Theory and numerical implementation of relaxed formulations of optimization problems involving coupled fields. 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 in the relaxed setting

Topology optimization of plates, shallow shells and 3D bodies. Numerical algorithms of filtering of composite domains in optimal solutions referring to fully relaxed settings, using specific features of meshless methods.

Optimization of layout of anisotropic elastic properties of inhomogeneous bodies with the method of controlling the parameters of the Hooke tensor as well as constitutive parameters of phenomenological models.

Expected results
Elaboration of comprehensive articles or a book on topology optimization of structures is planned. The authors are aware of the topic being widely discussed in the monographs by Bendsoe i 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 of the Michell family. In particular, the theory of surface nets was elaborated and new solutions concerning admissible stresses with unequal compression and tension were constructed;

Topology optimization of two-material plates with the plane elasticity problem. This classical topic has not been appropriately examined yet, 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). Numerical aspects and a thorough analysis of 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 used so far without appropriate relaxation of variation 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, new numerical algorithms are to be developed 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 the literature.

The final results of the project will be computer programs, descriptions of numerical algorithms, as well as detailed reports and publications. In our reports from the previous Research Project all the details of numerical algorithms have been revealed, thus making them useful; see e.g. the paper by Czarnecki and Lewiński (CAMES, 2003), where the 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 computer programs. A similar style of publishing will be kept, thus making our results well-documented as well as applicable for other researchers.

The final results are planned to be published in the proceedings (WCSMO, ECCOMAS, IFIP, SolMech, CMM) and journals, like Struct. Multidsic.Optimiz., the official journal of the ISSMO Society.