3-D WAVEGUIDE MODELING AND SIMULATION USING SBFEM Fabian Krome, Hauke Gravenkamp BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin, Germany email: Fabian.Krome@BAM.de The modeling of waveguides by means of the Scaled Boundary Finite Element Method (SBFEM) has recently been addressed and is considered an effective procedure for the simulation of ultrasonic guided waves in plates and uniform structures, as well as their interaction with defects. This work presents the extension of the known applications like uniform concrete foundation cylinders to structures with more complex shapes and defects. The main focus is the required modeling of 3-D structures in SBFEM to solve these efficiently. Furthermore the coupling of different models is discussed. This involves models like a mainly uniform foundation cylinder with varying geometry in certain areas, which has to be modeled in 3-D SBFEM or 3-D FEM. With the presentation of numerical examples the accuracy and performance of the modeling is discussed and the advantages in improving numerical stability are shown. 1. Introduction Elastic guided waves are used to examine materials in a variety of ways for a growing number of applications. Particularly guided waves in the ultrasonic frequency range are widely used. In the context of nondestructive testing [1],[2],[3], the combination of high frequencies and large geometries renders the modeling and simulation of wave propagation phenomena particularly important. Furthermore modern testing systems work with higher resolutions to describe smaller defects. Therefore even higher frequencies are needed. At the same time the desired complexity of geometries is increasing as well. Moreover many applications solve inverse problems and therefore have to repeatedly perform highly complex simulations. Modeling with sufficient accuracy is challenging, although established Finite Element Methods (FEM) already allow for modeling and simulating wave propagation and its phenomena. Small wavelengths in steadily growing dimensions of geometries require a very high number of degrees of freedom and therefore lead to tremendous computational costs. At the same time high frequencies require small time steps in transient simulations. If multiple simulations are needed to solve an inverse problem, the computational costs are often unacceptable. A growing number of methods have been introduced to increase efficiency while preserving accuracy. The particular problem of waveguides with long homogenous sections appears regularly in nondestructive testing applications. This rather trivial property is used in several methods, which range from purely analytical [4] to semi-analytical methods such as the Semi-Analytical Finite Element (SAFE) Method [5],[6]. This work is based on the Scaled Boundary Finite Element Method (SBFEM) [7],[8],[9],[10]. The SBFEM is a semi-analytical method discretising the boundary of the domain only. This method has ICSV22, Florence, Italy, 12-16 July 2015 1
been introduced recently to compute dispersion curves and mode shapes for infinitely long waveguides of constant cross-section [11],[12],[13]. Ever since the range of its applications has increased rapidly. The approach to compute dynamic stiffness matrices of homogenous plates proposed by Gravenkamp, Birk and Song [14] is one of the most promising developments in this field. They compute and couple stiffness matrices of domains with arbitrary plate shapes derived from SBFEM or FEM. This allows for very efficient computation of large geometries, since the stiffness matrix of a domain with constant plate thickness is computed in the same time regardless of the length. This even includes plates with infinite expansion in propagation direction. This paper extends the established concepts and approaches to three-dimensional domains of arbitrary cross-sections, which are constant in the direction of wave propagation. The results of this transformation will then be used to demonstrate the ability to couple domains with different geometries. Furthermore the formulation of irregular shapes in plates is considered for three dimensional irregularities as shown in Figure 1. Figure 1: Concept of modeling a three dimensional waveguide by separating domains of computation 2. Stiffness Matrix of homogeneous waveguides Figure 2: Domain with constant cross section: With length L and an unbounded domain This chapter describes the general approach to compute stiffness matrices of homogeneous waveguides of constant cross section with SBFEM. The presented method can be applied to waveguides of finite or infinite length and constant cross section Figure 2. The cross section is discretised by means of Finite Elements whereas the direction of propagation is described analytically. This approach corresponds to the case of a straight interface with scaling center at infinity. In [14] a one-dimensional line is discretised by Finite Elements, while now an arbitrary two-dimensional Finite Element discretisation is used to describe the cross-section. Following the same argumentation as in [14] high order elements are used to improve efficiency. 2 ICSV22, Florence, Italy, 12-16 July 2015
The stiffness matrix is derived by relating nodal displacements u n (z) and internal forces q n (z). The SBFEM formulation of homogeneous waveguides with constant cross computes (1) by solving the System (2) with (3) Z = [ Ψ = { un (z) q n (z) } Ψ,z = ZΨ. E 1 0 E T 1 ω 2 M 0 E 2 + E 1 E 1 0 E T 1 E 0 1 E 1 E 0 1 The matrix Z appears in several SBFEM applications and depends only on information derived from the cross-section. The solution process for stiffness matrices of bounded and unbounded domains differs in their consideration of direction of wave propagation. ]. 3. Numerical Examples Two examples are presented to illustrate the capability of this approach to describe wave propagation and its phenomena. Firstly a square rod (Figure 3) is excited harmonically by a point force and the frequency response is compared to standard FEM simulation. Secondly a more complex structure of a quadratic rod with a notch (Figure 4) is excited by a point force as well and is compared to standard FEM simulations. Material parameters are: Young s modulus: E = 76.923 GPa Density: ρ = 7850 kg/m 3 Poisson s ratio: v = 0.3 Figure 3: Cylinder with radius 0.5 mm and 1 mm length 3.1 Cylinder The cylinder in Figure 3 is exited by a force f = 10 9 N as indicated by the arrow. The resulting frequency response of the displacement in x-direction at point P1 was computed for an exemplary frequency range and is compared to an ANSYS FEM simulation. The comparison is shown in Figure 5. The frequency responses at this exemplary point can be considered a match. Small differences can be traced back to the mesh size. The FEM model had to be sized in context of computational time SBFEM simulation was computed within 5 Minutes for all frequencies versus 3 days of ANSYS FEM simulation for the same amount of frequencies. To render the solutions comparable the SBFEM mesh size was adjusted to the FEM mesh in context of accuracy. ICSV22, Florence, Italy, 12-16 July 2015 3
Figure 4: Rod with 3 Blocks 1x1x1 mm 3-1x0.5x1 mm 3-1x1x1 mm 3 3.2 Square rod with notch The rod in Figure 4 is also exited by a force f = 10 9 N as indicated by the arrow. Again the resulting frequency response of the displacement in x-direction at point P2 was computed for an exemplary frequency range and compared to an ANSYS FEM simulation. The comparison is shown in Figure 5. The frequency responses in this second example can also be considered as a match. Lastly Figure 5 also shows the results for a combination of FEM and SBFEM. The stiffness of the half cube (II) was calculated using a standard FEM approach and the remaining cubes (I + III) with the SBFEM approach. The resulting stiffness matrices were coupled. Again the frequency response can be considered a match. This result indicates that any defect in the FEM domain could now be included as long as the meshes on the connecting surfaces match. Moreover any two SBFEM mesh surfaces could be connected using a small FEM transformation domain. Figure 5: Results frequency response: Cylinder with radius 0.5 cm 4 ICSV22, Florence, Italy, 12-16 July 2015
Figure 6: Results frequency response: Rod with 3 Blocks 4. Conclusion This paper outlined the transition from two dimensional plate formulations to three dimensional waveguides of arbitrary cross-sections with SBFEM. The cases of finite and infinite domains were discussed and the coupling of different domains was introduced. Additionally coupling of the new method with domains described by classical SBFEM or FEM domains was explained. Therefore the method was made available for extension of arbitrary bounded domains. The validity of the proposed method was shown by two basic examples which agree with standard FEM analysis. Future work in this field will include improvement of coupling procedures, efficient defect representation, and coupling, as well as studies on numerical stability of the method. REFERENCES 1. Yam, L., Cheng, L., Wong, W., Numerical analysis of multi-layer composite plates with internal delamination, Computers & Structures 2004; 82:627 637 2. Su, Z., Ye, L., Lu, Y., Guided Lamb waves for identification of damage in composite structures: A review, Journal of Sound and Vibration 2006; 295:753 780 3. Baltazar, A., Hernandez-Salazar, CD., Manzanares-Martinez, B., Study of wave propagation in a multiwire cable to determine structural damage, NDT&E International 2010; 43:726 732 4. Nayfeh, AH., The general problem of elastic wave propagation in multilayered anisotropic media, Journal of the Acoustical Society of America 1991; 89:1521 1531 5. Kausel, E., Roesset, JM., Semianalytic hyperelement for layered strata, Journal of the Engineering Mechanics Division 1977; 8:569 588. 6. Kausel, E., Peek, R., Dynamic loads in the interior of a layered stratum: An explicit solution, Bulletin of the Seismological Society of America 1982; 72(5):1459 1481 7. Song, C, Wolf, JP., The Scaled boundary finite Element method alias consistent infinitesimal finiteelement cell method for elastodynamics, Computer Methods in Applied Mechanics and Engineering 1997; 147:329 355. ICSV22, Florence, Italy, 12-16 July 2015 5
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