Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

Codina, Ramon

doi:10.1016/s0045-7825(02)00337-7

Cited by 399 publications

(628 citation statements)

References 42 publications

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“…Some technical details may be omitted by considering that the subscale enhancements behave as bubble functions, vanishing on the elements boundaries. They can be viewed as a "high frequency" perturbation of the finite element field, which cannot be resolved in V h [12,21].…”

Section: Variational Multiscale Stabilizationmentioning

confidence: 99%

“…For this purpose, it is necessary to approximate the spatial and time differential operators that appear in the variational statement in discrete form. Regarding the spatial derivatives, Fourier's analysis of the problem for the subscales [21] shows that, within each element domain Ω e , the following algrebraic approximations can be used for their respective norms:…”

Section: Variational Multiscale Stabilizationmentioning

confidence: 99%

“…This is called the Orthogonal Sub-Scales (OSS) method [20,21,22]. In the present case, this means that the strain and displacement spaces are approximated as T ≃ T h ⊕ T ⊥ h and V ≃ V h ⊕ V ⊥ h , respectively.…”

Section: Orthogonal Subscale Stabilizationmentioning

confidence: 99%

“…Three field formulations ε− u − p have also been proposed for the incompressible limit [17,7]. All these mixed schemes are stabilized via the Variational Multi-Scale method (VMS) [21,31,32,33,41], and, specifically, the Orthogonal Subgrid Scales method, in order to gain control of all the variables while circumventing the restrictiveness of the inf-sup compatibility conditions on the choice of the interpolation spaces. Similar mixed approaches have been proposed using alternative stabilization techniques, like SUPG [4,5,29,36] or FIC [41].…”

Section: Introductionmentioning

confidence: 99%

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Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

et al. 2016

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K: explicit mixed finite elements, stabilization, incompressibility, plasticity, strain softening, strain localization, mesh independence. AbstractThis paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales (VMS). A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P 1P 1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr-Coulomb and Drucker-Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.

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Section: Variational Multiscale Stabilizationmentioning

confidence: 99%

Section: Variational Multiscale Stabilizationmentioning

confidence: 99%

Section: Orthogonal Subscale Stabilizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

et al. 2016

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“…This approach was proved to show better mesh independence properties while providing the formal guarantee of a strain field converging at the same rate as the displacement one, which manifests in enhanced stress/analysis accuracy in both linear and non-linear analyses [8]. Following the ideas in [9,10,11,12,13,14], a stabilization technique is needed to allow the use of the same order of interpolation for the two primary variables of interest. Specifically the Variational Multiscale Method (VMS) is employed in the current work.…”

Section: Introductionmentioning

confidence: 99%

Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

et al. 2015

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Aquesta és una còpia de la versió author's final draft d'un article publicat a la revista Computational mechanics. Although appealing for their simplicity, low-order finite elements face inherent limitations related to their poor convergence properties. Such difficulties typically manifest as mesh-dependent or excessively stiff behaviour when dealing with complex problems. A recent proposal to address such limitations is the adoption of mixed displacement-strain technologies which were shown to satisfactorily address both problems. Unfortunately, although appealing, the use of such element technology puts a large burden on the linear algebra, as the solution of larger linear systems is needed. In this paper, the use of an explicit time integration scheme for the solution of the mixed strain-displacement problem is explored as an alternative. An algorithm is devised to allow the effective time integration of the mixed problem. The developed method retains second order accuracy in time and is competitive in terms of computational cost with the standard irreducible formulation.

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Multiscale and Stabilized Methods

Hughes

Scovazzi

Franca

2004

Encyclopedia of Computational Mechanics

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This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite‐element methods.) The analytical ideas are first illustrated for time‐harmonic wave‐propagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the well‐known Dirichlet‐to‐Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented, which is applicable to advective–diffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid‐scale models and a posteriori error estimation. Hierarchical p ‐methods and bubble function methods are examined in order to understand and ultimately approximate the ‘fine‐scale Green's function’ that appears in the theory. Relationships among so‐called residual‐free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space–time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods (e.g. SUPG, GLS, etc.) that have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective–diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence is also described. This represents an alternative formulation of large eddy simulation (LES), which provides a simplified theoretical framework of LES with potential for improved modeling.

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Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

Cited by 399 publications

References 42 publications

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

Multiscale and Stabilized Methods

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