Theory and numerical implementation of relaxed formulations of optimization problems involving coupled fi elds. Designing of material layout within composite structures

aim of project
Theoretical solutions and numerical implementation of the following problems:
- Topology optimization of plates and shallow shells. Numerical implementation of the formulations based on the lower estimates of complementary energy within the relaxed setting;
- Topology optimization of plates, shallow shells and 3D bodies. Numerical algorithms of fi ltering composite domains in the optimal solutions referring to the fully relaxed settings, using specifi c features of the meshless methods;
- Optimization of layout of the anisotropic elastic properties of the inhomogeneous bodies with the method of controlling the parameters of the Hooke tensor as well as constitutive parameters of the phenomenological models.

expected results
Elaboration of comprehensive articles or a book on the topology optimization of structures is envisaged. The authors are aware of the topic being widely discussed in the monographs by Bendsoe and Sigmund (2003), Bendsoe (1995), Allaire (2002), Cherkaev (2000). Nonetheless, the articles to be prepared will encompass the most important problems of topology optimization not included in the available books. The emphasis is to be put on the following issues:
- Exact solutions for the Michell family. In particular, the theory of the surface nets was elaborated and new solutions concerning the case of admissible stresses in unequal compression and tension were constructed;
- Topology optimization of two-material plates in the plane elasticity problem. This classical topic is not appropriately examined until now, since the one-material shape optimization problem is usually dealt with in the publications available;
- Two-component topology optimization of plates in bending. The theory was given in the book by Lewiński and Telega (2000). The numerical aspects and the thorough analysis of the new optimal designs will be the subject of the present research;
- Topology optimization of two component plates subjected to simultaneous transverse and in-plane loadings. This topic has never been discussed;
- Topology optimization of two component shells, shallow shells — in particular. This topic has been till now used without appropriate relaxation of the variational setting of the minimum compliance formulation;
- Topology optimization of two component 3D bodies. By referring to the results of Czarnecki and Lewiński from the years 2001–2004 the new numerical algorithms are to be developed with using the Strongin-Sergeyev methods;
- Topology optimization of elastic bodies of arbitrary microstructure. The spectral representations of Hooke’s tensor will be used. This has never been considered in literature.

The final results of the project will be computer programs, descriptions of the numerical algorithms, as well as detailed reports and publications. In our reports from the previous research project, all the details of the numerical algorithms have been revealed, thus making them useful; see e.g. the paper by Czarnecki and Lewiński (CAMES, 2003), where a precise fl owchart of the topology optimization method for the two-component design of the stiff est 3D bodies was displayed, which can be the basis for the own computer programs. Similar style of publishing will be kept, thus making our results well documented as well as applicable for other researchers. The fi nal results are planned to be published in the proceedings (WCSMO, ECCOMAS, IFIP, SolMech, CMM) and journals, like Struct. Multidsic.Optimiz., the offi cial journal of the ISSMO Society.