Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

Abstract

Many applications are based on the use of materials with heterogeneous microstructure. Prominent examples are fiber-reinforced composites, multi-phase steels or soft tissue to name only a few. The modeling of structures composed of such materials is suitably carried out at different scales. At the micro scale, the detailed microstructure is taken into account, whereas the modeling at the macro scale serves to include sophisticated structural geometries with complex boundary conditions. The procedure is crucially based on an intelligent bridging between the scales. One of the methods derived for this purpose is the meanwhile well established FE 2 method which, however, leads to a very high computational effort. Unfortunately, this impedes the use of the FE 2 method and similar methodologies for practically relevant problems as they occur e.g. in production or medical technology. The goal of the present paper is to significantly improve computational efficiency by using model reduction. The suggested procedure is very generally applicable. It holds for large deformations as well as for all relevant types of inelasticity. An important merit of the work is the computation of the consistent tangent operator based on the reduced stiffness matrix of the microstructure. In this way a very fast (in most cases quadratic) convergence within the Newton iteration at macro level is achieved.

An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites

“…Based on the acquired information from pre-off-line full-field micro-scale analyzes, the full-order governing equations are projected into a suitably selected reduced order space. In order to avoid the use of a large number of displacement degrees of freedom, the microscale model is solved with a reduced number of unknown variables which are defined by means of proper orthogonal decomposition of the displacement field [34,35]. Furthermore, hyper-reduction techniques can be applied to reduce the computation cost of the internal forces resulting from the evaluation of the local constitutive equations [36,37].…”

Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations via Order-Reduction Techniques

“…While POD-based order-reduction techniques have been commonly used to solve problems in computational fluid dynamics [7][8][9], these techniques have also been extended to include nonlinear solid mechanics problems [6,[10][11][12][13][14]. For instance, Radermacher et al [10] were able to demonstrate improvements of the computational speed by a factor of 60-260 by employing a POD-based order-reduction technique in the analysis of an inelastic metal matrix composite.…”

“…For instance, Radermacher et al [10] were able to demonstrate improvements of the computational speed by a factor of 60-260 by employing a POD-based order-reduction technique in the analysis of an inelastic metal matrix composite. POD techniques have also been implemented within a multiscale framework.…”

“…Yvonnet and He [13] were able to achieve significant computational and memory savings for multiscale simulations of hyperelastic media. Radermacher et al [10] demonstrated two orders of magnitude speedup in the computational time of nonlinear multiscale simulations by implementing POD at the microscale. Similarly, Ricks et al [17] obtained significant computational savings by imbedding HFGMC within a macroscale linearly elastic FE model.…”

Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods

“…In [16], a POD reduced basis was used for the first time in multiscale analysis of nonlinear elasticity at finite strains, namely by reducing the micro-scale model. This was extended in [17] by introducing the computation of a consistent tangent operator based on the reduced model. For problems with nonlinearities or non-affine parameter dependence, the sole application of a reduced basis does not render the desired computational savings as the nonlinearity or non-affine parameter dependence has to be evaluated for the original model and subsequently projected onto the reduced basis.…”