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In analyzing large scale structures, it is necessary to take into account the material heterogeneity for accurate failure prediction. However, this greatly increases the degrees of freedom in the numerical method thus making it infeasible. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the fine scale on the coarse scale
has to be modelled directly.
This work aims to develop an efficient methodology to model heterogeneous structures combining the variational multiscale method and model order reduction techniques. Superposition-based methods assume a split of the solution field into coarse and fine scale contributions. In deriving practical methods, some form of localization is necessary to eliminate the fine-scale part from the coarse-scale equation. Hund and Ramm [2] discussed different locality constraints and in particular zero jump conditions enforced by a Lagrange-type method leading to a coupled solution scheme.
In this contribution, a combination of the variational multiscale method and model order reduction techniques is applied to model the influence of the fine scale on the coarse scale directly. First, possible coarse and fine scale solutions are exploited for a representative volume element (RVE), specific to the material of interest, to construct local approximation spaces. For the local fine scale spaces different choices are presented, which ensure continuity between adjacent coarse grid elements. Therefore,the resulting global system takes into account, the effect of the fine scale on the coarse scale, is sparse and has much lower dimensions compared to the full system in the direct numerical simulation.
The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870. This result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 818473).
In analyzing large scale structures, it is necessary to take into account the material heterogeneity for accurate failure prediction. However, this greatly increases the degrees of freedom in the numerical method making it infeasible. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the fine scale on the coarse scale has to be modelled directly.
This work aims to develop an efficient methodology to model heterogeneous structures combining the variational multiscale method and model order reduction techniques.
Superposition based methods assume a split of the solution field into coarse and fine scale contributions. In deriving practical methods some form of localization is necessary to eliminate the fine scale part from the coarse scale equation. Hund and Ramm discussed different locality constraints and resulting solution procedures in the context of solid mechanics. Particularly, zero jump conditions ensuring continuity of the fine scale solution which are enforced by a Lagrange type method lead to a coupled solution procedure.
In this contribution, a combination of the variational multiscale method and model order reduction techniques is applied to model the influence of the fine scale on the coarse scale directly. First, possible coarse and fine scale solutions are exploited for a representative volume element (RVE), specific to the material of interest, to construct local approximation spaces. The local spaces are designed such that local contributions of RVEs can be coupled in a conforming way. Therefore, the resulting global system takes the effect of the fine scale on the coarse scale into account, is sparse and reduced in size compared to the direct numerical simulation.
The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC Grant agreement No. 818473).
Multiscale modeling of heterogeneous structures based on a localized model order reduction approach
(2022)
Many of today’s problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Herein, it is necessary to take into account the heterogeneous structure on the lower scale, to capture the underlying physical phenomena. However, this poses a great challenge to the numerical solution as the computational cost is significantly increased by resolving the lower scale in the model. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the lower scale on the upper scale has to be modelled directly. This work aims to develop an efficient concurrent methodology to model heterogeneous structures combining the variational multiscale method (VMM) [1] and model order reduction techniques (e. g. [2]). First, the influence of the lower scale on the upper scale can be taken into account following the additive split of the displacement field as in the VMM. Here, also a decomposition of the global domain into subdomains, each containing a fine grid discretization of the lower scale, is introduced. Second, reduced approximation spaces for the upper and lower scale solution are constructed by exploring possible solutions for each subdomain based on a representative unit cell. The local reduced spaces are designed such that local contributions of each subdomain can be coupled in a conforming way. Thus, the resulting global system is sparse and reduced in size compared to the direct numerical simulation, leading to a faster solution of the problem. The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Grant agreement No. 818473).