Tensorized computational framework for stiffness matrix and its application to buckling optimization of multi-patch laminated shells via isogeometric analysis

The computational efficiency of stiffness matrix is commonly recognized as one of the primary challenges in mechanical analysis and optimization problems. In this paper, a tensorized framework is proposed to enhance the efficiency of stiffness matrix evaluations. The approach is validated through its application to isogeometric buckling optimization of laminated composite shells. Specifically, a matrix-oriented tensor multiplication (MOTM) is employed to facilitate parallel computation. Tensorized formulations for both stiffness matrix computation and sensitivity analysis are derived. Moreover, a comprehensive complexity analysis comparing the tensorized algorithm with conventional sequential algorithm is presented. Numerical examples illustrate that the proposed tensorized approach achieves a one-order-of-magnitude improvement in efficiency for stiffness matrix evaluations and a two-order-of-magnitude enhancement for sensitivity computations. Furthermore, this paper examines the elastic bound of lamination parameters (LPs), which are related to the positive definiteness of the elastic matrix. An artificial neural network (ANN) is integrated into the optimization process to enforce the elastic bound, thereby significantly reducing the number of indefinite elastic matrices at quadrature points.