Differential Quadrature and Differential Quadrature Based Element Methods (eBook)

Theory and Applications

Xinwei Wang (Autor)

eBook Download: PDF | EPUB

2015 | 1. Auflage
408 Seiten
Elsevier Science (Verlag)
978-0-12-803107-0 (ISBN)

Infos

Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications is a comprehensive guide to these methods and their various applications in recent years. Due to the attractive features of rapid convergence, high accuracy, and computational efficiency, the differential quadrature method and its based element methods are increasingly being used to study problems in the area of structural mechanics, such as static, buckling and vibration problems of composite structures and functional material structures. This book covers new developments and their applications in detail, with accompanying FORTRAN and MATLAB programs to help you overcome difficult programming challenges. It summarises the variety of different quadrature formulations that can be found by varying the degree of polynomials, the treatment of boundary conditions and employing regular or irregular grid points, to help you choose the correct method for solving practical problems. - Offers a clear explanation of both the theory and many applications of DQM to structural analyses - Discusses and illustrates reliable ways to apply multiple boundary conditions and develop reliable grid distributions - Supported by FORTRAN and MATLAB programs, including subroutines to compute grid distributions and weighting coefficients

Professor at Nanjing University of Aeronautics and Astronautics in China since 1993, Dr. Wang earned his MS degree in solid mechanics from the same university, and his PhD in mechanical engineering from the University of Oklahoma, USA. Prof. Wang has been a visiting scholar at the University of Michigan at Dearborn, the University of Maryland at Baltimore County, the University of Oklahoma and the University of California at Los Angeles. He has published over 200 papers in the areas of numerical methods and computational engineering, mechanics of composite materials and experimental finite plasticity, with over 50 of these related to the differential quadrature method.

Textauszug

Chapter 1

Differential Quadrature Method

Abstract

A variety of different quadrature (DQ) formulations exist in literatures. This has often caused confusion for researchers and engineers and led to a difficulty in making a choice of a different quadrature method (DQM) for solving practical problems. This chapter presents the basic principle of the DQM and summarizes various DQ formulations, including the original DQM, the modified DQM, harmonic differential quadrature method (HDQM), local adaptive differential quadrature method (LaDQM), and the DQ-based time integration scheme. Existing explicit formulas to compute the weighting coefficients are given. Various grid distributions are summarized and their discrete error is briefly discussed. Some recommendations are made. Although Grid III is the most widely used grid spacing in literature, however, research shows that Grid V is the most reliable grid spacing and thus recommended, especially for dynamic analysis by using the DQM. The LaDQM is recommended if the number of grid points is large.

Keywords

differential quadrature

weighting coefficients

harmonic differential quadrature

LaDQM

grid spacing

DQ-based time integration

explicit formulas

discrete error

1.1. Introduction

With the advance of computer technology, nowadays numerical simulations play an important role in science and engineering. Various numerical methods have been used in numerical analysis and are regarded as powerful tools for solving partial differential equations (PDEs). To name a few, finite element method (FEM) [1], finite difference method (FDM) [2], finite volume method (FVM) [3], and methods of weighted residuals (MWR) [4,5] such as Galerkin method and collocation method. Among all aforementioned methods, the most widely used methods are FEM and FDM. Perhaps due to its flexibility and ability in dealing with complex geometries and boundary conditions, FEM is more widely used in the area of structural mechanics [1].

It is known that none of the aforementioned numerical methods is versatile and can be used to solve all problems efficiently. Each method has its own merits and limitations. Even the most widely used FEM still suffers from difficulty in analyzing problems when phenomena such as singularities, steep changes, stress concentration, and large deformation exist. For example, the computational efficiency of the FEM is lost when the method is used to analyze problems of metal forming and elastoplasticity, high-velocity impact, dynamic crack propagation, explosion, and shock wave. Even with the modern computing machines, analysis for guided wave propagation in three-dimensional solids by the classical FEM would require impractical computational resources (the computational time and memory storage requirements) [6]. Therefore, along with the ever-growing advancement of faster computing machines, the research into the development of new efficient methods for numerical simulations is an ongoing parallel activity [7].

Motivated by the needs of modern science and technology, considerable efforts have been made in the development of new numerical methods, such as various meshless or mesh-free methods [8], wavelet-based numerical methods [9], the differential quadrature method (DQM) and the differential quadrature-based element method [7], the discrete singular convolution (DSC) algorithm [10], and the high-order FEM [11]. A few examples of the requirements of new methods are as follows: (1) simulations of many dynamic systems often require very fast numerical solution of the equations of the system mathematical models [7]; (2) the computer-aided design (CAD) process in which the database often requires large computer storage and the interpolative manipulations for the operating design parameters may be less accurate as well as quite timeconsuming [7]; and (3) structural health monitoring (SHM) applications call for both efficient and powerful numerical tools to predict the behavior of ultrasonic-guided waves since the existing well-known FEM would require impractical computational resources. The aforementioned methods try to resolve the limitations existing in the classical FEM and to fulfil the needs of modern science and technology.

To reduce the enormous computational costs in simulations of wave propagation in solids, higher- order FEMs with polynomial degrees p > 2 [11], time-domain spectral element methods (SEM) [12–14], and weak-form quadrature element method (QEM) [15] are proposed. Besides the merit of high rate of convergence, they also possess the advantages existing in the FEM and can be implemented into the commercial software such as ABAQUS, since these methods are essentially the FEMs. Obviously, these methods still have some limitations existing in the classical FEM. For example, they cannot be efficiently used in the large deformation analysis such as analysis of metal forming.

Aimed at resolving the limitations existing in the FEM for analyzing problems of metal forming and elastoplasticity, high-velocity impact, dynamic crack propagation, and explosion, various mesh-free methods [8] are proposed. Due to less mesh dependency, mesh-free methods can eliminate possible mesh distortion and entanglement encountered in FEM to analyze large deformation and explosion problems. For example, the method of smoothed particle hydrodynamic (SPH) [16], a mesh-free method, is proposed and implemented into the commercial finite element software LS-DYNA to complement the deficiency of the FEM in dealing with the explosion and high-velocity impact problems. Extended finite element method (XFEM) [17], another mesh-free method [8], is proposed to analyze the problem of dynamic crack propagations. Since the finite element mesh can remain unchanged during crack propagation with using the XFEM, the hybrid method of FEM together with XFEM is convenient to analyze such problems. It is seen that each method is proposed to overcome certain difficulty existing in FEM and to fulfil certain needs by the modern science and technology. Mesh-free methods have some disadvantages: their approximate functions are much complicated and larger computation effort is usually needed; and dealing with essential boundary conditions is far more complicated than the classical FEM [8].

Since wavelets possess multiresolution and localization properties, various wavelet-based methods are proposed to meet some needs of modern science and technology [9]. Similar to the mesh-free methods, wavelet-based methods can be more efficiently used to analyze problems with singularities, steep changes, and stress concentration. For example, the wavelet Galerkin method (WGM) is proposed and has been successfully used in solving a variety of PDEs in regular and irregular computational domains [18]. The drawback of the WGM is that the method can only handle simple boundary conditions and is complicated when nonhomogeneous boundary conditions are considered in two-dimensional cases [9].

The DSC algorithm [10] is efficient and robust for solving the Fokker–Planck equation describing various physical phenomena. The method employs compactly support wavelet interpolating functions. Banded differential matrices with well-behaved condition numbers are obtained thus the DSC is suitable for large-scale computations. Due to employing exterior grid points for treating boundary conditions, spurious eigenvalues are removed although the matrices are not symmetric. The unique advantage of the DSC is that it can also yield accurate high-order mode frequencies. Perhaps its drawback is not easy to apply the free boundary conditions, especially at free corners of anisotropic rectangular plates. Although with the aid of the iteratively matched boundary (IMB) method [19], this issue may be resolved to a certain degree but is very complicated and the existing problem has not completely solved yet.

Because of clear superiority of the FEM and FDM in engineering applications, some new or hybrid methods are proposed to improve the classical FEM and FDM: the aforementioned higher-order FEMs [11], SEM [12–14], QEM [15], mesh-free least-squares-based FDM [20], wavelet finite element method (WFEM) [21], and wavelet-optimized finite difference method (WOFD) [22]. These methods raise the computational efficiency and extend the application ranges of the classical FEM and FDM.

DQM was originated by Bellman and Casti in the early 1970s [23,24]. The method has a relatively recent origin in the later 1980s [25] and has been gradually emerging as a distinct numerical solution technique for the initial- and/or boundary-value problems of physical and engineering sciences since then [7]. In fact, the DQM can be formulated via the polynomial-based collocation method, one of the popular methods of MWR [26,27]. As a numerical method, the DQM can be applied in the fields of biosciences, transport processes, fluid mechanics, static and dynamic structural mechanics, static aeroelasticity, and lubrication mechanics. It has been shown that the DQM is simple and can yield highly accurate numerical solutions with minimal computational effort. The differential quadrature-based element methods are proposed to overcome some deficiency existing in the conventional DQM and extended the application range of the DQM in dealing with complex geometry...

Produkt-Details

• Erscheint lt. Verlag: 24.3.2015
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-12-803107-7 / 0128031077
• ISBN-13: 978-0-12-803107-0 / 9780128031070