Proper general decomposition (PGD) for the resolution of Navier–Stokes equations

In this work, the PGD method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions. In the first stage, the equations of Stokes and Burgers will be solved. Then, we will solve the Navier–Stokes problem in the case of the lid-driven cavity for different Reynolds numbers (Re = 100, 1000 and 10,000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.Région Poitou-Charente

A reduced order variational multiscale approach for turbulent flows

The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark. © 2019, Springer Science+Business Media, LLC, part of Springer Nature

Stabilized Multiscale Nonconforming Finite Element Method for the Stationary Navier-Stokes Equations

We consider a stabilized multiscale nonconforming finite element method for the two-dimensional stationary incompressible Navier-Stokes problem. This method is based on the enrichment of the standard polynomial space for the velocity component with multiscale function and the nonconforming lowest equal-order finite element pair. Stability and existence uniqueness of the numerical solution are established, optimal-order error estimates are also presented. Finally, some numerical results are presented to validate the performance of the proposed method

A variational subgrid scale model for transient incompressible flows

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

Adjoint-based methods for optimization and goal-oriented error control applied to fluid-structure interaction: implementation of a partition-of-unity dual-weighted residual estimator for stationary forward FSI problems in deal.II

[EN] In this work, we implement goal-oriented error control and spatial mesh adaptivity for stationary fluid-structure interaction (FSI). The a posteriori error estimator is accomplished using the dual-weighted residual method in which the adjoint equation arises. The fluid-structure interaction problem is formulated within a variational-monolithic framework using arbitrary Lagrangian-Eulerian coordinates. The overall problem is nonlinear and solved with Newton’s method. We specifically consider the FSI-1 benchmark problem in which quantities of interest include the elastic beam displacements, drag, and lift. The implementation is based on the deal.II finite element library and provided open-source published on github https://github.com/tommeswick/goal-oriented-fsi. Possible extensions are discussed in the source code and in the conclusions of this paper.This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy within the cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).Wick, T. (2022). Adjoint-based methods for optimization and goal-oriented error control applied to fluid-structure interaction: implementation of a partition-of-unity dual-weighted residual estimator for stationary forward FSI problems in deal.II. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 257-266. https://doi.org/10.4995/YIC2021.2021.12332OCS25726