A stress‐velocity least‐squares mixed finite element formulation for incompressible elastodynamics

Abstract

In the framework of solving elastodynamic problems using a least-squares mixed finite element method (LSFEM) the implementation of a stress-velocity formulation for small strains is introduced and discussed in the present contribution. The element formulation is based on a first-order div − grad system, with the balance equation of momentum and the constitutive law as the governing equations. Application of the L2-norm to the two residuals leads to a functional depending on stresses and velocities. Different time discretization schemes are considered, a scalar weighting is introduced and chosen in dependency of the different time discretization methods. In a numerical example the influence of the time integration method, the chosen time step width and the related weighing factor are investigated for a two-dimensional problem.

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

“…More details on the constructed LS formulation can be found e.g. in [7]. For the description of the time-dependent flow of an incompressible fluid, a first-order system can be constructed based on the Navier-Stokes equations by introducing the Cauchy stresses σ = 2ρ f ν f ∇ s v − p1.…”

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In this contribution an approach to model fluid-structure interaction (FSI) problems with monolithic coupling is presented. 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]. [2].

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“…[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.…”

“…[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.…”

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

“…The constitutive relation is given with the elasticity tensor C ijkl = λ δ ij δ kl + 2µ δ ik δ jl where λ and µ denote Lamé constants. The displacement u and the acceleration a are functions of the velocity field v. A detailed investigation on suitable time discretization schemes for the time-dependent solid formulation is given in [3]. Here, the Houbolt method is employed with reasonable small time steps of ∆t ∈ {0.001, 0.005}.…”