## 2 Numerical Approaches  
### Smooth Particle Hydrodynamics (SPH)  
Hydrocode is defined loosely as a code for solving large deformation, finite strain transient problems that occur over a short period of time, and AUTODYN hydrocode is capable of computing strains, stresses, velocities, and the propagation of shock waves as a function of time and position. The software can use both grid-based methods (Lagrange and Euler) and a mesh-free method (SPH) to describe the governing equations.  
The SPH technique is an N-body interaction scheme developed by [PERSON] [29] and [PERSON] [30]. The method was developed to avoid the limitations associated with the tangling and distortion of mesh that occur when the FEM is applied to problems that involve extremely large deformations. The main difference between the FEM and the SPH method is the absence of a grid [31]. The SPH method is a mesh-free, Lagrangian method in which the medium is divided into a set of discrete elements, which are referred to as particles, and there is no connectivity between adjacent particles. The contribution of each particle to a property at a point of interest is weighted according to the distance of the particle from the point of interest. The particle approximation in the SPH method is defined by Equation (2):  
\[\prod^{h}f(x)=\int f(y)\cdot W(x-y,\ h)dy \tag{2}\]  
where \(W\) is the kernel (or smoothing) function. The kernel function is defined using the function by the relation shown in Equation (3):  
\[W(x,\ h)=\frac{1}{h(x)^{d}}\theta(x) \tag{3}\]  
where \(d\) is the number of space dimensions and \(h\) is the so-called smoothing length that varies in time and space. The kernel function should be a centrally peaked function. The most common smoothing kernel function used with the SPH method is the cubic B-spline, which is defined by choosing \(\theta\), as in Equation (4).  
\[\theta(u)=\alpha\times\left\{\begin{array}{ll}1-\frac{3}{2}u^{2}+\frac{3}{4} u^{3},&\quad\text{for}\ |u|\leq 1\\ \frac{1}{4}(2-u)^{2},&\quad\text{for}\ 1\leq|u|\leq 2\\ 0,&\quad\text{for}\ 2<|u|\end{array}\right. \tag{4}\]  
where \(\alpha\) is a constant of normalization, which depends on the number of spatial dimensions, \(u=r/h\); \(r\) is the distance between two particles, and \(h\) is the smoothing length (Figure 4).  
To calculate the value of a function at particle I in Figure 4, it is necessary to sum the values of the function at all of the neighboring particles (interpolation points J1, J2, I, J3, and J4) multiplied by the weighting function. Hence, the SPH particles are not simply interesting mass points, but they are interpolation points from which the values of functions and their derivatives can be estimated at discrete points in a continuum. In the SPH method, the discrete points at which all of the quantities are evaluated are placed at the center of the SPH particles [16].  
Figure 5 shows the calculation cycle of the SPH method, as implemented in AUTODYN. The calculation cycle is similar to that of a Lagrange zone, except for additional steps in which a kernel approximation is used to compute the forces of the spatial derivatives of stress and velocity, which are required to compute strain rates. The SPH method also requires that the particles be sorted at least once every cycle in order to locate a given particle's current neighboring particles [16].  
Figure 4: Kernel function for the interpolation of smooth particle hydrodynamics (SPH) [16].