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1 The Semi-Analytical Finite Element Method

The content of the current section is primarily based on Dr. Ahmad’s research. The results of his investigations are published in a PhD thesis [1] and in two peer-reviewed journal articles [2, 3].

The semi-analytical finite element method (SAFE) is typically used to compute the dispersion curves for isotropic and composite plates [1]. More complex waveguides, such as rods, wires, and rail road tracks, have also been investigated [4, 9, 15, 27, 38, 44, 53]. If we want to consider the effect of initial axial loads on the dispersion curves, additional terms need to be added to the conventional formulation [43]. As in standard FE applications, both h- and/or p-refinements can be deployed to increase the accuracy of the simulations. This is especially important when higher order wave modes need to be evaluated [8]. The influence of external loads and a piezoelectric excitation can also be included in the SAFE formulation. For two-dimensional problems this is discussed in [9, 48] and three-dimensional (3D) cases are considered in [7, 57]. These simulations solve the wave propagation problem for plates with infinite (in-plane) dimensions, due to the fact that the SAFE method implicitly assumes the plate to be infinite in its formulation. Additional effects on the propagation of elastic guided waves, i.e. reflections and transmissions from damages, actuators or boundaries make the wave propagation behavior more complex and must be added separately. Therefore, the SAFE method can be combined with classical methods, such as the FEM, the boundary element method (BEM), and other computational approaches. The reflection and transmission behavior of ultrasonic guided waves at geometrical perturbations of the infinite plate have been investigated in [2, 3, 23, 35, 36, 39, 54, 55]. In these cases, the infinite plate section is modeled using the SAFE approach, while damages or boundaries are modeled using the FEM or the BEM, respectively [2, 3]. Leaky Lamb waves are also investigated with the help of SAFE to evaluate the energy loss into the surrounding medium [24, 29, 46].

1.1 Motivation

The SAFE method has its advantages in the computation of dispersion curves and in the analysis of wave interactions with obstacles. The solution to both problems benefits from the semi-analytical character of the SAFE. That is to say, unperturbed regions of a structure or infinite regions can be simulated at hardly any cost compared to classical FE approaches. The SAFE is not as severely affected by the need for a fine spatial discretization compared to the FEM because of its specifically tailored basis functions that account for typical properties of propagating waves. The details behind this behavior are explained in Sects. 7.1.2–7.1.6.

1.2 Theoretical Principles

In this section we provide the formulation of the semi-analytical finite element method according to [4, 13–15, 41, 45]. The SAFE method is basically a combination of the FEM in the cross section of the structure and analytical methods in the direction of wave propagation direction. Accordingly, we discretize the cross section of the waveguide by means of conventional finite elements and deploy a complex exponential function in the propagation direction [14], cf. Fig. 7.1. That is to say, we reduce the dimensionality of the problem by one which is also the reason for the decreased computational costs. Since the plate thickness can easily be resolved by only a few finite elements mesh requirements for the SAFE method are comparably low [1]. In the remainder of this chapter we will use the term in-plane to refer to the cross section plane of the waveguide, while out-of-plane denotes the wave propagation direction, cf. Fig. 7.1. Note that this is in contrast to the natural understanding of the words in- and out-of-plane.

Fig. 7.1
figure 1

Sketch of the cross section of a plate-like waveguide

Full size image

The point of departure for the derivation of the SAFE is the weak form of the equilibrium equations as discussed in Chap. 4 For the sake of clarity and simplicity, we only repeat the weak form of the mechanical equilibrium equations at this point

$$\displaystyle{ \int _{\varOmega }\mathbf{v}_{\mathrm{u}}^{T}\rho \ddot{\mathbf{u}}\mathrm{d}\varOmega +\int _{\varOmega }\left (\nabla _{\mathrm{ u}}\mathbf{v}_{\mathrm{u}}\right )^{T}\mathbf{C}\nabla _{\mathrm{ u}}\mathbf{u}\mathrm{d}\varOmega -\int _{\varOmega }\mathbf{v}_{\mathrm{u}}^{T}\mathbf{b}\mathrm{d}\varOmega -\int _{\varGamma _{\mathrm{ u}}}\mathbf{v}_{\mathrm{u}}^{T}\bar{\mathbf{t}}\mathrm{d}\varGamma = 0\;, }$$
(7.1)

with Ω and Γ denoting the computational domain and its boundary. The loading of the structure is given by b and t which are the vector of body forces and the surface traction vector, respectively. The overbar represents a prescribed value at the Neumann boundary. The mechanical displacement field is denoted by u and the test functions corresponding to the method of weighted residuals are v u. The material properties are given by C, the elasticity matrix, and by ρ, the mass density. In the framework of the SAFE we split the mechanical differential operator matrix ∇u into three parts to separate the derivatives with respect to the individual coordinates [45] as

$$\displaystyle{ \nabla _{\mathrm{u}} = \left [\begin{array}{*{10}c} \frac{\partial } {\partial x_{1}} & 0 & 0 & \frac{\partial } {\partial x_{2}} & 0 & \frac{\partial } {\partial x_{3}} \\ 0 & \frac{\partial } {\partial x_{2}} & 0 & \frac{\partial } {\partial x_{1}} & \frac{\partial } {\partial x_{3}} & 0 \\ 0 & 0 & \frac{\partial } {\partial x_{3}} & 0 & \frac{\partial } {\partial x_{2}} & \frac{\partial } {\partial x_{1}} \end{array} \right ]^{T} = \mathbf{L}_{ x_{1}} \frac{\partial } {\partial x_{1}}+\mathbf{L}_{x_{2}} \frac{\partial } {\partial x_{2}}+\mathbf{L}_{x_{3}} \frac{\partial } {\partial x_{3}}\;, }$$
(7.2)

with

$$\displaystyle{ \mathbf{L}_{x_{1}} = \left [\begin{array}{*{10}c} 1&0&0\\ 0 &0 &0 \\ 0&0&0\\ 0 &0 &0 \\ 0&0&1\\ 0 &1 &0 \end{array} \right ],\quad \mathbf{L}_{x_{2}} = \left [\begin{array}{*{10}c} 0&0&0\\ 0 &1 &0 \\ 0&0&0\\ 0 &0 &1 \\ 0&0&0\\ 1 &0 &0 \end{array} \right ],\quad \mathbf{L}_{x_{3}} = \left [\begin{array}{*{10}c} 0&0&0\\ 0 &0 &0 \\ 0&0&1\\ 0 &1 &0 \\ 1&0&0\\ 0 &0 &0 \end{array} \right ]\;. }$$

In the next step we discretize the displacement field in a similar fashion as known from the FEM. In the framework of the SAFE we assume that the displacement field is a harmonic function in x 1 and the cross section (x 2 − x 3 plane) of the plate is discretized by conventional two-dimensional finite elements. Therefore, we can express the displacement vector as [4]

$$\displaystyle{ \mathbf{u}(\mathbf{x},t) =\tilde{ \mathbf{N}}(x_{2},x_{3})\mathbf{U}_{\mathrm{e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\;, }$$
(7.3)

where \(j = \sqrt{-1}\) represents the imaginary unit and \(\tilde{\mathbf{N}}(x_{2},x_{3})\) is the two-dimensional basis function matrix. The wavenumber and the circular frequency are denoted by k and ω, respectively. Keep in mind that we use the same ansatz also for the test functions v u

$$\displaystyle{ \mathbf{v}_{\mathrm{u}}(\mathbf{x},t) =\tilde{ \mathbf{N}}(x_{2},x_{3})\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\;. }$$
(7.4)

The basis function matrix \(\tilde{\mathbf{N}}\) and the vector of nodal degrees of freedom U e have the following form

$$\displaystyle\begin{array}{rcl} \tilde{\mathbf{N}}& =& \left [\begin{array}{cccccccccccc} N^{\mathrm{N_{1}}} & & & N^{\mathrm{N_{2}}} & & & & & & N^{\mathrm{N_{n_{Node}}}} & & \\ & N^{\mathrm{N_{1}}} & & & N^{\mathrm{N_{2}}} & & & \ldots & & & N^{\mathrm{N_{n_{Node}}}} & \\ & & N^{\mathrm{N_{1}}} & & & N^{\mathrm{N_{2}}} & & & & & & N^{\mathrm{N_{n_{Node}}}} \end{array} \right ]\;,{}\end{array}$$
(7.5)
$$\displaystyle\begin{array}{rcl} \mathbf{U}_{\mathrm{e}} =\, \left [\begin{array}{cccccccccccc} u_{\mathrm{1}}^{\mathrm{N_{1}}} & u_{\mathrm{2}}^{\mathrm{N_{1}}} & u_{\mathrm{3}}^{\mathrm{N_{1}}} & u_{\mathrm{1}}^{\mathrm{N_{2}}} & u_{\mathrm{2}}^{\mathrm{N_{2}}} & u_{\mathrm{3}}^{\mathrm{N_{2}}} & \ldots & u_{\mathrm{1}}^{\mathrm{N_{n_{Node}}} } & u_{\mathrm{2}}^{\mathrm{N_{n_{Node}}} } & u_{\mathrm{3}}^{\mathrm{N_{n_{Node}}} } \end{array} \right ]^{T}\;,& &{}\end{array}$$
(7.6)

where n Node is the number of nodes per finite element. At this point we also introduce the linear strain–displacement relation

$$\displaystyle{ \boldsymbol{\varepsilon }_{\mathrm{u}} = \nabla _{\mathrm{u}}\mathbf{u}\;. }$$
(7.7)

If we now substitute Eqs. (7.2) and (7.3) into Eq. (7.7), we obtain [45]

$$\displaystyle{ \boldsymbol{\varepsilon }_{\mathrm{u}} = \left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{ u}}^{\#}\right )\mathbf{U}_{\mathrm{ e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\; }$$
(7.8)

with

$$\displaystyle\begin{array}{rcl} \mathbf{B}_{\mathrm{u}}^{{\ast}} = \mathbf{L}_{ x_{2}}\tilde{\mathbf{N}}_{,x_{2}} + \mathbf{L}_{x_{3}}\tilde{\mathbf{N}}_{,x_{3}}\;,& &{}\end{array}$$
(7.9)
$$\displaystyle\begin{array}{rcl} \mathbf{B}_{\mathrm{u}}^{\#} = \mathbf{L}_{ x_{1}}\tilde{\mathbf{N}}\;.& &{}\end{array}$$
(7.10)

The derivatives with respect to the variables x i are denoted by \((\cdot )_{,x_{i}}\). The semi-discrete equations of motion for one element can be derived from Eq. (7.1) and are given as

$$\displaystyle{ \begin{array}{ll} &\int _{\varOmega _{\mathrm{e}}}\left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\rho \; \frac{\partial ^{2}} {\partial t^{2}} \left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{U}_{\mathrm{e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]\mathrm{d}\varOmega + \\ &\int _{\varOmega _{\mathrm{e}}}\left [\left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{\#}\right )\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\mathbf{C}\left [\left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{\#}\right )\mathbf{U}_{\mathrm{e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]\mathrm{d}\varOmega - \\ &\int _{\varOmega _{\mathrm{e}}}\left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\mathbf{b}\mathrm{d}\varOmega - \\ &\int _{\varGamma _{\mathrm{u}}^{\mathrm{e}}}\left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\bar{\mathbf{t}}\mathrm{d}\varGamma = 0\;. \end{array} }$$
(7.11)

In the next steps we simplify the first two terms of Eq. (7.11) and derive the stiffness matrix K u and the mass matrix M u corresponding to the semi-analytical finite element method. The ansatz for the displacement field and the test functions has the inherent advantage that we can easily split the domain integration into two separate parts. First, we solve the one-dimensional integral with respect to x 1 and then we perform a domain integration over the x 2 − x 3 plane denoted by Ω × [4]. In order to transpose a complex matrix we need to compute the conjugate transpose matrix by negating the imaginary part. Therefore the first two terms in Eq. (7.11) are given by

$$\displaystyle\begin{array}{rcl} \begin{array}{ll} &\int _{\varOmega _{\mathrm{e}}}\left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\rho \; \frac{\partial ^{2}} {\partial t^{2}} \left [\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{U}_{\mathrm{e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]\mathrm{d}\varOmega = \\ &\int _{\varOmega _{\mathrm{e}}^{\times }}\int _{x_{1}} -\omega ^{2}\rho \left [\mathbf{V}_{\mathrm{ue}}^{T}\tilde{\mathbf{N}}(x_{2},x_{3})^{T}\tilde{\mathbf{N}}(x_{2},x_{3})\mathbf{U}_{\mathrm{e}}\right ]\mathrm{d}\,x_{1}\mathrm{d}\varOmega ^{\times } = \\ & -\omega ^{2}x_{1}\mathbf{V}_{\mathrm{ue}}^{T}\int _{\varOmega _{\mathrm{e}}^{\times }}\rho \left [\tilde{\mathbf{N}}(x_{2},x_{3})^{T}\tilde{\mathbf{N}}(x_{2},x_{3})\right ]\mathrm{d}\varOmega ^{\times }\mathbf{U}_{\mathrm{e}}\;,\end{array} & &{}\end{array}$$
(7.12)
$$\displaystyle\begin{array}{rcl} \begin{array}{ll} &\int _{\varOmega _{\mathrm{e}}}\left [\left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{\#}\right )\mathbf{V}_{\mathrm{ue}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]^{T}\mathbf{C}\left [\left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{\#}\right )\mathbf{U}_{\mathrm{e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\right ]\mathrm{d}\varOmega = \\ &\int _{\varOmega _{\mathrm{e}}^{\times }}\int _{x_{1}}\mathbf{V}_{\mathrm{ue}}^{T}\left [\mathbf{B}_{\mathrm{u}}^{{\ast}T} + jk\mathbf{B}_{\mathrm{u}}^{\#T}\right ]\mathbf{C}\left [\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{\#}\right ]\mathbf{U}_{\mathrm{e}}\mathrm{d}\,x_{1}\mathrm{d}\varOmega ^{\times } = \\ &x_{1}\mathbf{V}_{\mathrm{ue}}^{T}\int _{\varOmega _{\mathrm{e}}^{\times }}\left [\mathbf{B}_{\mathrm{u}}^{{\ast}T}\mathbf{C}\mathbf{B}_{\mathrm{u}}^{{\ast}} + jk\mathbf{B}_{\mathrm{u}}^{\#T}\mathbf{C}\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{u}}^{{\ast}T}\mathbf{C}\mathbf{B}_{\mathrm{u}}^{\#} + k^{2}\mathbf{B}_{\mathrm{u}}^{\#T}\mathbf{C}\mathbf{B}_{\mathrm{u}}^{\#}\right ]\mathrm{d}\varOmega ^{\times }\mathbf{U}_{\mathrm{e}}\;.\end{array}& &{}\end{array}$$
(7.13)

As mentioned before Eq. (7.11) represents the contribution of one finite element to the weak from. Therefore, we have to sum the contributions of all n el elements to obtain the semi-discrete (homogeneous) equations of motion for the system under investigation as

$$\displaystyle{ \sum \limits _{e=1}^{n_{\mathrm{el}} }\left [\mathbf{K}_{\mathrm{11u}}^{\mathrm{e}} + jk\mathbf{K}_{\mathrm{ 12u}}^{\mathrm{e}} + k^{2}\mathbf{K}_{\mathrm{ 22u}}^{\mathrm{e}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}^{\mathrm{e}}\right ]\mathbf{U}_{\mathrm{ e}} = \mathbf{0}\;, }$$
(7.14)

where K 11u e, K 12u e, K 22u e, and M u e denote the in-plane stiffness matrix, the coupling stiffness matrix, the out-of-plane stiffness matrix, and the mass matrix , respectively. Their definitions are given below

$$\displaystyle\begin{array}{rcl} \mathbf{K}_{\mathrm{11u}}^{\mathrm{e}} =\int _{\varOmega _{\mathrm{ e}}^{\times }}\mathbf{B}_{\mathrm{u}}^{{\ast}T}\mathbf{C}\mathbf{B}_{\mathrm{ u}}^{{\ast}}\;\mathrm{d}\varOmega ^{\times }\;,& &{}\end{array}$$
(7.15)
$$\displaystyle\begin{array}{rcl} \mathbf{K}_{\mathrm{12u}}^{\mathrm{e}} =\int _{\varOmega _{\mathrm{ e}}^{\times }}\mathbf{B}_{\mathrm{u}}^{\#T}\mathbf{C}\mathbf{B}_{\mathrm{ u}}^{{\ast}}\;\mathrm{d}\varOmega ^{\times }-\int _{\varOmega _{\mathrm{ e}}^{\times }}\mathbf{B}_{\mathrm{u}}^{{\ast}T}\mathbf{C}\mathbf{B}_{\mathrm{ u}}^{\#}\;\mathrm{d}\varOmega ^{\times }\;,& &{}\end{array}$$
(7.16)
$$\displaystyle\begin{array}{rcl} \mathbf{K}_{\mathrm{22u}}^{\mathrm{e}} =\int _{\varOmega _{\mathrm{ e}}^{\times }}\mathbf{B}_{\mathrm{u}}^{\#T}\mathbf{C}\mathbf{B}_{\mathrm{ u}}^{\#}\;\mathrm{d}\varOmega ^{\times },& &{}\end{array}$$
(7.17)
$$\displaystyle\begin{array}{rcl} \mathbf{M}_{\mathrm{u}}^{\mathrm{e}} =\int _{\varOmega _{\mathrm{ e}}^{\times }}\rho \tilde{\mathbf{N}}^{{\ast}T}\tilde{\mathbf{N}}\;\mathrm{d}\varOmega ^{\times }\;.& &{}\end{array}$$
(7.18)

Equation (7.14) constitutes the eigenvalue problem of the SAFE method which has to be solved to compute the propagation of elastic guided waves in infinite media. For further information on the SAFE method, the interested reader is referred to [22, 23, 26, 27, 32–37, 39, 40, 50] and the references cited therein.

1.3 Plate with Infinite Dimensions

Considering a plate with infinite dimensions in x 1- and x 2-directions we can further simplify the SAFE equations as there is no dependence of the displacement field on x 2 [45]. Therefore, only the thickness of the plate has to be discretized by one-dimensional finite elements. This is illustrated in Fig. 7.2. Due to this simplification the strain–displacement relation is given by

$$\displaystyle{ \boldsymbol{\varepsilon }_{\mathrm{u}} = \left (\mathbf{B}_{\mathrm{u}}^{{\ast}}- jk\mathbf{B}_{\mathrm{ u}}^{\#}\right )\mathbf{U}_{\mathrm{ e}}\mathrm{e}^{-j(kx_{1}-\omega t)}\; }$$
(7.19)

with

$$\displaystyle\begin{array}{rcl} \mathbf{B}_{\mathrm{u}}^{{\ast}} = \mathbf{L}_{ x_{3}}\tilde{\mathbf{N}}_{,x_{3}}\;,& &{}\end{array}$$
(7.20)
$$\displaystyle\begin{array}{rcl} \mathbf{B}_{\mathrm{u}}^{\#} = \mathbf{L}_{ x_{1}}\tilde{\mathbf{N}}\;.& &{}\end{array}$$
(7.21)

This also implies that the integration domain changes from dΩ × to d x 3. The integrals in Eqs. (7.15)– (7.18) can now be evaluated numerically by a standard one-dimensional Gaussian quadrature rule.

Fig. 7.2
figure 2

Sketch of the cross section of an infinite plate. The finite element discretization is indicated by its nodes (one-dimensional elements)

Full size image

In the context of the FEM it is advantageous to define the basis functions with respect to a reference domain [11, 28, 66]. Therefore, the finite elements need to be mapped from the reference frame to the global domain as discussed in Sect. 4.7. This procedure is exemplarily discussed for a 3-noded (quadratic) isoparametric finite element. The basis functions are [66]

$$\displaystyle\begin{array}{rcl} N_{\mathrm{u,3}}^{\mathrm{N_{1}} }(\xi _{1}) = \frac{1} {2}\xi _{1}(\xi _{1} - 1)\;,& &{}\end{array}$$
(7.22)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{u,3}}^{\mathrm{N_{2}} }(\xi _{1}) = (1 -\xi _{1}^{2})\;,& &{}\end{array}$$
(7.23)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{u,3}}^{\mathrm{N_{3}} }(\xi _{1}) = \frac{1} {2}\xi _{1}(\xi _{1} + 1)\;.& &{}\end{array}$$
(7.24)

For the infinite plate the domain integration over the cross section is simply given by

$$\displaystyle{ \int _{\varOmega _{\mathrm{e}}^{\times }}\ldots \mathrm{d}\varOmega =\int _{ \xi _{3}=-1}^{1}\ldots \det (\mathbf{J})\mathrm{d}\xi _{ 3}\;, }$$
(7.25)

where J denotes the Jacobian matrix of the mapping. In the one-dimensional case the evaluation of Eq. (4.59) reduces to a scalar value

$$\displaystyle{ J = \frac{\mathrm{d}\,x_{3}} {\mathrm{d}\xi _{1}} \;. }$$
(7.26)

Due to the isoparametric mapping concept discussed in Sect. 4.7.2 we can express x 3 within an element by a linear combination of the basis functions and the nodal coordinates X 3 i by

$$\displaystyle{ x_{3} =\mathrm{ Q}^{\mathrm{(e)}}(\xi _{ 1}) =\sum \limits _{ i=1}^{3}N_{\mathrm{ u,3}}^{\mathrm{N}_{i} }(\xi _{1})X_{3}^{i}\;. }$$
(7.27)

With the help of Eq. (7.27) we can compute the Jacobian matrix in terms of the nodal coordinates and the local coordinate ξ 1

$$\displaystyle{ J =\sum \limits _{ i=1}^{3}\frac{\mathrm{d}N_{\mathrm{u,3}}^{\mathrm{N_{i}}}(\xi _{ 1})} {\mathrm{d}\xi _{1}} X_{3}^{i} = \left [\begin{array}{*{10}c} \xi _{ 1} - 1&\;\;-2\xi _{1} & \;\;\xi _{1} + 1 \end{array} \right ]\left [\begin{array}{*{10}c} X_{3}^{1} \\ X_{3}^{2} \\ X_{3}^{3} \end{array} \right ]\;. }$$
(7.28)

1.4 Dispersion Curves for Undamped Media

When studying the propagation of elastic guided waves the knowledge of the group and phase velocity dispersion diagrams is essential. These curves provide information on the number of propagating modes for a specific combination of excitation frequency and plate thickness and naturally on the wave velocities of these wave modes. Therefore, we need to solve the SAFE eigenvalue problem formulated in Eq. (7.14). From the definition of the system matrices provided in Eqs. (7.15)– (7.18) we know that K 11u, K 22u, and M u are symmetric matrices, while K 12u is skew-symmetric [1].

Without loss of generality, we introduce a transformation matrix T to eliminate the imaginary unit in Eq. (7.14). To this end, we premultiply Eq. (7.14) by the transpose of T [4]

$$\displaystyle{ \mathbf{T}^{T}\left [\mathbf{K}_{\mathrm{ 11u}} + jk\mathbf{K}_{\mathrm{12u}} + k^{2}\mathbf{K}_{\mathrm{ 22u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}\right ]\mathbf{U} = \mathbf{0}\;. }$$
(7.29)

The transformation matrix is a (n dof × n dof)-diagonal matrix which is equal to 1 at the rows corresponding to displacements in x 2- and x 3-direction and equal to the imaginary unit j at those rows corresponding to displacements in x 1 direction

$$\displaystyle{ \mathbf{T} = \left [\begin{array}{*{10}c} j & & & & & & \\ &\quad 1& & &\mathbf{0}& &\\ & & \quad 1 & & &&\\ & & &\quad \ddots & && \\ & & & & \quad j & & \\ & &\mathbf{0}& & &\quad 1&\\ & & & & &&\quad 1 \end{array} \right ]\;. }$$
(7.30)

This matrix has some favorable properties, such as T T = T − and TT − = T − T = 1, where 1 denotes the identity matrix and (⋅ )− represents the conjugate transpose of a complex matrix. Therefore we can expand Eq. (7.29) resulting in

$$\displaystyle{ \mathbf{T}^{T}\left [\mathbf{K}_{\mathrm{ 11u}} + jk\mathbf{K}_{\mathrm{12u}} + k^{2}\mathbf{K}_{\mathrm{ 22u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}\right ]\mathbf{T}\mathbf{T}^{T}\mathbf{U} = \mathbf{0}\;. }$$
(7.31)

The proposed transformation does, however, not change the system matrices K 11u, K 22u, and M u which is due to the fact that in their definition provided in Eqs. (7.15), (7.17) and (7.18) the x 1 related terms do not interact with the x 2, x 3 related terms and consequently the following relations hold

$$\displaystyle\begin{array}{rcl} \mathbf{T}^{T}\mathbf{K}_{\mathrm{ 11u}}\mathbf{T} = \mathbf{K}_{\mathrm{11u}}\;,& &{}\end{array}$$
(7.32)
$$\displaystyle\begin{array}{rcl} \mathbf{T}^{T}\mathbf{K}_{\mathrm{ 22u}}\mathbf{T} = \mathbf{K}_{\mathrm{22u}}\;,& &{}\end{array}$$
(7.33)
$$\displaystyle\begin{array}{rcl} \mathbf{T}^{T}\mathbf{M}_{\mathrm{ u}}\mathbf{T} = \mathbf{M}_{\mathrm{u}}\;.& &{}\end{array}$$
(7.34)

Considering the coupling stiffness matrix K 12u the transformation has the desired effect that

$$\displaystyle{ \mathbf{T}^{T}\mathbf{K}_{\mathrm{ 12u}}\mathbf{T} = -j\hat{\mathbf{K}}_{\mathrm{12u}} }$$
(7.35)

where \(\hat{\mathbf{K}}_{\mathrm{12u}}\) is a symmetric matrix if we neglect material damping. In principle, the transformation can be thought of as a multiplication of \(u_{x_{1}}\) by the imaginary unit [4]. The final (real-symmetric) form of the SAFE eigenvalue problem is given as

$$\displaystyle{ \left [\mathbf{K}_{\mathrm{11u}} + k\hat{\mathbf{K}}_{\mathrm{12u}} + k^{2}\mathbf{K}_{\mathrm{ 22u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}\right ]\hat{\mathbf{U}} = \mathbf{0}\;, }$$
(7.36)

where the new nodal displacement vector is defined as

$$\displaystyle{ \hat{\mathbf{U}} = \mathbf{T}^{T}\mathbf{U}\;. }$$
(7.37)

If we now assign real values to k Eq. (7.36) constitutes a standard eigenvalue problem in ω(k); due to the fact that k is a real value all wave modes are propagating ones [1]. The number of Lamb wave modes that can be computed depends on the number of degrees of freedom n dof of the system. For each wavenumber k i we obtain n dof propagating modes for which we also determine the mode shape \(\hat{\mathbf{U}}_{i}\). Moreover, the phase velocity c p can be computed from the real part of the wavenumber and the angular frequency as

$$\displaystyle{ c_{\mathrm{p}} = \frac{\omega } {\mathfrak{R}\mathfrak{e}(k)}\,. }$$
(7.38)

On the other hand, if we are interested in the full spectrum of propagating and evanescent modes, we have to determine the unknown complex wavenumber k(ω) for a prescribed circular frequency ω. To this end, Eq. (7.36) is usually rewritten in the following form [4]

$$\displaystyle{ \left (\mathbf{A} - k\mathbf{B}\right )\mathbf{Q} = \mathbf{0}\;, }$$
(7.39)

where the matrices A and B are defined as

$$\displaystyle\begin{array}{rcl} \mathbf{A} = \left [\begin{array}{*{10}c} \mathbf{0} &\quad \mathbf{K}_{\mathrm{11u}} -\omega ^{2}\mathbf{M}_{\mathrm{u}} \\ \mathbf{K}_{\mathrm{11u}} -\omega ^{2}\mathbf{M}_{\mathrm{u}} & \quad \hat{\mathbf{K}}_{\mathrm{12u}} \end{array} \right ]\;,& &{}\end{array}$$
(7.40)
$$\displaystyle\begin{array}{rcl} \mathbf{B} = \left [\begin{array}{*{10}c} \mathbf{K}_{\mathrm{11u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}} & \quad \mathbf{0}\\ \mathbf{0} &\quad \mathbf{K} _{ \mathrm{22u}} \end{array} \right ]\;,& &{}\end{array}$$
(7.41)

and the vector Q is given as

$$\displaystyle{ \mathbf{Q} = \left [\begin{array}{*{10}c} \hat{\mathbf{U}}\\ k\hat{\mathbf{U}} \end{array} \right ]\;. }$$
(7.42)

Consequently, the size of the eigenvalue problem is doubled. The computed complex wavenumbers k = k Re + jk Im contain the phase velocity as the real part and their amplitude decay is described by the imaginary part [1]. The solution of Eq. (7.36) provides for each circular frequency ω i the 2n dof eigenvalues k i m and eigenvectors \(\hat{\mathbf{U}}_{i}\) (right eigenvector). We have to bear in mind that the formulation in Eq. (7.36) is preferred if only the propagating modes are considered [4].

1.4.1 Phase and Group Velocities of Guided Waves

After solving the eigenvalue problems in Eqs. (7.36) or (7.39) we can easily compute the phase velocity c p in terms of the real part of the complex wavenumber and the angular frequency, cf. Eq. (7.38). The group velocity c g can be determined if we compute the derivative of the phase velocity with respect to the wave vector k defined as

$$\displaystyle{ \mathbf{k} = k\mathbf{d}\;, }$$
(7.43)

where d is the propagation direction vector. The final result to compute the group velocity is [17]

$$\displaystyle{ c_{\mathrm{g}} = \frac{\partial \omega } {\partial k} = \frac{c_{\mathrm{p}}^{2}} {c_{\mathrm{p}} -\omega \frac{\mathrm{d}c_{\mathrm{p}}} {\mathrm{d}\omega } }\;. }$$
(7.44)

In the computation of the group velocity we follow the methodology published in [10, 21]. The procedure starts by evaluating the derivative of Eq. (7.36)

$$\displaystyle{ \frac{\partial } {\partial k}\left (\left [\mathbf{K}_{\mathrm{11u}} + k\hat{\mathbf{K}}_{\mathrm{12u}} + k^{2}\mathbf{K}_{\mathrm{ 22u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}\right ]\hat{\mathbf{U}}\right ) = \mathbf{0}\;. }$$
(7.45)

In the next step, we premultiply Eq. (7.45) with the transpose of the so-called left eigenvector \(\hat{\mathbf{U}}_{\mathrm{L}}^{T}\)

$$\displaystyle{ \hat{\mathbf{U}}_{\mathrm{L}}^{T}\left [\hat{\mathbf{K}}_{\mathrm{ 12u}} + 2k\mathbf{K}_{\mathrm{22u}} - 2\omega \frac{\partial \omega } {\partial k}\mathbf{M}_{\mathrm{u}}\right ]\hat{\mathbf{U}} = \mathbf{0}\;. }$$
(7.46)

The group velocity can now be computed as

$$\displaystyle{ c_{\mathrm{g}} = \frac{\hat{\mathbf{U}}_{\mathrm{L}}^{T}\left (\hat{\mathbf{K}}_{\mathrm{12u}} + 2k\mathbf{K}_{\mathrm{22u}}\right )\hat{\mathbf{U}}} {2\omega \hat{\mathbf{U}}_{\mathrm{L}}^{T}\mathbf{M}_{\mathrm{u}}\hat{\mathbf{U}}} \;. }$$
(7.47)

The left eigenvector is determined by solving the following eigenvalue problem

$$\displaystyle{ \hat{\mathbf{U}}_{\mathrm{L}}^{T}\left [\mathbf{K}_{\mathrm{ 11u}} + k\hat{\mathbf{K}}_{\mathrm{12u}} + k^{2}\mathbf{K}_{\mathrm{ 22u}} -\omega ^{2}\mathbf{M}_{\mathrm{ u}}\right ] = \mathbf{0}\;. }$$
(7.48)

From the result we can conclude that the left eigenvector \(\hat{\mathbf{U}}_{\mathrm{L}}\) is the complex conjugate of the right eigenvector \(\hat{\mathbf{U}}\)

$$\displaystyle{ \hat{\mathbf{U}}_{\mathrm{L}} =\hat{ \mathbf{U}}^{-}\;. }$$
(7.49)

1.4.2 Verification

In this paragraph we demonstrate the performance of the SAFE method by computing the dispersion diagrams for several plates made of different materials. The first simple benchmark problem constitutes the solution of the eigenvalue problem Eq. (7.36) for an aluminum plate (Young’s modulus E = 70 GPa, Poisson’s ratio ν = 0. 33, mass density ρ = 2700 kg/m3) . We deploy ten quadratic (one-dimensional) finite elements to discretize the thickness of the plate. In Fig. 7.3 we notice an excellent agreement with the analytical solution.

Fig. 7.3
figure 3

Dispersion curves for an aluminum plate. Comparison between SAFE results and analytical solutions

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The accuracy of the computed results naturally depends on the number of finite elements and the basis functions used. Analogous to the FEM we can always increase the accuracy by a h- or p-refinement. In this chapter we limit our analysis to quadratic basis function. However, the implementation of higher order polynomial functions introduced in Sect. 6.1 is straightforward.

Figure 7.4 illustrates the effect of a h-extension on the phase velocity. We merely assess the accuracy of the first four Lamb wave modes as this is a typical frequency range that is used in literature for damage detection purposes. The results highlight that three quadratic elements are sufficient to compute highly accurate dispersion curves with a relative error of well below 1% [1]. The number of elements over the thickness of the plate has to be increased if a higher frequency range is of interest.

Fig. 7.4
figure 4

Phase velocity dispersion curves for an aluminum plate for different finite element discretizations

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As a second example we investigate a composite plate made of 16 unidirectional (UD) layers with the following lay up [0∘∕45∘∕90∘∕ − 45∘]s2. The plate has an overall thickness of 3. 2 mm and the material properties for a UD layer are compiled in Table 7.1. Each ply is discretized with one quadratic finite element. We observe an excellent agreement with the results published in [14], cf. Fig. 7.5. We have to keep in mind that for a CFRP plate the Lamb and the SH wave modes are often coupled.

Table 7.1 Material properties of the UD layer at 0∘ [14]
Full size table
Fig. 7.5
figure 5

Dispersion curves for a 16-ply CFRP plate

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1.5 Interaction of Guided Waves with Perturbations

1.5.1 General approach

If we want to simulate arbitrary boundary conditions or geometries of plate edges, we have to couple the SAFE method with other numerical approaches . One possibility is to deploy the FEM to account for general perturbations of the plate-like structure [1–3]. The FEM is known for its versatility in modeling geometrically complex systems and therefore we define the boundary conditions, the plate edge, and/or other perturbations in the FE domain as shown in Figs. 7.6 and 7.7, respectively.

Fig. 7.6
figure 6

Illustration of the coupling between SAFE and FEM: Reflection of the wave at the edge of the plate

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Fig. 7.7
figure 7

Illustration of the coupling between SAFE and FEM: Interaction of the wave with a crack

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The continuity between the SAFE and the FE regions is ensured by using a conformal mesh at the coupling interface(s). In the SAFE method the displacements and the reaction forces at the interface have to be computed. If we now consider an incident wave mode interacting with the edge of a plate, the displacement vector of the reflected wave mode U reflected can be approximated by a modal sum of a finite number of modes n r [13]

$$\displaystyle{ \mathbf{U}^{\mathrm{reflected}} =\sum \limits _{ i=1}^{n_{\mathrm{r}} }b_{i}\mathbf{U}_{i}\mathrm{e}^{-jk_{i}x_{1} }\;, }$$
(7.50)

where b i denotes the amplitude of the ith wave mode and the vector U i is the ith eigenvector computed from Eq. (7.14) corresponding to the wavenumber k i . The approximation of the force is written analogously as

$$\displaystyle{ \mathbf{F}^{\mathrm{reflected}} =\sum \limits _{ i=1}^{n_{\mathrm{r}} }b_{i}\boldsymbol{\psi }_{i}\mathrm{e}^{-jk_{i}x_{1} }\;, }$$
(7.51)

where ψ i is the force eigenvector which represents the nodal force components due to stresses \(\sigma _{x_{1}}\) and \(\sigma _{x_{3}}\) [35]. According to [13] the force eigenvector is given as

$$\displaystyle{ \boldsymbol{\psi }_{i} = \left (\,jk_{i}\mathbf{K}_{\mathrm{22u}} + \mathbf{K}_{\mathrm{12u}}\right )\mathbf{U}_{i}\;. }$$
(7.52)

After having computed the displacement and force vectors due to the reflected wave we still need to determine the following quantities for the incident wave

$$\displaystyle\begin{array}{rcl} \mathbf{U}^{\mathrm{incident}} = a_{ i}\mathbf{U}_{i}^{(-)}\;,& &{}\end{array}$$
(7.53)
$$\displaystyle\begin{array}{rcl} \mathbf{F}^{\mathrm{incident}} = a_{ i}\boldsymbol{\psi }_{i}^{(-)}\;,& &{}\end{array}$$
(7.54)

where a i denotes the amplitude of the ith incident mode. The vectors U i (−) and ψ i (−) are obtained from their corresponding vectors U i and ψ i by negating each component related to the x 1-direction [1]. The overall displacement and force vectors are now computed as

$$\displaystyle\begin{array}{rcl} \mathbf{U} = \mathbf{U}^{\mathrm{incident}} + \mathbf{U}^{\mathrm{reflected}}\;,& &{}\end{array}$$
(7.55)
$$\displaystyle\begin{array}{rcl} \mathbf{F} = \mathbf{F}^{\mathrm{incident}} + \mathbf{F}^{\mathrm{reflected}}\;.& &{}\end{array}$$
(7.56)

These quantities can be included in a FEM software to compute the reflection coefficient C R ij for the different propagating wave modes [1, 3, 13]. According to Karunasena et al. [35] we define the reflection coefficient C R ij of the ith reflected mode due to the jth incident mode as

$$\displaystyle{ C_{\mathrm{R}}^{ij} = \frac{b_{i}} {a_{j}}\,.}$$
(7.57)

1.5.2 Verification

Benchmark examples to demonstrate the applicability of the discussed procedure are studied in the remainder of this section. The finite element stiffness matrix K u FE and mass matrix M u FE are computed by means of the software package Abaqus®;. At the boundary between the FE and SAFE regions a conformal discretization is generated. The plate under investigation is made of aluminum (Young’s modulus E = 70 GPa, Poisson’s ratio ν = 0. 33, mass density ρ = 2700 kg/m3) with a thickness of t = 1 mm. The maximal frequency of interest is 4. 5 MHz because in this frequency range only the first four higher order Lamb wave modes exist. In the numerical simulations we only excite the S 0-mode and observe its interaction with the boundary.

Figure 7.8 illustrates the wave reflection behavior at a symmetric plate edge. The term symmetric refers to the midplane of the plate. The infinite plate region is modeled with the help of SAFE, while the edge is discretized using the FEM. From the simulation we determine the edge reflection coefficient C R ∈ [0, 1] for the symmetric and antisymmetric modes. Here, one important advantage of the SAFE method can be exploited as each mode can be investigated separately.

Fig. 7.8
figure 8

SAFE-FE: Reflection of the S 0-mode at a symmetric plate edge. (a) Numerical model. (b) Reflection coefficient C R for the symmetric and antisymmetric guided wave modes

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As expected only the symmetric mode is reflected and no mode conversion takes place. In Fig. 7.8 we clearly observe the cut-off frequencies of the higher order symmetric Lamb wave modes. Note that the sum of the reflection coefficients of all propagating modes is equal to one.

Figure 7.9 illustrates the wave reflection behavior at an asymmetric plate edge . The plate edge is inclined at an angle of 45∘. Because of the asymmetry of the edge geometry mode conversion from the symmetric S 0-mode to the antisymmetric modes A i is observed. The reflection coefficient illustrates the existence of both symmetric and antisymmetric wave modes after the interaction with the boundary of the plate.

Fig. 7.9
figure 9

SAFE-FE: Reflection of the S 0-mode at an asymmetric plate edge. (a) Numerical model. (b) Reflection coefficient C R for the symmetric and antisymmetric guided wave modes.

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1.6 Force Response Analysis

1.6.1 General approach

In Sects. 7.1.4 and 7.1.5 we have computed the wave motion in an infinite plate and additionally we have included the interaction of the waves with perturbations in the platelike geomerty. This simulations have been executed without applying forces to the structure. In this section we, therefore, focus on the force response analysis. To this end, we include the effect of point forces. This is the most important case as all distributed forces can be decomposed into a set of (energetically equivalent) point forces in the framework of the FEM.

In this section we only discuss the two-dimensional implementation of the force response analysis. To this end, we follow the approach discussed in [9, 42, 44]. We consider the case of an arbitrarily distributed force f(t) as illustrated in Fig. 7.10. The frequency components of the external force are computed using the Fourier transform

$$\displaystyle{ \check{\mathbf{f}}(\omega ) =\int _{ -\infty }^{\infty }\mathbf{f}(t)\mathrm{e}^{-j\omega t}\mathrm{d}t\;. }$$
(7.58)
Fig. 7.10
figure 10

Sketch of the SAFE model for the force response analysis

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The homogeneous equation (7.39) can now be complemented by the force term

$$\displaystyle{ \left [\mathbf{A} - k\mathbf{B}(\omega )\right ]\mathbf{Q} =\check{ \mathbf{F}}(\omega )\;, }$$
(7.59)

with

$$\displaystyle{ \check{\mathbf{F}}(\omega ) = \left [\begin{array}{*{10}c} \check{\mathbf{f}}(\omega )\\ \mathbf{0} \end{array} \right ]\;. }$$
(7.60)

The solution of Eq. (7.59) can be expressed in terms of an eigenvector expansion [51]

$$\displaystyle{ \hat{\mathbf{U}}(k,\omega ) =\sum \limits _{ i=i}^{4n_{\mathrm{Node}} } \frac{1} {k_{i} - k}\;\mathbf{Q}_{i}\; \frac{\mathbf{Q}_{i}^{T}\check{\mathbf{F}}} {\mathbf{Q}_{i}^{T}\mathbf{B}\mathbf{Q}_{i}}\;. }$$
(7.61)

If n Node is the number of nodes corresponding to the finite element discretization of the cross section of the structure, we can compute 4n Node eigenvalue–eigenvector pairs. To determine the results in the spatio-temporal domain we additionally need to define the spatial as well as the inverse Fourier transforms [9, 42, 44] as

$$\displaystyle\begin{array}{rcl} \hat{\mathbf{U}}(k,\omega ) =\int _{ -\infty }^{\infty }\hat{\mathbf{U}}(x,\omega )\mathrm{e}^{\,jkx_{1} }\mathrm{d}\,x_{1}\;,& &{}\end{array}$$
(7.62)
$$\displaystyle\begin{array}{rcl} \hat{\mathbf{U}}(x,\omega ) = \frac{1} {2\pi }\int _{-\infty }^{\infty }\hat{\mathbf{U}}(k,\omega )\mathrm{e}^{-jkx_{1} }\mathrm{d}k\;.& &{}\end{array}$$
(7.63)

Accordingly, we obtain the solution in the x 1-domain by applying the inverse Fourier transform to Eq. (7.61) as

$$\displaystyle{ \hat{\mathbf{U}}(x_{1},\omega ) = \frac{1} {2\pi }\sum _{i=i}^{4n_{\mathrm{Node}} }\int _{-\infty }^{\infty } \frac{1} {k_{i} - k}\;\mathbf{Q}_{i}\; \frac{\mathbf{Q}_{i}^{T}\check{\mathbf{F}}} {\mathbf{Q}_{i}^{T}\mathbf{B}\mathbf{Q}_{i}}\mathrm{e}^{-jk_{i}x_{1} }\mathrm{d}k\;. }$$
(7.64)

Equation (7.64) is now solved with the help of Cauchy’s theorem of residues [42]. The solution depends on the location of the measurement point x 1M with respect to the excitation point x 1f, cf. Fig. 7.10. We consider the case where x 1M > x 1f

$$\displaystyle{ \hat{\mathbf{U}}(x_{1},\omega )\vert _{x_{\mathrm{1M}}>x_{\mathrm{1f}}} = -j\sum \limits _{i=i}^{2n_{\mathrm{Node}} }\mathbf{Q}_{i}\; \frac{\mathbf{Q}_{i}^{T}\check{\mathbf{F}}} {\mathbf{Q}_{i}^{T}\mathbf{B}\mathbf{Q}_{i}}\mathrm{e}^{-jk_{i}\breve{x}_{1} }\;, }$$
(7.65)

with

$$\displaystyle{ \breve{x}_{1} = x_{\mathrm{1M}} - x_{\mathrm{1f}}\;. }$$
(7.66)

Due to the requirement that the displacements should be bounded, we only sum over those modes that have real wavenumbers or complex wavenumbers with a positive imaginary part. The time-domain response is then computed by using the inverse Fourier transform as [1]

$$\displaystyle{ \hat{\mathbf{U}}(x_{1},t) = \frac{1} {2\pi }\int _{-\infty }^{\infty }\hat{\mathbf{U}}(x_{ 1},\omega )\mathrm{e}^{\,j\omega t}\mathrm{d}\omega \;. }$$
(7.67)

The extension to three-dimensional systems is discussed in [1, 42]. There, also an alternative approach, based on a convolution integral, is explained. Furthermore, an extension to account for edge reflections is also possible [1].

1.6.2 Verification

The excitation of the guided waves is achieved by means of a perfectly bonded thin actuator. Instead of modeling the transducer itself we apply the simplified force model developed in [16, 18]. Therefore, the actuator is modeled as two point forces acting in opposite directions at the end of the actuator edges.

The time response simulations are performed for a 1 mm thick aluminum plate (material properties: Young’s modulus E = 70 GPa, Poisson’s ratio ν = 0. 33, mass density ρ = 2700 kg/m3) with a 6 mm long actuator attached on the top surface, as shown in Fig. 7.11. For the current example we choose an excitation frequency of f ex = 250 kHz for the five cycle sine burst signal given in Eq. (6.37). At this frequency only the two fundamental Lamb wave modes are present in the plate. A comparison of the displacement history at the measurement point P M, which is located at a distance of 80 mm from the excitation source is obtained by using the SAFE method and Abaqus®; and shows an excellent agreement.

Fig. 7.11
figure 11

Force response analysis: Two-dimensional model of a perfectly bonded actuator. (a) Numerical model. (b) In-plane (u 1) and out-of-plane (u 3) displacements at the measurement point P M

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1.7 Summary

The SAFE method combines the versatility of FE-based approaches with the advantages of purely analytical solutions. Using this method the material properties can vary in the thickness direction of the plate, while infinite dimensions are assumed otherwise. To this end, the thickness is discretized using a conventional FE ansatz. Regarding the wave propagation direction we deploy complex-valued exponential functions. For plane waveguides, only one-dimensional (1D) elements are needed. Each material layer in the plate is represented by at least one 1D element. 2D elements, however, are needed for modeling complex, 3D waveguides. The SAFE method is formulated in the frequency domain similar to analytical approaches. The computational effort is significantly reduced compared to the fully three-dimensional FEM due to the fact that the dimensionality of the problem is reduced. However, if we want to recover the displacement field in the time-domain, an inverse Fourier transform has to be evaluated. One main advantage shared by both analytical and semi-analytical methods is that each mode can be considered individually. This allows a detailed analysis of the wave propagation behavior, since specific modes of interest can be considered separately in further stages of the analysis. Generally speaking these types of methods are applicable to [63]:

  1. 1.

    The investigation of the behavior of a single mode.

  2. 2.

    The efficient computation of dispersion diagrams.

  3. 3.

    The computation of the displacement fields if only a small area is of interest.

2 Coupling of Analytical Solutions and the Spectral Element Method in the Frequency Domain

The content of this section is primarily based on Dr. Vivar-Perez’ research. The results of his investigations are published in a PhD thesis [58] and in two peer-reviewed journal articles [59, 60].

2.1 Motivation

The application of purely analytical methods is very convenient when dealing with wave propagation problems in large homogeneous structures. From a computational point of view, the computation of analytical solutions is rather inexpensive. On the other hand, analytical methods are only suitable for certain geometries and they are derived mostly in the frequency domain. Therefore, many researchers favor the use of purely numerical methods, such as the finite element method (FEM) to deal with more general problems. The main advantage of the FEM lies in its flexibility to model arbitrarily shaped domains and to obtain solutions in the time-domain [58]. However, they are numerically more expensive and involve a higher number of degrees of freedom in comparison to the same computation using analytical methods. Consequently, we promote the development of hybrid methods that only deploy analytical approaches in large areas of the model that are unperturbed and suitable for this purpose. Therefore, only specific details that are of interest to the analyst need to be resolved by numerical approaches.

To attain our goal we seize ideas that were proposed by Karmazin [30, 31] and Glushkov et al. [19, 20]. They have analyzed the wave propagation in a plate in the frequency domain and obtained the solution to the system of partial differential equations in terms of the fundamental solution or Green’s tensor. To this end, Cauchy’s theorem of residues is applied. We then derive integral expressions which can be used to compute the response of the plate due to an arbitrary distribution of loads. This approach can naturally also be used to describe wave reflections at boundaries and defects. Similar ideas have already been presented in Sect. 7.1 in the context of the SAFE method.

The novelty of our approach lies in the fact that instead of using standard FEM to model the perturbations of the plate [6, 25] we deploy the spectral element method (SEM) which has a higher order of accuracy [5, 12, 56]. Furthermore, we determine the bonding conditions between the numerical model of the perturbation and the analytical model of the plate in the frequency domain by means of an analytical ansatz. With the help of quadrature formulae and the SEM we discretize the set of coupling equations and derive semi-analytical expressions for the propagation of guided waves. To this end, we exploit the concept of the dynamic reaction or response matrix of the plate with exact wavenumbers. The inherent advantage of using exact wavenumbers instead of approximated ones is to be seen in the increased accuracy of the final results. Additionally, no discretization through the thickness of the plate has been used in contrast to the works presented by Loveday [42], Ahmad [1], or Morvan et al. [47].

2.2 Definition of the Problem

In the following we derive the coupling equations to account for the bonding of a piezoelectric transducer to an isotropic (infinite) plate. Accordingly, the transducer and the plate share a common surface denoted by Γ C. A schematic representation of the problem under consideration is shown in Fig. 7.12.

Fig. 7.12
figure 12

Schematic representation and reference coordinate system of a piezoelectric patch bonded to a plate. The piezoelectric patch occupies a volume Ω piezo and shares a common interface Γ C with the plate

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The behavior of the piezoelectric transducer and of the plate are analyzed separately. To couple both systems we apply specific boundary conditions on the common interface Γ C. To this end, we first compute the displacements of the infinite plate due to mechanical loads acting on Γ C. Second, we analyze the mechanical displacements and the electric potential of the piezoelectric transducer due to certain boundary conditions applied on Γ C. These conditions describe the influence of the reaction forces exerted by the plate on the transducer. The basic idea is illustrated in Fig. 7.13.

Fig. 7.13
figure 13

Separate analyses of the dynamic response of the plate and the piezoelectric transducer are executed. The influence of the transducer on the plate is modeled as a transient load F piezo acting on Γ C. To account for the coupling of both systems specific boundary conditions are applied at the interface representing the reaction forces of the plate acting on the transducer

Full size image

In Sect. 7.2.3 we discuss how to analytically compute the displacements of an infinite plate due to an arbitrary distribution of loads. To this end, we consider the governing equations and boundary conditions presented in Sect. 4.1 and transform them into the frequency domain. Thereafter, we derive appropriate boundary conditions that need to be applied on the common interface Γ C, see Sect. 7.2.4.

2.3 Analytical Solution to the Wave Propagation Problem in Isotropic Plates

For the sake of clarity and completeness, we repeat Navier’s equation for a three-dimensional isotropic body

$$\displaystyle{ \left (\lambda +\mu \right )\nabla \nabla \cdot \mathbf{u} +\mu \varDelta \mathbf{u} =\rho \ddot{ \mathbf{u}}\;. }$$
(7.68)

The coefficients λ and μ denote Lame’s constants and ρ is the mass density. The displacement vector u is a function of the position vector x and the time t. The differential operators ∇ and Δ represent spatial derivatives according to the Nabla operator and Laplace’s operator, while an overdot \(\dot{(\cdot )}\) denotes a temporal derivative. In the following we assume that the plate has infinite dimensions in the x 1 − x 2-plane and regarding the x 3-direction it is bounded by two parallel surfaces at x 3 = ±d∕2. We account for the influence of external loads F ext (applied to the upper surface) by introducing Neumann boundary conditions

$$\displaystyle\begin{array}{rcl} \left [\mu \left (\nabla u_{3} + \mathbf{u}_{,3}\right ) +\lambda \nabla \cdot \mathbf{u}\mathbf{e}_{3}\right ]_{x_{3}=+d/2} = \mathbf{F_{\mathrm{ext}}}\;,& &{}\end{array}$$
(7.69)
$$\displaystyle\begin{array}{rcl} \left [\mu \left (\nabla u_{3} + \mathbf{u}_{,3}\right ) +\lambda \nabla \cdot \mathbf{u}\mathbf{e}_{3}\right ]_{x_{3}=-d/2} = \mathbf{0}\;,& &{}\end{array}$$
(7.70)

where e i stands for the unit normal vector in x i -direction and the notation (⋅ ),i represents the partial derivative with respect to x i .

2.3.1 Analytical Approach

As mentioned before we consider the solution to Navier’s equation in the frequency domain. Therefore, we have to apply the Fourier transform to the displacement vector and the load vector

$$\displaystyle\begin{array}{rcl} \check{\mathbf{u}}(\bar{\mathbf{x}},x_{3},\omega ) =\int _{ -\infty }^{\infty }\mathbf{u}(\bar{\mathbf{x}},x_{ 3},t)\mathrm{e}^{\,j\omega t}\mathrm{d}t\;,& &{}\end{array}$$
(7.71)
$$\displaystyle\begin{array}{rcl} \check{\mathbf{F}}_{\mathrm{ext}}(\bar{\mathbf{x}},\omega ) =\int _{ -\infty }^{\infty }\mathbf{F}_{\mathrm{ ext}}(\bar{\mathbf{x}},t)\mathrm{e}^{\,j\omega t}\mathrm{d}t\;.& &{}\end{array}$$
(7.72)

For reasons that will become apparent in the course of this section we divide the position vector into two parts that correspond to the in-plane components \(\bar{\mathbf{x}}\) and an out-of-plane component x 3 with respect to the infinite plate

$$\displaystyle\begin{array}{rcl} \bar{\mathbf{x}} = x_{1}\mathbf{e}_{1} + x_{2}\mathbf{e}_{2}\;,& &{}\end{array}$$
(7.73)
$$\displaystyle\begin{array}{rcl} \mathbf{x} =\bar{ \mathbf{x}} + x_{3}\mathbf{e}_{3}\;.& &{}\end{array}$$
(7.74)

Now the governing equations (7.68) and the boundary conditions (7.69) and (7.70) can be expressed in terms of the circular frequency ω

$$\displaystyle{ \left (\lambda +\mu \right )\nabla \nabla \cdot \check{\mathbf{u}} +\mu \varDelta \check{ \mathbf{u}} = -\rho \omega ^{2}\check{\mathbf{u}}\;. }$$
(7.75)

We have to bear in mind that any temporal derivation is replaced by a multiplication with jω when we operate in the frequency domain. Here, \(j = \sqrt{-1}\) denotes the imaginary unit. The boundary conditions are basically unchanged and are given as

$$\displaystyle\begin{array}{rcl} \left [\mu \left (\nabla \check{u}_{3} +\check{ \mathbf{u}}_{,3}\right ) +\lambda \nabla \cdot \check{\mathbf{u}}\mathbf{e}_{3}\right ]_{x_{3}=+d/2} =\check{ \mathbf{F}}_{\mathrm{ext}}\;,& &{}\end{array}$$
(7.76)
$$\displaystyle\begin{array}{rcl} \left [\mu \left (\nabla \check{u}_{3} +\check{ \mathbf{u}}_{,3}\right ) +\lambda \nabla \cdot \check{\mathbf{u}}\mathbf{e}_{3}\right ]_{x_{3}=-d/2} = \mathbf{0}\;.& &{}\end{array}$$
(7.77)

The solution to Eq. (7.75) satisfying the boundary conditions Eqs. (7.76) and (7.77) can be found in terms of the external load \(\check{\mathbf{F}}_{\mathrm{ext}}\) using the concept of Green’s tensor also known as response tensor

$$\displaystyle{ \check{\mathbf{u}}(\bar{\mathbf{x}},x_{3},\omega ) =\int _{\mathbb{R}^{2}}\check{\mathbf{E}}(\bar{\mathbf{x}} -\bar{\mathbf{x}}^{{\prime}},x_{ 3},\omega ) \cdot \check{\mathbf{F}}_{\mathrm{ext}}(\bar{\mathbf{x}}^{{\prime}},\omega )\mathrm{d}\bar{\mathbf{x}}^{{\prime}}\;, }$$
(7.78)

where \(\bar{\mathbf{x}}^{{\prime}}\) is the in-plane position vector of the applied load. Since it is possible to derive a closed-form analytical expression for the frequency-dependent response tensor \(\check{\mathbf{E}}\), we can also solve Eq. (7.78) for the displacement vector. A detailed derivation is found in [58] and similar procedures have also been published in [30, 31, 61]. In the remainder of this section we sketch the solution algorithm for the two-dimensional problem.

2.3.2 Two-Dimensional Problem

The statement of the problem is simplified by considering a distribution of loads that is only dependent on the spatial variable x ξ as illustrated in Fig. 7.14. The rotated coordinate system (x ξ , x ϕ , x 3) is obtained by a counterclockwise rotation about the x 3-axis. The rotation angle ϕ depends on the orientation of the distributed load. Therefore, all dependencies on the spatial variable x ϕ can be neglected and Eq. (7.78) is given as

$$\displaystyle{ \check{\mathbf{u}}(\bar{\mathbf{x}},x_{3},\omega ) =\int _{\mathbb{R}^{2}}\check{\mathbf{E}}(x_{\xi } - x_{\xi }^{{\prime}},x_{ 3},\omega ) \cdot \check{\mathbf{F}}_{\mathrm{ext}}(x_{\xi }^{{\prime}},\omega )\mathrm{d}\,x_{\xi }^{{\prime}}\;. }$$
(7.79)
Fig. 7.14
figure 14

Distribution of a line load and rotation of the coordinate system

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According to Vivar-Perez [58] Green’s tensor can be determined under these conditions using the following expression

$$\displaystyle{ \begin{array}{ll} \check{\mathbf{E}}(x_{\xi },x_{3},\omega ) =\sum \limits _{ n=0}^{\infty }&\left [\mathbf{E}^{\mathrm{A}}(x_{3},k_{\mathrm{n}}^{\mathrm{\,A}},\omega )\mathrm{e}^{\,jk_{\mathrm{n}}^{\mathrm{\,A}}x_{\xi } } + \mathbf{E}^{\mathrm{S}}(x_{3},k_{\mathrm{n}}^{\mathrm{S}},\omega )\mathrm{e}^{\,jk_{\mathrm{n}}^{\mathrm{S}}x_{\xi } }\right. \\ + &\;\;\left.\mathbf{E}^{\mathrm{a}}(x_{3},k_{\mathrm{n}}^{\mathrm{a}},\omega )\mathrm{e}^{\,jk_{\mathrm{n}}^{\mathrm{a}}x_{\xi } } + \mathbf{E}^{\mathrm{s}}(x_{3},k_{\mathrm{n}}^{\mathrm{s}},\omega )\mathrm{e}^{\,jk_{\mathrm{n}}^{\mathrm{s}}x_{\xi } }\right ]\;,\end{array} }$$
(7.80)

where E A, E S, E a, and E s denote the response matrices for the antisymmetric and symmetric Lamb wave (capital letter) and shear horizontal (SH) modes (lower case letter), respectively. These quantities depend on the material properties, the excitation frequency, the geometry of the structure, and the wavenumbers k n  A, k n S, k n a and k n s. It is important to notice that we are able to pre-compute all these parameters [58]. The two wavenumbers k n  A and k n S are solutions to the Rayleigh-Lamb dispersion equations for the symmetric and antisymmetric modes

$$\displaystyle\begin{array}{rcl} 4k^{2}pq\cos \frac{pd} {2} \sin \frac{qd} {2} + [k^{2} - q^{2}]^{2}\sin \frac{pd} {2} \cos \frac{qd} {2} = 0\;,& &{}\end{array}$$
(7.81)
$$\displaystyle\begin{array}{rcl} 4k^{2}pq\sin \frac{pd} {2} \cos \frac{qd} {2} + [k^{2} - q^{2}]^{2}\cos \frac{pd} {2} \sin \frac{qd} {2} = 0\;.& &{}\end{array}$$
(7.82)

The two wavenumbers k n a and k n s, on the other side, are solutions for the shear horizontal modes which can be computed explicitly using

$$\displaystyle\begin{array}{rcl} k_{\mathrm{n}}^{\mathrm{a}} = \sqrt{ \frac{\omega ^{2 } } {c_{2}^{2}} -\frac{(2n + 1)^{2}\pi ^{2}} {d^{2}}} \;,& &{}\end{array}$$
(7.83)
$$\displaystyle\begin{array}{rcl} k_{\mathrm{n}}^{\mathrm{s}} = \sqrt{ \frac{\omega ^{2 } } {c_{2}^{2}} -\frac{4n^{2}\pi ^{2}} {d^{2}}} \;,& &{}\end{array}$$
(7.84)

where p, q, c 1, and c 2 are defined as

$$\displaystyle\begin{array}{rcl} p = \sqrt{ \frac{\omega ^{2 } } {c_{1}^{2}} - k^{2}}\;,& &{}\end{array}$$
(7.85)
$$\displaystyle\begin{array}{rcl} q = \sqrt{ \frac{\omega ^{2 } } {c_{2}^{2}} - k^{2}}\;,& &{}\end{array}$$
(7.86)
$$\displaystyle\begin{array}{rcl} c_{1} = \sqrt{\frac{\lambda +\mu } {\rho }} \;,& &{}\end{array}$$
(7.87)
$$\displaystyle\begin{array}{rcl} c_{2} = \sqrt{\frac{\mu } {\rho }}\;.& &{}\end{array}$$
(7.88)

Equations (7.81) and (7.82) are implicit functions of the wavenumber and the circular frequency. These equations are numerically solved by means of tracking the individual wavenumber branches for each wave mode and each positive value of the frequency. The algorithm is based on a combination of Muller’s method [49], used to find the complex roots of Eqs. (7.81) and (7.82) and a procedure to trace implicit planar curves [65]. In the next step, we compute the response matrices E A, E S, E a, and E s for each single mode. In the two-dimensional case shown here the matrix functions are given as

$$\displaystyle\begin{array}{rcl} \mathbf{E}^{\mathrm{A}}(x_{ 3},k_{\mathrm{n}}^{\mathrm{A}},\omega ) = \frac{j} {2\mu \frac{\partial D_{\mathrm{A}}} {\partial k} }\left.\left [\begin{array}{*{10}c} N_{\mathrm{kk}}^{\mathrm{A}} & 0&N_{\mathrm{k3}}^{\mathrm{A}} \\ 0 &0& 0 \\ N_{\mathrm{3k}}^{\mathrm{A}} & 0&N_{\mathrm{33}}^{\mathrm{A}} \end{array} \right ]\right \vert _{k=k_{\mathrm{n}}^{\mathrm{\,A}}}\;,& &{}\end{array}$$
(7.89)
$$\displaystyle\begin{array}{rcl} \mathbf{E}^{\mathrm{S}}(x_{ 3},k_{\mathrm{n}}^{\mathrm{S}},\omega ) = \frac{j} {2\mu \frac{\partial D_{\mathrm{S}}} {\partial k} }\left.\left [\begin{array}{*{10}c} N_{\mathrm{kk}}^{\mathrm{S}} & 0&N_{\mathrm{k3}}^{\mathrm{S}} \\ 0 &0& 0 \\ N_{\mathrm{3k}}^{\mathrm{S}} & 0&N_{\mathrm{33}}^{\mathrm{S}} \end{array} \right ]\right \vert _{k=k_{\mathrm{n}}^{\mathrm{S}}}\;,& &{}\end{array}$$
(7.90)
$$\displaystyle\begin{array}{rcl} \mathbf{E}^{\mathrm{a}}(x_{ 3},k_{\mathrm{n}}^{\mathrm{a}},\omega ) = \frac{j(-1)^{n}} {\mu k_{\mathrm{n}}^{\mathrm{a}}d} \sin \frac{(2n + 1)\pi x_{3}} {d} \left [\begin{array}{*{10}c} 0&0&0\\ 0 &1 &0 \\ 0&0&0 \end{array} \right ]\;,& &{}\end{array}$$
(7.91)
$$\displaystyle\begin{array}{rcl} \mathbf{E}^{\mathrm{s}}(x_{ 3},k_{\mathrm{n}}^{\mathrm{s}},\omega ) = \frac{j(-1)^{n}} {\kappa _{\mathrm{n}}\mu k_{\mathrm{n}}^{\mathrm{s}}d} \cos \frac{2n\pi x_{3}} {d} \left [\begin{array}{*{10}c} 0&0&0\\ 0 &1 &0 \\ 0&0&0 \end{array} \right ]\;,& &{}\end{array}$$
(7.92)

where κ 0 = 2 and κ n = 1 for n ≥ 1. In the definitions of the response matrices E A and E S we introduced a set of functions N which is given as

$$\displaystyle\begin{array}{rcl} N_{\mathrm{kk}}^{\mathrm{A}}(x_{ 3},k,\omega ) = q\left [2k^{2}\sin \frac{qd} {2} \sin px_{3} - (k^{2} - q^{2})\sin \frac{pd} {2} \sin qx_{3}\right ]\;,& &{}\end{array}$$
(7.93)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{kk}}^{\mathrm{S}}(x_{ 3},k,\omega ) = q\left [-2k^{2}\cos \frac{qd} {2} \cos px_{3} + (k^{2} - q^{2})\cos \frac{pd} {2} \cos qx_{3}\right ]\;,& &{}\end{array}$$
(7.94)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{3k}}^{\mathrm{A}}(x_{ 3},k,\omega ) = -jk\left [2pq\sin \frac{qd} {2} \cos px_{3} + (k^{2} - q^{2})\sin \frac{pd} {2} \cos qx_{3}\right ]\;,& &{}\end{array}$$
(7.95)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{3k}}^{\mathrm{S}}(x_{ 3},k,\omega ) = -jk\left [2pq\cos \frac{qd} {2} \sin px_{3} + (k^{2} - q^{2})\cos \frac{pd} {2} \sin qx_{3}\right ]\;,& &{}\end{array}$$
(7.96)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{k3}}^{\mathrm{A}}(x_{ 3},k,\omega ) = jk\left [(k^{2} - q^{2})\cos \frac{qd} {2} \sin px_{3} + 2pq\cos \frac{pd} {2} \sin qx_{3}\right ]\;,& &{}\end{array}$$
(7.97)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{k3}}^{\mathrm{S}}(x_{ 3},k,\omega ) = jk\left [(k^{2} - q^{2})\sin \frac{qd} {2} \cos px_{3} + 2pq\sin \frac{pd} {2} \cos qx_{3}\right ]\;,& &{}\end{array}$$
(7.98)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{33}}^{\mathrm{A}}(x_{ 3},k,\omega ) = p\left [(k^{2} - q^{2})\cos \frac{qd} {2} \cos px_{3} - 2k^{2}\cos \frac{pd} {2} \cos qx_{3}\right ]\;,& &{}\end{array}$$
(7.99)
$$\displaystyle\begin{array}{rcl} N_{\mathrm{33}}^{\mathrm{S}}(x_{ 3},k,\omega ) = p\left [-(k^{2} - q^{2})\sin \frac{qd} {2} \cos px_{3} + 2k^{2}\sin \frac{pd} {2} \cos qx_{3}\right ]\;.& &{}\end{array}$$
(7.100)

Additionally, we need to define the functions D

$$\displaystyle\begin{array}{rcl} D_{\mathrm{A}} = 4k^{2}pq\cos \frac{pd} {2} \sin \frac{qd} {2} + [k^{2} - q^{2}]^{2}\sin \frac{pd} {2} \cos \frac{qd} {2} \;,& &{}\end{array}$$
(7.101)
$$\displaystyle\begin{array}{rcl} D_{\mathrm{S}} = 4k^{2}pq\sin \frac{pd} {2} \cos \frac{qd} {2} + [k^{2} - q^{2}]^{2}\cos \frac{pd} {2} \sin \frac{qd} {2} \;.& &{}\end{array}$$
(7.102)

Thus, each mode has its corresponding excitation matrix. This formulation is convenient for applications, where only the contribution of one single mode is needed [62].

In principle, the different wavenumbers can assume real or complex values. In case of complex wavenumbers the corresponding modes are evanescent meaning that their amplitude decays with the covered distance. On the other hand, real wavenumbers correspond to propagating modes. It can be shown that for each frequency there is a finite number of propagating modes and the rest are evanescent. In many applications we can neglect the evanescent modes as their contribution to the response matrix given in Eq. (7.80) is insignificant. Therefore, we can truncate the sum depending on a pre-defined error threshold [58]. Another point worth mentioning is that in isotropic plates the response matrix \(\check{\mathbf{E}}\) is independent of the wave propagation direction. Therefore, we assume that e ξ = e 1 without loss of generality. In anisotropic plates this is unfortunately not the case. The dispersion curves for an aluminum plate are exemplarily shown in Fig. 7.15.

Fig. 7.15
figure 15

Complex dispersion curves of the Lamb wave and SH modes for an aluminum plate (Young’s modulus E = 70 GPa, Poisson’s ratio ν = 0. 33, mass density ρ = 2700 kg/m3). Symmetric modes are denoted by black solid lines, while antisymmetric modes are marked with red solid lines. (a) Lamb wave modes. (b) SH wave modes

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2.4 Coupling Boundary Conditions

After having studied the analytical solution of an infinite plate under external loading we conduct a separate analysis of the bonded piezoelectric transducer. As already discussed in Chap. 4 piezoelectric materials exhibit a coupling of electrical and mechanical quantities [52, 64]. The mechanical state is characterized by the vector of displacements u, the strain tensor ɛ u, and the stress tensor σ u. On the other hand, the electrical state in a piezoelectric material is given by the electrical potential φ which is a scalar quantity, the electric field vector E, and the vector of electrical displacements D. The governing equations and the constitutive equations for such a material have been presented in Sects. 4.1 and 4.2, respectively.

In order to implement suitable boundary conditions on the interface Γ C between the infinite plate and the piezoelectric transducer we consider the problem in the frequency domain. We now apply the Fourier transform presented in Eqs. (7.71) and (7.72) to the governing equations (cf. Eqs. (4.1) and (4.2)) and take the constitutive equations (cf. Eqs. (4.14) and (4.15)) of the linear theory of piezoelectricity into account. Therefore, the formulation of the problem in the frequency domain is given in index notation as

$$\displaystyle\begin{array}{rcl} c_{\mathrm{ijkl}}\check{u}_{\mathrm{k,lj}} + e_{\mathrm{lij}}\check{\varphi }_{\mathrm{,lj}} = -\omega ^{2}\rho \check{u}_{ i}& &{}\end{array}$$
(7.103)
$$\displaystyle\begin{array}{rcl} e_{jkl}\check{u}_{\mathrm{k,lj}} -\kappa _{\mathrm{lj}}\check{\varphi }_{\mathrm{,lj}} = 0.& &{}\end{array}$$
(7.104)

Einstein’s summation convention regarding the sum over repeated indices k, l, and j is considered and an index j after a comma denotes a partial derivative with respect to the variable x j . Dirichlet and Neumann boundary conditions are prescribed in the usual way [58, 60].

The interaction between the plate and the transducer is now considered separately. We assume an ideal bonding and therefore both the displacements and the tractions are continuous. The forces in the interface exerted by the piezoelectric transducer on the plate/substrate and the reaction forces of the plate/substrate exerted on the transducer are denoted by \(\check{\mathbf{F}}_{\mathrm{piezo}}\) and therefore we obtain the bonding boundary condition as

$$\displaystyle{ \left (c_{\mathrm{ijkl}}\check{u}_{\mathrm{k,l}} + e_{\mathrm{lij}}\check{\varphi }_{\mathrm{,l}}\right )n_{j}\vert _{\varGamma _{\mathrm{C}}} +\check{ F}_{\mathrm{i,piezo}} = 0\;. }$$
(7.105)

The displacements of the piezoelectric transducer and the infinite plate are identical on Γ C. Hence, we can use Eq. (7.78) to obtain an expression to compute the displacements

$$\displaystyle{ \check{\mathbf{u}}(\bar{\mathbf{x}},d/2,\omega ) =\int _{\mathbb{R}^{2}}\check{\mathbf{E}}(\bar{\mathbf{x}} -\bar{\mathbf{x}}^{{\prime}},d/2,\omega ) \cdot \check{\mathbf{F}}_{\mathrm{ ext}}(\bar{\mathbf{x}}^{{\prime}},\omega )\mathrm{d}\bar{\mathbf{x}}^{{\prime}}\;,\quad \bar{\mathbf{x}}^{{\prime}}\in \varGamma _{\mathrm{ C}}. }$$
(7.106)

Equations (7.105) and (7.106) constrain the plate to account for the dynamical behavior of the piezoelectric transducer and vice versa. Once the system of governing equations restricted to these boundary conditions is solved, we obtain a solution for the behavior of an isolated piezoelectric patch when it is perfectly bonded to an infinite plate without the need to analyze the behavior of any other point in the plate except for those contained in Γ C.

The system of partial differential equations given by Eqs. (7.103) and (7.104) can be solved using a FE-based approach, see Chaps. 4 and 6 However, in [58, 60] a collocation approach has been deployed instead of a Bubnov-Galerkin method. In the following we only sketch the basic idea for the case of a transducer which is not bonded to the plate \(\check{\mathbf{F}}_{\mathrm{piezo}} = \mathbf{0}\), cf. Eq. (7.105). Therefore, we can set up a linear system of equations in \(\check{\mathbf{u}}\) for each frequency

$$\displaystyle{ \left (\mathbf{K} -\omega ^{2}\mathbf{M}\right )\check{\mathbf{u}} =\check{ \mathbf{F}}\;, }$$
(7.107)

where K and M denote the finite element stiffness and mass matrices, respectively. In the case where the piezoelectric transducer is perfectly bonded to the infinite plate we have to take the reaction forces \(\check{\mathbf{F}}_{\mathrm{piezo}}\) into account and their relation to the displacements \(\check{\mathbf{u}}\) (cf. Eq. (7.106)). A detailed discussion of the numerical procedure to solve the resulting linear system of equations is provided in [58].

2.5 Numerical Results

To demonstrate the properties of the proposed method we consider a two-dimensional problem. A piezoelectric transducer made of PIC-181 (the material data is compiled in Table 7.2) is perfectly bonded to an infinite aluminum plate. The thickness of the transducer is set to t p = 1 mm with a length of l p = 10 mm and the thickness of the aluminum plate is d = 2 mm, cf. Fig. 7.16. The surfaces of the plate are traction free, and the upper and the lower surface of the piezoelectric actuator are electroded. On the lower face of the actuator the potential is fixed (ϕ l = 0), while a time dependent voltage signal (cf. Eq. (6.37)) is applied at the upper surface. The center frequency is f ex = 200 kHz and three cycles are excited with an amplitude of 50 V.

Table 7.2 Material properties for PIC-181
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Fig. 7.16
figure 16

Two-dimensional model of the aluminum plate with a perfectly bonded piezoelectric transducer

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The piezoelectric transducer is discretized using one spectral element with 567 degrees of freedom (\(p_{x_{1}} = 20\), \(p_{x_{3}} = 8\)). To obtain the solution we only considered the first 16 terms in the infinite sum of Eq. (7.80). The simulation results are depicted in Figs. 7.17, 7.18 and 7.19.

Fig. 7.17
figure 17

Deformation of the perfectly bonded piezoelectric transducer depicted in Fig. 7.16. The coloring of the contour plots indicates the normalized distribution of the electric potential ϕ. (a) t 1 = 5 ms. (b) t 2 = 10 ms

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Fig. 7.18
figure 18

Horizontal and vertical displacement at the first observation point (x 1 = 200 mm, x 3 = 1 mm)

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Fig. 7.19
figure 19

Horizontal and vertical displacement at the second observation point (x 1 = 245 mm, x 3 = 1 mm)

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With the proposed method, the behavior of the bonded piezoelectric actuator can be analyzed by only considering points located on its domain. Nevertheless the effects of the reaction forces, caused by the elastic plate, on the behavior of the piezoelectric patch are observed. The impact on the bahavior of the plate in the simplified model of the bonded patch is introduced in terms of the response matrix and therefore, the application of our approach is limited to cases where the response matrix can be determined.

To verify the results we compute the displacements at two predefined observation points. Both points are located at the top surface of the plate at x 1 1 = 200 mm and at x 1 2 = 245 mm. We compare the obtained results with finite element computations. To this end, we set up a finite element model in Abaqus®; using 8-noded quadrilateral elements (plane strain conditions are assumed). The element size is h Abaqus = 2. 510−5 m. We observe that the curves are in good agreement.

This shows the capability of the method to describe the piezoelectric behavior of a sensor/actuator network bonded to a plate. Here, only points lying within the bonding surfaces are considered, i.e. we do not need to discretize the plate itself and we only include points in the interface in the computation. This is very advantageous regarding the simulations of Lamb waves excited by piezoelectric transducers with applications to non-destructive testing. The number of degrees of freedom of the system and the calculation effort are significantly reduced and additionally the analysis of the behavior of the sensor due to the effect of the signals emitted by the actuators is simplified.