On the Stabilization of Finite Element Approximations of the Stokes Equations

Brezzi, Franco; Pitkäranta, Juhani

doi:10.1007/978-3-663-14169-3_2

Cited by 302 publications

(354 citation statements)

References 8 publications

(5 reference statements)

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“…In particular we propose a modification of the technique used in [66] that penalizes the incompressibility equation with the gradient of the pressure. This approach recovers a strategy already proposed in [68] for finite elements.…”

Section: Introductionmentioning

confidence: 73%

“…Indeed, for low order finite elements, the first part of the consistent term is zero. A stabilization approach that involves only the gradient of the pressure has been also independently proposed in [68]. This simplification avoids the computation of the second derivatives of the basis functions, which can be cumbersome and ill-posed near the boundary [28,69].…”

Section: The Stabilized Viscoplastic Formulationmentioning

confidence: 99%

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A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

et al. 2014

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The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

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Section: Introductionmentioning

confidence: 73%

Section: The Stabilized Viscoplastic Formulationmentioning

confidence: 99%

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

et al. 2014

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“…This gives an inconsistently stabilized method similar to the one described in [14], and so some loss of accuracy is to be expected.…”

Section: Motivating Computational Experimentsmentioning

confidence: 99%

On stabilized finite element methods for the Stokes problem in the small time step limit

Bochev

Gunzburger

Lehoucq

2006

Numerical Methods in Fluids

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SUMMARYRecent studies indicate that consistently stabilized methods for unsteady incompressible flows, obtained by a method of lines approach, may experience difficulty when the time step is small relative to the spatial grid size. Using as a model problem the unsteady Stokes equations, we show that the semi-discrete pressure operator associated with such methods is not uniformly coercive. We prove that for sufficiently large (relative to the square of the spatial grid size) time steps, implicit time discretizations contribute terms that stabilize this operator. However, we also prove that if the time step is sufficiently small, then the fully discrete problem necessarily leads to unstable pressure approximations. The semi-discrete pressure operator studied in the paper also arises in pressureprojection methods, thereby making our results potentially useful in other settings.

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“…In order to precisely fulfill the kinematic interface condition (5) and apply the Master-Slave Relation, we need to reformulate the structure equation in terms of the structure velocityv s instead of its displacement u s in (15). We use the following relation ofv s andû ŝ (20) to reformulate (15), and rewrite the kinematic condition (5) in reference configuration aŝ…”

Section: Application Of Ale Methods To the Fsi Problemmentioning

confidence: 99%

“…We use the following relation ofv s andû ŝ (20) to reformulate (15), and rewrite the kinematic condition (5) in reference configuration aŝ…”

Section: Application Of Ale Methods To the Fsi Problemmentioning

confidence: 99%

Modeling and simulations for fluid and rotating structure interactions

Yang

Sun

Wang

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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In this paper, we study a dynamic fluid-structure interaction (FSI) model for an elastic structure that is immersed and spinning in the fluid. We develop a linear constitutive model to describe the motion of a rotational elastic structure which is suitable for the application of arbitrary LagrangianEulerian (ALE) method in FSI simulation. Additionally, a novel ALE mapping method is designed to generate the moving fluid mesh while the deformable structure spins in a non-axisymmetric fluid channel. The structure velocity is adopted as the principle unknown to form a monolithic saddle-point system together with fluid velocity and pressure. We discretize the nonlinear saddle-point system with mixed finite element method and Newton's linearization, and prove that the derived saddle-point problem is well-posed. The developed methodology is applied to a self-defined elastic structure and a realistic hydroturbine under a prescribed angular velocity. Both illustrate the satisfactory numerical results of an elastic structure that is deforming and rotating while interacting with the fluid. The numerical validation is also conducted to demonstrate the modeling consistency.

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On the Stabilization of Finite Element Approximations of the Stokes Equations

Cited by 302 publications

References 8 publications

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

On stabilized finite element methods for the Stokes problem in the small time step limit

Modeling and simulations for fluid and rotating structure interactions

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