On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials

Abstract

The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the $$C^1$$ C 1 -continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of $$C^1$$ C 1 -continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for $$C^0$$ C 0 -continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these $$C^1$$ C 1 -continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

An enriched virtual element method for 2D‐3C generalized membrane shell model on surface

“…Graphically, all mesh types produce similar results and demonstrate that the developed virtual plate elements can compute meaningful engineering solutions. The results of deflection are consistent with Reference 45, indicating that our model and algorithm can well characterize the clamping plate problem.…”

“…The material parameters selected for the study include Young's modulus E = 2 × 10 8 KPa, Poisson's ratio 𝜈 = 0.3, thickness 2𝜀 = 0.01m, and bending stiffness D = 18.315Nm. 45 Figure 15 depicts contour plots of the deflection w for the virtual element method with two kinds of polygonal meshes. Graphically, all mesh types produce similar results and demonstrate that the developed virtual plate elements can compute meaningful engineering solutions.…”

“…The plate is subjected to a uniform load $$$ \left({p}^1,{p}^2,{p}^3\right)=\left(0,0,-1.0\right)\mathrm{KPa} $$$. The material parameters selected for the study include Young's modulus $$$ E=2\times 1{0}^8\mathrm{KPa} $$$, Poisson's ratio $$$ \nu =0.3 $$$, thickness $$$ 2\varepsilon =0.01\mathrm{m} $$$, and bending stiffness $$$ D=18.315\mathrm{Nm} $$$ 45 …”

“…It doesn't need to define explicit expressions of shape functions and only uses degrees of freedom to calculate the stiffness matrix, which is very convenient and effective. In recent years, this method has been widely applied in various fields, such as Poisson problem, 28,33 elasticity problems, [34][35][36][37][38][39][40][41] plate bending problem, [42][43][44][45][46] Stokes problem, [47][48][49] Navier-Stokes problem, 50,51 contact problem, 52 Steklov eigenvalue problem, 53 Cahn-Hilliard problem, 54 Helmholtz problems, 55 others partial differential equations [56][57][58][59][60][61] and so on. It is worth noting that VEM has yet to be studied in the GMS model.…”

On a virtual element formulation for trusses and beams

“…1 Simple bar problem, geometry, data and discretization beam elements is provided by the Kirchhoff-Love theory for plate. Different formulations have already been discussed in [3], [5], [11] and [15] in this context .…”