A stabilized quasi and bending consistent meshfree Galerkin formulation for Reissner–Mindlin plates

Abstract

The state-of-the-art locking-free meshfree Galerkin formulation for modeling the Reissner–Mindlin plate problems is plagued by the following issues: (1) the requirement of a large enough kernel support size in avoiding kernel instability because of the quadratic basis in meeting the Kirchhoff mode reproducing condition, as well as (2) the tedious construction of the conforming representative domains and the smoothed strain in the stabilized conforming integration scheme. This study introduces an efficient and stabilized approach that circumvents the above-mentioned issues. A quasi-consistent reproducing kernel approximation is first developed to enable a smaller kernel support size to be used without the moment matrix singularity issue under a controllable loss of completeness; thus, the approximation construction is accelerated. Then, a bending consistent nodal integration method is proposed where the bending consistency in Galerkin formulation is achieved via an assumed strain approach without using the conforming cell. A variational multiscale stabilization method from our earlier research is implemented to avoid low energy instability while maintaining the locking-free property. The performance of the present formulation is validated in several benchmark problems.

Appendix A: Variational multiscale stabilization for the Reissner-Midndlin plate

In this study, the stabilization method in our earlier work [42] is employed. The variational multiscale stabilized Galerkin weak form for Reissner-Midndlin Plate reads: find $\left({w}^{h},{{\varvec{\theta}}}^{h}\right)\in {{\overline{\mathcal{S}} }^{w}}^{^h}\times {{\overline{\mathcal{S}} }^{\theta }}^{^h}$ such that $\forall \left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{^h}\right)\in {{\mathcal{V}}^{w}}^{^h}\times {{\mathcal{V}}^{\theta }}^{^h}$, where:

$$\begin{aligned} & a\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h};{w}^{h},{{\varvec{\theta}}}^{h}\right)+{a}_{VMS}\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h};{w}^{h},{{\varvec{\theta}}}^{h}\right)\\&\quad = L\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h}\right)+{L}_{VMS}\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h}\right)\end{aligned}$$

(71)

where $a\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h};{w}^{h},{{\varvec{\theta}}}^{h}\right)$ and $L\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h}\right)$ are respectively the standard plate bilinear form and linear functional that can be computed by SCNI (see Eq. (38)) or proposed BCI (See Eq. (64)); ${a}_{VMS}$ and ${L}_{VMS}$ are stabilization terms expressed as

$$\begin{aligned} & {a}_{VMS}\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h};{w}^{h},{{\varvec{\theta}}}^{h}\right)\\&\quad =\underset{\Omega }{\overset{}{\int }}{\left(\nabla \cdot \left({{\varvec{D}}}^{s}\delta {{\varvec{\gamma}}}^{h}\right)\right)}^{T}{\tau }_{w}\left(\nabla \cdot \left({{\varvec{D}}}^{s}\delta {{\varvec{\gamma}}}^{h}\right)\right)d\Omega\\&\quad +\underset{\Omega }{\overset{}{\int }}{\left(\nabla \cdot \left({{\varvec{D}}}^{b}\delta {{\varvec{\kappa}}}^{h}\right)\right)}^{T}{\tau }_{\theta }\left(\nabla \cdot \left({{\varvec{D}}}^{b}\delta {{\varvec{\kappa}}}^{h}\right)\right)d\Omega\end{aligned}$$

(72)

$${L}_{VMS}\left(\delta {w}^{h},\delta {{\varvec{\theta}}}^{h}\right)=-\underset{\Omega }{\overset{}{\int }}{\left(\nabla \cdot \left({{\varvec{D}}}^{s}\delta {{\varvec{\gamma}}}^{h}\right)\right)}^{T}{\tau }_{w}qd\Omega $$

(73)

where ${\tau }_{w}$ and ${\tau }_{\theta }$ are stabilization parameters and their values are picked based on a dimensional analysis (see [42]):

$${\tau }_{w}={c}_{w}\left(\frac{{h}^{2}}{k\mu }+\frac{{h}^{3}}{E}\right)$$

(74)

$${\tau }_{\theta }={c}_{\theta }\left(\frac{{h}^{2}}{E}+\frac{{t}^{*}}{k\mu }\right)$$

(75)

where $h$ is the nodal distance, ${t}^{*}=t/l$ with $l$ the dimension of the plate. The parameters ${c}_{w}\in \left[\mathrm{0,1}\right]$ and ${c}_{\theta }\in \left[\mathrm{0,1}\right]$ are stabilization control parameters and usually a value of 0.1 or lower can effectively controls the instability without locking occuring. For details in deriving the stabilization, readers may refer to [42] for more details.