B-spline wavelet discrete-continual finite element method for the local solution to the two-dimensional problem of the theory of elasticity

Abstract

Introduction. The distinctive article presents a local semi-analytical solution to the problem of the two-dimensional theory of elasticity. The corresponding structures, featuring the regularity (constancy) of physical and geometric parameters (the modulus of elasticity of the material of the structure, the Poisson’s ratio of the material of the structure, dimensions of the cross section of the structure) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction. Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial ope­rational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov. Results. Some topical issues of construction of normalized basis functions of a B-spline are considered, the approximation technique for corresponding vector functions and operators within DCFEM is described. Along the basic direction, the problem remains continual and an exact analytical solution can be obtained, while along the non-basic direction the finite element approximation is used in combination with a wavelet analysis apparatus. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, a two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method for the solution of such problems was developed, described and verified in numerous papers written by the authors. In particular, we consider the simplest sample analysis of a deep beam, fixed along the side faces and subjected to the load concentrated in the centre of the structure. Conclusions. The solution to the verification problem obtained using the proposed version of the wavelet-based DCFEM was in good agreement with the solution obtained using a conventional finite element method (corresponding solutions were constructed with localization and without localization; these solutions coincide almost completely, while the advantages of the numerical-analytical approach are quite obvious). It is shown that the use of B-splines of various degrees within the wavelet-based DCFEM leads to a significant reduction in the number of unknowns.