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Ultrasonic guided waves offer a wide range of applications in fields such as non-destructive testing, structural health monitoring or material characterization. They can be excited in thin-walled structures and propagate over comparably long distances. Due to their complex and dispersive propagation behavior, numerical methods are often required in order to analyze the guided wave modes that can be excited in a given structure and to simulate their interaction with defects. In the work presented in this thesis, highly efficient numerical methods have been developed that are specifically optimized for guided wave problems. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM). The SBFEM is a semi-analytical method which evolved from the concept of Finite Elements but requires the discretization of the boundary of the computational domain only. To compute dispersion curves and mode shapes of guided waves, only the cross-section of the waveguide is discretized in the Finite Element sense, while the direction of propagation is described analytically. The wavenumbers of guided wave modes and the corresponding mode shapes are obtained as the eigenvalues and eigenvectors of a frequency-dependent Hamiltonian matrix. For the discretization, higher-order spectral elements are employed, leading to very low computational costs compared to traditional Finite Elements. Particular formulations are presented for plate structures as well as axisymmetric waveguides, where only the throughthickness direction has to be discretized. For the cases where the waveguide is embedded in or coupled to a quasi-infinite medium, a dashpot boundary condition is proposed in order to account for the effect of waves being transmitted into the surrounding medium. Though this approach is not exact, it leads to sufficiently accurate results for practical applications, while the computational costs are typically reduced by several orders of magnitude compared to other Finite Element based approaches. As a particular application, an experimental set-up for material characterization is discussed, where the elastic constants of the waveguide’s material are obtained from the analysis of waves propagating through the waveguide. A novel solution procedure is proposed in this work, where each mode of interest is traced over the required frequency range. The solutions are obtained by means of inverse iteration. To demonstrate the potential of the SBFEM for non-destructive testing applications, the interaction of guided wave modes with cracks in plates is simulated in the time domain for several examples. Particularly for the modeling of cracked structures, the SBFEM is very well suited, since the side-faces of the crack do not require discretization and the stress-singularity at the crack tip does not introduce additional difficulties. Hence, the computational costs can be reduced by typically a factor 100 compared to traditional Finite Elements and the meshing is straightforward.
Numerical modelling of Lamb waves in cracked plates using the scaled boundary finite element method
(2012)
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.
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 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 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.
Numerical modelling of lamb waves in cracked plates using the scaled boundary finite element method
(2013)
Timber poles are commonly used for telecommunication and power distribution networks, wharves or jetties, piling or as a substructure of short span bridges. Most of the available techniques currently used for non-destructive testing (NDT) of timber structures are based on one-dimensional wave theory. If it is essential to detect small sized damage, it becomes necessary to consider guided wave (GW) propagation as the behaviour of different propagating modes cannot be represented by one-dimensional approximations. However, due to the orthotropic material properties of timber, the modelling of guided waves can be complex. No analytical solution can be found for plotting dispersion curves for orthotropic thick cylindrical waveguides even though very few literatures can be found on the theory of GW for anisotropic cylindrical waveguide. In addition, purely numerical approaches are available for solving these curves. In this paper, dispersion curves for orthotropic cylinders are computed using the scaled boundary finite element method (SBFEM) and compared with an isotropic material model to indicate the importance of considering timber as an anisotropic material. Moreover, some simplification is made on orthotropic behaviour of timber to make it transversely isotropic due to the fact that, analytical approaches for transversely isotropic cylinder are widely available in the literature. Also, the applicability of considering timber as a transversely isotropic material is discussed. As an orthotropic material, most material testing results of timber found in the literature include 9 elastic constants (three elastic moduli and six Poisson's ratios), hence it is essential to select the appropriate material properties for transversely isotropic material which includes only 5 elastic constants. Therefore, comparison between orthotropic and transversely isotropic material model is also presented in this article to reveal the effect of elastic moduli and Poisson's ratios on dispersion curves. Based on this study, some suggestions are proposed on selecting the parameters from an orthotropic model to transversely isotropic condition.
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 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.
Several ultrasonic approaches for material determination are formulated in terms of an (nonlinear) inverse problem, e.g. immersion technique (Castaings et al. (2000)) or plate-waveguide techniques (Marzani et al. (2012)). In this contribution we focus on cylindrical waveguides for ultrasonic material determination and especially on the sensitivity of recorded transmission signals to the material properties. We utilize composite scaled sensitivities to determine the information content that can be achieved by the setup to certain parameters and discuss the limitations of the approach.
When modeling the propagation of elastic guided waves in plates or cylinders, Finite Element based numerical methods such as the Scaled Boundary Finite Element Method (SBFEM) or the Semi-Analytical Finite Element (SAFE) Method lead to an eigenvalue problem to be solved at each frequency. For the particular case of shear horizontal modes in a homogeneous plate or torsional modes in a homogeneous cylinder, the problem can be drastically simplified. The eigenvalues become simple functions of the frequency, while the eigenvectors are constant. The current contribution discusses how this behavior is represented in the numerical formulation and derives the expressions for the eigenvalues and eigenvectors as well as the dynamic stiffness matrix of infinite elastic waveguides.
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.
Geführte Ultraschallwellen bieten eine Vielzahl von Einsatzmöglichkeiten in der Zerstörungsfreien Prüfung, der Zustandsüberwachung sowie der Materialcharakterisierung. Insbesondere für Rohrleitungen und ausgedehnte Plattenstrukturen ist eine Vielzahl von auf geführten Wellen basierenden Verfahren in der Entwicklung und teilweise bereits im Einsatz. Aufgrund des komplexen Ausbreitungsverhaltens geführter Wellen werden numerische Verfahren (etwa die Finite Elemente Methode (FEM) oder die Randelementemethode (BEM)) zur Simulation der Wellenausbreitung sowie der Wechselwirkung mit Defekten in Wellenleitern angewendet. Diese Methoden sind für große Strukturen extrem rechenintensiv und umständlich in der Anwendung. Ein ungleich effizienteres Verfahren wurde kürzlich von den Autoren auf Grundlage der Scaled Boundary Finite Element Method entwickelt. Ein semi-analytischer Ansatz erlaubt die Modellierung beliebig ausgedehnter Strukturen bei extrem kurzen Rechenzeiten. Die Wechselwirkung geführter Wellen mit Rissen kann auf besonders elegante und exakte Weise beschrieben werden. Mit dieser Methode können die komplexen Vorgänge in Wellenleitern innerhalb weniger Sekunden modelliert werden.
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.
Time-causal material modeling in the simulation of guided waves in circular viscoelastic waveguides
(2014)
Time-causal material modeling in the simulation of guided waves in circular viscoelastic waveguides
(2014)
For the description of linear viscoelasticity, the fractional Zener model may be used. Based on the spectral decomposition of the elasticity matrix as proposed by Theocaris, we generalize the one-dimensional analysis of the material model into three dimensions and discuss appropriate simplifications to reduce the amount of unknowns for the material description. Then, a decomposition approach that considers the real valued frequency dependence of the viscoelastic moduli and the real valued frequency dependence of their attenuation separately is proposed. The Scaled Boundary Finite Element Method is used for the efficient computation of the phase velocity dispersion and the modal wave fields given a frequency dependent but real valued viscoelasticity matrix. Utilizing the modal expansion approach, the transmitting and receiving transducer are taken into account to compute the modal amplitudes. Combining these modal amplitudes, the phase velocity dispersion and re-introducing the viscoelastic attenuation results in a transfer function of the viscoelastic waveguide including excitation and receiving conditions. The performance of the proposed simulation model is shown by comparison to measurements taken on a polypropylene sample.
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.
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.
This paper presents a mode-tracing approach for elastic guided waves based on analytically computed derivatives and includes a study of interesting phenomena in the dispersion curve representation. Numerical simulation is done by means of the Scaled Boundary Finite Element Method (SBFEM). Two approaches are used to identify the characteristics of the resulting wave modes: Taylor approximation and Padé approximation. Higher order differentials of the underlying eigenvalue problem are the basis for these approaches. Remarkable phenomena in potentially critical frequency regions are identified and the tracing approach is adapted to these regions. Additionally, a stabilization of the solution process is suggested.
An approach is presented to model elastic waveguides of arbitrary cross-section coupled to infinite solid media. The formulation is based on the scaled boundary-finite element method. The surrounding medium is approximately accounted for by a dashpot boundary condition derived from the acoustic impedances of the infinite medium. It is discussed under which circumstances this approximation leads to sufficiently accurate results. Computational costs are very low, since the surrounding medium does not require discretization and the number of degrees of freedom on the cross-section is significantly reduced by utilizing higher-order spectral elements.
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 presents recently developed approaches for the numerical simulation of guided elastic waves in structures that are embedded in infinite fluid or solid media. The waveguide is described by the Scaled Boundary Finite Element Method, which is a general semi-analytical method that requires discretization of the boundary only. The influence of the surrounding medium on the wave propagation inside the waveguide is accounted for by appropriate boundary conditions. It is demonstrated that for many practical applications a formulation based on simple dashpot boundary conditions yields sufficiently accurate results. To increase accuracy for fluids, an alternative formulation based on exact boundary conditions and inverse iteration is proposed. This approach is of use particularly if the acoustic properties of the waveguide and surrounding material are similar.
In this contribution, we present an efficient approach for the transient and time-causal modeling of guided waves in viscoelastic cylindrical waveguides in the context of ultrasonic material characterization. We use the scaled boundary finite element method (SBFEM) for efficient computation of the phase velocity dispersion. Regarding the viscoelastic behavior of the materials under consideration, we propose a decomposition approach that considers the real-valued frequency dependence of the (visco-)elastic moduli and, separately, of their attenuation. The modal expansion approach is utilized to take the transmitting and receiving transducers into account and to propagate the excited waveguide modes through a waveguide of finite length. The effectiveness of the proposed simulation model is shown by comparison with a standard transient FEM simulation as well as simulation results based on the exact solution of the complex-valued viscoelastic guided wave problem. Two material models are discussed, namely the fractional Zener model and the anti-Zener model; we re-interpret the latter in terms of the Rayleigh damping model. Measurements are taken on a polypropylene sample and the proposed transient simulation model is used for inverse material characterization. The extracted material properties may then be used in computer-aided design of ultrasonic systems.