Shape and topology optimization under nonlinear uncertainty

• Applicant: Professor Dr.-Ing. Benedikt Kriegesmann
• Subject Area: Mechanics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 508865334

This research project addresses structural optimization under uncertainty for large-scale problems with a large number of random variables. Many methods for probabilistic analysis (e.g. Monte Carlo methods, numerical integration, surrogate models) usually scale with the number of random variables, which leads to an impracticable computation time for high-dimensional problems. To overcome this problem, methods based on first-order Taylor approximations are used, which require only two function evaluations, regardless of the number of design or random variables. However, the low order of this approximation often leads to inaccurate results, which means that optimizations under uncertainty may hardly differ from deterministic optimizations. In many applications, the computing time is dominated by finite element analyses, while the preprocessing - for example the calculation of the Young's modulus from element densities in topology optimization - is almost negligible. On the other hand, these preprocessing steps often contain highly nonlinear functions that are implicitly linearized in the first-order Taylor approximation and thus contribute to significant inaccuracies in the probabilistic analysis. The new approach of the project consists of first calculating the stochastic moments of suitably selected intermediate parameters from the scattering input data and then performing the probabilistic analysis only from these intermediate parameters using first-order Taylor approximation. A simple example of this is the consideration of an uncertain projection parameter, from which the stochastic properties of the Young's modulus are calculated. This Young's modulus is then used as the new random parameter. This avoids errors caused by linear approximations in preprocessing. Since the preprocessing only requires a small computation time, the calculation of the stochastic moments of the intermediate parameters can be carried out efficiently. Hence, the additional computational effort remains manageable. The aim of the work is the development of analytical conversions from scattering primary variables to scattering intermediate parameters or, if this is not possible, the search for good approximations. The aspect of scalability is of crucial importance. The developed approach is to be applied to scattering load angles, projection parameters, material orientations of anisotropic materials and uncertain node coordinates.

• DFG Programme: Research Grants