# Shape and topology optimization of imperfection sensitive structures with random geometry

The project addresses the optimization of imperfection sensitive structures. For curved slender structures, the load carrying capacity depends on the shape and magnitude of geometric imperfections. These are unknown at the time of designing the structure but can be modeled as stochastic parameters. Embedding probabilistic methods into optimization is referred to as robust design optimization. Since Monte Carlo methods are extremely computationally costly, robust design optimization is often based on Taylor series expansions. In addition to derivatives with respect to design parameters, also first and second-order derivatives with respect to random parameters need to be determined. This requires the solution of additional equation systems, which typically increases the computational cost with increasing number of random parameters.

In the requested project, a method will be developed for which no additional equations systems need to be solved compared to a deterministic optimization. This method will make use of the fact that in shape optimization subject to random geometry the design parameters correspond to the random parameters. Hence, second-order derivatives can be estimated from the optimization history. The optimization using a second-order probabilistic approach however also requires third-order derivatives which shall not be computed. Furthermore, an estimation based on the optimization history does not work in the beginning of the optimization. Hence, a procedure is suggested which start with a first-order approximation. Estimated higher order terms gain influence during the optimization, while their accuracy increases.

In topology optimization subject to random geometry, the random parameters do not correspond to the design parameters. However, there is a relation that can be utilized to express the derivatives with respect to random parameters by derivatives with respect to design parameters. This function composition will be embedded into robust topology optimization.

The method to be developed will reduce the computational cost at the expense of the required memory. Thus, the scalability of the method will be investigated and measures will be developed that allow to optimize models of industrial relevance.

- Contact: Jan Krüger
- Duration: 27 Monate
- Funded by: DFG