An adaptive hyperreduced domain decomposition approach for nonlinear heterogeneous structures

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Almost all engineering materials are intrinsically multiscale. The often heterogeneous fine scales are important to describe the macroscopic behavior with physics based models. An increase of the accuracy by dissolving finer scales is accompanied with a significant increase of the computational cost. The resolution of the fine scale for a whole realistic building structure is not possible due to limited computational resources. The purpose of the proposal is to develop new computationally efficient discretization and solution techniques for complex, heterogeneous structures.Many multiscale methods are based on FE² type of approaches with a nested solution strategy. The finer scale is represented by a representative volume element (RVE) evaluated in each integration point of the coarse scale structure to compute the material response. As a result, the computational effort increases with the complexity of the RVE. Especially in case of concrete, where the RVE is relatively large and a scale separation can often not be achieved, these computational homogenization techniques are very expensive. Alternative approaches are domain decomposition techniques. In this case, the structure is divided into sub-domains resulting in a high number of small problems which can be solved in parallel. The FETI method couples each disconnected sub-domain using Lagrange multipliers. Using the advantage of parallel computing, these kind of methods are computationally very efficient. Furthermore, model reduction techniques have been developed in the last decades and are recently used in a variety of applications. Model reduction is a popular and powerful tool to decrease the computational effort of complex numerical simulations. The key idea is the projection of the system to a lower dimensional space which represent the overall behavior in a best possible way.The idea of the proposal is a combination of model reduction techniques with domain decomposition to solve complex and realistic heterogeneous mesoscale problems. The macroscopic domain is decomposed into representative subdomains. Based on offline simulation with periodic boundary conditions, a global set of basis functions for each subdomain is determined. In an adaptive scheme based on clustering techniques, a subset of these basis functions is determined to accurately approximate the displacement field. In this way, the order of reduction is directly linked to the nonlinearity of the solution - from linear elastic solutions with only six degrees of freedom per sub-domain up to a discretization with the complete set of basis functions. In addition, a hyperreduction approach is used to ensure an efficient speedup of the reduced order model for parallel implementations.The coupling of the reduced sub-domains is based on the FETI framework by enforcing a weak constraint of the displacement field at the interfaces. The applicability of the method will be shown for specific examples.