PGDrome

A FEniCS based python module of the Proper Generalized Decomposition (PGD) method.If you use this software, please cite it using the metadata from this file

Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction

In analyzing large scale structures it is necessary to take into account the fine scale heterogeneity for accurate failure prediction. Resolving fine scale features in the numerical model drastically increases the number of degrees of freedom, thus making full fine-scale simulations infeasible, especially in cases where the model needs to be evaluated many times. In this paper, a methodology for fine scale modeling of large scale structures is proposed, which combines the variational multiscale method, domain decomposition and model order reduction. To address applications where the assumption of scale separation does not hold, the influence of the fine scale on the coarse scale is modelled directly by the use of an additive split of the displacement field. Possible coarse and fine scale solutions are exploited for a representative coarse grid element (RCE) to construct local approximation spaces. The local spaces are designed such that local contributions of RCE subdomains can be coupled in a conforming way. Therefore, the resulting global system of equations takes the effect of the fine scale on the coarse scale into account, is sparse and reduced in size compared to the full order model. Several numerical experiments show the accuracy and efficiency of the method

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

The High-Fidelity Generalized Method of Cells (HFGMC) is one technique, distinct from traditional finite-element approaches, for accurately simulating nonlinear composite material behavior. In this work, the HFGMC global system of equations for doubly periodic repeating unit cells with nonlinear constituents has been reduced in size through the novel application of a Petrov-Galerkin Proper Orthogonal Decomposition order-reduction scheme in order to improve its computational efficiency. Order-reduced models of an E-glass/Nylon 12 composite led to a 4.8–6.3x speedup in the equation assembly/solution runtime while maintaining model accuracy. This corresponded to a 21–38% reduction in total runtime. The significant difference in assembly/solution and total runtimes was attributed to the evaluation of integration point inelastic field quantities; this step was identical between the unreduced and order-reduced models. Nonetheless, order-reduced techniques offer the potential to significantly improve the computational efficiency of multiscale calculations