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It can be difficult to efficiently model ultrasonic waves in 3D structures, especially when the computational model needs to account for complex geometries. This contribution presents a solution based on the Scaled Boundary Finite Element Method (SBFEM). It is a numerical tool suitable for elastodynamic problems. A space-tree discretisation, namely quad-trees, is used. This technique allows the decomposition of an image into quadrilaterals or quads, which are extruded to generate the 3D plate geometry. In particular, small quads resolve regions with discontinuities, allowing them to represent fine details in the structure. Moreover, this meshing technique allows for exploiting cell similarities, making the calculation procedure more efficient. The space-tree discretisations are generated from a high-resolution image containing all the information about damaged regions or boundary conditions. The resulting SBFEM polyhedral domains employ transition elements to ensure correct coupling between cells of different sizes. The analytical solution of a cylindrical scatterer serves as a reference to validate the proposed approach. Other examples also demonstrate the validity of the methodology and its flexibility.
The simulation of ultrasonic waves in a linearly elastic body can be computationally intensive. The reason is the relatively short wavelength compared to the body size for high frequencies. One possible approach to counteract the high computational costs is to decompose the domain into small parts and strive for parallelization. The Mortar Method is a well-established approach for domain decomposition.
A rather new approach to discretize the emerging subdomains is the Scaled Boundary Finite Element Method. This semi-analytical method has many attractive properties. Some of these properties are listed subsequently. The grid consists of polygonal elements, which leaves much freedom in the meshing process. A variety of material distributions, including anisotropic materials, can be considered. High-order shape functions can be used for optimal convergence properties. The approach treats singularities at crack tips and corners analytically. Especially in the frequency domain, the Scaled Boundary Finite Element Method reduces the dimension of the approximation because only degrees of freedom which are associated with the boundary of a polygonal element are necessary. Those desirable properties make the method particularly suitable for calculating the dynamic responses in bodies with cracks, as it is essential for many non-destructive testing and structural health monitoring applications.
In this contribution, we present a combination of the Scaled Boundary Finite Element Method with the Mortar Method in two dimensions. The presentation starts with a theoretical overview of both approaches. Subsequently, numerical examples demonstrate the stability of the combination for the polygonal boundary of the elements. The numerical examples increase in complexity and are compared to results computed on non-divided domains with the Finite Element Method.
The Scaled Boundary Finite Element Method is known as an efficient method for the simulation of ultrasonic wave propagation. As to investigate acoustic wave behavior in case of fluid‐structure interaction, a fluid model is implemented in the SBFEM for prismatic structures. To omit coupling terms a displacement‐based formulation is used. Spurious modes, which occur in the solution, are suppressed using a penalty parameter. To verify this formulation dispersion curves obtained with Comsol Multiphysics are compared to results of SBFEM. The results of both methods are in very good agreement
This paper presents an approach to the automatic enrichment of finite elements in the vicinity of a stress singularity. The enrichment consists of semi-analytical singular modes constructed using the Scaled Boundary Finite Element Method (SBFEM).
In contrast to analytical methods, the SBFEM provides modes for inhomogeneous and anisotropic materials without additional effort. The finite element basis can be of arbitrary order and remains unaltered by the enrichment. The approach requires enrichment in only one layer of elements around a node. Due to the compatibility of SBFEM with FEM, there is no Need for transitional elements, and there are no parasitic terms. The approach is tested for several benchmark problems. The stress intensity factors are computed based on techniques inspired by the SBFEM. The proposed procedure is compared to a Standard finite element implementation and shows a significant improvement in the error of the displacement field for problems involving singular stresses.
Acoustic-structure interaction in the Scaled Boundary Finite Element Method for primsatic geometries
(2019)
Due to the short wavelength compared to the dimensions of the structure, the simulation of ultrasonic waves is still a challenging task. A numerical method well suited for this purpose is the semi-analytical Scaled Boundary Finite Element Method (SBFEM). When applying this method, only the boundary of a computational domain is discretized using finite elements, while the interior is described by an analytical ansatz. Hence, the number of degrees of freedom is reduced significantly compared to the classical Finite Element Method (FEM).
In recent years, a particular formulation of the SBFEM for the simulation of ultrasonic guided waves was developed. The method constitutes an efficient algorithm for prismatic structures of arbitrary length, such as plates, pipes, or beams. Wave propagation phenomena in such structures can be modeled for isotropic and anisotropic inhomogeneous waveguides. Even though the method is an efficient tool for the simulation of guided waves in solid media, a reliable model for the simulation of acoustic wave propagation in fluids as well as acoustic-structure interaction in terms of SBFEM is still missing. In principle, the fluid can be described by a displacement-based formulation and thus be implemented in existing SBFEM algorithms for solid bodies. However, due to the discretization with classical finite elements, spurious modes occur, which cannot be separated from the physical modes straightforwardly. The spurious modes can be suppressed using a penalty parameter. Although very accurate results were achieved for some problems, this procedure has been proven unreliable for certain cases.
For this reason, we propose a different approach in this contribution. We employ a pressure model to simulate the acoustic behavior of fluids. The implementation of the pressure model results in a higher effort due to the necessity of incorporating coupling terms, but it presents a stable alternative without spurious modes. The accuracy of the method is demonstrated in comparison with analytical solutions and results obtained using the FEM.
Akustische Verfahren eignen sich hervorragend zur Bestimmung der Werkstoffeigenschaften. Die Verfügbarkeit derartiger Verfahren ist vor allem für Kunststoffe wichtig, da deren Eigenschaften stark abhängig vom jeweiligen Herstellungsprozess und vom Alterungszustand sind. Exakte und vollständige Werte sind daher in Datenbanken oder von Herstellern nur begrenzt zu finden. Insbesondere die Entwicklung von Methoden zur Charakterisierung faserverstärkter Kunststoffe (FKV) ist nach wie vor Gegenstand der Forschung. Hier müssen anisotropiebedingt mehrere Kennwerte bestimmt werden. Da FKV zumeist als dünnwandige Bauteile zum Einsatz kommen, können die Werkstoffparameter aus den Dispersionseigenschaften der Lamb-Wellen, die sich in diesen Strukturen ausbreiten, abgeleitet werden. Dazu ist eine räumliche Abtastung des sich ausbreitenden Schallfelds erforderlich.
In der vorliegenden Untersuchung wird dieser Ansatz für die relativ neue Werkstoffklasse der faserverstärkten Thermoplaste angewendet. Diese zeichnet ein ausgeprägtes Dämpfungsverhalten und eine Anisotropie der Materialparameter aus. Dazu wurde das Schallfeld im Ultraschallbereich mit einem Laser-Doppler-Vibrometers vermessen. Rechnerisch bestimmte Dispersionskurven wurden dann an die gemessenen Werte angepasst, womit die richtungsabhängigen Materialparameter bestimmt werden konnten.
Im Vortrag wird das Messverfahren vorgestellt und auf spezielle Probleme, die sich z.B. aus dem Dämpfungsverhalten des Werkstoffs ergeben, eingegangen. Ausgewählte gemessene Werte werden mit den Ergebnissen von Referenzverfahren verglichen.
Guided waves (GW) are of great interest for non-destructive testing (NDT) and structural health monitoring (SHM) of engineering structures such as for oil and gas pipelines, rails, aircraft components, adhesive bonds and possibly much more. Development of a technique based on GWs requires careful understanding obtained through modelling and analysis of wave propagation and mode-damage interaction due to the dispersion and multimodal character of GWs. The Scaled Boundary Finite Element Method (SBFEM) is a suitable numerical approach for this purpose allowing calculation of dispersion curves, mode shapes and GW propagation analysis. In this article, the SBFEM is used to analyse wave propagation in a plate consisting of an isotropic aluminium layer bonded as a hybrid to an anisotropic carbon fibre reinforced plastics layer. This hybrid Composite corresponds to one of those considered in a Type III composite pressure vessel used for storing gases, e.g., hydrogen in automotive and aerospace applications. The results show that most of the wave energy can be concentrated in a certain layer depending on the mode used, and by that damage present in this layer can be detected. The results obtained help to understand the wave propagation in multi-layered structures and are important for further development of NDT and SHM for Engineering structures consisting of multiple layers.
This paper addresses the computation of dispersion curves, mode shapes and propagation of elastic guided waves. It summarizes the approaches based on the Scaled Boundary Finite Element Method.
Descriptions for plates, rods, pipelines and waveguides with an arbitrary cross section are included. The important steps for the approximation of the displacement in bounded and unbounded domains are stated. The grid generation process is explained. It is highlighted that the Scaled Boundary Finite Element Method is very efficient, if large portions of the domain are either straight or with a constant curvature. The computation of dispersion curves for layered structures is presented.
The application of waveguides for acoustic measuring technologies and the development of non-destructive evaluation techniques with guided ultrasonic waves for plate like materials like carbon fiber reinforced plastic shells and layered structures require a good understanding of acoustic wave propagation inside the material. The well-known Finite Element Method can be used for simulations, however at least for higher frequencies, the ratio of wavelength and geometrical dimension demands a time-consuming fine grid. Using commercial simulation tools the computational costs increase considerably for ultrasonic frequencies.
In the recent years, the Federal Institute for Materials Research and Testing has developed a very efficient alternative for simulating acoustic wave propagation particularly in wave guides by extending the Scaled Boundary Finite Element Method (SBFEM). The SBFEM as a semi-analytical method has one main advantage over the classical Finite Element Method: It only demands a discretization of the boundary instead of the whole domain. This is pictured in the figures below. The method is still related to the Finite Element Method and uses their well-known solving strategies. SBFEM is shown to be highly efficient, especially in the frequency domain. Additionally, the efficiency can be increased by using higher-order spectral elements. In plates and cylinders, the SBFEM can be used to animate propagating modes and computes their wavenumber.
In this contribution, we present a short introduction into the basics of SBFEM formulation of the dynamic elastic wave equation. The applicability and efficiency of the approach is demonstrated by applying the method to layered structures and different wave guide geometries. As one example we present the wave propagation in a typical adhesive joint of different metal sheets as common in new designs in automotive industry. The analysis comprises the computation of dispersion curves as starting point of every development of non-destructive testing techniques for inspecting such structures as well as the analysis of the propagating modes. Additional examples presented handle special cases for axis-symmetric geometries, such as pipes and cylindrical rods which are common in various acoustic measurement applications.