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## Über dieses Buch

The subject of Partial Differential Equations (PDEs) which first emerged in the 18th century holds an exciting and special position in the applications relating to the mathematical modelling of physical phenomena. The subject of PDEs has been developed by major names in Applied Mathematics such as Euler, Legendre, Laplace and Fourier and has applications to each and every physical phenomenon known to us e.g. fluid flow, elasticity, electricity and magnetism, weather forecasting and financial modelling. This book introduces the recent developments of PDEs in the field of Geometric Design particularly for computer based design and analysis involving the geometry of physical objects. Starting from the basic theory through to the discussion of practical applications the book describes how PDEs can be used in the area of Computer Aided Design and Simulation Based Design. Extensive examples with real life applications of PDEs in the area of Geometric Design are discussed in the book.

## Inhaltsverzeichnis

### Chapter 1. Elementary Mathematics for Geometric Design

This chapter deals with some of the basic mathematical concepts that are required to fully understand the material discussed in the rest of this book. Particulary, this chapter presents, in a concise form, the basic concepts of vector algebra, matrices, systems of linear equations and mathematical properties of surfaces.
Hassan Ugail

### Chapter 2. Introduction to Geometric Design

This chapter provides an introduction to geometric design. It introduces various popular mathematical methods used for shape representation in geometric design. It also discusses the role of interactive design and parametric design to enhance the processes involved in a geometric design problem. Furthermore, this chapter discusses the use of design optimization to carry out automatic design for function.
Hassan Ugail

### Chapter 3. Introduction to Partial Differential Equations

This chapter provides an introduction to partial differential equations (PDEs) with the aim of introducing the reader with the mathematical concepts that are used in further chapters. The chapter first introduces the general concept of PDEs and discusses various types of PDEs. Special emphasis is given to elliptic PDEs since this type of equations form the basis for the development of geometric design techniques throughout this book.
Hassan Ugail

### Chapter 4. Elliptic PDEs for Geometric Design

This chapter deals with the use of elliptic PDEs for geometric design. The chapter introduces the common elliptic PDEs such as the Laplace equation and the Biharmonic equation and shows that they can be used as a tool for surface generation. This chapter also discusses the general elliptic PDEs for surface design. Solution schemes showing how to solve the chosen elliptic PDEs in analytic form is described. Several examples of surface generation using elliptic PDEs are also given in this chapter.
Hassan Ugail

### Chapter 5. Interactive Design

Using elliptic PDEs described in the previous chapter, especially using the Biharmonic equation, one can create the shape of an initial surface. This can be carried out through the interactive specification of curves which can be taken as the boundary conditions for the chosen PDE. Once this is done, it may be necessary to further manipulate the geometry in order to improve the shape. Hence, it is desirable to have as much control as possible over the shape of the surface once it has been defined.
Hassan Ugail

### Chapter 6. Parametric Design

This chapter provides details of how PDE based geometries can be efficiently parameterized. The geometry generated using PDEs has an efficient parametrization associated with it. That is, PDE based geometries are first of all characterized by boundary conditions. Furthermore, one can change the geometry easily by means of changing a small set of parameters.
Hassan Ugail

### Chapter 7. Functional Design

This chapter discusses how the PDE based approach to shape parametrization when combined with a standard method for numerical optimization is capable of setting up automatic design optimization problems allowing design for function to be more practical. The chapter first introduces the methodologies for design optimization. Afterwards several examples of how design for function can be carried out via PDE based shape parametrization along with optimization are discussed.
Hassan Ugail

### Chapter 8. Other Applications

This chapter presents a number of other application areas (which have not been discussed in previous chapters) which can benefit from using PDEs for geometric design. Particularly, in this chapter we show how PDEs can be effectively used for animation, data representation and compression. Furthermore, we discuss an emerging area of research where PDE based geometric design is being related to traditional spline based techniques.
Hassan Ugail

### Chapter 9. Conclusions

This book deals with the use of partial differential equations within the fast moving field of geometric design. In this book, we have attempted to describe how to use elliptic PDEs as one possible mechanism through which complex geometry can be generated, manipulated, parameterized, and furthermore optimized for specific design requirements. Due to the nature of the chosen PDEs, the geometric shapes generated using these types of PDEs are naturally fair. They also enable maintaining continuity between adjacent shapes and provide global control over the entire shape. PDEs are also capable of parameterizing complex shapes in terms of a small set of design parameters whilst maintaining sufficient flexibility in the range of generated shapes. Thus, in this book we describe a methodology for geometric design using PDEs whereby the same underlying PDE model can be used for the interactive definition and parameterization of the geometry, the integration of the design with analysis using the original parameterization, and the design optimization for a chosen design merit function with some imposed constraints.
Hassan Ugail

### Backmatter

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