In a broad scope, the project addresses to the problem of structure optimization and identification of defects with emphasize on cracks. The problem consists of proper modeling, theoretical analysis, and construction of efficient computational tools. Our concrete aim is to get a consistent mathematical description of singular geometric structures like cracks with respect to their topological properties. This task is motivated by bifurcation phenomena appearing in a wide range of real world applications.

The crucial point of all kind of structure design concerns topology changes of variable geometric objects. In the abstract sense, the generic topology change is produced by varying the degree of connectedness which is responsible for the number of connected components of a structure, or, respectively, voids inside a domain. A rough structure of the desired object is determined with a reasonable topology optimization procedure. This is a challenging mathematical problem. For refining the obtained structure, the design problem proceeds within the classic shape optimization context.

In the standard approach, the voids are generated typically by regular geometric objects with a boundary of codimension one. Opposite to this fact, cracks appear as high, i.e., more than one, codimension sets which are due to the presence of the crack tip or crack branches. As a consequence, we arrive at state problems given in a singular domain. This results generally in a lack of smoothness of its solutions. This key issue distinguishes the principal mathematical difficulties for analysis. Thus, the specialty of the topology optimization problem is focused on singular geometric objects. In the following we refer to topology changes represented by splitting or merging of cracks, and the kink phenomenon.

Our aims are to obtain the following results:

• To derive adequate kinematic description of singular geometric objects which represents the topology changes of bifurcation, branching, and alike.

• To obtain topological characteristics for sensitivity analysis of the energy-type objective functionals dependent on singular domains.

• To get the shape and topology optimization formulation for the problems describing bifurcation and identification of defects like cracks.

In order to attain these results we propose to use the velocity method relaxed on nonsmooth velocities and the implicit surface description, as well as the Lagrange approach for a generalized sensitivity analysis. Our theoretical investigations will be supported by developing the corresponding numerical algorithms and codes for computations on irregular domains.

PUBLICATIONS:

[1] I.I. Argatov and V.A. Kovtunenko, Generalization of the concept of the topological derivative for a kinking crack, in: Proc. 2009 ICCMME, Melbourne, Australia, P.228-233, Australian Inst. High Energetic Materials, 2010.

[2] I.I. Argatov and V.A. Kovtunenko, A kinking crack: generalization of the concept of the topological derivative, 2009 Ann. Bull. Australian Inst. High Energetic Materials, 1 (2010), 124-130.

[3] A.M. Khludnev, V.A. Kovtunenko and A. Tani, On the topological derivative due to kink of a crack with non-penetration, Research Report KSTS/RR-09/001, Math. Department, Keio University, 2009, 37pp.; J. Math.Pures Appl. 94 (2010), 571-596.

[4] H. Itou, V.A. Kovtunenko and A. Tani, The interface crack with Coulomb friction between two bonded dissimilar elastic media, Research Report KSTS/RR-10/001, Math. Department, Keio University, 2010, 22pp.; Appl. Math. 56 (2011), 1, 69-97.

[5] M. Hintermueller, V.A. Kovtunenko and K. Kunisch, Obstacle problems with cohesion: A hemi-variational inequality approach and its efficient numerical solution, MATHEON Report 681, DFG-Forschungszentrum, TU-Berlin, 2010, 32pp.; SIAM J. Opt 21 (2011), 2, 491-516.

[6] V.A. Kovtunenko, A hemivariational inequality in crack problems, Optimization (2010) 60 (2011), 8-9, 1071-1089.

[7] M. Hintermueller and V.A. Kovtunenko, From shape variation to topology changes in constrained minimization: a velocity method based concept, MATHEON Report 768, 826, DFG-Forschungszentrum, TU-Berlin, 2011, 26pp.; Optimization Meth. Software 26 (2011), 4-5, 513-532.

[8] V.A. Kovtunenko, State-constrained optimization for identification of small inclusions, Proc. Appl. Math. Mech. 11 (2011), 1, 721-722.