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A hyper reduced domain decomposition approach for modeling nonlinear heterogeneous structures
(2019)
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. However, an increase of accuracy by dissolving the fine scale inevitably leads to an increase in computational cost. In the context of multiscale simulations, the FE2 method is widely used. In a two-level computation, the fine scale is depicted by a boundary value problem for a representative volume element (RVE), which is then solved in each integration point of the macro scale to determine the macroscopic response. However, the FE2 approach in general is computationally expensive and problematic in the special case of concrete structures. Here rather large RVEs are necessary to sufficiently represent the meso-structure, such that separation of scales cannot be assumed.
Therefore, the aim is to develop an efficient approach to modeling nonlinear heterogeneous structures using domain decomposition and reduced order modeling.
Multiscale modeling of heterogeneous structures based on a localized model order reduction approach
(2023)
Many of today’s problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Thus, it is necessary to take into account the heterogeneous structure on the smaller scale, to capture the underlying physical phenomena. However, this poses a great challenge to the numerical solution since the computational cost is significantly increased by resolving the smaller scale in the model. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the smaller scale on the larger 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 smaller scale on the larger 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 smaller scale, is introduced. Second, local reduced approximation spaces for the smaller scale solution are constructed by exploring possible solutions for each subdomain based on the concept of oversampling [3]. The associated transfer operator is approximated by random sampling [4]. Herein, we propose to incorporate the actual physical behaviour of the structure of interest in the training data by drawing random samples from a multivariate normal distribution with the solution of a reduced global problem as mean. 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.
In this contribution, a methodology for fine scale modeling of large scale structures is proposed, which combines the variational multiscale method[1], domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modelled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Based on the work of Buhr and Smetana[2], local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.