Yu-Qiu Long

Introduction To The Third Form Of The Quadrilateral Area Coordinate Method (QACM‐III)

Two quadrilateral area coordinate methods (hereinafter referred to as QACM‐I and QACM‐II) have already been proposed in 1999 and 2008, respectively. The QACM‐I contains four components (L1, L2, L3, L4), among which only two are independent. And the QACM‐II contains only two independent coordinate components (Z1, Z2). Many successful applications demonstrate that these new coordinate systems are efficient tools for developing robust quadrilateral finite element models. In this paper, another new quadrilateral area coordinate method (hereinafter referred to as QACM‐III), containing two components T1 and T2, is systematically established. This new coordinate system (QACM‐III) not only keeps the most important advantages of the QACM‐I and QACM‐II, but also possesses other distinguished characters. For instances, it can be directly applied to quadrilateral elements with curved sides, and may have a simpler form in element shape functions. Furthermore, three different quadrilateral area coordinate methods, QACM...

Introduction — The Evolutive Finite Element Method

This chapter is an opening introduction to the entire book, and also an introduction to the evolutive Finite Element Method (FEM). Firstly, a brief review on the features of FEM is given. Then, a close relationship between FEM and variational principle is discussed according to the development history and categories of FEM. Thirdly, some research areas of FEM of significant interest are listed. Finally, the topics of the book are presented. The purpose of the above arrangement is to explain the background and main idea of this book.

Quadrilateral Area Coordinate Systems, Part I — Theory and Formulae

This chapter introduces new concepts for developing the quadrilateral finite element models. Firstly, the quadrilateral area coordinate system (QACM-I) with four coordinate components, which is a generalization of the triangular area coordinate method, is systematically established in detail. Then, on the basis of the QACM-I, another quadrilateral area coordinate system (QACM-II) with only two coordinate components is also proposed. These new coordinate systems provide the theoretical bases for the construction of new quadrilateral element models insensitive to mesh distortion, which will be introduced in Chap. 17.

The Sub-Region Variational Principles

This chapter focuses on the developments of the variational principles which are usually considered as the theoretical basis for the finite element method. In this chapter, we will discuss the sub-region variational principles which are the results by the combination of the variational principles and the concept of sub-region interpolation. Following the introduction, the sub-region variational principles for various structural forms, i.e., 3D elastic body, thin plate, thick plate and shallow shell, are presented respectively. Finally, a sub-region mixed energy partial derivative theorem is also given.

Variational Principles with Several Adjustable Parameters

This chapter also focuses on the development of the variational principles. Firstly, it introduces three patterns of functional transformation, i.e., pattern I, pattern II and pattern III. Then, on the basis of pattern III, some variational principles with several adjustable parameters are formulated. Finally, a variable-substitution-multiplier method is also proposed based on pattern I and pattern II[1,2].

A Hybrid Membrane Element Based on the Hamilton Variational Principle

The traditional hybrid element is usually formulated based on the complementary energy principle or the Hellinger-Reissner variational principle. This paper presents a new strategy for developing hybrid elements by the Hamilton variational principle with dual mixed variables.

Study of the Analytical Trial Functions

The analytical trial functions methods (ATFM) have been studied systematically in this paper. In the view of the weighted residual methods, the selection of the trial functions is very important to the effectivity of the computational methods. The analytical trial functions methods choose the analytical solutions of the problem as the trial functions of the unknown fields, such as the displacement or the stress in the elastic mechanics problems. However, due to the consistent of the trial functions in the analysis region, when the boundary condition is dealt with in the right way, the computation will be very fine. The generalized conforming methods and the hybrid elements methods are employed to formulate the elements based on the analytical trial functions. It is very successful in this new kind of finite element methods, and series plane elements are presented in this paper. More attractively, the elements can perform very well even in the seriously distorted meshes, which is still a challenge to some other finite element methods. The present ATFM possesses the advantages of both analytical and discrete method; its potential will be shown with the continued study.

Some Recent Advances on the Quadrilateral Area Coordinate Method

Since the Quadrilateral Area Coordinate Method (QACM) was systematically established at the end of last century [1], some successful applications of this new tool have been achieved by various researchers [2, 3]. Compared with the usual isoparametric coordinate method, the QACM can make a quadrilateral finite element model less sensitive to mesh distortion, and simplifies the copmputational procedures (such as no Jacobi inverse is needed).

Method of Volume Coordinates — from Tetrahedral to Hexahedral Elements

In the development history of finite element method, an element model with high performance possesses important significance. The three-dimension (3D) hexahedral isoparametric elements are widely used in scientific and engineering computations. However, their accuracy may drop obviously in an irregular mesh division. In order to improve the robustness of these elements, many researchers have made great efforts, and these jobs have never been stopped. On the other hand, for two-dimension (2D) problems, the area coordinate method has been successfully genrealized from triangular to quadrilateral elements. Compared with the models constructed by isoparametric coordinates, those quadrilateral elements by the area coordinate method are less sensitive to mesh distortion.

Computational Strategies for Curved-side Elements Formulated by Quadrilateral Area Coordinates (QAC)

The sensitivity problem to mesh distortion often occurs for a quadrilateral finite element model. In order to avoiding this trouble, a new Quadrilateral Area Coordinate (QAC) method has been systematically established by generalizing the area coordinate system from triangle to quadrangle [1]. Based on the QAC method, some membrane, plate/shell elements have already been successfully developed. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion [2].