Upscaling and reliable two-scale Fourier/finite element-based simulations
Mechanics
Mechanical Properties of Metallic Materials and their Microstructural Origins
Final Report Abstract
The broad aim of multi-scale computational engineering is to provide accurate predictions for physical quantities of interest by means of computer simulations of phenomena at different scales of resolution. Computational homogenisation the first focus of this project, it deals with heterogeneous materials with a complex microstructure. In the basic two-scale scenario, two widely separated length-scales are considered: the macro-scale, corresponding to structural engineering simulations, and the microscale, related to the characteristic size of material inhomogeneities or defects. Here in particular the focus was on elastic problems in the small strain regime, where perfect homogenisation is possible, i.e. one can find homogeneous elastic properties on the macro-scale which for macro-scale strain fields show the same stored energy as the heterogeneous micro-scale properties. This kind of homogenisation was first developed for periodic micro-scale arrangements, and this is where Fourier methods, and especially the fast Fourier transform (FFT) homogenisation (FFTH) approach, fit in perfectly as a fast computational tool. One first step is to develop this kind of homogenisation for materials which do not have a deterministic periodic microstructure, but instead are random but stochastically spatially homogeneous, i.e. the probabilistic description is the same everywhere in space. Mathematically, such homogeneous random fields (RFs) have correlations which again allow a series expansion in Fourier functions (trigonometric polynomials). This makes them ideal for FFT based computational methods such as FFTH. One additional feature which comes up already in the FFTH discretisation, and permeates the description of RFs, and especially their numerical treatment with Karhunen-Loève and polynomial chaos expansions, is the underlying tensor product structure — which is basically a possibility of a separation of variables. As in the classical separation of variables settings, this tensor product structure offers huge computational gains. This idea has been pursued successfully throughout the project. Hence from an algorithmic point of view, this introduction of low-rank tensor approximations has greatly accelerated the FFTH method, and given rise to a new generation of even faster solvers. The numerical discretisation framework for stochastic problems is theoretically also wedded to tensor formulations. Thus analogous low-rank tensor techniques have been used in the numerical treatment of RFs, and the solvers for stochastic problems. The idea of separation of variables extends even to the coupling of different scales in the form of strong coupling algorithms, and to appropriate computational procedures, both on the theoretical numerical as well as on the algorithmic and software-engineering level. In this context, methods have been developed to not only have random material properties on the micro-scale, but also stochastic micro-structural arrangements of different materials (or material phases). This is shown in the generation of micro-structures with inclusions of different shapes and roughness and porous materials with high volume rations, as well as for example random fibrous materials, both of these occur for example in bones. For nonlinear and especially irreversible inelastic material behaviour, thermodynamic theory indicates that not only stored energy (Helmholtz free energy) but also dissipation and entropy production have to be considered. The dissipation resp. entropy production has been taken as the criterion for upscaling, and it turns out that perfect homogenisation is not possible in this regime. The problem has therefore been extended to allow a stochastic description also on the macro-scale. The method of identification is based on Bayes’s theorem and is thus a very general approach, also underlying much of the machine learning methods. This allows the scale coupling of very diverse numerical models. The randomness on the macro-scale to some extent reflects the loss of resolution in the upscaling, and to another the inherent randomness on the micro-scale. These upscaling procedures are in principle independent of any numerical discretisation approaches, but were first developed in the context of the finite element method (FEM), and also for the combination of completely discrete micro-structural models and FEM. An efforts was made to carry all these developments over to Fourier discretisations, and additionally taking advantage of modern low-rank approximation and model reduction techniques.
Publications
- (2018). Tucker Tensor Analysis of Matérn Functions in Spatial Statistics, Computational Methods in Applied Mathematics, 19(1), 101–122
A. Litvinenko, D. Keyes, V. Khoromskaia, B. N. Khoromskij, and H.G. Matthies
(See online at https://doi.org/10.1515/cmam-2018-0022) - (2019). Double-grid quadrature with interpolation-projection (DoGIP) as a novel discretisation approach: An application to FEM on simplexes. Computers and Mathematics with Applications, 78(11), 3501–3513
J. Vondřejc
(See online at https://doi.org/10.1016/j.camwa.2019.05.021) - (2020). Energy-based comparison between the Fourier-Galerkin method and the finite element method, Journal of Computational and Applied Mathematics, 374, 112585
J. Vondřejc and T.W.J. de Geus
(See online at https://doi.org/10.1016/j.cam.2019.112585) - (2020). FFT-based homogenisation accelerated by lowrank tensor approximations, Computer Methods in Applied Mechanics and Engineering, 364, 112890
J. Vondřejc, D. Liu, M. Ladecký, and H.G. Matthies
(See online at https://doi.org/10.1016/j.cma.2020.112890) - (2020). Iterative algorithms for the post-processing of high-dimensional data, Journal of Computational Physics, 491, 109396
M. Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies, and E. Zander
(See online at https://doi.org/10.1016/j.jcp.2020.109396) - (2021). Computing f-Divergences and Distances of High-Dimensional Probability Density Functions — Low-Rank Tensor Approximations. Num. Lin. Alg. Appl.
A. Litvinenko, Y. Marzouk, H.G. Matthies, M. Scavino, and A. Spantini
(See online at https://doi.org/10.48550/arXiv.2111.07164)
