Weak-form Quadrature Element Analysis Of Three-dimensional Problems

In many engineering practices, especially for structural components with three comparable dimensions, the current beam and plate theories may lead to significant errors and even blunders. In this case, three-dimensional analysis becomes important and necessary.Functionally Graded Materials (FGM) are a new type of composite material, which have sprung up in recent years, to serve the needs in high technology. FGM are composed of two different materials with advanced manufacturing technology and there is continuous gradient of composition and structure between the two materials, without any explicit interfaces.The Weak Form Quadrature Element Method (QEM) is a new numerical method developed based on variational principles. The essence of the method is that the functionals involved in the variational principle are approximated using high order numerical integration and the derivatives at the sampling points are then approximated using the differential quadrature method. In the differential quadrature method, the derivative of a function with respect to a coordinate direction is expressed as a weighted linear sum of all the function values at sampling points. Introduction of the stationary condition yields a set of algebraic equations. In comparison with the p-version finite elements that introduce unknown variables without physical meanings, QEM choose field variables at sampling points. Especially for Functionally Graded Materials that pose challenge to other numerical methods, QEM can simulate the continuous change of material properties more easily and efficiently, highlighting its great potential in this application area.QEM inherits the global interpolations, high accuracy and efficiency of the differential quadrature method. Embracing the concept of domain decomposition and coordinate transformation, QEM can deal with problems with irregular geometry. Furthermore, due to the introduction of variational principles, the method enjoys many advantages of weak form description of a problem. In comparison with the finite element method, QEM weakens the concept of shape function and alleviates the burden of choosing element type. Moreover, the post-processing of quadrature element analysis is rather simple since function values at sampling points are taken as degrees of freedom.The present investigation is aimed to perform three-dimensional analysis of some structural problems. The work covers the implementation of QEM in the following areas: three-dimensional free vibrations of thick plates on Pasternak foundation, three-dimensional free vibrations of thick plates and three-dimensional free vibrations of curved panel. The theoretical basis of QEM and practical skills are elaborated at length in these examples, and the high computational efficiency is verified in three-dimensional analysis, especially in the vibration analysis of functionally graded components. A comparative study is made with the results of the finite element software ANSYS to gain some insights into QEM.