# Corrected SPH Approximation of a Function

Corrected SPH formulation 1 2 has been introduced in order to satisfy the so-called consistency conditions:(1)
$\underset{\Omega }{\int }W\left(y-x,h\right)=1,\forall x$
(2)
$\underset{\Omega }{\int }\left(y-x\right)W\left(y-x,h\right)=0,\forall x$

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:(3)

${\stackrel{^}{W}}_{j}\left(x,h\right)={W}_{j}\left(x,h\right)\alpha \left(x\right)\left[1+\beta \left(x\right)•\left(x-{x}_{j}\right)\right]$ 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:(4)
$\sum _{j}{V}_{j}{\stackrel{^}{W}}_{j}\left(x,h\right)=1,\forall x$
(5)
$\sum _{j}{V}_{j}\left(x-{x}_{j}\right){\stackrel{^}{W}}_{j}\left(x,h\right)=0,\forall x$
These equations enable the explicit evaluation of the correction parameters $\alpha \left(x\right)$ and $\beta \left(x\right)$ as:(6)
$\beta \left(x\right)={\left[\sum _{j}{V}_{j}\left(x-{x}_{j}\right)\otimes \left(x-{x}_{j}\right){W}_{j}\left(x,h\right)\right]}^{-1}\sum _{j}{V}_{j}\left({x}_{j}-x\right){W}_{j}\left(x,h\right)$
(7)
$\alpha \left(x\right)=\frac{1}{\sum _{j}{V}_{j}{W}_{j}\left(x,h\right)\left[1+\beta \left(x\right)•\left(x-{x}_{j}\right)\right]}$
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:(8)
${\stackrel{^}{W}}_{j}\left(x,h\right)={W}_{j}\left(x,h\right)\alpha \left(x\right)$
that is(9)
$\alpha \left(x\right)=\frac{1}{\sum _{j}{V}_{j}{W}_{j}\left(x,h\right)}$
(10)
$\sum _{j}{V}_{j}{\stackrel{^}{W}}_{j}\left(x,h\right)=1,\forall x$

1 Bonet J. and 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.