Efficient stress–velocity least-squares finite element formulations for the incompressible Navier–Stokes equations

et al. 2019

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In this contribution an approach to model fluid-structure interaction (FSI) problems with monolithic coupling is presented. The fluid as well as the structural domain are discretized using the least-squares finite element method (LSFEM), whose application results in a minimization problem with symmetric positive definite systems also for non self-adjoint operators, see e.g. [2]. In this study, the second-order systems are reduced to first-order systems by introducing new variables, which leads to least-squares formulations for both domains based on the stresses and velocities as presented in e.g.[5] and [7]. A conforming discretization of the unknown fields in H 1 and H(div) using Lagrange interpolation polynomials and vector-valued Raviart-Thomas interpolations functions, respectively, leads to the inherent fulfillment of the FSI coupling conditions. In more detail, a discretization in H 1 ensures continuity of the velocity field and a discretization in H(div) results in continuity of the normal stress components at the interface.

Section: Theoretical Frameworkmentioning

confidence: 99%

A least‐squares finite element approach to model fluid‐structure interaction problems

Averweg

et al. 2019

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“…with v denoting the velocity field, ρ f the density of the fluid, ν f the kinematic viscosity of the fluid and the Cauchy stress tensor σ. A detailed investigation of this stress-velocity formulation can be found in [6]. The stress-velocity formulation for linear elastodynamics is given with the scalar weighting factors ω 1 and ω 2…”

Section: A Least-squares Finite Element Based Fluid-structure Interacmentioning

confidence: 99%

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

Averweg

et al. 2018

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In the present contribution a fluid-structure interaction (FSI) approach based on a mixed least-squares finite element method (LSFEM) is introduced. The LSFEM is a variational approach for constructing finite element formulations, see e.g.[1] and [2]. In this work, the proposed FSI method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The conforming discretization of the stresses and velocities in the spaces H(div) and H 1 , respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. The idea of the LSFEM based FSI scheme is the solution of the full problem consisting of both domains in one triangulation at the same time. The general idea on this was firstly annotated in [8]. A numerical example considering an incompressible, linear elastic solid at small deformations (compare to [3]) together with an incompressible laminar fluid flow demonstrates the applicability of the LSFEM-FSI approach.

“…with || • || 2 B denoting the quadratic L 2 -norm. The discretization in space for the mixed least-squares formulations follows the assumptions in [4] and others. The approximation spaces are…”

Section: Theoretical Frameworkmentioning

confidence: 99%

“…Further details on both stress-velocity formulations can be found in [4], including detailed benchmarking and evaluations regarding efficiency and accuracy.…”

Section: Theoretical Frameworkmentioning

confidence: 99%

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Investigation of stress‐velocity LSFEMs for the incompressible Navier‐Stokes equations

Schröder

2017

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In this contribution three mixed least-squares finite element methods (LSFEMs) for the incompressible Navier-Stokes equations are investigated with respect to accuracy and efficiency. The well-known stress-velocity-pressure formulation is the basis for two further div-grad least-squares formulations in terms of stresses and velocities (SV). Advantage of the SV formulations is a system with a smaller matrix size due to a reduction of the degrees of freedom. The least-squares finite element formulations, which are investigated in this contribution, base on the incompressible stationary Navier-Stokes equations. The first formulation under consideration is the stress-velocity-pressure formulation according to [1]. Secondly, an extended stressvelocity formulation with an additional residual is derived based on the findings in [1] and [5]. The third formulation is a pressure reduced stress-velocity formulation based on a condensation scheme. Therefore, the pressure is interpolated discontinuously, and eliminated on the discrete level without the need for any matrix inverting. The modified lid-driven cavity boundary value problem, is investigated for the Reynolds number Re = 1000 for all three formulations.