Topology Optimization of  Linear Elastic Continuum Structures

                                   by

                              Lars A. Krog

                  JTB 320, 3:20pm, Monday May 12, 1997


                                Abstract

Shape optimization is a subject which has captured the minds of
researchers for many years, and boundary variation techniques for doing
shape optimization have today reached a level of maturity where they are
being incorporated into commercial finite element systems for solution
of structural analysis problems.

Traditional boundary variation methods for doing shape optimization rely
on the simple assumption that a structure is defined by the shape of its
boundaries, and that an optimal structure can be found by variation of
the shape of the boundaries of a given initial design. At a first glance
such an approach seems very general. However, as we are optimizing the
shapes of predefined boundaries such a formulation will neither allow us
to introduce new boundaries nor to remove existing boundaries, and the
topology of the finial optimized design is therefore given by the
initial design.

In structural mechanics the topology of a component, i.e., the
description of the number, position and connectivity of holes and
members in the structure is extremely important for its optimality.
Traditional boundary variation methods for doing shape optimization are
therefore severely limited in scope in the sense that the success of
such an approach rely on the skills and intuition of an experienced
designer to come up with a good initial design, i.e., a design with the
right topology. There is therefore a general need for optimization
methods which are able to predict the optimal topology of continuum
structures.

The present lecture deals with the description of such an optimization
method. Especially, we will consider formulations for finding the
optimal topologies of planar structures and for finding the optimal
layouts of stiffener reinforcements on existing plate and shell
structures. As design criteria for the optimization we will consider
both the structural stiffness and the eigenfrequencies of vibration for
the structure, and a special formulation for solution of optimization
problems in the case of existence of non-differentiable multiple
eigenfrequencies will be presented.


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