Speaker
Description
Fully coupled high-fidelity simulations using the multiscale finite element (FE2) method are prohibitively expensive for many relevant multiscale problems. The computational cost of FE2 simulations is dominated by the assembly and solution of large linear equation systems which are required by a nonlinear solver on the microscale [1,2]. Additionally, obtaining homogenised macroscopic stresses and stiffnesses from microscale representative volume element solutions also requires the assembly and solution of large linear equation systems [3].
Projection-based model order reduction (MOR) methods address the cost of solving linear equation systems by searching for solutions in a low-dimensional approximation space [2]. Existing MOR methods utilise linear, piecewise linear, or specific nonlinear Ansatz approximation spaces [4,5,6]. Additionally, hyperreduction techniques efficiently estimate quantities appearing in the reduced linear equation systems by integrating over a reduced integration domain [4,5,6].
In a recent publication [2], we proposed a nonlinear MOR scheme which uses manifold learning techniques to obtain a flexible, continuously nonlinear approximation space. This facilitates the construction of smaller reduced order models than those obtained by alternative methods, while obtaining similar levels of accuracy. In this contribution, we discuss a tailored hyperreduction methodology and the resulting computationally efficient nonlinear MOR algorithm. We investigate the performance of the proposed scheme for hyperelastic microstructures within a homogenisation framework and conduct performance comparisons.
[1] F. Fritzen, M. Hodapp, The finite element square reduced (FE2R) method with GPU acceleration: Towards three-dimensional two-scale simulations, International Journal for Numerical Methods in Engineering 107 (10) (2016) 853–881. https://doi.org/10.1002/nme.5188
[2] L. Scheunemann, E. Faust, A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics (2024). arXiv:2408.12415
[3] C. Miehe, J. Schotte, J. Schröder, Computational micro–macro transitions and overall moduli in the analysis of polycrystals at large strains, Computational Materials Science 16 (1-4) (1999) 372–382. https://doi.org/10.1016/S0927-0256(99)00080-4
[4] A. Radermacher, S. Reese, POD-based model reduction with empirical interpolation applied to nonlinear elasticity, International Journal for Numerical Methods in Engineering 107 (6) (2016) 477–495. https://doi.org/10.1002/nme.5177
[5] D. Amsallem, M. J. Zahr, C. Farhat, Nonlinear model order reduction based on local reduced-order bases, International Journal for Numerical Methods in Engineering 92 (10) (2012) 891–916. https://doi.org/10.1002/nme.4371
[6] J. Barnett, C. Farhat, Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction, Journal of Computational Physics 464 (2022) 111348. https://doi.org/10.1016/j.jcp.2022.111348
