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This note studies the accuracy of Isogeometric Analysis (IGA) applied in the simulation of incompressible flows around a cylinder in two and three dimensions. Quantities of interest, like the drag coefficient, the lift coefficient, and the difference of the pressure between the front and the back of the cylinder are monitored. Results computed with standard finite element methods are used for comparison.
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.
In Non-Destructive Testing, ultrasonic waves are commonly used to identify flaws and cracks.
In plates, shells, pipes and other geometries guided waves can be used to test the whole structure at once. In these tests, the input and response signal can have a nonlinear relationship due to cracks. At least for higher deflections, the propagating wave excites each side of the crack in such a way that it hits the other side. This clapping generally leads to a generation of higher harmonic waves and is referred to Contact Acoustic Nonlinearity (CAN). To get a better insight into the salient physics of the effect numerical simulations are necessary.
In the recent years, the Scaled Boundary Finite Element Method (SBFEM) was introduced to efficiently simulate wave propagation. The main advantage of the method is an easy grid generation process because the domain is discretized with arbitrary polygons instead of the triangles and rectangles. Another advantage is the possibility to model crack tips elegantly without additional workload. The SBFEM approach is still related to the Finite Element Method and uses similar techniques. The method is very efficient using high-order-spectral elements.
In this contribution, we present a short introduction into the basics of SBFEM formulation of the dynamic elastic wave equation. The SBFEM is then extended for modeling the non-linear behavior of crack clapping. Different approaches with increasing complexity are presented and assessed with respect to numerical stability.
The Scaled Boundary Finite Element Method (SBFEM) is a semi-analytical method that shows promising results in modelling of guided ultrasonic waves. Efficiency and low computational cost of the method are achieved by a discretisation of the boundary of a computational domain only, whereas for the domain itself the analytical solution is used. By means of the SBFEM different types of defects, e.g. cracks, pores, delamination, corrosion, integrated into a structure consisting of anisotropic and isotropic materials can be modelled.
In this contribution, the SBFEM is used to analyse the propagation of guided waves in a structure consisting of an isotropic metal bonded to anisotropic carbon fibre reinforced material. The method allows appropriate wave types (modes) to be identified and to analyse their interaction with different defects. Results obtained are used to develop a structural health monitoring system for composite pressure vessels used in automotive and aerospace industries.
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
The Spectral Element Method (SEM) has been proven to be an efficient numerical method for solving the wave equation in linear elastic bodies. This efficiency is reduced if a stress singularity is present in the body. For example, re-entrant corners, material interfaces, fixed boundaries, and especially crack tips can cause stress singularities. To preserve the efficiency of the SEM, special solution strategies are required. There are many approaches which consider stress singularities, but comparisons are rare for dynamic problems. Finding an efficient model is an important step for many applications. In particular, applications for structural health monitoring and non-destructive evaluation rely on accurate and efficient crack models.
In this contribution, we present several models to capture singularities in combination with the SEM. The theory behind the models are shortly summarized and we show results for different benchmark problems in two dimensions. The first model deploys a new class of singular elements. For these singular elements, the crack tip approximation is computed based on the static Scaled Boundary Finite Element Method (SBFEM). The second model uses the continued‐fraction‐based SBFEM, while the third model uses an hp-refinement near the singularity. The models are compared among each other and evaluated in terms of their respective efficiency and accuracy.
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.
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.