Fluid Mechanical and Multidisciplinary Applications

• Published in: Introduction to Shape Optimization p. 1
• Main Authors: Mäkinen R. A. E, Haslinger J
• Format: Book Chapter
• Language: English
• Published: Society for Industrial and Applied Mathematics (SIAM) 2003; Society for Industrial and Applied Mathematics
• Series: Advances in Design and Control
• Subjects: General Engineering & Project Administration; Manufacture & Processing; Structural optimization—Mathematics
• ISBN: 0898715369; 9780898715361
• DOI: 10.1137/1.9780898718690.ch8

Traditionally shape optimization has been restricted to one discipline, linear elasticity, only. Recently, there has been much interest in shape optimization of systems governed by equations of both fluid mechanics and electromagnetics. In this chapter we shall consider shape optimization with state problems related to fluid mechanics or combined fluid mechanics and electromagnetics problems. Finite element and nonlinear programming methods in shape optimization of compressible subsonic flows have been used by Angrand [Ang83], and later by Beux and Dervieux [BD92]. Angrand computed optimal shapes of a nozzle and a lifting airfoil by using the full potential equation. Beux and Dervieux optimized the shape of a nozzle in the case of subsonic Euler flow. A slightly different approach has been used by Jameson [Jam88]. For further study on numerical methods in shape optimization problems governed by fluid flow or in multidisciplinary problems we refer to [MP01]. 8.1 Shape optimization of a dividing tube This section deals with a shape optimization problem governed by the Navier—Stokes equations for viscous incompressible fluids. 8.1.1 Introduction The quality of paper produced is largely determined by phenomena taking place in the device of a paper machine called the “headbox.” For example, the basis weight and the fiber orientation variations depend on the fluid dynamics in the headbox. The first flow passage in the headbox is a dividing tube (the “header”). It is designed to distribute a fiber suspension (wood fibers, filler clays, and chemicals mixed in water) in such a way that the paper produced will have a uniform thickness and an optimal fiber orientation across the width of the paper machine.