as far as the objective mechanical performance index is concerned. Then the robustness–based preferential guidelines are proposed for realizing the direct ranking of various design vectors.
4.1 Feasibility Robustness Index and its Calculation
 (1)
For a feasible robust design vector x, there is \(I_{\text{FR}} \left( {\varvec{x}} \right) = 1\).
 (2)
For a design vector x that is not feasible robust, there is \(0 \le I_{\text{FR}} \left( {\varvec{x}} \right) < 1\).
 (3)
The larger feasibility robustness index indicates the better acceptability of a design vector as far as the constraints are concerned.
4.2 Objective Robustness Index and its Calculation
It is obvious from Eq. (11) that the objective robustness index is positive for any design vector, and the larger objective robustness index indicates the better and more robust of a design vector as far as the objective mechanical performance index is concerned.
4.3 Preferential Guidelines for Ranking Various Design Vectors
 (1)
A feasible robust design vector is always superior to an infeasible one. That is, design vector x_{1} is superior to design vector x_{2} when \(I_{\text{FR}} \left( {{\varvec{x}}_{1} } \right) = 1\) and \(I_{\text{FR}} \left( {{\varvec{x}}_{2} } \right) < 1\).
 (2)
The infeasible design vectors are ranked according to their corresponding values of feasibility robustness indices. Specifically, infeasible design vector x_{1} is superior to infeasible design vector x_{2} when \(I_{\text{FR}} \left( {{\varvec{x}}_{1} } \right) > I_{\text{FR}} \left( {{\varvec{x}}_{2} } \right)\).
 (3)
The feasible robust design vectors are ranked according to their objective robustness indices. Specifically, feasible design vector x_{1} is superior to feasible design vector x_{2} when \(I_{\text{OR}} \left( {{\varvec{x}}_{1} } \right) > I_{\text{OR}} \left( {{\varvec{x}}_{2} } \right)\).
5 Integrated Algorithm for Directly Solving the Interval Robust Optimization Model
A robust optimization algorithm integrating Kriging models and nested GA is proposed to directly solve the constrained interval robust optimization model of the uncertain structure. The Kriging model is utilized here to replace finite element analysis (FEA) for efficiently computing the mechanical performance indices of the uncertain structure. For the mechanical performance index influenced by ndimensional design vector x and qdimensional interval vector U, such as \(f\left( {{\varvec{x}},{\varvec{U}}} \right)\) and \(g_{i} \left( {{\varvec{x}},{\varvec{U}}} \right)\) in Eq. (1) is concerned, the sample points for constructing the Kriging model should be generated in the (n+q)dimensional space determined by n design variables and q interval parameters based on Latin hypercube sampling (LHS). To ensure the prediction accuracy of Kriging models, the adaptive resampling technology proposed in our previous work [29] is also adopted. Specifically, the construction of every Kriging model is an iterative process until the achievement of satisfactory local and global precision evaluated by multiple correlation coefficient R^{2} and relative maximum absolute error (RMAE). The inner layer GAs integrated with Kriging models calculate in parallel the intervals of the mechanical performance indices under the influence of uncertain parameters while the outer layer GA realizes the direct sorting of various design vectors according to the robustnessbased preferential guidelines and locates the optimal solution to the constrained interval robust optimization model.
 Step 1::

Construct the robust optimization model of an uncertain structure with interval parameters. The mechanical performance indices of the uncertain structure are described as the functions of design variables and interval parameters. The center and halfwidth of the most important mechanical performance index are described as objective functions while the requirements of the other mechanical performance indices are described as constraint functions.
 Step 2::

Construct the Kriging models for efficiently computing the mechanical performance indices of the uncertain structure based on finite element (FE) model, LHS and adaptive resample technology.
 Step 3::

Initialize the GA parameters involved in the nested optimization, including the population sizes, maximum iteration numbers, crossover and mutation probabilities of the inner and outer layer GAs. Set the iteration number of the outer layer GA as 1 and generate the initial population.
 Step 4::

Rank the individuals in the current population of outer layer GA according to the robustnessbased preferential guidelines and calculate their fitness values, during the process of which inner layer GAs integrated with Kriging models constructed in Step 2 are implemented in parallel for computing the left and right bounds of the mechanical performance indices.
 Step 5::

Output the design vector with the largest fitness value if the convergent threshold or maximum iteration number of the outer layer GA is reached. Otherwise, increase the iteration number of the outer layer GA by 1 and go to Step 4.
6 Illustrative Examples
Two illustrative examples are investigated in this section to verify the effectiveness of the proposed approach for directly solving interval robust optimization problems and its applicability in engineering practice. The construction of Kriging models is unnecessary for the first example since its objective and constraint functions are analytical. It is obvious that the proposed robust interval optimization algorithm has the same advantages as our previous one in realizing the direct solution of interval optimization problems and avoiding the complicated model transformation process. Consequently, the optimization results obtained by the proposed algorithm are only compared with those obtained by the direct one hereinafter.
6.1 Numerical Example
GA parameters for numerical example in Eq. ( 12 )
GA  Population size  Crossover probability  Mutation probability  Maximum iteration number 

Outer layer  100  0.90  0.05  200 
Inner layer  50  0.99  0.05  150 
The optimization results of numerical example obtained by proposed and previous algorithms
Algorithm  Optimal solution  Objective functions  Constraint functions  

[f ^{ L }, f ^{ R }], \(\left\langle {f^{C} ,f^{W} } \right\rangle\)  [g _{1} ^{ L } , g _{1} ^{ R } ]  [g _{2} ^{ L } , g _{2} ^{ R } ]  
Proposed  x^{ o } = (3.17, 0.02, 2.00)  [7.31,10.92], \(\left\langle {9.11,\;1.80} \right\rangle\)  [10.00, 12.41]  [7.55, 9.94] 
Previous  x^{ * } = (3.11, 0.20, 2.08)  [7.62,11.31], \(\left\langle {9.47,\;1.84} \right\rangle\)  [9.58, 12.00]  [7.98, 10.53] 
It is also clear from Table 2 that both objective functions of the optimal solution obtained by the proposed algorithm are smaller than those obtained by our previous one, which demonstrates that the proposed algorithm can obtain a better and more robust solution than the previous one as far as the objective functions are concerned. Moreover, the convergent curves in Figures 3, 4 demonstrate that the proposed algorithm can locate the optimal solution more efficiently than the previous one.
6.2 Engineering Example
Mechanical performance indices of the initial design scheme of the upper beam
[w^{ L }, w^{ R }] kg  [δ^{ L }, δ^{ R }] MPa  [d ^{ L }, d ^{ R }] mm, \(\left\langle {d^{C} ,d^{W} } \right\rangle \;{\text{mm}}\) 

[5216.2, 5232.3]  [45.21, 50.00]  [0.1716, 0.2114], \(\left\langle {0.1915,\;0.0199} \right\rangle\) 
6.2.1 Optimization Results Obtained by Proposed Algorithm
GA parameters for the robust optimization of the upper beam
GA  Maximum iteration number  Population size  Crossover probability  Mutation probability 

Inner layer  150  60  0.99  0.05 
Outer layer  300  120  0.90  0.01 
6.2.2 Comparison with Previous Algorithm
Comparison of the optimization results of the upper beam obtained by the proposed and previous algorithms
Algorithm  Optimal solution (h_{1}, h_{2}, l_{1}, l_{2}, l_{3}) mm  Mechanical performance indices of the upper beam  

[w^{ L }, w^{ R }] kg  [δ^{ L }, δ^{ R }] MPa  [d^{ L }, d^{ R }] mm, \(\left\langle {d^{C} ,d^{W} } \right\rangle \;{\text{mm}}\)  
Proposed  (238.89,280.22,81.61,32.73,386.21)  [4983.9, 5000.0]  [40.16, 44.95]  [0.1834, 0.2220], \(\left\langle {0.2027,\;0.0193} \right\rangle\) 
Previous  (248.03,296.28,85.31,30.31,387.89)  [4992.3, 5008.9]  [38.64, 43.56]  [0.1976, 0.2366], \(\left\langle {0.2171,\;0.0195} \right\rangle\) 
7 Conclusions
To improve the mechanical performance indices of an uncertain structure with interval parameters and ensure their satisfaction with performance requirements when fluctuating under uncertainties, a constrained interval robust optimization model was constructed with both the center and halfwidth of the most important mechanical performance index described as objective functions and the other mechanical performance indices included in constraint functions. A novel concept of interval violation vector was proposed for evaluating the feasibility robustness of a design vector, the mathematical formulae for calculating the interval violation vectors of various constraint functions were also provided. Then the robustness–based preferential guidelines were proposed for directly ranking various design vectors and an algorithm integrating Kriging technique and nested GA was put forward to realize the direct solution of the constrained interval robust optimization problem.
The proposed direct interval robust optimization algorithm has the same advantage as our previous one [27] in avoiding the complex model transformation process from interval to deterministic. The optimization results of the numerical example demonstrated that the proposed algorithm was more efficient and effective than our previous one. The robust optimization of the upper beam in a highspeed press with interval material density and elastic modulus demonstrated the feasibility and validity of the proposed method in engineering practice.
Declarations
Authors’ Contributions
JC, ZYL and JRT were in charge of the whole trial; JC wrote the manuscript; YYZ, MYT and GFD assisted with sampling and laboratory analyses. All authors have read and approved the final manuscript.
Authors’ Information
Jin Cheng, born in 1978, is currently an associate professor at State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, China. She received her PhD degree from Zhejiang University, China, in 2005. Her research interests include uncertainty modeling, structural optimization and intelligent design. Tel: +8657187951273; Email: cjinpjun@zju.edu.cn.
ZhenYu Liu, born in 1974, is currently a professor at State Key Laboratory of CAD & CG, Zhejiang University, China. He received his PhD degree from Zhejiang University, China, in 2002. His research interests include CAD, virtual prototyping, virtualrealitybased simulation and robotics. Email: liuzy@zju.edu.cn.
JianRong Tan, born in 1954, is currently an academician of the Chinese Academy of Engineering, professor at State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, China. His research interests include CAD &CG, mechanical design and theory, digital design and manufacture. Email: egi@zju.edu.cn.
YangYan Zhang, born in 1993, is currently a master candidate at State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, China. Her research interest is uncertainty optimization of engineering structures. Email: zyyzjz1116@163.com.
MingYang Tang, born in 1991, received his master’s degree from Zhejiang University, China, in 2017. His research interest is uncertainty optimization of engineering structures. Email: 861658082@qq.com.
GuiFang Duan, born in 1979, is currently an associate professor at State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, China. His research interests include CAD &CG, digital design and manufacture. Email: gfduan@zju.edu.cn.
Competing Interests
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Not applicable.
Funding
Supported by National Natural Science Foundation of China (Grant Nos. 51775491, 51475417, U1608256, 51405433).
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