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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.
The simulation of Lamb waves in a cracked plate using the scaled boundary finite element method
(2012)
The scaled boundary finite element method is applied to the simulation of Lamb waves for ultrasonic testing applications. With this method, the general elastodynamic problem is solved, while only the boundary of the domain under consideration has to be discretized. The reflection of the fundamental Lamb wave modes from cracks of different geometry in a steel plate is modeled. A test problem is compared with commercial finite element software, showing the efficiency and convergence of the scaled boundary finite element method. A special formulation of this method is utilized to calculate dispersion relations for plate structures. For the discretization of the boundary, higher-order elements are employed to improve the efficiency of the simulations. The simplicity of mesh generation of a cracked plate for a scaled boundary finite element analysis is illustrated.
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.
In this paper, a method to determine the complex dispersion relations of axially symmetric guided waves in cylindrical structures is presented as an alternative to the currently established numerical procedures. The method is based on a spectral decomposition into eigenfunctions of the Laplace operator on the cross-section of the waveguide. This translates the calculation of real or complex wave numbers at a given frequency into solving an eigenvalue problem. Cylindrical rods and plates are treated as the asymptotic cases of cylindrical structures and used to generalize the method to the case of hollow cylinders. The presented method is superior to direct root-finding algorithms in the sense that no initial guess values are needed to determine the complex wave numbers and that neither starting at low frequencies nor subsequent mode tracking is required. The results obtained with this method are shown to be reasonably close to those calculated by other means and an estimate for the achievable accuracy is given.
Für alle Anwendungen geführter Wellen, beispielsweise in
der zerstörungsfreien Materialprüfung, ist die exakte und
effiziente Berechnung von Dispersionseigenschaften erforderlich.
Dabei müssen für eine gegebene Frequenz die Anzahl
der ausbreitungsfähigen Moden und deren Wellenzahlen
sowie Phasen- und Gruppengeschwindigkeiten berechnet
werden. Für den Fall von Lambwellen in homogenen
isotropen Platten existieren analytische Gleichungen
für die Wellenzahlen, die sich mit numerischen Nullstellensuchverfahren
lösen lassen. Für komplexere Strukturen
oder dreidimensionale, nicht rotationssymmetrische
Wellenleiter ist die Entwicklung numerischer Methoden
erforderlich. In der vorliegenden Arbeit wird ein numerisches
Verfahren, basierend auf der Scaled Boundary Finite
Element Method (SBFEM) [1] vorgestellt. Mit diesem
lassen sich Dispersionseigenschaften von beliebigen Wellenleitern
sehr effizient berechnen. Ergebnisse werden für
den Fall von Wellen in Platten mit komplexer Materialzusammensetzung
präsentiert.
In this paper the Scaled Boundary Finite Element Method (SBFEM) is applied for the simulation of Lamb waves in cracked plates. This method is highly advantageous to study the interaction of different Lamb wave modes with cracks as the crack is not discretized and no refinement is required around the crack tip. Numerical examples are presented for the reflection of the fundamental symmetric and antisymmetric modes from cracks of different depth. The spatial Fourier transformation is employed to calculate the amplitudes of reflected Lamb wave modes. The results reveal possibilities to obtain details of the crack geometry in non-destructive testing and structural health monitoring applications.
Der Einsatz geführter Wellen für die
zerstörungsfreie Prüfung mit Ultraschall eröffnet neue Möglichkeiten,
räumlich ausgedehnte Bauteile mit begrenzter Zugänglichkeit
auf ihre Integrität zu prüfen und gewinnt daher
zunehmend an Bedeutung. Dieser Artikel behandelt die
physikalischen Grundlagen der Schallausbreitung. Deren Verständnis
bildet die Grundlage für die Entwicklung geeigneter
Prüfsysteme. An Beispielen werden verschiedene Möglichkeiten
zur Simulation der Schallausbereitung vorgestellt. Aktuelle
Lösungsansätze zur Prüfung von plattenförmigen Strukturen
und von Rohrleitungen werden beschrieben, wobei besonders
auf die Sensortechnik und die speziellen Anforderungen
an die Prüfhardware eingegangen wird. -----------------------------------------------------------------------------------------------------------------------------------
Guided waves are widely used for non-destructive
testing using ultrasound. Recently, the method has become increasingly
important for integrity tests of spatially extended
components with limited accessibility. This article discusses the
basic physics of the sound propagation of guided waves. Their
understanding forms the basis for the successful development
of adapted inspection systems. Examples for simulating the
wave propagation using different methods are presented. Current
approaches for the inspection of plate-like structures and
pipelines are described with focus on sensor technology and the
specific requirement on the test hardware.
In this paper, a method is presented for the numerical computation of dispersion properties and mode shapes of guided waves in plate structures. The formulation is based on the Scaled Boundary Finite Element Method. The through-thickness direction of the plate is discretized in the finite element sense, while the direction of propagation is described analytically. This leads to a standard eigenvalue problem for the calculation of wave numbers. The proposed method is not limited to homogeneous plates. Multi-layered composites as well as structures with continuously varying material parameters in the direction of thickness can be modeled without essential changes in the formulation. Higher-order elements have been employed for the finite element discretization, leading to excellent convergence for complex structures. It is shown by numerical examples that this method provides highly accurate results with a small number of nodes while avoiding numerical problems and instabilities.
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.