Regularized least squares for the building of a sizing criterion based on damage mechanics

Utilization of the building-block approach for the design of composite parts emerged from the need to develop a true understanding of the complex composite structural response through experimentation and analysis. The present work aims to provide a numerical counterpart to the expensive experimental testing of advanced composite laminate structures. This detailed analysis can be fully integrated into the building-block approach to enhance its capability to examine full-scale structural behavior and improve its reliability. To this end, a Virtual Testing Analysis (VTA) based on the structural decomposition into a number of complicated laminates is introduced for determining premature damage evolution. VTA augment FEM with a hierarchical procedure which goes down to the meso-scale where a detailed modeling of each component is examined using non-linear FEM. Design of Experiments is utilized for an optimal sampling of the design space related to the laminate subcomponents. Linear least squares regression using Tikhonov regularization (L2-norm penalty function) and Truncated Singular Value Decomposition (TSVD) are used to combat overfitting and stabilize possible ill-conditioned matrix inverse problems. Furthermore, a Multi-Stage Least Squares methodology is implemented as an alternative approximation which utilizes a separation of the design variables into sets. This method yields optimal solutions in each subspace in a hope to obtain the final global optimal solution from the composition of locally optimal solutions. Performance and efficiency are evaluated both for training and validation data using error measures based on the residuals. Preliminary results showed acceptable accuracy for the linear least squares and Multi-Stage Least Squares methods. An important key aspect of VTA is that off-line computations for the advanced laminates can be stored and used for different full-scale structures as a simple failure criterion.