A semi-incremental model order reduction approach for fatigue damage computations

Nowadays, there is an increasing need, and interest, in model order reduction (MOR) techniques that make it feasible to approximate complex high fidelity models (HFM) in real-time and many-query scenarios using limited computational resources. The development of model order reduction techniques suitable for structural problems with nonlinear material behaviour is investigated in this research. A semi-incremental framework based on a large time increment (LATIN) approach is proposed to tackle fatigue damage computations subjected to variable amplitude and frequency loadings. Due to the nonlinear damage growth, the damage accumulation driven by variable loads should reflect the load sequence effect. Experiments have revealed that such an effect becomes more crucial as the difference in amplitudes increases \parencite{lemaitre2005engineering}. %ignore: experiments show that the deviation is more and more important The proposed implementation approximates the structural response within a material-independent framework, i.e., different material models may be incorporated straightforwardly. Loads with variable amplitudes and frequencies are addressed in a semi-incremental manner, where full cycles are simulated consecutively, and convergence is ensured using a hybrid approach. A low-rank approximation, in terms of proper generalised decomposition (PGD) of the solution, is sought directly in the online phase of the proposed scheme where the optimality of the generated PGD basis and its growth are controlled using different orthogonalisation schemes. PGD bases can be interpreted as a set of linear subspaces adapted on-the-fly to the problem settings. Different orthonormalisation techniques were tested to ensure the optimality of the PGD generated modes. Following the assessment, a randomised singular value decomposition (SVD) approach that exploits the outer-product format of the PGD solution was selected. The SVD scheme resulted in a considerable computational time saving by limiting the number of modes compared to a Gram-Schmidt procedure. The whole numerical scheme is realised in the online phase, and no offline phase is considered so far. Improvements to the introduced reduced order model (ROM) are further investigated by exploiting low-rank approximations in an arithmetic operation toolbox that allows for faster simulations with lower memory footprints. Then, a data assisted approach that combines machine learning techniques such as artificial neural networks (ANN) with MOR is examined to show the promising results of data recycling, i.e., reusing previously generated data. The semi-incremental scheme and a displacement formulated standard finite element incremental framework are implemented to illustrate their differences in terms of computational time and memory footprint. Numerical examples with variable loadings that show speedups in the order of 10-100 are discussed, and a typical implementation is provided as open-source code, available at https://gitlab.com/shadialameddin/romfem