Ultrasonic guided waves offer a wide range of applications in fields such as non-destructive testing, structural health monitoring or material characterization. They can be excited in thinwalled structures and propagate over comparably long distances. Due to their complex and dispersive propagation behavior, numerical methods are often required in order to analyze the guided wave modes that can be excited in a given structure and to simulate their interaction with defects.
In the work presented in this thesis, highly efficient numerical methods have been developed that are specifically optimized for guided wave problems. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM). The SBFEM is a semi-analytical method which evolved from the concept of Finite Elements but requires the discretization of the boundary of the computational domain only. To compute dispersion curves and mode shapes of guided waves, only the cross-section of the waveguide is discretized in the Finite Element sense, while the direction of propagation is described analytically. The wavenumbers of guided wave modes and the corresponding mode shapes are obtained as the eigenvalues and eigenvectors of a frequency-dependent Hamiltonian matrix. For the discretization, higher-order spectral elements are employed, leading to very low computational costs compared to traditional Finite Elements. Particular formulations are presented for plate structures as well as axisymmetric waveguides, where only the through-thickness direction has to be discretized. For the cases where the waveguide is embedded in or coupled to a quasi-infinite medium, a dashpot boundary condition is proposed in order to account for the effect of waves being transmitted into the surrounding medium. Though this approach is not exact, it leads to sufficiently accurate results for practical applications, while the computational costs are typically reduced by several orders of magnitude compared to other Finite Element based approaches.
As a particular application, an experimental set-up for material characterization is discussed, where the elastic constants of the waveguide’s material are obtained from the analysis of waves propagating through the waveguide. A novel solution procedure is proposed in this work, where each mode of interest is traced over the required frequency range. The solutions are obtained by means of inverse iteration.
To demonstrate the potential of the SBFEM for non-destructive testing applications, the interaction of guided wave modes with cracks in plates is simulated in the time domain for several examples. Particularly for the modeling of cracked structures, the SBFEM is very well suited, since the side-faces of the crack do not require discretization and the stress-singularity at the crack tip does not introduce additional difficulties. Hence, the computational costs can be reduced by typically a factor 100 compared to traditional Finite Elements and the meshing is straightforward.

Nomenclature xi

1 Introduction 1

2 Fundamentals 11

2.1 Linear elastodynamics 11

      2.1.1 Governing equations 11

      2.1.2 Plane strain and plane stress 14

2.2 The Finite Element Method 15

2.3 The Scaled Boundary Finite Element Method 20

3 Derivation for plate structures 25

3.1 SBFEM formulation for guided waves in plates 27

3.2 Discretization using high-order spectral elements 34

      3.2.1 Shape functions 35

      3.2.2 Numerical integration 36

      3.2.3 Element order 37

      3.2.4 Plates with varying material parameters 38

3.3 Properties of the eigenvalue problem 39

      3.3.1 Properties of the coefficient matrices 39

      3.3.2 Hamiltonian structure 39

      3.3.3 Algorithms 40

3.4 Group velocity 41

3.5 Shear-horizontal modes 43

3.6 Anisotropy 45

3.7 Details on the implementation 46

      3.7.1 Non-dimensionalization 46

      3.7.2 Parallelization 46

3.8 Numerical examples 47

      3.8.1 Homogeneous plate 47

      3.8.2 Layered composite 51

      3.8.3 Functionally graded material 53

4 Extension to arbitrary cross-sections 57

4.1 Three-dimensional waveguides 58

4.2 Discretization of three-dimensional waveguides 61

      4.2.1 Two-dimensional higher-order elements 61

      4.2.2 Symmetry 62

4.3 Mode-tracing 64

4.4 Numerical examples 68

      4.4.1 Isotropic circular rod 68

      4.4.2 Square pipe 71

5 Axisymmetric waveguides 75

5.1 Axisymmetric formulation of the SBFEM 77

5.2 Real coefficient matrices 81

5.3 Longitudinal and torsional modes 84

      5.3.1 Longitudinal modes 84

      5.3.2 Torsional modes 85

5.4 Solid cylinders 86

5.5 Numerical examples 87

      5.5.1 Isotropic homogeneous pipe 88

      5.5.2 Layered rod 90

      5.5.3 Anisotropic pipes 92

5.6 Convergence and adaptive meshing 94

6 Embedded waveguides 99

6.1 SBFEM formulation for embedded waveguides 101

      6.1.1 One-dimensional dashpot boundary condition 101

      6.1.2 Application to plate structures 103

      6.1.3 Application to axisymmetric waveguides 106

      6.1.4 Absorbing region 107

6.2 Numerical examples 109

      6.2.1 Plate attached to infinite medium 109

      6.2.2 Varying acoustic impedances 111

      6.2.3 Timber pole embedded in soil 115

7 Novel solution procedure 117

7.1 Background 117

      7.1.1 Motivation 117

      7.1.2 Fundamental equations 119

7.2 Solution procedure 121

      7.2.1 Linear approximation 121

      7.2.2 Initial values 122

      7.2.3 Mode-tracing 126

      7.2.4 Inverse iteration 126

7.3 Details of the implementation 128

      7.3.1 Dimensionless parameters 128

      7.3.2 Adaptive meshing 129

      7.3.3 Parallelization 129

      7.3.4 The L(0,0) mode 129

7.4 Numerical examples 130

      7.4.1 Natural polypropylene (PPN) 130

      7.4.2 Polyphenylene oxide (PPO-GF30) 135

8 Simulations in the time domain 137

8.1 SBFEM formulation in the time domain 138

8.2 Problem definition 141

8.3 Discretization 143

8.4 Results 143

      8.4.1 Comparison with Finite Element Analysis 143

      8.4.2 Reflection of the fundamental Lamb wave modes from a crack 148

9 Concluding remarks 155

Appendix A 157

A.1 Parameters used for the simulation in Section 1 157

A.2 Equations related to Section 2.3 158

A.3 Equations related to Section 5.2 159

A.4 Inverse iteration for a generalized eigenvalue problem 162

Bibliography 163

List of Figures 187

List of Tables 191

Acknowledgements 193

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