Abstract
We propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders as the projection bases. The introduced formulation is much cheaper computationally than the classical \(\bar{B}\) method. We show the numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulation alleviates locking and yields good accuracy even for slenderness ratios of \(10^5\), and has the ability to capture deformations of thin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to locking with irregular meshes.
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Acknowledgements
Q. Hu is thankful for Prof. Gengdong Cheng for the valuable suggestions of this research subject. Y. Xia is funded by National Natural Science Foundation of China (Nos. 11702056, 61572021). Stéphane Bordas thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) “RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery”. We also thank the funding from the Luxembourg National Research Fund (INTER/MOBILITY/14/8813215/CBM/Bordas and INTER/FWO/15/10318764).
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Appendices
Appendix A: Using an approximated normal vector field
In Eqs. 8 and 9, the normal vector field \(\varvec{n}(\xi ,\eta )\) should be commonly built as \(\sum R_A \varvec{n}_A\), where \(\varvec{n}_A\) needs to be specified for each control point: it denotes the normal vector of a mid-surface point associated with the control point \(\varvec{x}_A\). Although no approximation error occurs, it is time-consuming to solve for the mid-surface projecting points of \(\varvec{x}_A\). In this paper the normal vectors at the Greville abscissae \((\tilde{\xi },\tilde{\eta })\) are adopted to construct a normal vector field approximately [21], the field is re-written as \(\sum R_A(\xi ,\eta )\tilde{\varvec{n}}_A\). Following Eq. 2, these normal vectors \(\tilde{\varvec{n}}_A\) at the Greville abscissae points can be easily calculated. Thus we have
and
Appendix B: Detailed formulation of degenerated Reissner–Mindlin shell element
In the following we introduce some necessary aspects to obtain the element stiffness matrix:
-
(a)
Using Voigt notation, the relation between the strain vector and the stress vector is expressed as
$$\begin{aligned} \varvec{\sigma } =\varvec{D}_g \pmb {\varepsilon }, \end{aligned}$$(B.1)
where \(\varvec{D}_g\) is the global constitutive matrix, and
here \(\varvec{D}_l\) is a given local constitutive matrix. To fulfill the plane stress state \(\sigma _{33}=0\), the local constitutive matrix is given by
in which \(\kappa \) is the shear correction factor and it is set to be 5/6 in this research. To fit the physical shape of the mid-surface, a transformation is employed for \(\varvec{D}_l\), specifically
At each Gauss quadrature point, vectors in Eqs. 2 and 3 are calculated, then the transformation in Eq. B.2 is performed.
-
(b)
Based on Eq. A.1, the Jacobi matrix is calculated by
then the physical gradient of the shape functions are
Moreover for \(\zeta \frac{h}{2}R_A:=\tilde{R}_A\), their physical gradients are given by
-
(c)
Taking a look at the strain formula Eq. 5, we form the strain matrix as
Finally the element stiffness matrix is calculated by
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Hu, Q., Xia, Y., Natarajan, S. et al. Isogeometric analysis of thin Reissner–Mindlin shells: locking phenomena and B-bar method. Comput Mech 65, 1323–1341 (2020). https://doi.org/10.1007/s00466-020-01821-5
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DOI: https://doi.org/10.1007/s00466-020-01821-5