Multiresolution Finite Element Method Based on a New Locking-Free Rectangular Mindlin Plate Element

Yiming Xia

doi:10.4236/wjm.2016.66016

Abstract

A locking-free rectangular Mindlin plate element with a new multi-resolution analysis (MRA) is proposed and a new finite element method is hence presented. The MRA framework is formulated out of a mutually nesting displacement subspace sequence whose basis functions are constructed of scaling and shifting on the element domain of basic full node shape function. The basic full node shape function is constructed by extending the split node shape function of a traditional Mindlin plate element to other three quadrants around the coordinate zero point. As a result, a new rational MRA concept together with the resolution level (RL) is constituted for the element. The traditional 4-node rectangular Mindlin plate element and method is a mono-resolution one and also a special case of the proposed element and method. The meshing for the monoresolution plate element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the rigorous mathematical basis. The analysis clarity of a plate structure is actually determined by the RL, not by the mesh. Thus, the accuracy of a plate structural analysis is replaced by the clarity, the irrational MRA by the rational and the mesh model by the RL that is the discretized model by the integrated.

Highlights

Multi-resolution analysis (MRA) is a popular technique that has been applied in many domains such as the signal and image processing, the damage detection and health monitoring, the differential equation solution, etc.How to cite this paper: Xia, Y.M. (2016) Multiresolution Finite Element Method Based on a New Locking-Free Rectangular Mindlin Plate Element
The great efforts have been made over the past thirty years to overcome the drawbacks of the FEM with many improved methods to come up, such as wavelet finite element method (WFEM), meshfree method (MFM) and natural element method (NEM) etc., which open up a transition from the monoresolution finite element method to the multiresolution finite element method featured with adjustable element node number
Due to the basic full node shape function, the stiffness matrix and the loading column vectors of a proposed element can be automatically acquired through quadraturing around nodes in the element matrix formation step while those of the traditional 4-node rectangular Mindlin plate element obtained through complex artificially reassembling of the element matrix around the node-related elements in the re-meshing process for their 1/4 split nodes in a conventional element, which contributes a lot to computation efficiency improvement of the Central deflection

Summary

Introduction

Multi-resolution analysis (MRA) is a popular technique that has been applied in many domains such as the signal and image processing, the damage detection and health monitoring, the differential equation solution, etc. The great efforts have been made over the past thirty years to overcome the drawbacks of the FEM with many improved methods to come up, such as WFEM, MFM and NEM etc., which open up a transition from the monoresolution finite element method to the multiresolution finite element method featured with adjustable element node number These MRA methods have illustrated their powerful capability and computational efficiency in dealing with some problems, they always have such major inherent deficiencies as the complexity of shape function construction, the absence of the Kronecker delta property of the shape function and the lack of a rigorous mathematical basis for the MRA, which make the treatment of element boundary condition complicated and the selection of element node layout empirical that substantially reduce computational efficiency. The proposed element method can bring about substantial improvement of the computational efficiency in the structural analysis when compared with the corresponding FEM or other MRA methods

Basic Full Node Shape Function

Displacement Subspace Sequence

Multiresolution Rectangular Mindlin Plate Element Formulation

Transformation Matrix

Numerical Example and Discussion

Conclusion and Prospective