Pseudospectral collocation method for viscoelastic guided wave problems in generally anisotropic media
Creators: name: Hernando Quintanilla, Francisco
Publisher: name: Imperial College London
Year: 2016

description: In Non-Destructive Evaluation (NDE) applications guided waves are attractive to perform rapid inspections of long lengths and large areas. However, they are complicated, therefore it is important to have as much information and understanding about their physical properties as possible in order to design the most efficient and robust inspection process as well as to draw the correct conclusions from the measurement results. The main piece of information to gain insight into the guided wave's properties is dispersion curves which, for isotropic structures such as plates and cylinders, have been available for many years. There are many robust algorithms which are currently used to compute them: finite element simulations, partial wave based root finding routines (PWRF) and semi-analytical finite element simulations (SAFE). These methodologies have been generalized and also used to study and compute dispersion curves of more complicated anisotropic materials though the range of tractable cases was limited. Although robust, all these approaches present several challenges, mostly computational, such as missing modes (PWRF), the so called "large-fd" problem (PWRF), artificially increased stiffness (FE, SAFE) or improvement of dispersion curve tracing routines (FE, PWRF, SAFE). In addition, when studying complicated anisotropic materials with a low degree of symmetry or unusual axes configurations where propagation does not take place along any of the principal axes, PWRF routines are frequently unreliable and one must resort to specific SAFE simulations which also present their own challenges and, depending on the SAFE scheme used, can yield spurious modes which need to be carefully filtered. Recently, Pseudospectral Methods (Galerkin and Collocation schemes), were introduced in the field of elastic guided waves, providing a powerful, yet strikingly and conceptually simple alternative to the above algorithms by successfully finding the dispersion curves in isotropic structures and in some simple anisotropic problems. However, a systematic and general approach for accurately and robustly computing dispersion curves of guided waves in anisotropic media, up to the most general case of triclinic symmetry, has not yet been developed. The goal of the work presented in this thesis is to develop such a tool by means of the Pseudospectral Collocation Method (SCM) and to take advantage of its particular features to make it as robust as possible. Firstly, a PSCM scheme is developed for computing dispersion curves of guided waves in anisotropic elastic media by finding all the frequencies for a given value of the real wavenumber. The results are validated with the existing literature as well as with the results provided by the software DISPERSE developed in the NDT group at Imperial College London. Many of the most remarkable features of the PSCM (spectral accuracy, speed, and its failure to miss modes for instance) are already observed in this simple, yet important, class of problems in elastic media. Secondly, guided waves in viscoelastic anisotropic media are studied. In this case, modes present attenuation due to material damping which is reflected in the wavenumber being complex. In order to handle complex wavenumbers the PSCM schemes developed for elastic materials are appropriately extended by means of the Companion Matrix Method. It will be seen that, apart from lowly attenuated propagating modes, all the other highly attenuated modes are found, yielding the full three-dimensional spectrum of the problem under consideration. Moreover, when the PSCM schemes for viscoelastic media are used to compute the dispersion curves of guided waves in an elastic medium, all the remaining, imaginary as well as complex, roots of the elastic problem which were not computed by the simpler PSCM elastic schemes are found, providing the full three-dimensional picture of the dispersion curves. These PSCM schemes, as any other of the aforementioned approaches, only find pairs (\omega,k). If dispersion curves are to be plotted, those pairs must be linked correctly in order to plot the desired dispersion curves, which is non-trivial when crossings amongst modes occur. Motivated by this, an investigation of the parity and coupling properties of guided wave solutions is carried out in detail for all crystal classes. This investigation provides a robust alternative to conventional tracing routines and avoids the problem of mode crossings by exploiting the parity and coupling properties of the solutions. Finally, the most complicated problems involving embedded structures are investigated by including a Perfectly Matched Layer (PML) in the previously developed PSCM schemes for viscoelastic media. The dispersion curves for leaky and trapped modes in an isotropic elastic plate and in a similar cylinder immersed in an infinite ideal fluid are found, showing very good agreement with the results given by PWRF routines in a large range of frequencies. Last, but not least, an illustration of a two-dimensional PSCM scheme is presented to study a vibrating membrane. The results are compared with the available analytical solution showing again excellent agreement.