Project heads | Martin Eigel, Dietmar Hömberg, Rene Henrion and Reinhold Schneider |

Staff | Thomas Petzold |

Internal Cooperation | D-OT1 (Hintermüller, Mielke, Surowiec, Thomas) B-MI2 (Schmidt) C-SE9 (Schneider, Tröltzsch) |

External Cooperation | Ch. Ruprecht, L. Blank (Universität Regensburg) H. Matthies, E. Zander (TU Braunschweig) |

Industrial Cooperation | TEMBRA GmbH Wind Turbine Development |

*Introduction:*
The application focus of this project is the topology optimization of the main frame of wind turbines. This is the central assembly platform at the tower head accommodating the drive train, the generator carrier, the azimuth bearing and drives and a lot of small components.
Topology optimization should not be mistaken for legally mandated structural analysis computations. For the latter, it is standard to solicit a number of single load scenarios based on available time series data.
While this approach is questionable already for stress analysis, it is prohibitive for topology optimization.
Disregarding the multivariate distribution of the random loads would not provide any probabilistic certificate for bounding stresses.
Moreover, the natural way to choose weights is to derive a stochastic load from available time series data.
The main frame is made of cast iron which is prone to a number of material impurities like shrink holes, dross, and chunky graphite.
This motivates the additional consideration of randomness for the material stiffness.
Structures resulting from topology optimization often exhibit unacceptably high stresses necessitating costly subsequent shape design works.
To avoid this already during the optimization, state constraints have to be included in the optimization problem.
The main novelty of this project is that it combines a phase field relaxed topology optimisation problem not only with uncertain loading and material data but also with chance state constraints.
Even in the finite-dimensional case, the derivation of optimality conditions including gradient formulas is completely open. In the long run, including an appropriate damage model as additional state equation will be a further task of great practical importance.

*Topology Optimization:* The goal in topology optimization is to
partition a given domain into regions occupied by either void or
material by minimizing a given cost functional such that the displacement u solves a mechanical equilibrium problem. Here, the material distribution is described with the help of a
phase field variable, taking values in [0,1], correspondingg to the phases void and material, respectively.

*Uncertainty Quantification:*
The inclusion of uncertain parameters in the modelling and simulation with PDEs has evolved as one of the most fruitful and recognized emerging fields in applied mathematics in recent years. Apart from classical Monte Carlo techniques and perturbation approaches, numerical methods based on the expansion of stochastic fields in some polynomial basis of a related probability space (the so called polynomial chaos expansion), have received great interest in the scientific community.

The aim of the Matheon project C-SE13 is to develop a tool for topology optimization taking into account randomness in material data and loads.

We are organizing the Matheon-WIAS workshop "Direct and Inverse Problems for PDEs with Random Coefficients". The workshop takes place from November 9-13, 2015 at WIAS Berlin.

We are developing a software tool for topology optimization of 2D and 3D structures with uncertainties. This is based on the toolbox WIAS-PDELib2 and ALEA.

Relating phase field and sharp interface approaches to structural topology optimization
ESAIM: Control, Optimisation and Calculus of Variations, 20, pp 1025-1058
DOI: 10.1051/cocv/2014006

Shape optimization for a sharp interface model of distortion compensation
Preprint 1792, WIAS, Berlin, 2013
Link: WIAS Preprint 1792

Adaptive stochastic Galerkin FEM
Computer Methods in Applied Mechanics and Engineering, 270, pp 247–269, 2014
DOI: 10.1016/j.cma.2013.11.015

A gradient formula for linear chance constraints under Gaussian distribution
Mathematics of Operations Research, 37, pp 475–488, 2012
DOI: 10.1287/moor.1120.0544