*Article* **A Novel Surrogate Model-Based Solving Framework for the Black-Box Dynamic Co-Design and Optimization Problem in the Dynamic System**

**Qi Zhang, Yizhong Wu \* and Li Lu**

National Center of Technology Innovation for Intelligent Design and Numerical Control, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China **\*** Correspondence: cad.wyz@hust.edu.cn

**Abstract:** When encountering the black-box dynamic co-design and optimization (BDCDO) problem in the multidisciplinary dynamic system, the finite difference technique is inefficient or even infeasible to provide approximate numerical gradient information for the optimization algorithm since it requires numerous original expensive evaluations. Therefore, a solving framework based on the surrogate model of the state equation is introduced to optimize BDCDO. To efficiently construct the surrogate model, a sequential sampling method is presented on the basis of the successive relative improvement ratio. Meanwhile, a termination criterion is suggested to quantify the convergence of the solution. Ultimately, the newly proposed sampling strategy and termination criterion are incorporated into the BDCDO solving framework to optimize two numerical examples and two engineering examples. The results demonstrate that the framework integrating the proposed sampling strategy and termination criterion has the best performance in terms of the accuracy, efficiency, and computational budget compared to the existing methods.

**Keywords:** black-box dynamic system; co-design and optimization; surrogate model; sequential sampling; termination criterion

**MSC:** 93-10
