A general solution procedure for the scaled boundary finite element method via shooting technique

摘要

The scaled boundary finite element method (SBFEM) is known for its inherent ability to simulate unbounded domains and singular fields, and its flexibility in the meshing procedure. Keeping the analytical form of the field variables along one coordinate intact, it transforms the governing partial differential equations of the problem into a system of one-dimensional (initial-)boundary value problems. However, closed-form solution of the said system is not available for most cases (e.g. transient heat transfer, acoustics, ultrasonics, etc.) since the system cannot be diagonalized in general. This paper aims to establish a numerical tool within the context of the shooting technique to evaluate the coefficient matrices of the subdomains without a priori knowledge of the analytical solution of the semi-discretized system. With proper choice of boundary conditions, the technique uses the strong form of the scaled boundary finite element equations to pass the required information and with the desired accuracy from one boundary to another. Due to generality of the technique, its procedure can be adjusted for any field equations. Since this technique is presented here for the first time, linear elastostatics, for which the closed-form solution is well-established, is formulated to provide valid comparisons. In addition, any direct solution method can be used for integrating the scaled boundary equations. Thus, without loss of generality, a Nystrom extension of the classical fourth-order Runge-Kutta method is employed. A quantitative sensitivity analysis is also conducted, and efficiency of the classical and proposed solution techniques is compared in terms of computational time. Finally, some numerical examples, including bounded and unbounded domains, as well as singular stress fields are simulated based on the classical and proposed solution techniques. (C) 2021 Elsevier B.V. All rights reserved.