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.

The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space for the displacement. We define the space for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the and a discrete norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).

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In this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal-normal continuous symmetric tensor-valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements.ANISOTROPIC MIXED FINITE ELEMENTS FOR ELASTICITY 197 anisotropic structures within a unified framework is provided in [12]. Analysis for the hierarchical approach can be found in [13][14][15], a posteriori error estimates in [16][17][18].In [19,20], we first introduced a mixed, Hellinger-Reissner type method, where the stresses are considered as separate unknowns. We searched for the displacement in H.curl/, using tangentialcontinuous Nédélec finite elements. For the stresses, we proposed the space H.div div/, discretized by symmetric tensor-valued elements with continuous normal component of the normal stress vector. The degrees of freedom for these elements are then the tangential component of the displacement and the normal component of the normal stress vector, which shall be abbreviated by normal-normal stress. In the present paper, we apply this approach to a prismatic, tensor-product mesh. We see that these elements do not suffer from shear locking: In the discrete setting, we can use a 'broken norm' of piecewise strains for the displacement. Employing appropriate transformations of the finite element shape functions from the reference element to an element in the mesh, we overcome the difficulties arising from Korn's inequality. We provide anisotropic interpolation operators for the stress and displacement spaces with respect to the broken norms. There, we make use of the tensor product structure and Clément-type interpolators [21,22], which satisfy a commuting diagram property. Basic notationsFor some Hilbert space X , let .., ./ X be the inner product, and k.k X the induced norm. By angles h., .i X , we denote the duality product between the dual space X and X . For some sufficiently regular domain R 3 , let L 2 . / be the Lebesgue space. As a short hand for the L 2 norm k.k L 2 . / , we also use k.k . By H k . / integer k > 0, we denote the standard Sobolev space, with semi-norm j.j H k . / and norm k.k H k . / defined via A tensor field has a normal vector n D n. The normal and tangential parts nn , n of this vector are defined by n D nn n C n , nn D n T n.Here the vector field u is the unknown displacement, ".u/ D 1 2 .ru C .ru/ T / is the strain, represents the symmetric stress tensor, and A is the compliance tensor. The boundary @ consists of a non-trivial part D Â @ , where displacement boundary co...

Many widely used beam finite element formulations are based either on Reissner's classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper.In many applications, a nonlinear large deformation beam element with bending, axial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner's virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of conventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully parameterized ANCF element.