# Numerical Study of Mode Conversion between Fundamental Lamb and Rayleigh Waves
## 2 Numerical Model  
The various model geometries used for the wave propagation analysis are shown in Figure 2. Figure 2a,b shows a thin plate (thickness of 1 mm) attached to a quarter space of dimensions 0.5 m X 0.5 m, which will be referred to as the \"quarter-space model\" throughout this article. Figure 2c shows a thin plate attached to a half space, which will be referred to as the \"half-space model\". The quarter-space model has a discontinuity \(D_{1}\), which is defined as the point where the abrupt transition in thickness occurs. The thickness transition plane in which the discontinuity occurs is termed as the discontinuity face. The half-space model has two discontinuities, \(D_{1}\) and \(D_{2}\), as shown in Figure 2c. The finite element model was constrained at the bottom of the thick plate (half-space) with fixed boundary conditions, and all other faces were given traction free boundary  
Figure 1: Dispersion curves showing the phase velocity as a function of frequency for a 1 mm plate. The modes of interest are \(A_{0}\) and \(S_{0}\) at 1 MHz-mm. The properties of aluminum are \(E=69.8\,\mathrm{GP}\), \(\
u=0.33\), density: \(\rho=2700\,\mathrm{kg/m^{3}}\) and were used to generate the dispersion curve in Disperse software. conditions to satisfy guided wave propagation as shown in the schematic. The model was meshed using 2D, 8 node quadrilateral elements with a quadratic shape function and plane strain conditions. For convergence of a numerical solution, it is important to have at least 10 elements per wavelength of a specific mode. The present work considers the following modes with their respective velocities: \(A_{0}:2335\) m/s, \(S_{0}:5346\) m/s and \(R:2865\) m/s. The present work considers narrow-band 1 MHz excitation; therefore, the wavelengths will be \(A_{0}:2.335\) mm, \(S_{0}:5.346\) mm and \(R:2.865\) mm. Considering the lowest mode wavelength (2.335 mm) and the highest frequency of interest (1 MHz), we require the element size to be 0.23 mm. The final element size chosen for the study such that it captures all the modes was 0.0833 mm. The number of elements near the 90\({}^{\circ}\) corner of the quarter-space and half-space models was increased to model the sharp discontinuity. The equation of motion is given by:  
\[[M]\{\ddot{u}\}+[C]\{\dot{u}\}+[K]\{u\}=\{F(t)\} \tag{1}\]  
where \(M\) is the mass matrix, \(C\) is the damping matrix and \(K\) is the stiffness matrix. \(\ddot{u}\), \(\dot{u}\) and \(u\) are the nodal acceleration, velocity and displacement vectors, respectively, and \(F(t)\) is the load vector. The elemental stiffness matrix was obtained by using the elastic properties of aluminum, \(E=69.8\) GPa and \(\
u=0.33\), and the mass matrix was obtained using the density \(\rho=2700\) kg/m\({}^{3}\). For the present case, the damping matrix was set to 0. The load vector is the excitation source and is a function of time for the transient analysis. This was modeled using a 1 MHz, 7-cycle sinusoidal toneburst subjected to a Hanning window as shown in Figure 3a. Time integration was performed using [PERSON]'s time integration scheme [18], with an integration time of 50 nanoseconds and a total simulation time of 50 microseconds. The integration time step was chosen such that it satisfies the Nyquist criteria. The finite element problem was solved in ANSYS using a direct sparse solver. Due to the enormous size of the model, graphic processing unit (GPU) acceleration using a Tesla C2075 [19] was utilized. Additionally, the double precision computation and error correcting code of the Tesla were retained to improve simulation accuracy [19]. Selective mode generation of Lamb waves was achieved by applying \(U_{y}\) (for \(A_{0}\) mode) and \(U_{x}\) (for \(S_{0}\) mode) nodal displacements along the vertical edge of the plate, as shown in Figure 2. The nodal displacement amplitude determined as a function of time shows the disturbance as it propagates along the plate.