Empirical Hyper Element Integration Method (EHEIM) with Unified Integration Criteria for Efficient Hyper Reduced FE${}^2$ Simulations

Abstract

Numerical homogenization for mechanical multiscale modeling by means of the finite element method (FEM) is an elegant way of obtaining structure-property relations, if the behavior of the constituents of the lower scale is well understood. However, the computational costs of this so-called FE${}^2$ method are so high that reduction methods are essential. While the construction of a reduced basis for the microscopic nodal displacements using proper orthogonal decomposition (POD) has become a standard technique, the reduction of the computational effort for the projected nodal forces, the so-called hyper reduction, is an additional challenge, for which different strategies have been proposed in the literature. The empirical cubature method (ECM), which has been proven to be very robust, implemented the conservation of the total volume is used as a constraint in the resulting optimization problem, while energy-based criteria have been proposed in other contributions. The present contribution presents a unified integration criteria concept, involving the aforementioned criteria, among others. These criteria are used both with a Gauss point-based as well as with an element-based hyper reduction scheme, the latter retaining full compatibility with the common modular finite element framework. The methods are combined with a previously proposed clustered training strategy and a monolithic solver. Numerical examples empirically demonstrate that the additional criteria improve the accuracy for a given number of modes. Vice verse, less modes and thus lower computational costs are required to reach a given level of accuracy.

References in corpus (7)

• FE${}^{\text{ANN}}$ $-$ An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining
• A hybrid approach to simulate the homogenized irreversible elastic-plastic deformations and damage of foams by neural networks
• A comparative study on different neural network architectures to model inelasticity
• On-the-fly adaptivity for nonlinear twoscale simulations using artificial neural networks and reduced order modeling
• High performance reduction technique for multiscale finite element modeling (HPR-FE2): towards industrial multiscale FE software
• Reduced-order modeling for second-order computational homogenization with applications to geometrically parameterized elastomeric metamaterials
• A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics