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J. Chen et al.
the available thin shell element is either a nonconforming element or a conforming
element with complicated theoretical derivation and difficult numerical implementation
[10].
Based on the Reissner/Mindlin assumption, the construction of the thick/thin shell
element considering the shear effect is relatively easy because only C 0 continuity is
required [11]. However, the thick/thin shell element often encounters the shear and
membrane locking when applied to thin shells, for example, the three-node triangular
flat shell element by Zhang et al. [12], the four-node quadrilateral element by Wang
et al. [13], the EXG element by Kulikov et al. [14], the DKMQ24 element by Katili
et al. [15], the three-dimensional superparameter shell element by Hahn et al. [16],
and the three-dimensional relative DOFs shell element by Li et al. [17]. Numerical
expediencies, such as the reduced/selected integration, the assumed shear strain/stress
approach, the hybrid/mixed formulation and the refined nonconforming element method,
have to be employed to eliminate the shear and membrane locking. The employment of
these numerical expediencies will lead to new problems, such as spurious zero energy
modes, complicated theoretical derivation, cumbersome numerical implementation and
the numerical oscillation of the calculation results [18].
Great effort has been put into the research of thick/thin shell elements using isogeometric analysis for the past few years [19–22]. Within the theoretical framework
of isogeometric analysis, the error introduced by the geometric approximation is eliminated at the source, and the accuracy of the structural response analysis is improved
[23–25]. The isogeometric shell element can also avoid the shear and membrane locking
by simply increasing the order of the displacement function with no need for numerical expediencies [26]. Although it has the above-mentioned advantages, the geometric
modeling system of isogeometric analysis is mainly based on the nonuniform rational
B-spline, and the complexity for simulating the discontinuities or multi-patches in shell
structures hinders its application by researchers to a certain extent [27].
In summary, given the complexity of the mechanical property and the difficulty in the
mathematical processing, the construction of a simple, efficient, robust and locking-free
thick/thin shell element remains one of the hottest and most challenging research topics
in the field of mechanical analysis and engineering computation [28].
Based on the Reissner/Mindlin theory, a new locking-free thick/thin shell element,
the SEIA, is proposed in this paper. The third order polynomial is employed to define
an incompatible displacement approximation in each independent element and fictitious
thin layers are utilized to ensure the displacement conformity between adjacent elements, which makes the SEIA avoids the shear and membrane locking and maintain
the displacement conformity, simple formulation and easy numerical implementation.
is insensitive to element distortion, has a good convergence rate and high accuracy and
provides reliable solutions for thick/thin shells.
The paper is organized as follows. The geometric description of the shell in a general
orthogonal curvilinear coordinate system is briefly introduced in Sect. 2. The theoretical
formulation of the SEIA is deduced in detail in Sect. 3. Section 4 demonstrates the high
performance of the present element with six typical numerical examples. Finally, in
Sect. 5, the work in this paper is concluded, and the future steps are proposed.
J. Chen et al.
the available thin shell element is either a nonconforming element or a conforming
element with complicated theoretical derivation and difficult numerical implementation
[10].
Based on the Reissner/Mindlin assumption, the construction of the thick/thin shell
element considering the shear effect is relatively easy because only C 0 continuity is
required [11]. However, the thick/thin shell element often encounters the shear and
membrane locking when applied to thin shells, for example, the three-node triangular
flat shell element by Zhang et al. [12], the four-node quadrilateral element by Wang
et al. [13], the EXG element by Kulikov et al. [14], the DKMQ24 element by Katili
et al. [15], the three-dimensional superparameter shell element by Hahn et al. [16],
and the three-dimensional relative DOFs shell element by Li et al. [17]. Numerical
expediencies, such as the reduced/selected integration, the assumed shear strain/stress
approach, the hybrid/mixed formulation and the refined nonconforming element method,
have to be employed to eliminate the shear and membrane locking. The employment of
these numerical expediencies will lead to new problems, such as spurious zero energy
modes, complicated theoretical derivation, cumbersome numerical implementation and
the numerical oscillation of the calculation results [18].
Great effort has been put into the research of thick/thin shell elements using isogeometric analysis for the past few years [19–22]. Within the theoretical framework
of isogeometric analysis, the error introduced by the geometric approximation is eliminated at the source, and the accuracy of the structural response analysis is improved
[23–25]. The isogeometric shell element can also avoid the shear and membrane locking
by simply increasing the order of the displacement function with no need for numerical expediencies [26]. Although it has the above-mentioned advantages, the geometric
modeling system of isogeometric analysis is mainly based on the nonuniform rational
B-spline, and the complexity for simulating the discontinuities or multi-patches in shell
structures hinders its application by researchers to a certain extent [27].
In summary, given the complexity of the mechanical property and the difficulty in the
mathematical processing, the construction of a simple, efficient, robust and locking-free
thick/thin shell element remains one of the hottest and most challenging research topics
in the field of mechanical analysis and engineering computation [28].
Based on the Reissner/Mindlin theory, a new locking-free thick/thin shell element,
the SEIA, is proposed in this paper. The third order polynomial is employed to define
an incompatible displacement approximation in each independent element and fictitious
thin layers are utilized to ensure the displacement conformity between adjacent elements, which makes the SEIA avoids the shear and membrane locking and maintain
the displacement conformity, simple formulation and easy numerical implementation.
is insensitive to element distortion, has a good convergence rate and high accuracy and
provides reliable solutions for thick/thin shells.
The paper is organized as follows. The geometric description of the shell in a general
orthogonal curvilinear coordinate system is briefly introduced in Sect. 2. The theoretical
formulation of the SEIA is deduced in detail in Sect. 3. Section 4 demonstrates the high
performance of the present element with six typical numerical examples. Finally, in
Sect. 5, the work in this paper is concluded, and the future steps are proposed.