stability of SPH have allowed for more widespread applications in Solid Mechanics. Other examples of applications and developments of the method include: Metal forming simulations. SPH-based method SPAM (Smoothed Particle Applied Mechanics) for impact fracture in solids by William G. Hoover. Modified SPH (SPH/MLSPH) for fracture and fragmentation. Taylor-SPH (TSPH) for shock wave propagation in solids. Generalized coordinate SPH (GSPH) allocates particles inhomogeneously in the Cartesian coordinate system and arranges them via mapping in a generalized coordinate system in which the particles are aligned at a uniform spacing. Numerical tools Interpolations The Smoothed-Particle Hydrodynamics (SPH) method works by dividing the fluid into a set of discrete moving elements , referred to as particles. Their Lagrangian nature allows setting their position by integration of their velocity as: These particles interact through a kernel function with characteristic radius known as the "smoothing length", typically represented in equations by . This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles that lie within the range of the kernel, the latter being used as a weighting function . This can be understood in two steps. First an arbitrary field is written as a convolution with : The error in making the above approximation is order . Secondly, the integral is approximated using a Riemann summation over the particles: where the summation over includes all particles in the simulation. is the volume of particle , is the value of the quantity for particle and denotes position. For example, the density of particle can be expressed as: where denotes the particle mass and the particle density, while is a short notation for . The error done in approximating the integral by a discrete sum depends on , on the particle size (i.e. , being the space dimension), and on the particle arrangement in space. The latter effect is still poorly known. Kernel functions commonly used include the Gaussian function, the quintic spline and the Wendland kernel. The latter two kernels are compactly supported (unlike the Gaussian, where there is a small contribution at any finite distance away), with support proportional to . This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles. Although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of