### Doctoral Thesis Defense, Optimization and Systems Theory

Friday, March 7, 2003, 10.00, Kollegiesalen, Administration building,
Valhallavägen 79, KTH

**Mathias Stolpe**

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On models and methods for global optimization of structural topology

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Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan
framlägges till offentlig granskning för avläggande av teknologie
doktorsexamen fredagen den 7:e mars 2003 kl 10.00 i
Kollegiesalen, Administrationsbyggnaden, Kungliga Tekniska
Högskolan, Valhallavägen 79.
*
This thesis consists of an introduction and seven independent,
but closely related, papers which all deal with problems in structural
optimization. In particular, we consider models and methods for global
optimization of problems in topology design of discrete and
continuum structures.

In the first four papers of the thesis the nonconvex problem of
minimizing the weight of a truss structure subject to stress
constraints is considered. First it is shown that a certain subclass
of these problems can equivalently be cast as linear programs and thus
efficiently solved to global optimality. Thereafter, the behavior of
a certain well-known perturbation technique is studied. It is concluded
that, in practice, this technique can not guarantee that a global
minimizer is found. Finally, a convergent continuous branch-and-bound
method for global optimization of minimum weight problems with stress,
displacement, and local buckling constraints is developed. Using this
method, several problems taken from the literature are solved with a
proof of global optimality for the first time.

The last three papers of the thesis deal with topology optimization
of discretized continuum structures. These problems are usually
modeled as mixed or pure nonlinear 0-1 programs. First, the behavior
of certain often used penalization methods for minimum compliance
problems is studied. It is concluded that these methods
may fail to produce a zero-one solution to the considered problem.
To remedy this, a material interpolation scheme based on a rational
function such that compliance becomes a concave function is proposed.
Finally, it is shown that a broad range of nonlinear 0-1 topology
optimization problems, including stress- and displacement-constrained
minimum weight problems, can equivalently be modeled as *linear*
mixed 0-1 programs. This result implies that any of the standard
methods available for general linear integer programming can now be used on
topology optimization problems.

Calendar of seminars

*Last update: February 4, 2003 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*