An adaptive hyperreduced domain decomposition approach for nonlinear heterogeneous structures
Mechanics
Final Report Abstract
The aim of the project was to develop a methodology for the multiscale modeling of nonlinear heterogeneous structures. In this context, it is necessary to take into account the materials’ fine scale heterogeneity to accurately model the structure’s behaviour. However, capturing the underlying physical phenomena, e. g. , the formation of macrocracks, by resolving the fine scales in the numerical model also leads to a significant increase in the computational cost. Moreover, in the presence of such localization phenomena, the assumption of separation of scales which is at the basis of classical homogenization approaches is no longer valid. Therefore, it was proposed to model the nonlocal influence of the fine scale on the coarse scale directly by fully resolving the fine scale features in the numerical model. This was achieved by an additive decomposition of the displacement field in line with the variational multiscale method. Usually, in superposition-based multiscale methods one aims at static condensation (elimination of the fine scale term from the coarse scale equation) of the fine scale solution to enable an efficient solution procedure. However, this necessitates the fine scale solution to vanish along boundaries of coarse grid cells or leads to a Lagrange multiplier method for constraint enforcement (zero jump conditions). While the former does not depict the real physical situation, the latter introduces additional unknowns and poses difficulties in the solution of the global system of equations with state-of-the-art solvers. In contrast to this, the proposed methodology aimed at displacement compatibility between subdomains without the need of additional unknowns (Lagrange multipliers) or constraint equations. The main challenges with this approach were the efficient reduction in computational complexity without loss of accuracy and the conforming coupling of subdomains ensuring continuity in the displacement field. To address the computational complexity projection-based model order reduction techniques were used. The reduction was achieved by projecting the original system of equations onto a subspace of the solution manifold. In this context, this was done locally (on subdomain level), obtaining the local contribution of each subdomain. The coarse scale basis functions are defined locally by extending standard finite element (FE) shape functions on the boundary of local subdomains into the interior of the respective subdomains. The local reduced approximation space for the fine scale part was constructed by use of an oversampling problem, in which possible solutions on the target subdomain contained in the oversampling domain are explored by varying boundary conditions. Here, a coarse grid parametrization of the boundary conditions was introduced to incorporate the deformation states of the actual structure of interest (by solving a reduced global problem beforehand) into the construction of the reduced spaces. The solution obtained on the target domain was further restricted to the boundary of the target domain (edges of the coarse grid cell) which allowed to construct fine scale edge basis functions using proper orthogonal decomposition (POD). These fine scale edge basis functions were extended into the interior of the target subdomain to obtain the final fine scale basis. Moreover, the basis functions for the coarse and fine scale solution were designed to enable a conforming coupling. In the domain decomposition inherent to the variational multiscale ansatz this enabled an assembly procedure as in standard finite element assembly procedures and hence an easy implementation. The extension of the method to the nonlinear case is still subject of ongoing work. In this regard, informing the boundary conditions in the oversampling problem with the reduced global solution of the structure of interest is expected to be important for a high quality reduced basis. Additionally, the similarity of the constructed reduced basis with hierarchical FE shape functions should serve as the basis for an adaptive method.
Publications
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“A hyperreduced domain decomposition approach for modeling nonlinear heterogeneous structures”. In: ECCOMAS YIC 2019 5th Young Investigators Conference, Krakow (Poland). 2019
Philipp Diercks, Annika Robens-Radermacher, and Jörg F. Unger
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“A variational multiscale method to model linear elastic heterogeneous structures”. In: 14th WCCM and ECCO-MAS Congress 2020. 2020
Philipp Diercks, Karen Veroy, Annika Robens-Radermacher, and Jörg F. Unger
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“A Variational Multiscale Method for Heterogeneous Structures”. In: 16th US National Congress on Computational Mechanics. 2021
Philipp Diercks, Karen Veroy, Annika Robens-Radermacher, and Jörg F. Unger
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“Multiscale modeling of linear elastic heterogeneous structures based on a localized model order reduction approach”
Philipp Diercks, Karen Veroy, Annika Robens-Radermacher, and Jörg F. Unger
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“Reduced order approximations of fine scale edge basis functions within a variational multiscale approach”. In: 16th International Conference on Computational Plasticity. 2021
Philipp Diercks, Karen Veroy, Annika Robens-Radermacher, and Jörg F. Unger
