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## Über dieses Buch

Almost all physical phenomena can be mathematically described in terms of differential equations. The finite element method is a tool for the appro- mate solution of differential equations. However, despite the extensive use of the finite element method by engineers in the industry, understanding the principles involved in its formulation is often lacking in the common user. As an approximation process, the finite ele~ent method can be for- lated with the general technique of weighted residuals. This technique has the advantage of enhancing the essential unity of all processes of approxi- tion used in the solution of differential equations, such as finite differences, finite elements and boundary elements. The mathematics used in this text, though reasonably rigorous, is easily understood by the user with only a basic knowledge of Calculus. A common problem to the courses of Engineering is to decide about the best form to incorporate the use of computers in education. Traditional c- pilers, and even integrated programming environments such as Turbo Pascal, are not the most appropriate, since the student has to invest much time in developing an executable program that, in the best of cases, will be able to solve only one definitive type of problems. Moreover, the student ends up learning more about programming than about the problem that he/she wants to solve with the developed executable program.

## Inhaltsverzeichnis

### 1. Introduction to Maple

Abstract
Maple is a symbolic computational system. This means that it does not require numerical values for all variables, as numerical systems do, but manipulates information in a symbolic or algebraic manner, maintaining and evaluating the underlying symbols, like words and sentence-like objects, as well as evaluates numerical expressions. As a complement to symbolic operations, Maple provides the user with a large set of graphic routines, numerical algorithms and a comprehensive programming language.
Artur Portela, Abdellatif Charafi

### 2. Computational Mechanics

Abstract
Mathematical modelling of a physical system is an iterative process that basically involves the definition of a continuous model, equivalent to the physical system, and a discrete model.
Artur Portela, Abdellatif Charafi

### 3. Approximation Methods

Abstract
Once a continuous model of a physical problem is set up and a formal solution of this model cannot be obtained, a discrete model is then formulated, in order to obtain an approximate solution of the differential equation that describes the continuous model. Choosing an approximation method is an important task in the formulation, since it affects the accuracy of the solution and the efficiency of the process.
Artur Portela, Abdellatif Charafi

### 4. Interpolation

Abstract
The finite element method considers an approximation function defined in each finite element in terms of a set of interpolation functions. This feature, already introduced in the previous chapter through linear interpolation functions, plays a key role in the finite element method. Hence, it requires a special approach in this introductory text.
Artur Portela, Abdellatif Charafi

### 5. The Finite Element Method

Abstract
Discretization is an essential engineering tool for the analysis of physical problems. Formal solutions of continuum models, defined in the first step of the mathematical modelling process, are generally not available for practical problems and, consequently discretization must be used in order to obtain an approximate solution.
Artur Portela, Abdellatif Charafi

### 6. Fluid Mechanics Applications

Abstract
The application of the finite element method to fluid mechanics problems is not as advanced as it is in solid mechanics. The main reason seems to be that the finite difference method has proved very successful in solving fluid flow problems. Furthermore, the large investment of time and money made in the development of finite difference software naturally led to a reluctance to consider the application of other methods. This situation is gradually changing and the finite element method is now becoming a standard approximation tool used in the solution of fluid mechanics problems.
Artur Portela, Abdellatif Charafi

### 7. Solid Mechanics Applications

Abstract
Mathematical modelling in Solid Mechanics uses the theory of elasticity as the fundamental generating model which is thus considered a mathematicallyexact model. Based on this exact model and considering simplifying assumptions, asymptotic models, still continuous, can be generated. These continuous asymptotic models are eventually discretized in order to obtain numerical approximate solutions.
Artur Portela, Abdellatif Charafi

### Backmatter

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