Abstract

We introduce in this paper a variational subgrid scale model for the solution of the incompressible Navier-Stokes equations. With respect to classical multiscale-based stabilisation techniques, we retain the subgrid scale effects in the convective term and integrate the subgrid scale equation in time. The method is applied to the Navier-Stokes equations in an accelerating frame of reference and with Dirichlet (essential), Neumann (natural) and mixed boundary conditions. The concrete objective of the paper is to test a numerical algorithm for solving the non-linear subgrid scale equation and the introduction of the subgrid scale into the grid scale equation. The performance of the technique is demonstrated through the solution of two numerical examples: one to test the tracking of the subgrid scale in the convection term and the other to investigate the effects of considering the subgrid scale transient.

Abstract

Fiber network modeling can be used for studying mechanical properties of paper [1]. The individual fibers and the bonds in-between constitute a detailed representation of the material. However, detailed microscale fiber network models must be resolved with efficient numerical methods. In this work, a numerical multiscale method for discrete network models is proposed that is based on the localized orthogonal decomposition method [4]. The method is ideal for these network problems, because it reduces the maximum size of the problem, it is suitable for parallelization, and it can effectively solve fracture propagation. The problem analyzed in this work is the nodal displacement of a fiber network given an applied load. This problem is formulated as a linear system that is solved by using the aforementioned multiscale method. To solve the linear system, the multiscale method constructs a low-dimensional solution space with good approximation properties [5, 2]. The method is observed to work well for unstructured fiber networks, with optimal rates of convergence obtainable for highly localized configurations of the method.

Abstract

This work aims at deriving special types of one-dimensional Finite Elements (1D FE) for efficiently modeling heterogeneous prismatic structures, in the small strains regime, by means of reduced-order modeling (ROM) and domain decomposition techniques. The employed partitioning framework introduces “fictitious” interfaces between contiguous subdomains, leading to a formulation with both subdomain and interface fields. We propose a low-dimensional parameterization at both subdomain and interface levels by using reduced-order bases precomputed in an offline stage by applying the Singular Value Decomposition (SVD) on solution snapshots. In this parameterization, the amplitude of the fictitious interfaces play the role of coarse-scale displacement unknowns. We demonstrate that, with this partitioned framework, it is possible to arrive at a solution strategy that avoids solving the typical nested local/global problem of other similar methods (such as the FE2method). Rather, in our approach, the coarse-grid cells can be regarded as special types of finite elements, whose nodes coincides with the centroids of the interfaces, and whose kinematics are dictated by the modes of the “fictitious” interfaces. This means that the kinematics of our coarse-scale FE are not pre-defined by the user, but extracted from the set of “training” computational experiments. Likewise, we demonstrate that the coarse-scale and fine-scale displacements are related by inter-scale operators that can be precomputed in the offline stage. Lastly, a hyperreduced scheme is considered for the evaluation of the internal forces, allowing us to deal with possible material nonlinearities.

Abstract