scholarly journals RECTANGULAR MIXED FINITE ELEMENTS FOR ELASTICITY

2005 ◽  
Vol 15 (09) ◽  
pp. 1417-1429 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
GERARD AWANOU
Keyword(s):  
Finite Elements ◽  
Vector Fields ◽  
Mixed Finite Elements ◽  
Symmetric Tensor ◽  
Plane Elasticity ◽  
Discrete Version ◽  
Tensor Fields ◽  
Piecewise Polynomials ◽  
The Family ◽  
Differential Complex

We present a family of stable rectangular mixed finite elements for plane elasticity. Each member of the family consists of a space of piecewise polynomials discretizing the space of symmetric tensor fields in which the stress field is sought, and another to discretize the space of vector fields in which the displacement is sought. These may be viewed as analogues in the case of rectangular meshes of mixed finite elements recently proposed for triangular meshes. As for the triangular case the elements are closely related to a discrete version of the elasticity differential complex.

TANGENTIAL-DISPLACEMENT AND NORMAL–NORMAL-STRESS CONTINUOUS MIXED FINITE ELEMENTS FOR ELASTICITY

2011 ◽  
Vol 21 (08) ◽  
pp. 1761-1782 ◽  
Author(s):  
ASTRID PECHSTEIN ◽  
JOACHIM SCHÖBERL
Keyword(s):  
Finite Elements ◽  
Normal Stress ◽  
Poisson Ratio ◽  
Mixed Finite Elements ◽  
Symmetric Tensor ◽  
Numerical Results ◽  
Tangential Displacement ◽  
Shear Locking ◽  
Thin Structures ◽  
Vector Valued

In this paper, we introduce new finite elements to approximate the Hellinger Reissner formulation of elasticity. The elements are the vector-valued tangential continuous Nédélec elements for the displacements, and symmetric tensor-valued, normal–normal continuous elements for the stresses. These elements do neither suffer from volume locking as the Poisson ratio approaches ½, nor suffer from shear locking when anisotropic elements are used for thin structures. We present the analysis of the new elements, discuss their implementation, and give numerical results.

scholarly journals QUANTIZATION OF THE FREEDMAN–TOWNSEND MODEL OF MASSIVE VECTOR MESONS

1991 ◽  
Vol 06 (36) ◽  
pp. 3359-3363 ◽  
Author(s):  
M. LEBLANC ◽  
D. G. C. McKEON ◽  
A. REBHAN ◽  
T. N. SHERRY
Keyword(s):  
Vector Fields ◽  
Symmetric Tensor ◽  
Vector Mesons ◽  
Tensor Fields ◽  
Gauge Invariant ◽  
Massive Vector ◽  
Four Dimensions ◽  
Vector Gauge ◽  
Invariant Coupling ◽  
The Masses

We examine a model for massive vector mesons in four dimensions proposed by Freedman and Townsend, where the masses for non-Abelian vector gauge fields are generated without symmetry breaking through a gauge invariant coupling to anti-symmetric tensor fields. The model is quantized using the formalism of Batalin and Vilkovisky. While the Abelian version immediately gives a renormalizable model for massive vector fields, it is shown that in the non-Abelian case the addition of an extra gauge invariant term in the initial Lagrangian leads to an ultraviolet behavior consistent with power-counting renormalizability.

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

2016 ◽  
Vol 26 (09) ◽  
pp. 1649-1669 ◽  
Author(s):  
Jun Hu ◽  
Shangyou Zhang
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Mixed Finite Elements ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Three Dimensions ◽  
Symmetric Tensors ◽  
Element Space ◽  
Finite Element Approximations ◽  
Order Case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

scholarly journals Stabilization of low-order mixed finite elements for the plane elasticity equations

2017 ◽  
Vol 73 (3) ◽  
pp. 363-373 ◽  
Author(s):  
Zhenzhen Li ◽  
Shaochun Chen ◽  
Shuanghong Qu ◽  
Minghao Li
Keyword(s):  
Finite Elements ◽  
Mixed Finite Elements ◽  
Plane Elasticity ◽  
Elasticity Equations ◽  
Low Order

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

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