Toward the robust establishment of variable-fidelity surrogate models for hierarchical stiffened shells by two-step adaptive updating approach

Since the high-fidelity model (HFM) of hierarchical stiffened shells is time-consuming, the sampling points based on HFM are generally few, which would result in a certain randomness of the sampling process. In some cases, the prediction accuracy of the variable-fidelity surrogate model (VFSM) is prone to be not robust and reliable. In order to improve the robustness of the prediction accuracy of VFSM, a two-step adaptive updating approach is proposed for the robust establishment of VFSM. In the first step, the leave-one-out (LOO) cross validation is carried out for sampling points of the low-fidelity model (LFM), aiming at finding out those with large prediction error. Then, these points are evaluated by HFM and then added into the original HFM set. In the second step, another LOO cross validation is performed on sampling points of the hybrid bridge function linking HFM and LFM. Based on the Voronoi diagram method, new updating points are chosen from where the largest prediction error of the bridge function lies, and then the VFSM is updated. After above two-step updating process, the VFSM is established. Three simple examples of test functions are firstly presented to verify the effectiveness and efficiency of the proposed method. Further, the proposed method is applied to an engineering example of hierarchical stiffened shells. In order to provide evaluation indexes for prediction accuracy and robustness of VFSM, the VFSM is established by multiple times, and the mean value and the standard deviation of the relative root mean square error ( RRMSE ) values of the multiple sets of VFSM are calculated. Results indicate that, under the similar computational cost, the mean value and the standard deviation of the RRMSE values of the proposed method decrease by 24.1% and 82.0% than those of the traditional VFSM based on the direct sampling method, respectively. Therefore, the high prediction accuracy and robustness of the proposed method is verified. Additionally, the total computational time of the proposed VFSM decreases by 70% than that of the surrogate model based on HFM when achieving the similar prediction accuracy, indicating the high prediction efficiency of the proposed VFSM.

[1]  M. Tyan,et al.  Improving variable-fidelity modelling by exploring global design space and radial basis function networks for aerofoil design , 2015 .

[2]  Adrian Murphy,et al.  Aerospace Stiffened Panel Initial Sizing With Novel Skin Sub-Stiffening Features , 2012 .

[3]  Raphael T. Haftka,et al.  Multi-fidelity design of stiffened composite panel with a crack , 2002 .

[4]  Wei Sun,et al.  A radial basis function-based multi-fidelity surrogate model: exploring correlation between high-fidelity and low-fidelity models , 2019, Structural and Multidisciplinary Optimization.

[5]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[6]  Yunlong Ma,et al.  Grid-pattern optimization framework of novel hierarchical stiffened shells allowing for imperfection sensitivity , 2017 .

[7]  R. Haftka,et al.  Surrogate-based Optimization with Parallel Simulations using the Probability of Improvement , 2010 .

[8]  Daining Fang,et al.  Equivalent analysis and failure prediction of quasi-isotropic composite sandwich cylinder with lattice core under uniaxial compression , 2013 .

[9]  Paulo Moura Oliveira,et al.  Particle swarm optimization with fractional-order velocity , 2010 .

[10]  Peng Hao,et al.  A hybrid descent mean value for accurate and efficient performance measure approach of reliability-based design optimization , 2018, Computer Methods in Applied Mechanics and Engineering.

[11]  John E. Renaud,et al.  Variable Fidelity Optimization Using a Kriging Based Scaling Function , 2004 .

[12]  Stefan Görtz,et al.  Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function , 2013 .

[13]  Jae-Sang Park,et al.  Postbuckling analyses and derivations of Knockdown factors for hybrid-grid stiffened cylinders , 2018, Aerospace Science and Technology.

[14]  Masahiro Kanazaki,et al.  Hybrid surrogate-model-based multi-fidelity efficient global optimization applied to helicopter blade design , 2017 .

[15]  Bo Wang,et al.  Proper-Orthogonal-Decomposition-Based Buckling Analysis and Optimization of Hybrid Fiber Composite Shells , 2018 .

[16]  D. Fang,et al.  Fabrication and testing of composite hierarchical Isogrid stiffened cylinder , 2018 .

[17]  Bo Wang,et al.  A high-fidelity approximate model for determining lower-bound buckling loads for stiffened shells , 2017, International Journal of Solids and Structures.

[18]  Alexander I. J. Forrester,et al.  Multi-fidelity optimization via surrogate modelling , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Gang Xu,et al.  An adaptive parameter tuning of particle swarm optimization algorithm , 2013, Appl. Math. Comput..

[20]  Paul M. Weaver,et al.  Stiffness tailoring of elliptical composite cylinders for axial buckling performance , 2016 .

[21]  Hui Zhou,et al.  An active learning variable-fidelity metamodelling approach based on ensemble of metamodels and objective-oriented sequential sampling , 2016 .

[22]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[23]  Peng Hao,et al.  Experimental validation of cylindrical shells under axial compression for improved knockdown factors , 2019, International Journal of Solids and Structures.

[24]  Fan Yang,et al.  Optimal design of hierarchical grid-stiffened cylindrical shell structures based on linear buckling and nonlinear collapse analyses , 2017 .

[25]  Michael D. Shields,et al.  The generalization of Latin hypercube sampling , 2015, Reliab. Eng. Syst. Saf..

[26]  Peng Hao,et al.  An adaptive response surface method and Gaussian global-best harmony search algorithm for optimization of aircraft stiffened panels , 2018, Appl. Soft Comput..

[27]  Stefan Görtz,et al.  Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling , 2012 .

[28]  Hamda Chagraoui,et al.  Multidisciplinary design optimization of stiffened panels using collaborative optimization and artificial neural network , 2018 .

[29]  Ke Zhang,et al.  Tailoring the optimal load-carrying efficiency of hierarchical stiffened shells by competitive sampling , 2018, Thin-Walled Structures.

[30]  Daining Fang,et al.  Fabrication and testing of composite orthogrid sandwich cylinder , 2017 .

[31]  D. Hodges,et al.  Mechanics of structure genome-based global buckling analysis of stiffened composite panels , 2019, Acta Mechanica.

[32]  Nam H. Kim,et al.  Issues in Deciding Whether to Use Multifidelity Surrogates , 2019, AIAA Journal.

[33]  Hao Wu,et al.  Carbon fiber reinforced hierarchical orthogrid stiffened cylinder: Fabrication and testing , 2018 .

[34]  Chao-Chao Wang,et al.  An active learning metamodeling approach by sequentially exploiting difference information from variable-fidelity models , 2016, Adv. Eng. Informatics.

[35]  Patrick R. Palmer,et al.  Multi-fidelity simulation modelling in optimization of a submarine propulsion system , 2010, 2010 IEEE Vehicle Power and Propulsion Conference.

[36]  Doo-Hyun Choi,et al.  Cooperative mutation based evolutionary programming for continuous function optimization , 2002, Oper. Res. Lett..

[37]  Peng Hao,et al.  Multilevel Optimization Framework for Hierarchical Stiffened Shells Accelerated by Adaptive Equivalent Strategy , 2016, Applied Composite Materials.

[38]  Yuanming Xu,et al.  A new effective smeared stiffener method for global buckling analysis of grid stiffened composite panels , 2016 .

[39]  Hao Hu,et al.  A Hybrid Reliability-Based Design Optimization Approach with Adaptive Chaos Control Using Kriging Model , 2016 .

[40]  Jae-Sang Park,et al.  Derivations of Knockdown Factors for Cylindrical Structures Considering Different Initial Imperfection Models and Thickness Ratios , 2018, International Journal of Aeronautical and Space Sciences.

[41]  Shinobu Yoshimura,et al.  An Improved Contact Formulation for Impact Crack Simulations in a Laminated Glass Beam , 2018, International Journal of Computational Methods.

[42]  Yu Sun,et al.  Hybrid analysis and optimization of hierarchical stiffened plates based on asymptotic homogenization method , 2015 .

[43]  Bo Wang,et al.  Hybrid optimization of hierarchical stiffened shells based on smeared stiffener method and finite element method , 2014 .

[44]  Bo Wang,et al.  Optimum design of hierarchical stiffened shells for low imperfection sensitivity , 2014 .

[45]  Stefan Görtz,et al.  Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling , 2012 .

[46]  Zeng Meng,et al.  A Hybrid Framework of Efficient Multi-Objective Optimization of Stiffened Shells with Imperfection , 2020 .

[47]  R. Haftka,et al.  Review of multi-fidelity models , 2016, Advances in Computational Science and Engineering.

[48]  M. Hojjati,et al.  Use of curvilinear fibers for improved bending-induced buckling capacity of elliptical composite cylinders , 2017 .