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DQ Based Methods: Theory and Application to Engineering and Physical Sciences

Copyright © 2012. 31 pages.
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DOI: 10.4018/978-1-61350-116-0.ch014
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Tomasiello, Stefania. "DQ Based Methods: Theory and Application to Engineering and Physical Sciences." Handbook of Research on Computational Science and Engineering: Theory and Practice. IGI Global, 2012. 316-346. Web. 1 Aug. 2014. doi:10.4018/978-1-61350-116-0.ch014

APA

Tomasiello, S. (2012). DQ Based Methods: Theory and Application to Engineering and Physical Sciences. In J. Leng, & W. Sharrock (Eds.) Handbook of Research on Computational Science and Engineering: Theory and Practice (pp. 316-346). Hershey, PA: Engineering Science Reference. doi:10.4018/978-1-61350-116-0.ch014

Chicago

Tomasiello, Stefania. "DQ Based Methods: Theory and Application to Engineering and Physical Sciences." In Handbook of Research on Computational Science and Engineering: Theory and Practice, ed. J. Leng and Wes Sharrock, 316-346 (2012), accessed August 01, 2014. doi:10.4018/978-1-61350-116-0.ch014

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Abstract

Though relatively unknown, the Differential Quadrature Method (DQM) is a promising numerical technique that produces accurate solutions with less computational effort than other numerical methods such as the Finite Element Method. There are different versions of the former method, so one can refer to a class of methods based on the differential quadrature (DQ) approach. This chapter provides systematic steps to understand DQ based methods: how they work, how to use and develop them, as well as exemplary problems. It is divided into two sections: the first provides fundamentals and theories related to the DQ approach, while the second presents the application of DQ based methods to three significant problems, i.e. free vibrations of a multi-span beam (a typical model for bridge decks and floors), time dependent heat transfer, and vibrations of a rectangular membrane. The main focus is on the application of the method to space-time domains as a whole, an issue not well covered by the current literature.
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Introduction

A variety of numerical methods are available for solving problems in science and engineering, when analytical solutions (so-called closed-form solutions) cannot be achieved. Numerical methods are part of the field called scientific computing, which can be defined as (Golub and Ortega, 1992, p. 2):

“The collection of tools, techniques, and theories required to solve on a computer mathematical models of problems in science and engineering”.

Mathematical models very often take the form of differential equations, with initial and/or boundary conditions. The variables of the problem may depend on time and/or n-space coordinates respectively (in this latter case the problem is called n-dimensional, where n=1,2,3 usually). For example, a boundary condition can represent restraints i.e. the supports of the multi-span beam (see the first example in the final section) and an initial condition can be the initial heat profile in the time-dependent problem of heat transfer (see the second example in the final section).

Numerical computations are affected by several types of error, rounding errors are due to machine precision. Discretization errors are created when the “continuous” problem domain is substituted by a “discrete” one. The convergence error is due to the iterative nature of some numerical methods, where a finite sequence of approximations to a solution is generated. An acceptable degree of error depends on the problem being solved: for many problems in industry or engineering three digit accuracy is quite acceptable.

Discretization is the core of almost all the numerical methods (including the methods discussed in this chapter), by introducing a set of points into the problem or computational spatial domain (i.e. where the solution will be computed). These points, often equally spaced, can have different names, e.g. grid points, in the Finite Difference Method (FDM), collocation points in the collocation methods or nodes in the Finite Element Method (FEM). Choosing the points in the computational domain is achieved by constructing the approximate solution of the values at these points. This has the benefit of converting the differential equations into an algebraic system of equations that can be easily solved on a computer.

Replacing the differential equations with an algebraic system of equations can be done by approximating the derivatives of the unknown function (the solution of the problem) by formulas based on the values of the function itself at discrete points and is generally achieved in a systematic way (e.g in FDM) or by using polynomials which interpolate the function at the given points (e.g. FEM or collocation methods). How these interpolating polynomials are chosen and used is the main difference between some numerical methods. In collocation methods a finite-dimensional space of candidate solutions (usually, polynomials up to a certain degree) are chosen, so that the solution satisfies the given equation at the collocation points. The FEM approximates the solution as a linear combination of piecewise functions (usually, polynomials), that are nonzero on small sub-domains. In fact, the basic idea of FEM is to divide the spatial domain of the problem into smaller parts (or sub-domains) called finite elements or elements, connected by points called nodes, giving a topological map which is called a mesh, whereas the process of making the mesh is called mesh generation. The main advantage of the FEM is its ability to reproduce non-smooth solutions, in conjunction with the ability to handle arbitrarily shaped domains. For example, stress analysis is an important process in engineering that requires the solution of a system of partial differential equations that are very difficult to solve by analytical methods except for very simple shapes, such as rectangles; however, engineering problems seldom involve such simple shapes. For linear problems, the solution is determined by solving a system of linear equations, for a certain number of unknowns, which are the nodal values of the problem function. With regard to engineering problems, in order to obtain a reasonably accurate solution, thousands of nodes are usually needed. Generally, the accuracy of the solution improves as the number of elements (and nodes) increases, but the computing time (and hence the cost) also rises.

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Complete Chapter List

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Table of Contents
Preface
Joanna Leng, Wes Sharrock
Chapter 1
Gabriele Jost, Alice E. Koniges
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Hardware Trends and Implications for Programming Models
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Chapter 2
Ivan Girotto, Robert M. Farber
This chapter focuses on the technical/commercial dynamics of multi-threaded hardware architecture development, including a cost/benefit account of... Sample PDF
Multi-Threaded Architectures: Evolution, Costs, Opportunities
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Chapter 3
Domingo Benitez
Many accelerator-based computers have demonstrated that they can be faster and more energy-efficient than traditional high-performance multi-core... Sample PDF
High-Performance Customizable Computing
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Chapter 4
Rasit O. Topaloglu, Swati R. Manjari, Saroj K. Nayak
Interconnects in semiconductor integrated circuits have shrunk to nanoscale sizes. This size reduction requires accurate analysis of the quantum... Sample PDF
High-Performance Computing for Theoretical Study of Nanoscale and Molecular Interconnects
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Chapter 5
Prashobh Balasundaram
This chapter presents a study of leading open source performance analysis tools for high performance computing (HPC). The first section motivates... Sample PDF
Effective Open-Source Performance Analysis Tools
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Chapter 6
David Worth, Chris Greenough, Shawn Chin
The purpose of this chapter is to introduce scientific software developers to software engineering tools and techniques that will save them much... Sample PDF
Pragmatic Software Engineering for Computational Science
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Chapter 7
Diane Kelly, Daniel Hook, Rebecca Sanders
The aim of this chapter is to provide guidance on the challenges and approaches to testing computational applications. Testing in our case is... Sample PDF
A Framework for Testing Code in Computational Applications
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Chapter 8
Judith Segal, Chris Morris
There are significant challenges in developing scientific software for a broad community. In this chapter, we discuss how these challenges are... Sample PDF
Developing Software for a Scientific Community: Some Challenges and Solutions
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Chapter 9
Fumie Costen, Akos Balasko
The computational architecture of Enabling Grids for E-sciencE is introduced as it made our code porting very challenging, and the discussion... Sample PDF
Opportunities and Challenges in Porting a Parallel Code from a Tightly-Coupled System to the Distributed EU Grid, Enabling Grids for E-sciencE
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Chapter 10
Abid Yahya, Farid Ghani, R. Badlishah Ahmad, Mostafijur Rahman, Aini Syuhada, Othman Sidek, M. F. M. Salleh
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Development of an Efficient and Secure Mobile Communication System with New Future Directions
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Chapter 11
Hubertus J. J. van Dam
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Parallel Quantum Chemistry at the Crossroads
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Chapter 12
Marc Hafner, Heinz Koeppl
With the advances in measurement technology for molecular biology, predictive mathematical models of cellular processes come in reach. A large... Sample PDF
Stochastic Simulations in Systems Biology
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Chapter 13
C. T. J. Dodson
Many real processes have stochastic features which seem to be representable in some intuitive sense as `close to Poisson’, `nearly random’, `nearly... Sample PDF
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Chapter 14
Stefania Tomasiello
Though relatively unknown, the Differential Quadrature Method (DQM) is a promising numerical technique that produces accurate solutions with less... Sample PDF
DQ Based Methods: Theory and Application to Engineering and Physical Sciences
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Chapter 15
Marco Evangelos Biancolini
Radial Basis Functions (RBF) mesh morphing, its theoretical basis, its numerical implementation, and its use for the solution of industrial... Sample PDF
Mesh Morphing and Smoothing by Means of Radial Basis Functions (RBF): A Practical Example Using Fluent and RBF Morph
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Chapter 16
Joanna Leng, Theresa-Marie Rhyne, Wes Sharrock
This chapter focuses on state of the art at the intersection of visualization and CSE. From understanding current trends it looks to future... Sample PDF
Visualization: Future Technology and Practices for Computational Science and Engineering
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Chapter 17
Peter Sarlin
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Visualizing Indicators of Debt Crises in a Lower Dimension: A Self-Organizing Maps Approach
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Chapter 18
Iain Barrass, Joanna Leng
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Improving Computational Models and Practices: Scenario Testing and Forecasting the Spread of Infectious Disease
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Chapter 19
Eldon R. Rene, Sung Joo Kim, Dae Hee Lee, Woo Bong Je, Mirian Estefanía López, Hung Suck Park
Sequencing batch reactor (SBR) is a versatile, eco-friendly, and cost-saving process for the biological treatment of nutrient-rich wastewater, at... Sample PDF
Artificial Neural Network Modelling of Sequencing Batch Reactor Performance
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Chapter 20
Joanna Leng, Wes Sharrock
Computational Science and Engineering (CSE) is an emerging, rapidly developing, and potentially very significant force in changing scientific... Sample PDF
The State of Development of CSE
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Chapter 21
Kerstin Kleese van Dam, Mark James, Andrew M. Walker
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Integrating Data Management and Collaborative Sharing with Computational Science Research Processes
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Chapter 22
Jens Jensen, David L. Groep
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Chapter 23
Matt Ratto
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CSE as Epistemic Technologies: Computer Modeling and Disciplinary Difference in the Humanities
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Chapter 24
Phillip L. Manning, Peter L. Falkingham
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Key Terms in this Chapter

Boundary Conditions: The conditions that are satisfied on the boundaries of the spatial domain of the problem. The governing differential equation with the boundary conditions represents a boundary value problem (BVP)

Nodes: Points in space where finite elements are connected and where the unknowns of the algebraic equation system are computed.

Grid Points: Points are selected in the problem spatial domain where the numerical solution is computed.

FDM: The Finite Difference Method computes approximate values for the solution at grid points. To compute these values, derivatives are replaced by divided differences. This method can be useful for solving heat transfer problems, fluid mechanics problems and works well for two-dimensional regions with boundaries parallel to coordinates axes.

Meshfree Methods: Numerical method which compute the approximate solution without a predefined mesh.

Initial Conditions: For time dependent problems, the conditions that must be satisfied at the initial/start time. The governing equations with the initial conditions represent the initial value problem (IVP).

Collocation Points: see collocation methods

Collocation Methods: The approximate solution is produced by means of the values it assumes in some locations, called collocation points, where the governing differential equation is satisfied.

FEM: The Finite Element Method, computes the solution by piecewise functions being computed in smaller parts of the problem domain, called finite elements. A set of algebric equations are established and solved. It is employed extensively in the analysis of solids, structures, heat transfer and fluids. It can handle arbitrarily shaped domains.

MeSH: Topological map of the finite element spatial domain as discrete points rather than a continuous field.