Poli-USP recebe palestra de influente pesquisador internacional da área de matemática aplicada à Engenharia, Alemdar Hasanoglu, no dia 16 de novembro, às 10h30

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In this talk, typical inverse problems and related mathematical models of solid mechanics with applications in nanotechnology are analyzed in detail. All these models are governed by the damped dynamic Euler-Bernoulli beam equation subject to various types of boundary conditions, and available boundary observations (measured output), namely, the measured deflection, slope or moment. Three types of basic inverse problems – inverse source problems, inverse boundary value problems and inverse coefficient problems, are discussed separately with an indication of which areas of engineering sciences they arise from. After a brief history and review of the literature, the main principles of the proposed new approach are discussed.

As an important application of the proposed approach, the dynamic model of reconstruction of the shear force in the transverse dynamic force microscopy (TDFM)-cantilever tip-sample interaction is proposed.  For this inverse problem, an input-output operator is introduced and then compactness of this operator, thus, ill-posedness of the inverse problems is proved. Least square solution of the inverse problem is introduced through the Tikhonov functional. The Lipschitz continuity of the input-output operator is proved. As a consequence of this, existence of the least square solution is proved. Explicit formula for the Fréchet gradient is derived by making use of the unique solution of the corresponding  adjoint problem. This allows us to construct an effective and fast reconstruction algorithm, as the presented computational experiments show.