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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 often heterogeneous structure on the fine scale, to capture the underlying physical phenomena. Despite ever increasing Computational resources, dissolving the fine scales in a direct numerical simulation is prohibitive. This work aims to develop an efficient approach to modeling nonlinear heterogeneous structures using the variational multiscale method (VMM) and model order reduction (MOR).
The VMM, introduced in, assumes an additive split of the solution into coarse and fine scale contributions. In, the VMM is applied to a damage mechanics–based material model for concrete-like materials. Herein, suitable boundary conditions for the fine scale which enable localization phenomena to evolve are discussed. As such, zero jump conditions between fine scale solutions are proposed which are enforced pointwise by a Lagrange type method leading to a coupled solution procedure.
In this contribution, possible extensions of the VMM with reduced order modeling are presented. In the linear case, assuming the fine scale solution to be zero on coarse scale element boundaries allows for static condensation and a decoupled solution procedure. Based on this, an efficient localized Training strategy will be developed. For the nonlinear case, the situation of coupled non-conforming spaces, i. e. finite element and reduced order spaces for the fine scales, arises. Thus the imposition of suitable fine scale interface conditions in the weak sense by the use of Lagrange multipliers is investigated. Specific problems in solid mechanics are used to illustrate the performance of the above approaches.
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).