Development of high-order doubly asymptotic open boundaries for wave propagation in unbounded domains by extending the scaled boundary finite element method

摘要

This thesis presents the development of high-order doubly asymptotic open boundaries used for the numerical simulation of wave propagation problems in unbounded domains, including homogeneous semi-infinite layers with a constant depth, homogeneous full-planes with a circular cavity and semi-infinite layered systems. The proposed open boundaries are necessary for dynamic and seismic analyses of large-scale structures such as dams, nuclear power plants etc. The theoretical framework of the research in the thesis is extended by employing the scaled boundary finite element method, which is a semi-analytical fundamental-solution-less boundary-element method based on finite elements.To avoid the computationally expensive task of numerically integrating the scaled boundary finite element equation in dynamic stiffness, the doubly asymptotic continued fraction solution for dynamic stiffness matrices is developed in the frequency domain using the technique of continued fraction. Factor coefficients or matrices are introduced in the continued fraction solution to improve the stability of the solution. As the continued fraction orders increase, the doubly asymptotic continued fraction solution converges to the exactness at both high- and low-frequency limits.By introducing auxiliary variables and the doubly asymptotic continued fraction solution to the force-displacement relationship in the frequency domain, a high-order doubly asymptotic open boundary condition is obtained. The open boundaries are expressed as systems of first-order ordinary differential equations in the time domain which are similar to the equation of motion with time-independent matrices in structural dynamics.The high-order doubly asymptotic open boundaries can be coupled seamlessly with standard finite elements. The accuracy of the results in the frequency and time domains depends on the orders of continued fraction selected by the user. Standard time-step schemes e.g. the Newmark's method etc. in structural dynamics are directly applicable to the high-order doubly asymptotic open boundaries for the implementation in the time domain. No convolution integral, which is the expensive task in the time-domain analysis, is required.