Random matrix theory for robust topology optimization with material uncertainty

• Published in: Structural and multidisciplinary optimization Vol. 66; no. 11; p. 240
• Main Authors: Li, Linxi, Steeves, Craig A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2023; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Boundary conditions; Compliance; Computational efficiency; Computational Mathematics and Numerical Analysis; Design; Engineering; Engineering Design; Fields (mathematics); Mathematical analysis; Mathematical models; Matrix theory; Methods; Modulus of elasticity; Optimization; Random variables; Research Paper; Robustness (mathematics); Statistical analysis; Statistical inference; Statistical methods; Stiffness matrix; Stochastic processes; Structural design; Theoretical and Applied Mechanics; Topology optimization; Uncertainty; Intrusive method; Random matrix theory; Material uncertainties; Stochastic mechanics; Robust topology optimization
• ISSN: 1615-147X; 1615-1488
• DOI: 10.1007/s00158-023-03665-y

This paper presents an efficient computational method for optimal structural design in the presence of uncertain Young’s modulus modeled using discretized random fields. To quantify and propagate the uncertainty, random matrix theory is employed to quantify uncertainty in the context of robust topology optimization (RTO) for the minimization of compliance. Random matrix theory employs statistical inference methods to model the matrix-variate probability distribution of the finite element stiffness matrix. This provides analytical expressions for the mean and the standard deviation of the compliance, a combination of which is minimized in RTO. The novel random matrix theory-based RTO is computationally efficient due to the intrusive nature of the method, and is flexible as its computational performance and robustness remain consistent regardless of the correlation lengths or the variance of the random field, as demonstrated through numerical cases. The random matrix RTO method is applied to several two-dimensional numerical problems where the random fields of the modulus are assigned with ranges of correlation lengths and variances to illustrate the versatility of the method. The performance of random matrix RTO is compared with Monte Carlo RTO and stochastic collocation RTO to explore the efficiency and accuracy of the method.