Fictitious Domain Methods in Shape Optimization

• Published in: Introduction to Shape Optimization pp. 1 - 2
• 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.ch6

Sizing and shape optimization problems are typical bilevel problems. The upper level consists of minimizing a cost functional by using appropriate mathematical programming methods. Some of these have been presented in Chapter 4. The lower level provides solutions of discretized state problems needed to evaluate cost and constraint functions and their derivatives at the upper level. Typically, discrete state problems are given by large-scale systems of algebraic equations arising from finite element approximations of state relations. Usually the lower level is run many times. Thus it is not surprising that the efficiency of solving discrete state problems is one of the decisive factors of the whole computational process. This chapter deals with a type of method for the numerical realization of linear elliptic state equations that is based on the so-called fictitious domain formulation. A common feature of all these methods is that all computations are carried out in an auxiliary simply shaped domain Ω^ (called fictitious) in which the original domain Ω representing the shape of a structure is embedded. There are different ways to link the solution of the original problem to the solution of the problem solved in the fictitious domain. Here we present a clas s of methods based on the use of boundary Lagrange (BL) and distributed Lagrange (DL) multipliers. Just the fact that the new problem is solved in a domain with a simple shape (e.g., a rectangle) enables us to use uniform or “almost” uniform meshes for constructing finite element spaces yielding a special structure of the resulting stiffness matrix. Systems with such matrices can be solved by special fast algorithms and special preconditioning techniques. In this chapter we confine ourselves to fictitious domain solvers that use the so-called nonfitted meshes, i.e., meshes not respecting the geometry of Ω. Besides the advantages we have already mentioned there is yet another one, which makes this type of state solver interesting: programming shape optimization problems is easier and “user friendly.” To see this let us recall the classical approach based on the so-called boundary variation technique widely used in practice. Suppose that a gradient type minimization method at the upper level and a classical finite element approach at the lower level are used. The program realizes a minimizing sequence {Ω(k)} of domains, i.e., a sequence decreasing the value of a cost functional. Each new term Ω(k+1) of this sequence is obtained from Ω(k) by an appropriate change in ∂Ω(k).