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

Semi-incremental model order reduction approach for fatigue damage computations

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Toward Optimality of Proper Generalised Decomposition Bases

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method

A kinetic two-scale damage model for high-cycle fatigue simulation using multi-temporal Latin framework

International audienceThe goal of this paper is to introduce a model order reduction method for high-cycle fatigue simulations using a kinetic damage model, i.e. a constitutive model in which the damage evolution law is defined as a rate form for the damage variable D. In the framework of continuum mechanics, high-cycle fatigue simulation involves a two-scale damage model, which includes macroscopic elastic and microscopic plastic behaviours, for a very large number of cycles. Unlike the classical usage of the two-scale damage model by Lemaitre and co-workers, where damage is calculated as a post-process of an elastic or elasto-plastic macroscopic analysis, in this work, a fully coupled analysis is conducted assuming a macroscopic damage feedback from its microscopic counterpart. Damage is considered to be isotropic with micro-defect closure effect on both macroscopic and microscopic scales. To overcome the numerical expense, the large time increment (LATIN) method is used as a linearisation framework, where the constitutive behaviour is separated from the global admissibility which in turn is solved through separation of variables using a proper generalised decomposition (PGD)-based model reduction method. A multi-temporal discretisation approach is henceforth used based on finite element like description in time for the quantities of interest, providing a sophisticated numerical approach suitable for high-cycle fatigue simulation under complex loading