Tölkes, Sascha: Numerical methods in stochastic and two-scale shape optimization. - Bonn, 2018. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-50560
@phdthesis{handle:20.500.11811/7556,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-50560,
author = {{Sascha Tölkes}},
title = {Numerical methods in stochastic and two-scale shape optimization},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2018,
month = apr,

note = {In this thesis, three different models of elastic shape optimization are described. All models use phase fields to describe the elastic shapes and regularize the interface length on some level or scale to control fine-scale structures.
First, the paradigm of stochastic dominance is transferred from finite dimensional stochastic programming to elastic shape optimization under stochastic loads. The shapes are optimized under the constraint, that they dominate a given benchmark shape in a certain stochastic order. This allows for a flexible risk aversion comparison. Risk aversion is handled in the constraint rather than the objective functional, which results in an optimization over a subset of admissible shapes only. First and second order stochastic dominance constraints are examined and compared. An (adaptive) Q1 finite element scheme is used, that was implemented for two of the models described in this thesis and is introduced here. Several stochastic loads setups and benchmark variables are discretized and optimized.
Starting with the observation that unregularized elastic shape optimization methods create arbitrarily fine micro-structures in many scenarios, domains composited of a number of geometrical subdomains with prescribed boundary conditions are considered in the second model. A reference subdomain is mapped to each type of geometrical subdomains to optimize computational complexity. These are suitable to model fine-scale elastic structures, that are widespread in nature. Examples are fine-scale structures in bones or plants, resulting from the need for a stiff and low-weight structure. The subdomains are coupled to simulate fine-scale structures as they appear e. g. in bones (branching periodic structures). The elastic shape is optimized only for those reference subdomains, simulating periodically repeating structures in one or more coordinate directions. The stress is supposed to be continuous over the domain. A stress-based finite volume discretization and an alternating optimization algorithm are used to find optimal elastic structures for compression and shear loads.
Finally, a model considering a fine-scale material in which the elastic shape is modeled by a phase field on the microscale is introduced. This approach further investigates the fine-scale structures mentioned above and allows for a comparison with laminated materials and previous work on homogenization. A short introduction into homogenization is given and the two-scale energies required for the optimization are derived and discretized. An estimation of the scale between macro- and microscale is derived and a finite element discretization using the Heterogeneous Multiscale Method is introduced. Numerical results for compression and shear loads are presented.},

url = {https://hdl.handle.net/20.500.11811/7556}
}

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