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Über dieses Buch

Numerical simulation methods in all engineering disciplines gains more and more importance.

The successful and efficient application of such tools requires certain basic knowledge about the underlying numerical techniques.

The text gives a practice-oriented introduction in modern numerical methods as they typically are applied in mechanical, chemical, or civil engineering. Problems from heat transfer, structural mechanics, and fluid mechanics constitute a thematical focus of the text.

For the basic understanding of the topic aspects of numerical mathematics, natural sciences, computer science, and the corresponding engineering area are simultaneously important. Usually, the necessary information is distributed in different textbooks from the individual disciplines. In the present text the subject matter is presented in a comprehensive multidisciplinary way, where aspects from the different fields are treated insofar as it is necessary for general understanding. Overarching aspects and important questions related to accuracy, efficiency, and cost effectiveness are discussed.

The topics are presented in an introductory manner, such that besides basic mathematical standard knowledge in analysis and linear algebra no further prerequisites are necessary.

The book is suitable either for self-study or as an accompanying textbook for corresponding lectures. It can be useful for students of engineering disciplines as well as for computational engineers in industrial practice.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
In this introductory chapter, we elucidate the value of using numerical methods in engineering applications. Also, a brief overview of the historical development of computers is given, which, of course, are a major prerequisite for the successful and efficient use of numerical simulation techniques for solving complex practical problems.
Michael Schäfer

Chapter 2. Modeling of Continuum Mechanical Problems

Abstract
A very important aspect when applying numerical simulation techniques is the “proper” mathematical modeling of the processes to be investigated. If there is no adequate underlying model, even a perfect numerical method will not yield reasonable results.
Michael Schäfer

Chapter 3. Discretization of Problem Domain

Abstract
Having fixed the mathematical model for the description of the underlying problem to be solved, the next step in the application of a numerical simulation method is to approximate the continuous problem domain (in space and time) by a discrete representation (i.e., nodes or subdomains), in which the unknown variable values are determined.
Michael Schäfer

Chapter 4. Finite-Volume Methods

Abstract
Finite-volume methods (FVM)—sometimes also called box methods—are mainly employed for the numerical solution of problems in fluid mechanics, where they were introduced in the 1970s by McDonald, MacCormack, and Paullay. However, the application of the FVM is not limited to flow problems.
Michael Schäfer

Chapter 5. Finite-Element Methods

Abstract
The techniques known today as finite-element methods (FEM) date back to work conducted between 1940 and 1960 in the field of structural mechanics. The term finite element was introduced by Clough (1960). Nowadays, the FEM is widely used primarily for numerical computations in solid mechanics and can be regarded as a standard tool there.
Michael Schäfer

Chapter 6. Other Discretization Methods

Abstract
In addition to the standard methods introduced in the previous chapters, there are some interesting alternative discretization techniques that might be useful for special classes of continuum mechanics problems in order to enhance the accuracy, efficiency, and/or flexibility of the numerical predictions. A selection of such methods is discussed in this chapter.
Michael Schäfer

Chapter 7. Time Discretization

Abstract
In many practical applications, the processes under consideration are unsteady and thus require for their numerical simulation the solution of time-dependent model equations.
Michael Schäfer

Chapter 8. Solution of Algebraic Systems of Equations

Abstract
The discretization of steady or unsteady problems with implicit time integration, either by finite-volume or finite-element methods, results in large sparse systems of algebraic equations. The solution procedure for these equation systems is an important part of a numerical method.
Michael Schäfer

Chapter 9. Properties of Numerical Methods

Abstract
In this chapter, we summarize characteristic properties of numerical methods, which are important for the functionality and reliability of the corresponding methods as well as for the “proper” interpretation of the achieved results and, therefore, are most relevant for their practical application.
Michael Schäfer

Chapter 10. Finite-Element Methods in Structural Mechanics

Abstract
The investigation of deformations and stresses in solids belongs to the most frequent tasks in engineering applications. In practice nowadays the numerical study of such problems involves almost exclusively finite-element methods. Due to the great importance of these methods, in this chapter we will address in more detail the particularities and the practical treatment of corresponding problems. In particular, the important concept of isoparametric finite elements will be considered. We will do this exemplarily by means of linear two-dimensional problems for a 4-node quadrilateral element. However, the formulations employed allow in a very simple way an understanding of the necessary modifications if other material laws, other strain-stress relations, and/or other types of elements are used.
Michael Schäfer

Chapter 11. Finite-Volume Methods for Incompressible Flows

Abstract
In this chapter we will specially address the application of finite-volume methods for the numerical computation of flows of incompressible Newtonian fluids. This subject matter is of particular importance because most flows in practical applications are of this type and nearly all commercial codes that are available for such problems are based on finite-volume discretizations. Special emphasis will be given to the coupling of velocity and pressure which constitutes a major problem in the incompressible case.
Michael Schäfer

Chapter 12. Lattice-Boltzmann Methods for Flow Simulation

Abstract
As outlined in the previous chapter, standard methods for flow simulations are based on the numerical solution of the (continuum mechanical) Navier-Stokes equations. The so-called Lattice-Boltzmann methods, which became popular, in recent years, also industrial practice, offer an alternative approach. The basis of these methods is a particle-based description of a fluid and—as one knows today—they are in close relation to the fundamental Boltzmann equation of statistical physics. However, the original starting point was in the field of theoretical computer science, namely, the theory of cellular automata. In order to understand the principles, it is helpful to consider this approach first.
Michael Schäfer

Chapter 13. Computation of Turbulent Flows

Abstract
Flow processes in practical applications are in most cases turbulent. Although the Navier-Stokes equations introduced in Sect. 2.​5 are valid for turbulent flows as well—as we will see in the following section—due to the enormous computational effort that would be related to this, it usually is not possible to compute the flows directly on the basis of these equations. Therefore, it is necessary to introduce special modeling techniques to achieve numerical results for turbulent flows. In this section, we will consider this subject in an introductory way. In particular, we will address statistical turbulence models, the usage of which mostly constitutes the only way to compute practically relevant turbulent flows with “reasonable” computational effort.
Michael Schäfer

Chapter 14. Acceleration of Computations

Abstract
For complex practical problems the numerical simulation of the corresponding continuum mechanical model equations usually is highly demanding with respect to the efficiency of the numerical solution methods as well as to the performance of the computers. In order to achieve sufficiently accurate numerical solutions, in particular for flow simulations, in many practically relevant cases a very fine resolution is required and consequently results in a high computational effort and high memory requirements. Thus, in recent years intensive efforts have been undertaken to develop techniques to improve the efficiency of the computations.
Michael Schäfer

Backmatter

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