Stress constrained shape and topology optimization with fixed mesh: A B-spline finite cell method combined with level set function

In this paper, we develop an efficient and flexible design method that integrates the B-spline finite cell method (B-spline FCM) and the level set function (LSF) for stress constrained shape and topology optimization. Any structure of complex geometry is embedded within an extended, regular and fixed Eulerian mesh no matter how the structure is optimized. High-order B-spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis. Meanwhile, level set functions, i.e., implicit functions are used to enable topological changes of the considered structure through smooth boundary variations. Involved parameters rather than the conventional discrete form of LSF are directly taken as design variables to facilitate the numerical computing process. To be specific, the LSF is constructed by means of R-functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh, while the compactly supported radial basis functions (CS-RBFs) are employed as implicit functions for stress constrained topology optimization. It is shown the proposed FCM/LSF method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision. Representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the FCM/LSF method.