Carolin Birk

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

In this paper, an approach is presented to model the propagation of elastic waves and their interaction with defects in plate structures. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM), a general semi-analytical method requiring the discretization of boundaries only. For a homogeneous finite or infinite plate section, only the through-thickness direction of the plate is discretized. To describe a defect, the full boundary of a short plate section of irregular shape is discretized. High-order spectral elements are employed for the discretization. The formulation for infinite plates can model the transmission into an unbounded domain exactly. Results are compared with conventional Finite Element Analyses in both time domain and frequency domain. The presented approach allows for the simulation of complex reflection and scattering phenomena using a very small number of degrees of freedom while the mesh consists of one-dimensional elements only.

The computation of dispersion relations for axisymmetric waveguides using the Scaled Boundary Finite Element Method

This paper addresses the computation of dispersion curves and mode shapes of elastic guided waves in axisymmetric waveguides. The approach is based on a Scaled Boundary Finite Element formulation, that has previously been presented for plate structures and general three-dimensional waveguides with complex cross-section. The formulation leads to a Hamiltonian eigenvalue problem for the computation of wavenumbers and displacement amplitudes, that can be solved very efficiently. In the axisymmetric representation, only the radial direction in a cylindrical coordinate system has to be discretized, while the circumferential direction as well as the direction of propagation are described analytically. It is demonstrated, how the computational costs can drastically be reduced by employing spectral elements of extremely high order. Additionally, an alternative formulation is presented, that leads to real coefficient matrices. It is discussed, how these two approaches affect the computational efficiency, depending on the elasticity matrix. In the case of solid cylinders, the singularity of the governing equations that occurs in the center of the cross-section is avoided by changing the quadrature scheme. Numerical examples show the applicability of the approach to homogeneous as well as layered structures with isotropic or anisotropic material behavior.

Computation of dispersion curves for embedded waveguides using a dashpot boundary condition

In this paper a numerical approach is presented to compute dispersion curves for solid waveguides coupled to an infinite medium. The derivation is based on the scaled boundary finite element method that has been developed previously for waveguides with stress-free surfaces. The effect of the surrounding medium is accounted for by introducing a dashpot boundary condition at the interface between the waveguide and the adjoining medium. The damping coefficients are derived from the acoustic impedances of the surrounding medium. Results are validated using an improved implementation of an absorbing region. Since no discretization of the surrounding medium is required for the dashpot approach, the required number of degrees of freedom is typically 10 to 50 times smaller compared to the absorbing region. When compared to other finite element based results presented in the literature, the number of degrees of freedom can be reduced by as much as a factor of 4000.

A local high-order doubly asymptotic open boundary for diffusion in a semi-infinite layer

A high-order open boundary for transient diffusion in a semi-infinite homogeneous layer is developed. The method of separation of variables is used to derive a relationship between the modal function and the flux at the near field/far field boundary in the Fourier domain. The resulting equation in terms of the modal impedance coefficient is solved by expanding the latter into a doubly asymptotic series of continued fractions. As a result, the open boundary condition in the Fourier domain is represented by a system of algebraic equations in terms of i@w. This corresponds to a system of fractional differential equations of degree @a=0.5 in the time-domain. This temporally global formulation is transformed into a local description by introducing internal variables. The resulting local high-order open boundary condition is highly accurate, as is demonstrated by a number of heat transfer examples. A significant gain in accuracy is obtained in comparison with existing singly-asymptotic formulations at no additional computational cost.

Coupled acoustic response of two-dimensional bounded and unbounded domains using doubly-asymptotic open boundaries

A high-order doubly-asymptotic open boundary for modelling scalar wave propagation in two-dimensional unbounded media is presented. The proposed method is capable of handling domains with arbitrary geometry by using a circular boundary to divide these into near field and far field. The original doubly-asymptotic continued-fraction approach for the far field is improved by introducing additional factor coefficients. Additionally, low-order modes are approximated by singly-asymptotic expansions only to increase the robustness of the formulation. The scaled boundary finite element method is employed to model wave propagation in the near field. Here, the frequency-dependent impedance of bounded subdomains is also expanded into a series of continued fractions. Only three to four terms per wavelength are required to obtain accurate results. The continued-fraction solutions for the bounded domain and the proposed high-order doubly-asymptotic open boundary are expressed in the time-domain as coupled ordinary differential equations, which can be solved by standard time-stepping schemes. Numerical examples are presented to demonstrate the accuracy and robustness of the proposed method, as well as its advantage over existing singly-asymptotic open boundaries.

Dynamic response of foundations on three-dimensional layered soil using the scaled boundary finite element method

This paper is devoted to the dynamic analysis of arbitrarily shaped three-dimensional foundations on layered ground using a coupled FEM-SBFEM approach. A novel scaled boundary finite element method for the analysis of three-dimensional layered continua over rigid bedrock is derived. The accuracy of the new method is demonstrated using rigid circular foundations resting on or embedded in nonhomogeneous soil layers as examples.

The Scaled Boundary Finite Element Method for Transient Wave Propagation Problems

A high-order time-domain approach for wave propagation in bounded and unbounded domains is developed based on the scaled boundary finite element method. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued-fraction expansions. The coefficient matrices of the expansions are determined recursively. This approach leads to accurate results with only about 3 terms per wavelength. A scheme for coupling the proposed 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 frequencyindependent coefficient matrices, which can be solved efficiently using standard time-integration schemes. A numerical example is presented.

Numerical investigation of axisymmetric thin-walled shells using the scaled boundary finite element method

Over the last decades thin-walled shell structures have been a topic of ongoing interest in research activities. The main reason for the sustained interest is the wide scope of application of thin-walled shells throughout numerous fields of engineering. The efficient load-carrying behaviour of thin-walled shells lead to more complex designs. While it is desirable to have less constraints in the design process, limitations of analytical solutions demand for the use of numerical analyses.

A Temporally Local Absorbing Boundary for Diffusion in 3D Unbounded Domains

The scaled boundary finite element method is used to model diffusion in unbounded domains. A time‐domain representation is derived expanding the stiffness into a series of continued fractions and using fractional derivatives. A method for transforming the resulting system of fractional differential equations into a local formulation is presented.

A high-order doubly asymptotic open boundary for scalar waves in semi-infinite layered systems

Wave propagation in semi-infinite layered systems is of interest in earthquake engineering, acoustics, electromagnetism, etc. The numerical modelling of this problem is particularly challenging as evanescent waves exist below the cut-off frequency. Most of the high-order transmitting boundaries are unable to model the evanescent waves. As a result, spurious reflection occurs at late time. In this paper, a high-order doubly asymptotic open boundary is developed for scalar waves propagating in semi-infinite layered systems. It is derived from the equation of dynamic stiffness matrix obtained in the scaled boundary finite-element method in the frequency domain. A continued-fraction solution of the dynamic stiffness matrix is determined recursively by satisfying the scaled boundary finite-element equation at both high- and low-frequency limits. In the time domain, the continued-fraction solution permits the force-displacement relationship to be formulated as a system of first-order ordinary differential equations. Standard time-step schemes in structural dynamics can be directly applied to evaluate the response history. Examples of a semi-infinite homogeneous layer and a semi-infinite two-layered system are investigated herein. The displacement results obtained from the open boundary converge rapidly as the order of continued fractions increases. Accurate results are obtained at early time and late time.