# Corrected SPH Approximation of a Function

Corrected SPH formulation ^{1}
^{2} has been introduced in order to satisfy the
so-called consistency conditions:

These equations insure that the integral approximation of a function f coincides with f for constant and linear functions of space.

CSPH is a correction of the kernel functions:

with ${W}_{j}\left(x,h\right)=W\left(x-{x}_{j},h\right)$

Where the parameters $\alpha \left(x\right)$ and $\beta \left(x\right)$ are evaluated by enforcing the consistency condition, now given by the point wise integration as:

These equations enable the explicit evaluation of the correction parameters $\alpha \left(x\right)$ and $\beta \left(x\right)$ as:

Since the evaluation of gradients of corrected kernel (which are used for the SPH integration of continuum equations) becomes very expensive, corrected SPH limited to order 0 consistency has been introduced. Therefore, the kernel correction reduces to the following equations:

^{1}Bonet J., TSL Lok,

Variational and Momentum Preservation Aspects of Smooth Particle Hydrodynamic Formulations, Computer Methods in Applied Mechanics and Engineering, Vol. 180, pp. 97-115 (1999).

^{2}Bonet J. and Kulasegram S.,

Correction and Stabilization of Smooth Particle Hydrodynamics Methods with Applications in Metal Forming Simulations, Int. Journal Num. Methods in Engineering, Vol. 47, pp. 1189-1214, 2000.