In this paper a numerical approach, based on the Scaled Boundary Finite Element Method (SBFEM), is described to obtain dispersion relations for propagating modes in wave guides. While the formulation is developed for plate structures, it can easily be extended to wave guides with arbitrary cross-section. The cross-section is discretized in the Finite Element sense while all equations remain analytical in the direction of propagation. The wave numbers of all propagating modes are obtained as the solutions of a standard eigenvalue problem. The group velocities can be calculated accurately as the eigenvalue derivatives. The use of higher-order elements drastically increases the efficiency and accuracy of the computation. This approach can be used for wave guides with arbitrary distribution of material parameters.
The simulation of guided waves in plate structures and cylinders coupled to infinite fluids is addressed. The approach is based on the Scaled Boundary Finite Element Method. Only a straight line is discretized that represents the through-thickness direction or the radial direction. The surrounding fluid is accounted for by employing a damping boundary condition that is based on the analytical description of the radiation impedance. Since the radiation impedance is a function of the wavenumber in the waveguide, an iterative solution procedure is applied. The algorithm is highly efficient while the results are in agreement with the Global Matrix Method.
Numerical simulation of ultrasonic guided waves using the scaled boundary finite element method
(2012)
The formulation of the Scaled Boundary Finite Element Method is applied for the computation of dispersion properties of ultrasonic guided waves. The cross-section of the waveguide is discretized in the Finite Element sense, while the direction of propagation is described analytically. A standard eigenvalue problem is derived to compute the wave numbers of propagating modes. This paper focuses on cylindrical waveguides, where only a straight line has to be discretized. Higher-order elements are utilized for the discretization. As examples, dispersion curves are computed for a homogeneous pipe and a layered cylinder.
In this paper we propose an algorithm to compute specific parts of the dispersion curves for elastic waveguides. The formulation is based on an axisymmetric representation of the Scaled Boundary Finite Element Method, where the wavenumbers of propagating modes are obtained as solutions of a Hamiltonian eigenvalue problem. The novel solution procedure involves tracing selected modes over a given frequency range and computing the corresponding solutions by means of inverse iteration. The resulting algorithm is applied in the context of material characterization, where the efficiency of the computation is crucial.
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.
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.
In this paper, a numerical approach for the computation of dispersion relations for three-dimensional waveguides with arbitrary cross-section is proposed. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM). It is an extension of the approach previously derived for plate structures. It is shown that the wavenumbers of guided waves in a waveguide can be obtained as the eigenvalues of the Z matrix, which is well known in the SBFEM. The Hamiltonian properties of this matrix are utilized to derive an efficient way to compute the group velocities of propagating waves as eigenvalue derivatives. The cross-section of the waveguide is discretized using higher-order spectral elements. It is discussed in detail how symmetry axes can be utilized to reduce computational costs. In order to sort the solutions at different frequencies, a mode-tracking algorithm is proposed, based on the Padé expansion.