Mean-strain 8-node hexahedron with optimized energy-sampling stabilization

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Abstract

A method for stabilizing the mean-strain hexahedron was described by Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule, which was shown to guarantee consistency and stability. The stabilization energy was assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The stabilization required user-selected values of the stabilization parameters. In the present work we eliminate the arbitrariness of the stabilization parameters. We formulate the technique more precisely as an assumed-strain method, and we express the stabilization energy in terms of input parameters of the real material. Finally, we fix the value of the stabilization parameters in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element. For simplicity the developments are limited to linear elasticity, but with an arbitrarily anisotropic elasticity matrix. The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in Abaqus and with a number of successful hexahedral and shell elements and we demonstrate that the present element performs very well when used with large aspect ratios for thin structures such as plates or shells.

Introduction

First-order bricks (hexahedra) tend to be exploited for 3-D analysis for their efficiency, robustness, and ease-of-use. Eight-node mean-strain hexahedra seem to be capable of providing both coarse-mesh accuracy and reliable convergence, and significant progress has been made over the years in this technology.

We refer to [1], [2] for overview of the recent developments. The main issues for the mean-strain hexahedra are how to achieve at the same time (i) locking-free response, (ii) good coarse-mesh accuracy, and (iii) stability. Strictly mean-strain hexahedra achieve locking-free response, but lose stability. Adding stability, for instance by treating the so-called hourglassing modes, would tend to affect locking and accuracy. Coarse-mesh accuracy requires of the stabilization to not to deteriorate the response of the element but rather to enhance the ability of the element to accurately respond to deformations in the hourglassing modes such as bending or torsion. Puso׳s hexahedron is a good example of a successful approach [3].

A method for stabilizing the mean-strain hexahedron that differed from the then-known approaches was described by Krysl [4]. The technique relied on a sampling of the stabilization energy using two quadrature rules, the mean-strain quadrature and the full Gaussian integration rule. The use of two quadrature rules was shown to guarantee both consistency and elimination of the hourglassing modes. The stabilization energy was assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The stabilization required user-selected values of stabilization parameters, which is in general undesirable.

In the present work we eliminate the arbitrariness of the stabilization parameters. Firstly, in Section 2 we formulate the technique more precisely as an assumed-strain method. The stabilization energy is then introduced in Section 3 as a quadratic form which is added and subtracted at the same time: added for strains linked to the displacements and subtracted for the assumed strains. We develop an argument for the resulting hexahedral element being convergent by establishing consistency and positive-semi-definiteness of the strain energy. In Section 4 the parameters of the stabilization material are expressed in terms of input parameters of the real material in a way that avoids locking due to volumetric and other constraints (such as for strongly anisotropic materials). The value of the remaining stabilization parameter (Young׳s modulus) is fixed in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element. For simplicity the developments are limited to linear elasticity, but with an arbitrarily anisotropic elasticity matrix.

Section 5 illustrates the performance of the proposed approaches on a variety of benchmark problems, for isotropic and anisotropic material models. Importantly, the coarse-mesh response is significantly improved by the choice of the stabilization parameter. This proves important especially for thin shells and plates, where the present element is shown to match the performance of specialized shell and plate elements. The performance of the stabilization is also tested for highly distorted elements in a vibration problem. The present stabilization technique is also shown to work for anisotropic materials.

The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in Abaqus. In addition, we compare with a number of successful hexahedral and shell elements and we demonstrate that the present element performs very well for thin structures such as plates or shells. Crucially, the hexahedron formulation presented here eliminates the need for user-selected values of the stabilization parameters. Together with excellent performance this makes the present element a good general-purpose hexahedron.

Section snippets

Assumed-strain formulation

We will re-derive the mean-strain eight-node hexahedra [5], [6] taking the variational approach. This will make the inclusion of the stabilization particularly illuminating. We shall take as the starting point the strain-displacement (de Veubeke) functionalΠ(ϵ¯,u)=Ψ(ϵ¯)+Ωσ¯·(ϵϵ¯)dΩW.HereΨ(ϵ¯)=ΩU(ϵ¯)dΩ,is the strain energy, where the energy density U(ϵ¯) is generated by the assumed strains ϵ¯. For instance in linear elasticity we define a quadratic form of the assumed strains ϵ¯U(ϵ¯)=12ϵ¯TDϵ¯

Stabilization

Our approach to the stabilization differs from the well-known approaches of Flanagan and Belytschko [5], [2] (perturbation hourglassing stabilization); of Belytschko and Bindemann [6] (assumed-strain stabilization), and Puso [3] and Reese [8]; of Reese [9] (enhanced-strain (incompatible mode) stabilization); and of Belytschko and Bachrach [10] and Liu et al. [11] (physical stabilization based on the Taylor-series expansion of the strains). We propose the so-called stabilization by energy

Form of the stabilization energy

In this section we will develop a quasi-optimal expression for the stabilization energy. For the stress analysis of linearly elastic anisotropic solids Krysl [4] proposed the constitutive matrix D^ that induces the stabilization strain energy U^ in the formD^=ϕi=1nmψiKiviviT+ϕj=nm+16KjvjvjT,where Ki and vi are the eigenvalues and eigenvectors of the material stiffness matrix D, and 0<ϕ1 and 0<ψi1 are suitable scalar multipliers. The multipliers ψi may be taken at a magnitude that scales the

Numerical examples

In the numerical examples in this section we compare the performance of the present formulation with the mean-strain elements implemented in the Abaqus finite element software. The element C3D8RH is the mean-strain element with default hourglass stabilization (i.e. the artificial stiffness stabilization) as proposed by Flanagan and Belytschko [5]. The element denoted C3D8RH(enh) in the present work is the mean-strain element with the enhanced (assumed strain) hourglass stabilization as proposed

Conclusions

A technique for stabilizing the mean-strain hexahedron was developed as a modification of the energy-sampling approach of Krysl [4]. The hourglassing modes of deformation are never explicitly separated from the rigid-body and mean-strain modes. Instead, stabilizing energy is postulated that is both added and subtracted from the strain energy of the element. The sampling of these two contributions is carried out by two different quadrature rules: the mean-strain quadrature is combined with the

Acknowledgments

Partial support from U.S. Navy CNO-N45, project management Frank Stone and Ernie Young, and continued support of Mike Weise (grant number N00014-09-1-0611) (Office of Naval Research), is gratefully acknowledged.

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