In order to solve a wide range of structural optimization problems numerical methods are applied. The design function S, which describes the general layout of the structure like the shape, and the state variables u, which describe the structural response like the deflections, are discretized. The approximated solution for the design function at the optimum s* is determined by mathematical programming or optimality criteria methods. The structural response in equilibrium u* is approximately determined by e.g. the finite element method. Conventionally, the discretizations of S and u are fixed during the optimization process. However, this leads to some shortcomings with respect to the reliability of the results and the numerical efficiency of the optimization process.

In the first part of this paper some essential aspects of an adaptive discretization of the design function S are discussed. Different refinement indicators and strategies are presented. The adaptive procedure is illustrated by some examples of shape and topology optimization. In the second part the adaptive discretization of the state variables is considered. In addition to standard adaptive finite element methods the discretization is adapted with respect to the error in the state variables and at the same time with respect to the error in their sensitives with respect to the design variables. The costs and benefits of an extended adaptive finite element method are illustrated by some examples.

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