nce and Modelling
Structural optimisation of multiscale problems
Franz-Joseph Barthold, TU Dortmund, Dortmund, GermanyWojciech Kijanski, Numerische Methoden und Informationsverarbeitung / TU Dortmund, Dortmund, Germany
Nowadays profitability and efficiency of practical and industrial applications and therefore of materials design is becoming a more important issue. Due to different well-known and established approaches for analysis and simulation of complex heterogeneous materials on multiple scales based on numerical homogenisation techniques, see [1,2], development and production of high performance materials and as a consequence optimisation and design of materials is reality. The objective is to find optimal structures with optimal materials distribution under given constraints related to posed problems and tailoring applications to their special requirements.
Sensitivity analysis of structural response due to changes of input values provides the basis for improving structures and mechanical components. Especially the variational derivation and formulation of necessary sensitivity relations, which are discussed in [3,4] for single scales in detail, seems to be the most promising way referred to correctness and efficient computations. The extension to multiscale problems in terms of FE2 methods, which allow a direct coupling of referred scales, requires the formulation of objective functions, constraints and design variables on both scales.
The essential steps for variational sensitivity analysis for multiscale problems on continuous level as well as its discrete representation will be presented. A compilation with effective techniques for numerical realisations will give a final overview. The results are part of the project „Strukturoptimierung von Mehrskalenproblemen basierend auf der numerischen Homogenisierung mittels FE2- Analyse“ in cooperation with Prof. J. Schröder (UDE) and funded by the DFG.
 J. Schröder. Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Universität Stuttgart, 2000.
 C. Miehe, C. G. Bayreuther. On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers. Int. J. of Numerical Methods in Engineering (2007, Vol. 71), 1135-1180.
 F.-J. Barthold. Zur Kontinuumsmechanik inverser Geometrieprobleme. Habilitation, TU Braunschweig, 2002.
 D. Materna, F.-J. Barthold. Theoretical aspects and applications of variational sensitivity analysis in the physical and material space. Nova Science Pub. (2010), 397-444.