A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

Abstract

Keywords:3D elastostatic problems Semi-analytical method Diagonal coefficient matrices Chebyshev polynomials Subparametric elements Clenshaw-Curtis quadrature Decoupled ordinary differential equations a b s t r a c tIn this paper, a new semi-analytical method is presented for modeling of three-dimensional (3D) elastostatic problems. For this purpose, the domain boundary of the problem is discretized by specific subparametric elements, in which higher-order Chebyshev mapping functions as well as special shape functions are used. For the shape functions, the property of Kronecker Delta is satisfied for displacement function and its derivatives, simultaneously. Furthermore, the first derivatives of shape functions are assigned to zero at any given node. Employing the weighted residual method and implementing Clenshaw-Curtis quadrature, coefficient matrices of equations' system are converted into diagonal ones, which results in a set of decoupled ordinary differential equations for solving the whole system. In other words, the governing differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. To evaluate the efficiency and accuracy of the proposed method, which is called Decoupled Scaled Boundary Finite Element Method (DSBFEM), four benchmark problems of 3D elastostatics are examined using a few numbers of DOFs. The numerical results of the DSBFEM present very good agreement with the results of available analytical solutions.

Development of a new semi-analytical method in fracture mechanics problems based on the energy release rate

“…In other words, emphasis is basically devoted to those important aspects of the method which are subjected to remarkable modifications in comparison with the previous works of the authors [32][33][34][35][36][37][38]. For more details on the formulation of the method, the readers are referred to Ref.…”

“…(34) and (35), the set of differential equations (30) may be expressed as a set of decoupled differential equations regarding to a specific node i as…”

“…In other words, the present method offers an efficient procedure for solving various problems as already reported in Refs. [32][33][34][35][36][37][38]. In this section, the required procedure for solving Euler's differential equations is illustrated, and the extension of the procedure to derive the SIFs is extensively discussed in the next section.…”

“…A new semi-analytical method has been developed by Khaji and coworkers [32][33][34][35][36][37][38]. In this method, only boundaries of problems are discretized using specific subparametric elements and higher-order Chebyshev mapping functions.…”

The Use of Enriched Base Functions in the Three-Dimensional Scaled Boundary Finite Element Method

“…[40,42,15,30,34,43]) and 3D applications (e.g. [5,32,24,13,31,9,20,26,18]). The FEM eigenanalysis is based on the approximation of the displacements by a separation of variables representation with FE-shape functions in the circumferential direction, i.e.…”