A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

Abstract

A high-order time-domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued-fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued-fraction approach for bounded domains is proposed, which yields numerically more robust time-domain formulations. The coefficient matrices of the corresponding continued-fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high-order time-domain formulation for bounded domains with a high-order transmitting boundary suggested previously is also proposed. In the time-domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency-independent coefficient matrices, which can be solved efficiently using standard time-integration schemes. Numerical examples for modal and time-domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method.

Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis

“…The computation of the higher‐order terms , has been detailed in . In , matrices X ( i ) are introduced for preconditioning: Based on this expansion, an equation of motion in the frequency domain …”

High performance of the scaled boundary finite element method applied to the inclined soil field in time domain

“…As both the Eqs. (38) and (45) are with respect to fuðtÞg, the bounded and unbounded domains can be coupled seamlessly by using the same displacement shape functions. Assuming the system is linear, and combining Eqs.…”

“…Assuming the system is linear, and combining Eqs. (38) and (45), the coupled global equation of motion can be obtained…”

“…Birk et al [34,35] developed a high-order doubly asymptotic boundary by using doubly asymptotic continued fraction method in acoustic and diffusion problems. Chen [36,37] and Birk [38] proposed the continued fraction solution in time domain both in bounded and unbounded domains. Standard procedure in structural dynamic can be directly applied in time domain.…”

Simulation of elastic guided waves interacting with defects in arbitrarily long structures using the Scaled Boundary Finite Element Method

“…The integer M cf denotes the order of the continued fraction expansion that has to be chosen according to the highest frequency of interest. More recently, an improved formulation has been proposed [77], introducing scaling factors X (i) for preconditioning in order to enhance numerical stability:…”