## 2 Materials and Methods  
The ejection involves free-surface (interface) large deformation, tensile, and fracture. The Lagrangian method has the advantage of clearly describing material interfaces, but in regions of large deformation, elements tend to twist, entangle, or even flip, leading to the termination of the calculation. The Eulerian method has the advantage of being able to handle large material deformations, but it is difficult to track the interface history. The SPH method, originally introduced to address astrophysical problems in three-dimensional open space [21, 22], has been widely adopted for solving various scientific and engineering problems. Its Lagrangian and mesh-free characteristics make it suitable for applications such as fluid mechanics [23, 24, 25, 26, 27], solid mechanics [28, 29, 30, 31, 32], fluid-structure interactions [33, 34, 35, 36, 37], and explosion and shock dynamics [38, 39, 40, 41, 42]. In this paper, we study the elastic-plastic ejection from grooved aluminum surfaces under double supported shocks using a self-developed SPH program that has been validated in previous research [9, 43, 44, 45, 19]. The governing equations of continuum mechanics in the Lagrangian framework [46] are expressed as follows:  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}\rho}{\mathrm{d}t}=-\rho\frac{ \partial\rho^{\delta}}{\partial x^{\delta}}\\ \frac{\mathrm{d}\tau^{\delta}}{\mathrm{d}t}=\frac{1}{\rho}\frac{\partial\tau ^{\delta}}{\partial x^{\delta}}\\ \frac{\mathrm{d}\epsilon}{\mathrm{d}t}=\frac{\sigma^{\delta}\delta}{\rho} \frac{\partial\tau^{\delta}}{\partial x^{\delta}}\end{array} \right., \tag{1}\]  
where \(\rho\) denotes the density, \(e\) represents the specific internal energy, \(v^{\alpha}\) denotes the velocity component, \(x^{\delta}\) corresponds to the spatial coordinate component, \(\sigma^{\alpha\beta}\) represents the stress tensor component, and \(\alpha\) and \(\beta\) are the dimension indices. The stress tensor component \(\sigma^{\alpha\beta}\) can be expressed as the sum of hydrostatic and deviatoric components:  
\[\sigma^{\alpha\beta}=-p\delta^{\alpha\beta}+S^{\alpha\beta}, \tag{2}\]  
where \(p\) denotes the pressure, \(\delta^{\alpha\beta}\) represents the second-order identity tensor, and \(S^{\alpha\beta}\) corresponds to the deviatoric stress tensor component. The pressure \(p\) and deviatoric stress tensor component \(S^{\alpha\beta}\) are solved by the equation of state and the constitutive model, respectively. The SPH equations can be obtained by applying the kernel and particle approximations to discretize Equation (1) as follows:  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}\rho_{i}}{\mathrm{d}t}=\rho_{i}\sum \limits_{j=1}^{N}\frac{m_{j}}{\rho_{j}}\left(v_{i}^{\beta}-v_{j}^{\beta} \right)\frac{\partial W_{ij}^{C}}{\partial x_{i}^{\beta}}\\ \frac{\mathrm{d}v_{i}^{\alpha}}{\mathrm{d}t}=\sum\limits_{j=1}^{N}m_{j}\left( \frac{\sigma_{i}^{\alpha\beta}}{\rho_{i}^{2}}+\frac{\sigma_{j}^{\alpha\beta} }{\rho_{j}^{2}}-\Pi_{ij}\delta^{\alpha\beta}+f^{n}R_{ij}^{\alpha\beta}\right) \frac{\partial W_{ij}^{C}}{\partial x_{i}^{\beta}}\\ \frac{\mathrm{d}e_{i}}{\mathrm{d}t}=\frac{1}{2}\sum\limits_{j=1}^{N}m_{j}\left( \frac{\sigma_{i}^{\alpha\beta}}{\rho_{i}^{2}}+\frac{\sigma_{j}^{\alpha\beta} }{\rho_{j}^{2}}-\Pi_{ij}\delta^{\alpha\beta}\right)\left(v_{j}^{\alpha}-v_{i} ^{\alpha}\right)\frac{\partial W_{ij}^{C}}{\partial x_{i}^{\beta}}\end{array}  \right., \tag{3}\]  
where \(N\) denotes the number of neighboring particles, \(m_{j}\) represents the mass of the neighboring particle, \(\Pi_{ij}\) denotes the artificial viscosity, \(f^{n}R_{ij}^{\alpha\beta}\) denotes the artificial stress, and \(W_{ij}^{C}\) denotes the kernel function obtained using the kernel gradient correction technique. In this study, the Wendland kernel function is used. The artificial viscosity \(\Pi_{ij}\)[47] is used as follows:  
\[\Pi_{ij}=