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Mixed finite elements for elasticity

Mixed finite elements for elasticity,10.1007/s002110100348,Numerische Mathematik,Douglas N. Arnold,Ragnar Winther

Mixed finite elements for elasticity   (Citations: 89)
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Summary.   There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.
Journal: Numerische Mathematik - NUMER MATH , vol. 92, no. 3, pp. 401-419, 2002
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    • ...Not until the year 2002, was the first family of stable mixed finite elements [6] proposed for Hellinger-Reissner formulation, using polynomial shape functions with respect to a single arbitrary triangular mesh for both the stress and the displacement spaces...
    • ...As shown in [6], it is unavoidable to use vertex degrees of freedom for symmetric stress spaces and for conforming finite elements with continuous shape functions on an arbitrary triangulation...
    • ...The approach of the analysis comes from [2, 6]. In Sect...
    • ...The elasticity complex in two dimensions [6] takes the form:...
    • ...An analogous exact sequence with less smoothness [6 ]i s:...
    • ...To this end, we invoke [6] the fact that for each v ∈ L 2 (Ω, R 2 ) there exists a τ ∈ H 1 (Ω, S) such that...

    Shao-Chun Chenet al. Conforming Rectangular Mixed Finite Elements for Elasticity

    • ...The following two conditions are sufficient for (A1) and (A2) [6]: (A1) � divhΣh ⊂ Vh. (A2) � There exists a linear operator Πh : H 1 (Ω, S) ∩ H0,N(div, Ω, S) → Σh, bounded in L(H 1 , L 2 )...
    • ...In [6], Arnold and Winther proposed a family of stable conforming triangular elements satisfying the stability conditions (A1) � and (A2) � , and derived optimal order approximation...
    • ...We adopt some plane elasticity mixed finite element spaces, including Arnold and Winther’s conforming and nonconforming families [6, 7], and the nonconforming elements we propose in this contribution, for the approximation of the stress and velocity, whereas continuous piecewise polynomials for the approximation of the pressure...
    • ...T, S). There hold the following properties [6]...
    • ...Thus, by following a discussion similar to that in [6, 7], i.e., using the matrix Piola transformation and a standard scaling argument, we can show (A3) holds for Πh =Π (1) with m =1 , 2,...
    • ...They are: Arnold-Winther’s conforming element family [6], Arnold-Winther’s nonconforming elements [7], and the nonconforming elements proposed in Section 3...

    Xiaoping Xieet al. New mixed finite elements for plane elasticity and Stokes equations

    • ...We consider the nite element pairs ( Sh;Uh) of symmetric tensors and vectors constructed by Arnold and Winther [7,1] which satisfy the inf-sup condition...
    • ...In [7,1] an interpolation operator h : S\H1( )d d! Sh is constructed for which...

    Jason S. Howellet al. Inf–sup conditions for twofold saddle point problems

    • ...Conversely, the complementary energy principle (equilibrium method) that uses stresses as variables will give an upper bound on the global strain energy [11,12]...

    Z. C. Xuanet al. On computing upper and lower bounds on the outputs of linear elasticit...

    • ...This difficulty, if not circumvented, is severely restrictive (see [21] and [22, 23] for the analysis of admissible elements in linear elasticity)...

    M. Cerveraet al. Mixed stabilized finite element methods in nonlinear solid mechanics

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