Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben

2000 | Buch

Mathematical Fundamentals of Differential Quadrature Method: Linear Vector Space Analysis and Function Approximation Kapitel lesen Erstes Kapitel lesen

Differential Quadrature and Its Application in Engineering

verfasst von: Chang Shu, PhD

Verlag: Springer London

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

In the past few years, the differential quadrature method has been applied extensively in engineering. This book, aimed primarily at practising engineers, scientists and graduate students, gives a systematic description of the mathematical fundamentals of differential quadrature and its detailed implementation in solving Helmholtz problems and problems of flow, structure and vibration. Differential quadrature provides a global approach to numerical discretization, which approximates the derivatives by a linear weighted sum of all the functional values in the whole domain. Following the analysis of function approximation and the analysis of a linear vector space, it is shown in the book that the weighting coefficients of the polynomial-based, Fourier expansion-based, and exponential-based differential quadrature methods can be computed explicitly. It is also demonstrated that the polynomial-based differential quadrature method is equivalent to the highest-order finite difference scheme. Furthermore, the relationship between differential quadrature and conventional spectral collocation is analysed.
The book contains material on:
- Linear Vector Space Analysis and the Approximation of a Function;
- Polynomial-, Fourier Expansion- and Exponential-based Differential Quadrature;
- Differential Quadrature Weighting Coefficient Matrices;
- Solution of Differential Quadrature-resultant Equations;
- The Solution of Incompressible Navier-Stokes and Helmholtz Equations;
- Structural and Vibrational Analysis Applications;
- Generalized Integral Quadrature and its Application in the Solution of Boundary Layer Equations.
Three FORTRAN programs for simulation of driven cavity flow, vibration analysis of plate and Helmholtz eigenvalue problems respectively, are appended. These sample programs should give the reader a better understanding of differential quadrature and can easily be modified to solve the readers own engineering problems.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Mathematical Fundamentals of Differential Quadrature Method: Linear Vector Space Analysis and Function Approximation
Abstract
Most engineering problems are governed by a set of partial differential equations (PDEs) with proper boundary conditions. For example, Newtonian fluid flows are modeled by the Navier-Stokes equations; the vibration of thin plates is governed by a fourth order partial differential equation; whereas acoustic waves and microwaves can be simulated by the Helmholtz equation. In general, it is very difficult for us to obtain the closed-form solution of these equations. On the other hand, the solution of these PDEs is always demanded due to practical interests. For example, when we design an aircraft, we need to know the curve of cl (lift coefficient) versus cd (drag coefficient) for a given airfoil shape. The values of cl and cd can be obtained from the solution of Navier-Stokes equations. Therefore, it is important for us to develop some approximate solutions to the given PDEs.
Chang Shu
Chapter 2. Polynomial-based Differential Quadrature (PDQ)
Abstract
The fundamentals of the differential quadrature (DQ) method, that is, linear vector space analysis and function approximation, have been described in the last chapter. In this chapter, we will show the details of determining the weighting coefficients in the DQ approximation when the solution of a partial differential equation (PDE) is approximated by a polynomial of high degree. Since for this case, the DQ approximation is related to the polynomial approximation to the solution of a PDE, for simplicity, it is termed the polynomial-based differential quadrature (PDQ) method.
Chang Shu
Chapter 3. Fourier Expansion-based Differential Quadrature (FDQ)
Abstract
In Chapter 1, we have shown that the solution of a PDE can be approximated by a polynomial of high degree or by the Fourier series expansion, depending on the feature of the problem. It is noted that the expressions for polynomial approximation and Fourier series expansion are quite different. Thus, when the DQ approximation is applied to these two cases, the formulations to compute the weighting coefficients in the DQ approximation may be different. In Chapter 2, the computation of the weighting coefficients for polynomial-based differential quadrature (PDQ) was described in detail. In PDQ, the solution of a PDE is approximated by a high degree polynomial. The polynomial approximation is suitable for most engineering problems. However, for some problems, especially for those with periodic behaviors such as the Helmholtz problems, polynomial approximation may not be the best choice for the true solution. In contrast, Fourier series expansion could be the best approximation. In this chapter, we will demonstrate that using Fourier series expansion and linear vector space analysis, the weighting coefficients in the Fourier expansion-based differential quadrature (FDQ) can also be calculated by explicit formulations.
Chang Shu
Chapter 4. Some Properties of DQ Weighting Coefficient Matrices
Abstract
It has been shown in Chapters 2 and 3 that the DQ approximation for a derivative of any order has a similar form. The difference in the approximation for the respective derivatives lies only in the weighting coefficients. Consider a one-dimensional problem. It is supposed that there are N grid points in the whole domain with coordinates x1, x2, …, xN. At any location xi, the nth order derivative of a function u(x,t) with respect to x can be approximated by the DQ method as
$$ u_x^{\left( n \right)}\left( {{x_i},t} \right) = \sum\limits_{j = 1}^N {w_{ij}^{\left( n \right)}} u\left( {{x_j},t} \right), $$
(4.1)
for i = 1.2 ...,N
Chang Shu
Chapter 5. Solution Techniques for DQ Resultant Equations
Abstract
In most applications of the DQ method to engineering problems, which are governed by time-dependent partial differential equations (PDEs), the spatial derivatives are discretized by the DQ method whereas the time derivatives are discretized by low order finite difference schemes. For the general case, we consider a time-dependent PDE as follows
$$ \frac{{\partial w}}{{\partial t}} + \ell \left( w \right) = g $$
(5.1)
where l (w) is a differential operator containing all the spatial derivatives and g is a given function. Equation 5.1 should be specified with proper initial and boundary conditions for the solution to a specific problem.
Chang Shu
Chapter 6. Application of Differential Quadrature Method to Solve Incompressible Navier-Stokes Equations
Abstract
The basic theory and properties of the DQ method have been described in the previous chapters. In this chapter, we will show how the DQ method can be applied to solve the incompressible Navier-Stokes equations.
Chang Shu
Chapter 7. Application of Differential Quadrature Method to Structural and Vibration Analysis
Abstract
As shown in the previous chapters, the differential quadrature method has a feature in that it can obtain very accurate numerical results by using just a few grid points. This feature has a particular merit in its application to structural and vibration analysis. For example, the vibration of a thin plate is governed by a fourth order partial differential equation. When a numerical method is applied to discretize the spatial derivatives, the partial differential equation can be reduced to a set of algebraic equations. The eigenvalues of the resultant algebraic equation system provide the vibrational frequencies of the problem. Usually, the number of interior grid points is equal to the dimension of the resultant algebraic equation system, thus providing the same number of eigenfrequencies. Among all the computed eigenfrequencies, only low frequencies are of practical interest. As we know, low order methods such as finite differences and finite elements are only capable of obtaining accurate numerical results by using a large number of grid points. So, when low order methods are applied, they need to use a large number of grid points to obtain highly accurate values for low frequencies. As a result, a lot of virtual storage and computational effort are required. On the other hand, when the DQ method is applied, the low frequencies can be obtained very accurately by using a considerably smaller number of grid points due to the feature of the method. As a consequence, very little computational effort and virtual storage are needed when the DQ method is used.
Chang Shu
Chapter 8. Miscellaneous Applications of Differential Quadrature Method
Abstract
In the previous chapters, we have described the applications of the DQ method to fluid mechanics and solid mechanics. Apart from these two areas, the DQ method has been applied in many other areas. For the solution of general partial differential equations, Mansell et al. (1993) compared the DQ method with conventional numerical approaches. Tomasiello (1998) also discussed the application of the DQ method for general initial-boundary-value problems. Another example is the application of the DQ method to solve ocean engineering problems by Gutierrez, Laura and Rossi (1994). In this chapter, we will briefly describe applications of the DQ method to heat transfer, chemical rector, lubrication, waveguide analysis problems, and to the solution of Helmholtz equation. The effect of mesh point distribution on the accuracy of the DQ results will also be discussed in this chapter. It should be indicated that some of these applications are still under active investigation, so the results reported here are very incomplete.
Chang Shu
Chapter 9. Application of Differential Quadrature to Complex Problems
Abstract
In the previous chapters, we have described applications of the DQ method for problems involving simple domains. These applications have proven to be very successful.
Chang Shu
Chapter 10. Generalized Integral Quadrature (GIQ) and Its Application to Solve Boundary Layer Equations
Abstract
In practice, for some problems such as the area of a surface and the volume of a body, it is necessary to know the integration of a function over a domain. In most cases, it is difficult to obtain the value of the integration analytically. As a result, numerical integration is of great interest in engineering. The numerical integration of a function f(x) over a domain [a, b] can usually be written in the form.
Chang Shu
Backmatter
Metadaten
Titel
Differential Quadrature and Its Application in Engineering
verfasst von
Chang Shu, PhD
Copyright-Jahr
2000
Verlag
Springer London
Electronic ISBN
978-1-4471-0407-0
Print ISBN
978-1-4471-1132-0
DOI
https://doi.org/10.1007/978-1-4471-0407-0