Analytische Chemie
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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.
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.
An essential task in many industries, e.g. food, petrol or chemical industry, is the precise and accurate characterization of liquids. Therefore, the development of innovative in-line sensors is of great interest. New concepts based on periodic structures, so-called phononic crystals (PnCs), are an interesting idea for the design of innovative sensors.
A PnC-based sensor can be designed by introducing a resonance inside a bandgap, a frequency region where no wave propagation is allowed. High-Q measurement systems using PnCs are already reported in the literature. However, existing designs cannot be implemented into a piping system directly, but need special fittings, openings or by-passes to be in contact with the liquid.
To circumvent this issue, we develop a new sensor based on PnCs, which can be directly implemented as part of the piping system. For this purpose, we use a PnC consisting of hollow cylinders with a periodic change of the outer diameter.
A bandgap could be found for the described geometry without fluid in simulation and measurement. However, simulations show, that a bandgap for fluid-filled cylinders can only be obtained for quasi-longitudinal modes. Hence, we propose a mode selective excitation for the sensor.
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.