Topology Optimization in Structural and Continuum Mechanics

June 18, 2012 — June 22, 2012


  • Tomasz Lewinski (Warsaw University of Technology, Poland)
  • George Rozvany (Budapest University of Technology and Economics, Hungary)

Topology optimization is a relatively new but extremely rapidly expanding research field of structural and continuum mechanics. It has interesting theoretical implications in mathematics, mechanics, multi-physics and computer science, but also important practical applications in the manufacturing (in particular, car, aerospace and machine) industries, and is likely to have a significant role in micro- and nano-technologies. Topology optimization achieves much higher savings than cross-section or shape optimization. The proponent of this course was coordinator for two CISM courses on topology optimization in 1990 and 1996.

The objective of the course is to review new developments in structural topology optimization that occurred since the last CISM course on this subject. After revisiting briefly the earlier history of exact topology optimization, including the optimal layout theory, recent developments in the theory of Michell structures will be examined. This will cover the review of some errors in the classical literature, and difficulties in obtaining certain types of exact analytical solutions. Moreover, some general principles (symmetry, non-uniqueness, domain incrementation) will be discussed, and methods for verifying numerical solutions presented.

Further lectures on exact topology optimization will examine the basic properties of Michell-Hencky networks, with a number of applications to new classes of solutions for both plane and surface structures. Moreover, two-material optimization for plane stress, plate bending, shells, 3D bodies and sandwich plates will be discussed. Finally, free material optimization will be briefly reviewed.

Another series of lectures will show that, from a mathematical viewpoint, topology optimization problems are often ill-posed, even if this difficulty can be overcome by two methods. One is called “relaxation” and the main tool involved is the homogenization theory. It will be discussed in detail, both from theoretical and numerical points of view. The second approach consists of restricting the set of admissible solutions using the level set representation, that may be seen as a constraint on the set of admissible solutions. The level set algorithm will be explained extensively through various applications and examples.

Turning to numerically oriented topology optimization, the theory of band-gap structures will be presented. The problem of maximizing single and multiple eigenfrequencies and frequency gaps, as well as minimizing the dynamic compliance will also be examined. Finally, the bifurcation and post-buckling analysis of bi-modal optimum columns will be considered.
Yet another series of lectures will look into topology optimization using the Extended Finite Element Method (XFEM), diffusive transport problems, topology optimization of flows, as well as coupled multi-physics, meso-scale transport and nano-scale problems. These lectures will finally discuss topology optimization under uncertainty.

The lectures will also cover an alternative material interpolation scheme (RAMP), trajectories of penalization methods, some fundamental properties of discrete topology optimization problems, global topology optimization by branch-and-cut methods (algorithm and example) and global topology optimization by local branching.

This course is recommended to Ph. D. and post-doctoral scholars and other research staff who intend working on any aspect of topology optimization of structures and continua.
KEYWORDS: Topology Optimization, Shape Optimization, Sensitivity Analysis, Penalization Methods.


See also