TY - JOUR

T1 - Sequential mixture of Gaussian processes and saddlepoint approximation for reliability-based design optimization of structures

AU - Do, Bach

AU - Ohsaki, Makoto

AU - Yamakawa, Makoto

N1 - Funding Information:
Financial support from the Japan International Cooperation Agency (JICA) for the first author and JSPS KAKENHI No. JP19H02286 for the second author is fully acknowledged.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - This paper presents an efficient optimization procedure for solving the reliability-based design optimization (RBDO) problem of structures under aleatory uncertainty in material properties and external loads. To reduce the number of structural analysis calls during the optimization process, mixture models of Gaussian processes (MGPs) are constructed for prediction of structural responses. The MGP is used to expand the application of the Gaussian process model (GPM) to large training sets for well covering the input variable space, significantly reducing the training time, and improving the overall accuracy of the regression models. A large training set of the input variables and associated structural responses is first generated and split into independent subsets of similar training samples using the Gaussian mixture model clustering method. The GPM for each subset is then developed to produce a set of independent GPMs that together define the MGP as their weighted average. The weight vector computed for a specified input variable contains the probability that the input variable belongs to the projection of each subset onto the input variable space. To calculate the failure probabilities and their inverse values required during the process of solving the RBDO problem, a novel saddlepoint approximation is proposed based on the first three cumulants of random variables. The original RBDO problem is replaced by a sequential deterministic optimization (SDO) problem in which the MGPs serve as surrogates for the limit-state functions in probabilistic constraints of the RBDO problem. The SDO problem is strategically solved for exploring a promising region that may contain the optimal solution, improving the accuracy of the MGPs in that region, and producing a reliable solution. Two design examples of a truss and a steel frame demonstrate the efficiency of the proposed optimization procedure.

AB - This paper presents an efficient optimization procedure for solving the reliability-based design optimization (RBDO) problem of structures under aleatory uncertainty in material properties and external loads. To reduce the number of structural analysis calls during the optimization process, mixture models of Gaussian processes (MGPs) are constructed for prediction of structural responses. The MGP is used to expand the application of the Gaussian process model (GPM) to large training sets for well covering the input variable space, significantly reducing the training time, and improving the overall accuracy of the regression models. A large training set of the input variables and associated structural responses is first generated and split into independent subsets of similar training samples using the Gaussian mixture model clustering method. The GPM for each subset is then developed to produce a set of independent GPMs that together define the MGP as their weighted average. The weight vector computed for a specified input variable contains the probability that the input variable belongs to the projection of each subset onto the input variable space. To calculate the failure probabilities and their inverse values required during the process of solving the RBDO problem, a novel saddlepoint approximation is proposed based on the first three cumulants of random variables. The original RBDO problem is replaced by a sequential deterministic optimization (SDO) problem in which the MGPs serve as surrogates for the limit-state functions in probabilistic constraints of the RBDO problem. The SDO problem is strategically solved for exploring a promising region that may contain the optimal solution, improving the accuracy of the MGPs in that region, and producing a reliable solution. Two design examples of a truss and a steel frame demonstrate the efficiency of the proposed optimization procedure.

KW - Frame structure

KW - Gaussian mixture model

KW - Mixture of Gaussian processes

KW - Reliability-based design optimization

KW - Saddlepoint approximation

KW - Sequential deterministic optimization

UR - http://www.scopus.com/inward/record.url?scp=85105399270&partnerID=8YFLogxK

U2 - 10.1007/s00158-021-02855-w

DO - 10.1007/s00158-021-02855-w

M3 - Article

AN - SCOPUS:85105399270

VL - 64

SP - 625

EP - 648

JO - Structural Optimization

JF - Structural Optimization

SN - 1615-147X

IS - 2

ER -