Methods of fast Fourier transform in diffraction problems of elastic and acoustic waves with applications to crack mechanics

The activity is connected with a development of modern fast computational methods applied to problems in wave dynamics, acoustics, and boundary-value problems of mechanics with mixed boundary conditions. This includes: 1) Development of fast methods for integral equations with convolution kernels arising in these fields of application. Such integral equations, after discretization, may be reduced to linear algebraic systems with matrix of Toepliz or circulant form. For both the types there can be applied fast iteration methods founded on Conjugate Gradient method with a preconditioning. This leads to a quasi-linear numerical algorithm. 2) Applications are constructed in crack mechanics. Two problems are studied with their mechanical conclusions: (i) Diffraction of a planar acoustic wave by a planar crack in the classical linear elastic isotropic space; (ii) static problem for linear cracks in the non-classical porous material of a Cowin-Nunziato type. 3) In the case when the diffraction is happen by a general-form object whose shape is neither linear nor circular, the problem can be reduced to an integral equation with a general-form kernel. There is developed a new approach, which permits an iteration scheme with a convolution kernel at each iteration. This admits again a quasi-linear numerical algorithm. 4) The same idea is applicable to wave processes with obstacles which represent an arbitrary set of linear rigid screen of finite length. The iteration process is proposed, when at each iteration step one needs only solution of the problem for every isolated single screen. All equations in this case are of convolution type, and they are reduced again to Toepliz-like matrix equations in a discrete form.