"The art of structure is where to put the holes" Robert Le Ricolais, 1894-1977 This is a completely revised, updated and expanded version of the book titled "Optimization of Structural Topology, Shape and Material" (Bends0e 1995). The field has since then developed rapidly with many new contributions to theory, computational methods and applications. This has that a simple editing of Bends0e (1995) had to be superseded by what meant is to a large extent a completely new book, now by two authors. This work is an attempt to provide a unified presentation of methods for the optimal design of topology, shape and material for continuum and discrete structures. The emphasis is on the now matured techniques for the topology design of continuum structures and its many applications that have seen the light of the day since the first monograph appeared. The technology is now well established and designs obtained with the use of topology optimization methods are in production on a daily basis. The efficient use of materials is important in many different settings. The aerospace industry and the automotive industry, for example, apply sizing and shape optimization to the design of structures and mechanical elements.

This volume contains the edited versions of papers presented at the IUTAM-Symposium Topological design optimization of structures, machines and materials - status and perspectives, held at Rungstedgaard, near Copenhagen, Denmark, in October 2005. The Symposium was attended by scientists in mechanics, optics, and applied mathematics from 19 countries.

It is now more than 15 years ago that the so-called homogenization method was proposed as a basis for computational means to optimize the topology and shape of continuum structures. From initially being capable mainly of treating minimum compliance design we now see the basic material distribution idea of the methodology applied to a wide range of structural and mechanical problems as well as to problems that couple structural response to other physical responses. Also, the method has provided insight for micro-mechanical studies, meaning that the method has given feedback to the area which provided impetus to the field of topological design optimization in its creation. Finally, topological design is now an integral part of most FEM software systems and it has become a standard industrial tool in some fields.

The IUTAM Symposium provided a forum for the exchange of ideas for future developments in the area of topological design optimization. This encompassed the application to fluid-solid interaction problems, acoustics problems, and to problems in biomechanics, as well as to other multiphysics problems. New basic modelling paradigms, covering new geometry modelling such as level-set methods and topological derivatives, as well as developments in computational approaches were also focus areas.

The topology optimization method solves the basic engineering problem of distributing a limited amount of material in a design space. The first edition of this book has become the standard text on optimal design, which is concerned with the optimization of structural topology, shape and material. This edition has been substantially revised and updated to reflect progress made in modelling and computational procedures. It also encompasses a comprehensive and unified description of the state of the art of the so-called material distribution method, based on the use of mathematical programming and finite elements. Applications treated include not only structures but also MEMS and materials.

This volume contains the edited versions of papers presented at the IUTAM-Symposium Topological design optimization of structures, machines and materials - status and perspectives, held at Rungstedgaard, near Copenhagen, Denmark, in October 2005. The Symposium was attended by scientists in mechanics, optics, and applied mathematics from 19 countries. It is now more than 15 years ago that the so-called homogenization method was proposed as a basis for computational means to optimize the topology and shape of continuum structures. From initially being capable mainly of treating minimum compliance design we now see the basic material distribution idea of the methodology applied to a wide range of structural and mechanical problems as well as to problems that couple structural response to other physical responses.

In the last week of June 1996 the 9th conference of the European Consortium for Math- ematics Industry, ECMI 96, took place at the Technical University of Denmark. The present volume of papers is a selection among the almost 200 contributions to the con- ference. As a logo on the announcements of the conference the organising committee chose a picture of the connection between Denmark and Sweden which is currently under construction. We chose this picture primarily because of the elegant and decorative lines of the bridge, but for other reasons as well: Denmark is a country of islands, and the art of building bridges has a long tradition here. Danish civil engineers have built bridges all over the world and have been rated among the most competent experts in their field. Many have acted as consultants as well as professors at the Technical University of Denmark, and one of them once said: To build a bridge you need steel and mathematics. We think that this selection of papers with its broad spectrum of industrial top- ics proves that the importance of mathematics is continuously growing. Mathematics has penetrated subjects far beyond traditional engineering applications and in more and more places it becomes natural to add mathematics to the list of things you obviously need to succeed.