- Research article
- Open Access
An extension of assumed stress finite elements to a general hyperelastic framework
- Nils Viebahn^{1}Email authorView ORCID ID profile,
- Jörg Schröder^{1} and
- Peter Wriggers^{2}
https://doi.org/10.1186/s40323-019-0133-z
© The Author(s) 2019
- Received: 23 January 2019
- Accepted: 17 May 2019
- Published: 31 May 2019
Abstract
Assumed stress finite elements are known for their extraordinary good performance in the framework of linear elasticity. In this contribution we propose a mixed variational formulation of the Hellinger–Reissner type for hyperelasticity. A family of hexahedral shaped elements is considered with a classical trilinear interpolation of the displacements and different piecewise discontinuous interpolation schemes for the stresses. The performance and stability of the new elements are investigated and demonstrated by the analysis of several benchmark problems. In addition the results are compared to well known enhanced assumed strain elements.
Keywords
- Mixed FEM
- Hellinger–Reissner formulation
- Hyperelasticity
- Hourglassing
- Locking free
- Assumed stress element
- Nearly incompressibility
Introduction
An enormous effort was invested in the development of finite element methods based on the variational approach going back to Galerkin [28], which are in general considering the approximation of one basic variable. It is well known that low order elements, based on this variational approach yield poor results in the framework of nearly incompressibility and bending dominated problems. The reason for this are locking effects, see [12]. Developments considering the approximation of additional fields in the variational setup are for example given in Reissner [44] (compare also [29] and [43]), here an independent stress approximation is applied in addition to the displacements, which acts as a Lagrange multiplier. We refer to this type of formulation, which is based on a complementary stored energy function, as Hellinger–Reissner (HR) formulation. A few years later, [30] and [57] proposed independently a variational principle related to displacements, stresses and strains, based on the so-called Hu-Washizu functional. In the framework of finite element analysis, mixed formulations lead to saddle-point problems and therefore a major constraint of these approaches is the restriction to the so-called LBB-conditions, see [11, 19, 20, 35] and [10]. Mathematical aspects concerning the mixed finite element formulation for elasticity based on the Hellinger–Reissner (HR) principle are discussed in Arnold and Winther [4], Auricchio et al. [10], Lonsing and Verfürth [37], Arnold et al. [3], Boffi et al. [18] and Cockburn et al. [25]. In contrast to the Stokes problem, where the enrichment of the discrete space of the primal variable leads always to stable elements in the linear range, the situation is more difficile in the framework of the HR formulation. Here, the discrete spaces have to be carefully balanced.
Within the principle of HR two methods can be distinguished. A shift of the derivatives from the displacements to the stresses, using integration by parts, leads to a formulation with the displacements in the \(L^2\) and the stresses in the \(H(\mathrm{Div})\) Sobolev spaces. In the literature, this is often denoted as the dual HR formulation. Stable elements in the linear range based on the dual HR formulation with a strong enforcement of the symmetry can only be achieved at high polynomial order, see e.g. [6, 31] in 2D and [1, 7] in 3D. A reduction of the symmetry constraint leads to some flexibility in the construction of the finite elements, see [5, 18, 53, 54] and [32].
In the primal HR formulation the derivatives are correlated to the displacements, such that the displacements are in the \(H^1\) and the stresses in \(L^2\) Sobolev spaces. Elements, based on this formulation, are called assumed stress elements going back to the pioneering work of Pian [39]. In 2D, often a discontinuous stress approximation using a 5-parameter ansatz proposed by Pian and Sumihara [40] is used. The advantages of this approach are characterized by a remarkable insensitivity to mesh distortion, locking free behavior for plane strain quasi-incompressible elasticity and superconvergent results for bending dominated problems, see e.g. [40, 52] and [23]. Its stability with regard to the LBB-conditions and an a posteriori error estimation has been shown by Yu et al. [61] and Li et al. [36]. A well known extension to the 3D case is the element by Pian and Tong [41]. The family of assumed stress elements is closely related to the family for enhanced assumed strains elements, see e.g. [2, 60] and [16].
The “direct” extension of the HR principle, which requires a complementary energy function in terms of stress measures, is in general not possible. Some effort has been pursued in the extension of the dual version of the HR principle considering small-strain elasto-plasticity, [46, 48]. Considering a large-strain setup the workgroup of Atluri (see [8, 9, 49]), has been proposed an incremental variational formulation, involving the discretization of the displacement, rotations and the hydrostatic pressure considering the primal HR formulation. In addition [42] extended the underlying variational formulation to a Hu-Whashizu type form, closely related to the family of enhanced assumed strains. Another approach has been discussed in Wriggers [58], where the complementary constitutive relation has been derived in an explicit form for a special Neo-Hookean type.
In the proposed work, a general framework is introduced, which extends the family of assumed stress elements to hyperelasticity. It is based on an iterative solution procedure of the constitutive law at the level of the element integration points. In this work, the framework is adopted to a family of 3D hexahedral elements, with a varying interpolation scheme of the stresses. It should be noted that in the framework of hyperelasticity, the corresponding solution spaces are shifted from the classical Lebesgue spaces to the more general framework of the Hilbert spaces, see [24] for a detailed discussion. Thus, the related solution spaces for the proposed formulations are represented for the displacements and stresses by \(W^{1,p}\) and \(W^{0,p}\), whereas \(p\ge 2\) depends on the constitutive relation. The stress interpolation is discussed with respect to volumetric locking, shear locking and hourglassing. In addition the related EAS elements are mentioned and the properties and characteristics of the various elements are discussed on the example of different numerical benchmarks.
Kinematics and variational formulation
Kinematics, constitutive quantities and stresses
Symbol | Continuum mechanical description |
---|---|
\({{\varvec{u}}}\) | Displacement vector |
\(\varvec{F}={{\varvec{I}}}+ {\mathrm{Grad}}{{\varvec{u}}}\) | Deformation gradient |
\(\varvec{C}={{\varvec{F}}}^T{{\varvec{F}}}\) | Right Cauchy-Green tensor |
\(\varvec{E} = \frac{1}{2}({{\varvec{C}}}-{{\varvec{I}}})\) | Green-St. Venant strain tensor |
\(\chi \) | Complementary stored energy |
\(\psi \) | Helmholtz free energy |
\(\varvec{S}\) | Second Piola-Kirchhoff stress tensor |
\(\varvec{P} = {{\varvec{F}}}{{\varvec{S}}}\) | First Piola-Kirchhoff stress tensor |
\(\varvec{\tau } = {{\varvec{P}}}{{\varvec{F}}}^T \) | Kirchhoff stress tensor |
\(\varvec{\sigma } = (\det {{\varvec{F}}})^{-1}{\varvec{\tau }}\) | Cauchy stress tensor |
Discrete formulation and interpolation
Stress interpolation
Numerical simulations
Overview of considered elements
AS-39 | Assumed stress element with 39 stress modes, see Eq. (23) |
H1 | Isoparametric eight-node hexahedral element |
AS-18 | Assumed stress element with 18 stress modes, see Eq. (24) |
EAS-21 | Enhanced assumed strain element with 21 modes, see [2] |
AS-30 | Assumed stress element with 30 stress modes, see Eq. (25) |
EAS-9 | Enhanced assumed strain element with 9 modes, see [34] |
AS-24 | Assumed stress element with 24 stress modes, see Eq. (26) |
EAS-15 | Enhanced assumed strain element with 15 modes, see [38] |
H1P0 | Displacement-pressure approach with piecewise constant pressure, see [50] |
Equivalence of the formulations in linear elasticity
Inhomogeneous compression block in linear elasticity
\(\mathbf n_{\text {ele}} \) | \(\varvec{2^3}\) | \(\varvec{4^3}\) | \(\varvec{8^3}\) | \(\varvec{16^3}\) |
---|---|---|---|---|
AS-39 | 29.2568 | 27.106 | 27.0202 | 26.9881 |
H1 | 29.2568 | 27.106 | 27.0202 | 26.9881 |
AS-18 | 31.3466 | 27.2494 | 27.0808 | 27.0046 |
EAS-21 | 31.3466 | 27.2494 | 27.0808 | 27.0046 |
AS-30 | 29.9399 | 27.1645 | 27.042 | 26.9938 |
EAS-9 | 29.9399 | 27.1645 | 27.042 | 26.9938 |
AS-24 | 31.1904 | 27.2326 | 27.0806 | 27.0046 |
EAS-15 | 31.1904 | 27.2326 | 27.0806 | 27.0046 |
H1P0 | 31.7699 | 27.1882 | 27.0856 | 27.0072 |
Patch test
The Patch test is a necessary condition for the convergence of finite elements. It demands that an arbitrary patch of assembled elements is able to reproduce a constant state of stress and strain if subjected to boundary displacements consistent with constant straining. This condition is necessary since with respect to mesh refinement, where \(h\rightarrow 0\), all boundary value problems tend to constant stress and strains in each element. This test is mainly attributed to the work of Bruce Iron, first presented in Bazeley et al. [15]. A summary on its theory, practice and possible conclusions on its satisfaction can be found in Taylor et al. [55]. Following Korelc et al. [33], two different load scenarios are considered, described in Figs. 2 and 3. Load case (A) prescribes a pure rigid body motion by a rotation around the z-axis. All proposed elements are free of resulting stresses and strains and thus fulfill the first patch test. Load case (B) prescribes a combined deformation of shear and uniaxial strain, which analytical solution leads to a constant strain and stress over the whole domain. All proposed elements result in the expected constant stress and strain field and therefore fulfill the second patch test. Note that the patch tests verify only the consistency of the finite element and thus represents only a necessary but not a sufficient condition for the stability of the formulation.
Inf-sup test
Figure 4 shows the development of the inf-sup value for the chosen boundary value problem over the number of elements considering a regular structured mesh refinement. The value seems to be bounded from below for the AS-39, AS-30 and AS-24. In case of the AS-18 the inf-sup value shows asymptotical convergence, which also indicates a distinct lower bound and thus passes the inf-sup test. Both described behavior are in sharp contrast to the results of the H1P0. Here, the inf-sup value decreases with respect to mesh refinement by an almost constant rate, clearly indicating the failure of the inf-sup test. It should be mentioned, that such a numerical verification does not replace the need of an analytical investigation in order to ensure the statements on the stability. Nonetheless, to the best knowledge of the authors, not a single example has been found, where the inf-sup test does not give similar results as the analytical proofs.
Eigenvalue analysis of initial element matrix
Eigenvalue spectrum for incompressible square element
AS-18 / EAS-21 | AS-24/EAS-15 | AS-30/*EAS-9 | H1/AS-39 | H1P0 | |
---|---|---|---|---|---|
1 | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) |
2 | 0.33 | 0.33 | 0.38 | \(\infty \) | 0.33 |
3 | 0.33 | 0.33 | 0.38 | \(\infty \) | 0.33 |
4 | 0.33 | 0.33 | 0.38 | \(\infty \) | 0.33 |
5 | 0.33 | 0.33 | 0.33 | \(\infty \) | 0.33 |
6 | 0.33 | 0.33 | 0.33 | \(\infty \) | 0.33 |
7 | 0.33 | 0.33 | 0.33 | \(\infty \) | 0.22 |
8 | 0.33 | 0.33 | 0.33 | 0.33 | 0.167 |
9 | 0.33 | 0.33 | 0.33 | 0.33 | 0.167 |
10 | 0.22 | 0.22 | 0.22 | 0.33 | 0.167 |
11 | 0.11 | 0.11 | 0.167 | 0.33 | 0.09 |
12 | 0.11 | 0.11 | 0.167 | 0.33 | 0.09 |
13 | 0.11 | 0.11 | 0.167 | 0.22 | 0.09 |
14 | 0.056 | 0.093 | 0.093 | 0.16 | 0.06 |
15 | 0.056 | 0.093 | 0.093 | 0.16 | 0.06 |
16 | 0.056 | 0.093 | 0.093 | 0.16 | 0.06 |
17 | 0.056 | 0.056 | 0.056 | 0.056 | 0.056 |
18 | 0.056 | 0.056 | 0.056 | 0.056 | 0.056 |
Hyperelastic nearly incompressible Cook’s membrane
Consideration of the necessary load steps, depicted in Fig. 8b, indicate that the assumed stress elements are able to deal with large load steps in case of nearly incompressibility. For this boundary value problem the considered elements require only a single load step, independent of the mesh size. In contrast, the enhanced assumed strain elements and the H1P0 element require a significant higher number of load increments, leading to a substantially larger computation time. The level of necessary load steps in case of the H1 element is moderate but it should kept in mind that their performance by means of displacement accuracy is insufficient.
Hyperelastic bending of a compressible clamped plate
The elements can be mainly distinguished into three groups, whereas the close relationship between the assumed stress and enhanced assumed strain elements can be recognized again. In case of the structured meshing the AS-18, EAS-21, AS-24 yield EAS-15 yield optimal results for the tip-displacement already for the coarsest meshing. However, taking into account the unstructured meshes the quality of the results is weakened, especially for the coarsest level. However, the AS-18 and EAS-21 depict the best mesh convergence of all considered elements, independent of the discretization strategy. The AS-24 and EAS-15 are slightly weaker in case of unstructured coarse meshes. In contrast to these four elements the H1P0, H1, AS-39, AS-30 and *EAS-9 demonstrate a distinct locking behavior in this numerical example. Note that these observations are consent with the results of the eigenvalue study related to Fig. 6, where locking behavior has been predicted for the latter elements in case of boundary value problems with slender domains. Interestingly, their tip-displacement convergence is slightly better for the unstructured case, which is in contrast to the behavior of the remaining elements.
In addition the number of necessary load steps are depicted on the right of Fig. 11. It can be recognized that also in this bending dominated problem, the number of necessary load steps is significantly smaller for the proposed family of assumed stress elements. Especially the enhanced assumed strain elements, which do not show locking effects (EAS-15, EAS-21) suffer due to the need of smaller load increments compared to their AS counterparts (AS-18, EAS-24).
Hyperelastic bending of a clamped plate—nearly incompressible
Considering the displacement convergence, shown on the left of Fig. 12, it can be noted that the qualitatively response of the elements is equivalent to the compressible case, except for the AS-39 and H1. These elements clearly depict a conspicuous additional volumetric locking. A similar picture as for the compressible case is obtained considering the number of necessary load steps, shown in Fig. 12b. Even if the displacement results for the EAS-21 and EAS-15 seem to be satisfactory, they suffer due to the need of smaller load steps (a factor of 20), compared to their assumed stress counterparts.
Hyperelastic hourglassing test
Hyperelastic fiber reinforced Cook’s membrane problem
Concluding remarks
We proposed an extension of a mixed finite element formulation based on the Hellinger–Reissner principle to the framework of hyperelasticity and investigated it numerically. This principle requires an independent interpolation of the stresses and displacements. The displacements are interpolated by classical trilinear shape functions, whereas for the stresses four different interpolation strategies are discussed. The numerical results for each interpolation can be summarized by the following. The AS-39 correlates to the H1 element also in the nonlinear regime and shows volumetric as well as shear locking. The AS-18 represents the hyperelastic extension of the element proposed by Pian and Tong [41]. A close relationship can be drawn to the EAS-21. The AS-18 performs very well in bending dominated and nearly incompressible problems. Unfortunately, it suffers due to hourglassing modes which could lead to instabilities also in more complex boundary value problems. The AS-30 element is inspired by the *EAS-9 enhanced formulation, proposed in Krischok and Linder [34]. It is free of volumetric locking, does not show hourglass instabilities in the discussed numerical example. Unfortunately, it is not free of shear locking and therefore behaves poor in the bending dominated problems. The AS-24 element is closely related to the EAS-15 proposed by Pantuso and Bathe [38]. It is free of volumetric and shear locking. However, it suffers due to hourglass instabilities.
Even if non of the investigated interpolation schemes are free of all drawbacks, the proposed procedure of Hellinger–Reissner formulations for large deformations has emerged as a promising approach. The proposed elements seem to be significantly improved in terms of robustness and large load increments. This leads to an enormous gain in terms of computational cost comparing to the widely used EAS formulations.
Declarations
Author's contributions
All authors contributed in the derivation and development of the idea for the proposed mixed finite element method. The implementation, numerical studies and the draft of the manuscript have been mainly performed by NV. JS and PW supervised the study and corrected the manuscript. All authors read and approved the final manuscript.
Acknowledgements
Not applicable
Availability of data and materials
Not applicable
Competing interests
The authors declare that they have no competing interests.
Funding
The authors appreciate the support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Novel finite elements - Mixed, Hybrid and Virtual Element formulations at finite strains for 3D applications” under the project “Reliable Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis” (SCHR 570/23-2) (WR 19/50-2). Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 255432295.
We acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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