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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.
In this contribution, we present three models to capture singularities in combination with the Spectral Element Method. The first model, the continued-fraction-based Scaled Boundary Finite Element Method, the second model, a new approach based on enrichment with static modes, and the third model, which uses an hp-refinement near the singularity, are compared among each other and evaluated in terms of their respective efficiency and accuracy.
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.
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.