Optimierung von mechanischen Strukturen und Systemen


Klassische kontinuierliche Ansätze berücksichtigen die besondere atomare oder molekulare Struktur von Materialien nicht explizit. Somit sind sie für die korrekte Beschreibung hochgradig multiskaliger Phänomene wie beispielsweise Rissausbreitung oder Interphaseneffekte in Polymerwerkstoffen nicht gut geeignet. Um die atomare Auflösungsebene zu integrieren, wurde die „Capriccio“-Methode als eine neuartige Multiskalentechnik entwickelt. Sie wird z.B. für…

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This project targets the formulation and implementation of a method for structural shape and topology optimization within an embedding domain setting. Thereby, the main consideration is to embed the evolving structural component into a uniform finite element mesh which is then used for the structural analyses throughout the course of the optimization. A boundary tracking procedure based on adaptive (or hierarchical) mesh refinement is used to identify interior and exterior elements, as well as such elements that are intersected by the physical domain boundary of the structural component. By this mechanism, we avoid the need to provide an updated finite element mesh that conforms to the boundary of the structural component for every single design iteration. Further, when considering domain variations of the structural component, its material points are not attached to finite element nodal points but rather move through the stationary finite element mesh of the embedding domain such that no mesh distortion is observed. Hence, one circumvents the incorporation of time consuming mesh smoothing operations within the domain update procedure. In order to account for the geometric mismatch between the boundary of the structural component and its non-conforming finite element representation within the embedding domain setting, a selective domain integration procedure is employed for all elements that are intersected by the physical domain boundary. This is to distinguish the respective element area fractions interior and exterior to the structural component. We rely on an explicit geometry description for the structural component, and an adjoint formulation is used for the derivation of the design sensitivities in the continuous setting.

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We consider local refinements of finite element triangulations as continuous graph operations, for instance by splitting nodes and inflating edges to elements. This approach allows for the derivation of sensitivities for functionals depending on the finite element solution, which may in turn be used to define local refinement indicators. Thereby, we develop adaptive algorithms exploiting sensitivities for both hierarchical and non-hierarchical mesh changes, and analyze their properties and performance in comparison with established methods.

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