In the engineering field, dealing with black-box problems poses numerous challenges. Existing methods such as the deterministic safety factor method and traditional uncertainty methods (e.g., Gaussian distribution) have drawbacks like high costs and the "black-box in, black-box out" issue. This paper focuses on the uncertainty quantification of structural responses under constraint conditions and proposes a new solution approach. The research provides a mathematical description of uncertain propagation with constraints and establishes a numerical algorithm, the Key Conditional Quotient (KCQ), based on the generalized quasi-Monte Carlo method. By introducing a reference value coefficient to judge the value of key constraints, more accurate uncertainty propagation analysis can be achieved. This method has a wide range of applications and performs well in static and dynamic response estimation, heat conduction, navigation state estimation, and structural response interval analysis. It can effectively reduce the degree of uncertainty, provide an accurate statistical basis for structural safety and reliability analysis, and has significant advantages in real-time computing scenarios such as digital twins, contributing to solving complex black-box problems.

Wu Feng is a full professor and doctoral supervisor in the School of Mechanics and Aerospace Engineering, Dalian University of Technology. He was on the editorial board of China Offshore Platform, Journal of Applied Mechanics and was a member of International Association of Computational Mechanics. He has published more than 80 journal articles (including IJMS, SEC, CMAME, MSSP, SIAM and other international top journals), 2 academic monographs, and has realized the achievement transformation of 2 million yuan. He has also won the second prize of national defense science and technology progress ; Dalian Youth Science and Technology Star ; Qian Weichang 's excellent thesis award, etc. He has been engaged in the research of fluid mechanics and computational mechanics for a long time. He has won 1 excellent teaching quality award from Dalian University of Technology, 3 excellent teaching quality awards, 1 second prize from Liaoning University Teaching Innovation Competition, 1 special prize from Liaoning Province Graduate Teaching Achievements, and 1 second prize from China National Graduate Teaching Achievements (2nd). His current research interests include computational symplectic mechanics and numerical calculation of uncertainty quantitative analysis.