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Abstract:
Mixed eight-node (hexahedron) solid-shell elements based on the standard or partial version of the three-field Hu–Washizu (HW) functionals are developed for Green strain. Three reduced representations of the assumed stress/strain fields are selected. They improve effectiveness, yet retaining good accuracy and convergence properties. At the outset, the standard HW functional and the assumed stress/strain representations of the 3D solid element B8-15P (Weissman in Int J Numer Methods Eng 39:2337–2361, 1996) are used to derive a solid-shell element with 51 parameters. To eliminate locking, the ANS method is applied to the thickness strain (Betsch and Stein in Commun Numer Methods Eng 11:899–909, 1995) and to the transverse shear strain (Dvorkin and Bathe in Eng Comput 1:77–88, 1984). It is a correct element which, however, yields too large displacements for coarse meshes and trapezoidal through-thickness shapes. To improve the above formulation, the ζ-independent reduced representations of the assumed stress/ strain fields are selected and the transformations to Cartesian components are modified. The thickness strain is enhanced by the EAS method. The element with 35 parameters is derived from the standard/enhanced HW functional, but, to further reduce the assumed fields, partial/enhanced HW functionals are constructed from the 3D potential energy by applying the Lagrange multiplier method only to selected strain components. In the element with 27 parameters, this is applied to the constant in-plane strain and to the transverse shear strain while in the element with 19 parameters, to the constant in-plane strain only.Two other modifications are implemented to enhance the behavior of these elements: (A) the skew coordinates are used in the reduced representations of the in-plane stress/strain (Wisniewski and Turska in Int J Numer Methods Eng 90:506–536, 2012), and (B) the Residual Bending Flexibility correction of the transverse shear stiffness (MacNeal in Comput Struct 8(2):175–183, 1978) is adapted. Finally, the performance of the proposed solid-shell HW elements is demonstrated on several linear and non-linear examples for the linear elastic material and the hyper-elastic material. The proposed elements are compared to each other and to the best existing elements of this class.

Keywords:
Eight-node (hexahedron) solid-shell elements , Standard or partial Hu–Washizu functionals, Reduced representations of assumed stress/strain , RBF correction

Abstract:
The 9-node quadrilateral shell element MITC9i is developed for the Reissner-Mindlin shell inematics, the extended potential energy and Green strain. The following features of its formulation ensure an improved behavior: 1. The MITC technique is used to avoid locking, and we propose improved ransformations for bending and transverse shear strains, which render that all patch tests are passed for the regular mesh, i.e. with straight element sides and middle positions of midside nodes and a central node. 2. To reduce shape distortion effects, the so-called corrected shape functions of Celia and Gray (Int J Numer Meth Eng 20:1447–1459, 1984) are extended to shells and used instead of the standard ones. In effect, all patch tests are passed additionally for shifts of the midside nodes along straight element sides and for arbitrary shifts of the central node. 3. Several extensions of the corrected shape functions are proposed to enable computations of non-flat shells. In particular, a criterion is put forward to determine the shift parameters associated with the central node for non-flat elements. Additionally, the method is presented to construct a parabolic side for a shifted midside node, which improves accuracy for symmetric curved edges. Drilling rotations are included by using the drilling Rotation Constraint equation, in a way consistent with the additive/multiplicative rotation update scheme for large rotations. We show that the corrected shape functions reduce the sensitivity of the solution to the regularization parameter γ of the penalty method for this constraint. The MITC9i shell element is subjected to a range of linear and non-linear tests to show passing the patch tests, the absence of locking, very good accuracy and insensitivity to node shifts. It favorably compares to several other tested 9-node elements.

Keywords:
9-node shell element MITC9i, Two-level approximation of strains, Patch tests, Corrected shape functions, Node shift parameters, Coarse mesh accuracy, Drilling rotations

Abstract:
The paper concerns two-dimensional nine-node quadrilateral elements based on the Green strain and the two-level approximations of strains. These approximations reduce locking well for regular meshes but cannot prevent the drop of accuracy when the side and central nodes are shifted from the middle positions.

To reduce the deterioration of accuracy when nodes are shifted, we assess the corrected shape functions of Celia and Gray (1984 [10]) as a replacement for the standard isoparametric ones. In Celia and Gray (1984 [10]), the corrected shape functions were tested for an eight-node element, the heat conduction equation and the 4×4 integration. Here, we test their applicability to nine-node elements for plane elasticity and the 3×3-point integration.

We modify and examine four elements: QUAD9⁎⁎ (Huang and Hinton, 1986 [15]), MITC9 [1] and ours 9-AS (Panasz and Wisniewski, 2008 [21]) and MITC9i (Wisniewski and Panasz, 2012 [26]). The elements are subjected to a range of tests involving several types of mesh distortions, to confirm passing of various forms of the patch test, to prove the absence of locking as well as to establish their coarse mesh accuracy and sensitivity to mesh distortions. We show that all the tested elements benefit from using the corrected shape functions, but still remain significant differences in their performance.

Keywords:
Two-dimensional nine-node elements, Corrected shape functions, Two-level approximations of strains, Patch tests, Shape distortions

Abstract:
In this paper, enhanced four-node shell elements with six DOFs/node based on the Hu–Washizu (HW) functional are developed for Green strain. The drilling rotation is included through the drilling rotation constraint equation. The key features of the approach are as follows.

The shell HW functional is derived from the shell potential energy functional, which is an alternative to the derivation from the three-dimensional HW functional. This method is more versatile as it enables the derivation of the so-called partial HW functionals, with different treatment of the bending/twisting part and the transverse shear part of strain energy.
For the membrane part of HW shell elements, a seven-parameter stress, a nine-parameter strain and a two-parameter enhanced assumed displacement gradient enhancement are selected as optimal. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. This improves accuracy and has positive theoretical consequences.
The drilling rotation constraint equation is treated by the perturbed Lagrange method. The faulty term resulting from the equal-order approximations of displacements and the drilling rotation is eliminated, and one spurious mode is stabilized using the gamma method. The proposed formulation is insensitive to the element's distortions and yields a large radius of convergence in the examples involving in-plane bending.

The performance of 4 four-node shell HW elements, having different bending/twisting and transverse shear parts, is analyzed on several numerical examples. Such aspects are considered as: accuracy, radius of convergence, required number of iterations of the Newton method or the arc-length method and time of computations. The element with 29 parameters (HW29) is selected as the best performer.

Keywords:
four-node mixed shell element with six DOFs/node, pure or partial Hu–Washizu functionals, drilling rotation, optimal representations, skew coordinates

Abstract:
Mixed 4-node elements based on the Hu–Washizu (HW) functional are developed for the representation of the assumed strain in the natural basis at the element’s center, i.e. for the contravariant transformation rule. In other aspects, the formulation is identical as in our previous paper [9], to which this note is an addendum.

Two mixed HW elements based on the 5-parameter stress are developed; they use either the 7-parameter or 9-parameter strain representation. The stress and strain representations are assumed in terms of skew coordinates, see [10]. The numerical tests involving coarse distorted meshes are used to assess the effects of using the contravariant strain representation.

The tests show that both elements pass the discrete inf-sup test. Accuracy of the element based on the 9-parameter strain, designated as HW14-S and selected as the best in [9], remains unaffected. Accuracy of the element based on the 7-parameter strain is significantly improved.

Keywords:
Four-node finite elements, Hu–Washizu functional, Plane stress, Mixed elements, Skew coordinates, Contravariant assumed strain

Abstract:
Mixed 4-node elements based on the Hu–Washizu (HW) functional are developed for stress and strain representations in various coordinates, including the skew, natural and Cartesian ones. The HW functional is used in incremental form, suitable for non-linear materials. The key features of our approach are as follows.

(1) The representations of stress and strain are assumed in skew coordinates associated with the natural basis at the element’s center, which implies that, for a linear elastic case, the homogenous equilibrium equations and the compatibility condition are satisfied point-wise. For stress, the same 5- and 7-parameter representations as for the Hellinger–Reissner (HR) elements by Wisniewski and Turska [Wisniewski K, Turska E. Improved four-node Hellinger–Reissner elements based on skew coordinates. Int J Numer Methods Eng 2008;76:798–836] are used. For strain, a 9-parameter linear representation is selected.
(2) A mixed element HW14-S using a 5-parameter representation of stresses assumed in skew coordinates is developed from the non-enhanced HW functional. This element is equally accurate as our HR5-S element of Wisniewski and Turska (1998), the HR element by Yuan et al. [Yuan K-Y, Huang Y-S, Pian THH. New strategy for assumed stress for 4-node hybrid stress membrane element. Int J Numer Methods Eng 1993;36:1747–63], and the HW elements by Piltner and Taylor [Piltner R, Taylor RL. A quadrilateral mixed finite element with two enhanced strain modes. Int J Numer Methods Eng 1995;38:1783–808; Piltner R, Taylor RL. A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int J Numer Methods Eng 1999;44:615–39], and Piltner [Piltner R. An implementation of mixed enhanced finite elements with strains assumed in Cartesian and natural element coordinates using sparse View the MathML sourceB¯-matrices. Eng Comput 2000;17(8):933–49]. Compared to these HW elements, our element uses a smaller number of parameters.
(3) A mixed/enhanced element HW18 using a 7-parameter representation of stress is developed from the enhanced HW functional. For the elements based on this stress representation, the strain representation has to be enriched; we use a 2-parameter EADG enhancement. Various combinations of the natural, skew and Cartesian coordinates are tested, and these for which this element performs best are selected.
(4) A specific modification of the FTFFTF product, consisting of the expansion of FF and the selection of meaningful terms in the product, was applied to selected elements. With this modification, the element HW14-S performs better for coarse distorted meshes than the HW elements described in the literature.

The developed elements are based on the Green strain, and are tested for linear and non-linear constitutive laws modified by the zero normal stress condition, because they will be used as a membrane part of a shell element. Several numerical tests show their performance, in particular, their robustness to the element’s shape distortion for coarse meshes.

Keywords:
4-Node finite elements, Plane stress, Incremental Hu–Washizu functional, Mixed elements, Mixed/enhanced elements, Skew coordinates

Abstract:
Mixed four-node elements based on the Hellinger–Reissner (HR) functional are developed for stress representations in various coordinates, including the skew, natural and Cartesian ones. The two-field HR functional is used in the classical form and in the incremental form suitable for non-linear materials.

We argue that the skew coordinates, not the natural ones, should be associated with the natural basis at the element's center. If 5- and 7-parameter stress representations are assumed in these coordinates, then, for a linear elastic case, the homogenous equilibrium equations and the stress form of compatibility equation are satisfied point-wise.

Two mixed four-node elements are developed and tested:
1. An assumed stress element (HR5-S) is developed from the non-enhanced HR functional, for a 5-parameter representation of stresses, formally identical as the one used, for example, in Pian and Sumihara [Int. J. Numer. Meth. Engng 1984; 20:1685–1695], but in terms of skew coordinates. This element is very simple and uses a smaller number of parameters, but is equally accurate as the elements by Yuan et al. [Int. J. Numer. Meth. Engng 1993; 36:1747–1763] and by Piltner and Taylor [Int. J. Numer. Meth. Engng 1995; 38:1783–1808].
2. An assumed stress/enhanced strain element (HR9) is developed from the enhanced HR functional, for a 7-parameter representation of stress and a 2-parameter enhanced assumed displacement gradient or enhanced assumed strain enhancement. Various forms of 7-parameter representations appearing in the literature are reviewed, and we prove that they are linked by a linear onto transformation. The choice of coordinates for the stress and the enhancement turns out to be the crucial factor, and four combinations of coordinates for which the element performs the best are identified.

Both elements are based on the Green strain, and several numerical tests show their good accuracy, in particular, their robustness to shape distortions for coarse meshes. Two update schemes for the multipliers of modes and the incremental constitutive procedure accounting for the plane stress condition for non-linear materials are tested for large deformation problems.

Keywords:
four-node finite elements, incremental Hellinger–Reissner functional, assumed stress element, assumed stress/enhanced strain element, skew coordinates

Abstract:
The paper presents an analytical constitutive design sensitivity analysis (DSA) algorithm for explicit dynamics of elastic-plastic finite rotation shells. Two explicit dynamical algorithms for finite rotation shells are presented, and the DSA is developed for the one formulated in terms of the rotation vector and its time derivatives, {ψ,ψ˙,ψ¨}. The hypo-elastic constitutive model based on the Green-McInnis-Naghdi stress rate is used to derive an incremental algorithm in terms of ‘back-rotated’ objects. The associative deviatoric Huber-Mises plasticity modified by plane stress conditions is implemented in the form suitable for finite rotation/small elastic strain increments. The analytical DSA is developed for the above-specified problem, with the design derivatives calculated w.r.t. material parameters. Design-differentiation of the dynamic algorithm and the scheme of handling the history data and the predicted values in differentiation, which is crucial in computing correct derivatives, are described. Besides, we show how to avoid Newton loops in the DSA algorithm, when such a loop is present in the constitutive algorithm. Numerical examples show that, despite a great complexity of the solution algorithm for the finite-rotation elastic-plastic shells, it is feasible to compute analytical design derivatives of very good accuracy.

Keywords:
Explicit dynamics, Finite rotation shell, Elastic-plastic material, Analytical Design Sensitivity Analysis for constitutive parameters

Abstract:
The paper concerns a four-node quadrilateral element based on Allman shape functions undergoing finite (unrestricted) drilling rotations, and aims at improving its accuracy and facilitating its implementation.

Firstly, the classical Allman shape functions are valid only for small in-plane rotations, and must be used with a co-rotational frame, which embeds finite rotations. We derive a new form of Allman shape functions, which is valid for finite drilling rotations, and allows to avoid the use of such a frame.

Secondly, the classical Allman quadrilateral shows locking in the in-plane shear test. We study this problem, identify its source, and remove it by enhancing the element with two additional modes, via the Enhanced Assumed Displacement Gradient (EADG) method. To accomplish this, we extend the original version of the method to mixed functionals including rotations.

Two variational formulations including the drilling rotation via the rotation constraint (RC) equation are considered; one based on the Green strain, and the other on the relaxed non-symmetric right stretch strain. Numerical tests of the corresponding finite elements show that the improved Allman elements are as exact in linear tests as the EADG4 element, and perform very well in a severe in-plane shear test for one layer of elements undergoing large rotations.

Keywords:
New Allman shape functions for finite drilling rotations, Enhanced Assumed Displacement Gradient method for formulations with rotations, Enhanced Allman finite elements

Abstract:
This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber–Mises deviatoric plasticity with non‐linear isotropic/kinematic hardening, one which is applicable to metals.

The standard radial return algorithm for linear hardening is generalized to account for non‐linear hardening functions. Two generalizations are presented; in both the non‐linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design differentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation.

The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of verification tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for verification of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with finite difference computations and (b) the effect of the finite element formulation on the design derivatives for an isochoric plastic flow.

Keywords:
design sensitivity analysis, elasto–plastic material with non‐linear hardening, parallel finite element code

Abstract:
A method of approximate analysis is presented concerning the state of stress, and the stress intensity factors in particular, in elastic media subject to a plane state of strain and containing arbitrary arrays of cracks. In the case when ·the crack distribution is not too dense, the method proposed makes possible the determination of the required stress parameters in a manner resembling that used in solving the statically indeterminate systems of structural mechanics.

Abstract:
The paper describes our improved 9-node quadrilateral shell element MITC9i, which is derived for the Reissner-Mindlin shell kinematics, the extended potential energy functional and Green strain. 1. The MITCi technique is used to avoid locking and it is based on the improved transformations proposed in (Wisniewski & Panasz 2013) for a membrane element. Here, these transformations are extended to bending/twisting and transverse shear shell strains. 2. To reduce the shape distortion effects, the so-called corrected shape functions (CSF) of (Celia & Gray 1984) are used instead of the isoparametric ones, and we propose the method of computation the shift parameters for non-flat shell elements. 3. The drilling rotations are included via the drilling Rotation Constraint and the penalty method. This rotation is used in the multiplicative/additive update scheme valid for large (unrestricted) rotations. The effect of the MITC9i technique and the CSF is that all three patch tests are passed, also for shifted side nodes along the straight edges and for arbitrary shifts of an interior node. The MITC9i shell element was subjected to a range of linear and non-linear numerical tests described in (Wisniewski & Turska 2017); here we provide additional examples illustrating its accurate and robust behavior.

Keywords:
9-node shell element MITC9,two-level approximation of strains, corrected shape functions, node shift parameters, drilling rotations

Keywords:
finite element method, shell elements

Abstract:
Mixed/enhanced four-node shell elements with six dofs/node based on the Hu-Washizu (HW) functional are developed for Green strain. The shell HW functional is derived from the shell potential energy functional instead of from the three-dimensional HW functional. Partial HW functionals, differing in the bending/twisting part and the transverse shear part, are obtained. For the membrane part of HW shell elements, a 7-parameter stress, a 9-parameter strain and a 2-parameter EADG enhancement are selected as performing best. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. The drilling rotation is included through the drilling Rotation Constraint (RC) equation and the Perturbed Lagrange method. The spurious mode is stabilized using the gamma method. Several versions of shell HW elements are tested using several benchmark examples and the optimally performing element is selected (HW29) in (Wisniewski & Turska 2012).

Keywords:
design sensitivity analysis, finite element method, shell structures, elasto-plasticity

Abstract:
The paper describes a constitutive algorithm for elastic-plastic finite rotation shells and explicit dynamics with design derivatives calculated w.r.t. We show that despite a great complexity of the solution algorithm for the finite-rotation elastic-plastic shells, it is feasible to compute analytical design derivative of this algorithm, and the yielded sensitivities are of very good accuracy.

Keywords:
design sensitivity analysis, finite elment method, shell structures, dynamics, finite rotations

Keywords:
Design sensitivity analysis, finite element method, finite rotations, shell elements