Novosibirsk, Russia, May, 30 – June, 4, 2011

"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Миренков В.Е. Красновский А.А.## Inverse problem to restore boundary conditions## Reporter: Миренков В.Е.Classical inverse problems on identification of boundary efforts require the knowledge of displacement components along the entire boundary of the study area. The said components are found experimentally with some errors. This class of problems also involves incorrect problems, resulted from a priori assumption relative to deformation processes: perfectly stiff bodies, displacement jumps, rigid and soft covering plates, the conformality disturbance in the finite number of points, perfect slip, and so on. The regulation methods are employed to obtain an approximated solution, durable to small variations in initial data (A.N. Tikhonov, A.N. Kolmogorov). The term “regulation” is treated as an attempt to correct the “consciously”-introduced inaccuracies. Incorrectness can be overcome either through regulation or exact equations for boundary stress and displacement component values. The singular integral equation sets to relate values of the stress and displacement components along the entire boundary are derived to solve the problem under consideration. Solutions are presented for the displacement components in quadratures, identifying them at contacts and through the rest boundary conditions similar to stress functions. The first approximation for the direct problem is formulated on the basis of quantitative behavior of displacement components at lateral faces in the form of, say, the first primal problem, which solution gives the complete estimate of the stress-strain state at the boundary. Taking the direct problem solution and redefined conditions at lateral faces into account, it is possible to calculate values of stress functions at contacts. Just now we state direct problems in terms of stress for the plate. The solution to these problems is determined by the second approximation for plate ends in terms of displacements. Thus, the first cycle of approximation is over at this very stage and the second cycle starts from these calculated displaements, and so on until the preset precision level is gained. The study was conducted with financial support from the Russian Foundation for Basic Research, Project No. 09-05-00133.
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