A family of mixed finite elements for the elasticity problem

We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We first apply dual-mixed approaches in both domains, and then employ the governing equations to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the Cauchy stress tensor and the rotation of the solid, together with the gradient of p and the traces of u and p on the boundary of the fluid, constitute the unknowns of the coupled problem. Next, we show that suitable decompositions of the spaces to which the stress and the gradient of p belong, allow the application of the Babuška-Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart-Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Finally, some numerical results illustrating the theory are presented.

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“…Indeed, following the usual procedure from linear elasticity (see [1], [6] and [19]), we first introduce the rotation…”

Section: The Continuous Variational Formulationmentioning

confidence: 99%

A priorierror analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

et al. 2013

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“…Stable mixed finite element for the elasticity problem are usually obtained by relaxing the symmetry of the stress tensor [28,10] which leads to the use of the so-called Hellinger-Reissner modified functional of the elasticity system. This yields to modify the elasticity system (1.1)-(1.6) as follows: We seek the displacement u : Ω → R 2 , the stress field σ : Ω → R 2 and γ : Ω → M 2×2 skew := {η ∈ R 2×2 : η + η t = 0} such that…”

Section: Variational Formulationmentioning

confidence: 99%

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Belhachmi¹,

Sac-Epee²,

Tahir³

2009

Math. Model. Nat. Phenom.

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Abstract. We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

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“…Additionally, following Gil and Ortigosa [1,2], we propose an extended Hu-Washizu [44,52,54,54,[69][70][71][71][72][73][74][75][76][77][78][79][80] mixed variational principle for nearly and truly incompressible scenarios. Considering now interpolation spaces which satisfy the LBB condition, a comparison of this enhanced methodology is carried out in this paper against the three-field formulation, where both LBB compliant and non-compliant interpolation spaces are considered.…”

Section: Introductionmentioning

confidence: 99%

“…The off-diagonal contributions K xp and K px in above (72) for the variational principle Π W N IC (40) yields…”

mentioning

confidence: 99%

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

Ortigosa

Gil

Lee

2016

Computer Methods in Applied Mechanics and Engineering

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Please cite this article as: R. Ortigosa, A.J. Gil, C.H. Lee, A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10. 1016/j.cma.2016.06.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies. AbstractThe series of papers published by Gil and Ortigosa [1-3] introduced a new convex multi-variable variational and computational framework for the numerical simulation of Electro Active Polymers (EAPs) in scenarios characterised by extreme deformations and/or extreme electric fields. Building upon this body of work, five key novelties are incorporated in this paper. First, a generalisation of the concept of multi-variable convexity to energy functionals additively decomposed into isochoric and volumetric components. This decomposition is typical of nearly and truly incompressible materials, group which represents the majority of the most relevant EAPs. Second, convexification or regularisation strategies are applied to a priori non-convex multi-variable isochoric functionals to yield physically meaningful convex multi-variable functionals. Third, based on the mixed variational principles introduced in Reference [1] in the context of compressible electro-elasticity, a novel extended Hu-Washizu mixed variational principle for nearly and truly incompressible scenarios is presented. From the computational standpoint, a static condensation procedure is applied in order to condense out the element-wise extra fields, the resulting formulation having a comparable cost to the more standard three-field displacement-potential-pressure mixed formulation. Fourth, the computational framework for the three-field mixed variational principle in nearly and truly incompressible scenarios is also presented. In this case, the novelty resides in the consideration of convex multi-

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A family of mixed finite elements for the elasticity problem

Cited by 287 publications

References 24 publications

A priorierror analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

A priorierror analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

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