Bending of Nonconforming Thin Plates Based on the Mixed-Order Manifold Method with Background Cells for Integration

Xin Qu,Xingqian Xu,Lijun Su,Zhijun Liu,Fangfang Diao,Wei Li,Georgios I Giannopoulos

doi:10.1155/2020/6681214

Abstract

As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed-order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed-order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

Highlights

Cells for IntegrationFor the domains with irregular physical boundaries, especially with curved boundaries, the traditional meshes are difficult to match, which leads to an inaccurate approximation and deteriorates the accuracy of solutions. us, locally refined meshes are often adopted to approximate these irregular physical boundaries.the traditionally local refinement technology must generate more degrees of freedom in the finite element method (FEM), which deteriorates the efficiency
Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. e results indicated that the solutions achieved with the proposed numerical manifold method (NMM) rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM
Since the NMM creates two completely independent cover systems, considerably refined meshes designated as background cells can be used to approximate the domains with irregular physical boundaries, while relatively coarse meshes were employed for interpolation

Summary

The Penalized Variational Formulation

The integration over the entire problem is divided into a summation of integrations over all the elements, and the solution is constructed by an interpolation polynomial. The continuity of the normal rotation on element surfaces in a nonconforming thin plate cannot be achieved. If an element can pass the “patch test”, its convergence can be achieved [34]. E existing studies [12, 17, 35] and numerical practices have shown that the elements formed by regular meshes, as proposed by the NMM, are able to pass the “patch test”. The regular meshes cannot be available for the FEM in some cases, in the domain with irregular physical boundaries, which creates convergence issues for the nonconforming elements. Us, the total potential energy based on Kirchhoff’s thin plate theory can be given as follows:. When imposing essential boundary conditions, the FEM requires that all nodes must be on the domain boundaries. Where Sw S1∪S2 is the essential boundary and kw and kM are the penalty parameters, varying little for the range of 103 E–105 E

Element Analysis

Local Displacement Function

Background

Numerical Examples

Conclusions