Two ways of organizing scalar product in the boundary state method

Introduction. The influence of two ways of organizing scalar product on the convergence rate of the solution in the energy method of boundary states is considered. The method is based on the spaces of internal and boundary states which are conjugated through isomorphism. Both spaces are orthonormalized using one scalar product or another. The desired state is expanded in the Fourier series according to the elements of the orthonormalized basis; and the coefficients of this linear combination are determined. The two methods differ in the assignment of scalar products and the calculation of the Fourier coefficients.

Materials and Methods. In relation to the method of boundary states, a new theory of organizing a scalar product in the spaces of internal and boundary states is proposed. Computational algorithms are constructed for its practical implementation. In the traditional (first) approach, the internal energy of elastic deformation is used as an orthogonalizer in the space of internal states. Here, the Fourier coefficients are the work of given forces on the basis vectors of displacement of the boundary points. In the studied (second) approach, scalar products are integrals of the cross products of the basis force vectors at the boundary. Accordingly, the Fourier coefficients are calculated as integrals of the product of the given forces at the body boundary by the basic force vectors.

Results. A numerical study of the first primal axisymmetric problem of the elasticity theory for a transversely isotropic cylinder in the absence and presence of mass forces is conducted. In the absence of mass forces, an analysis of the elas-tic fields obtained for the same number of used basic elements has shown that the second method has the greatest accuracy of the results. Under solving the problem with the presence of mass forces, the second method did not show effi-ciency in terms of the uniqueness of the solution; however, it is quite suitable for constructing a multitude of elastic fields used to solve more complex problems.

Discussion and Conclusions. The results obtained can be used to solve boundary-value problems of mechanics of not only an anisotropic body, but also an isotropic one. When solving more complex problems, such as contact and mixed ones, the issue of the rate of convergence requires a separate study.

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