**6. Conclusions**

The BDCDO solving framework based on the surrogate models of the derivative functions in the state equation is an effective approach to solve the black-box dynamic co-design

and optimization problem. For efficient construction of the surrogate models for the right hand-side functions of the state equation, a novel adaptive sequential sampling strategy, called SRIRMD, was proposed in this work. This strategy refines the surrogate models by selecting suitable sample points from the trajectory discrete points. At the same time, to quantify the convergence and intuitively reflect the convergence trend of the solution during the solving process, a new termination criterion, called the state trajectory overlap ratio (STOR), was also introduced. Finally, the BDCDO solving framework combined with SRIRMD and STOR was utilized to address two numerical optimization problems and two engineering co-design and optimization problems. The numerical examples indicate that the SRIRMD sampling strategy proposed in this work was superior to the existing sampling strategies with respect to both the solution accuracy and robustness. The 3-DOF robot co-design and optimization problem and the horizontal axis wind turbine co-design and optimization problem revealed that the BDCDO solving framework combined with SRIRMD and STOR is a feasible and efficient tool to optimize the black-box dynamic systems. In summary, the proposed sampling strategy and the termination criterion in this research not only improve the efficiency of the BDCDO solving framework but also save the computational budget.

**Author Contributions:** Methodology, Q.Z. and Y.W.; Software, Q.Z.; Writing—original draft, Q.Z.; and Writing—review and editing, Y.W. and L.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Key Research and Development Program of China (Grant No. 2018YFB1700905) and National Natural Science Foundation of China (Grant No. 51575205).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Researchers interested in this method can access the code through the corresponding author.

**Conflicts of Interest:** The authors declare no potential conflict of interest.
