Corrected SPH Approximation of a Function

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

\int_{Ω} W (y - x, h) = 1, \forall x

\int_{Ω} (y - x) W (y - x, h) = 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:

{\hat{W}}_{j} (x, h) = W_{j} (x, h) α (x) [1 + β (x) • (x - x_{j})]

with $W_{j} (x, h) = W (x - x_{j}, h)$

Where the parameters $α (x)$ and $β (x)$ are evaluated by enforcing the consistency condition, now given by the point wise integration as:

\sum_{j} V_{j} {\hat{W}}_{j} (x, h) = 1, \forall x

\sum_{j} V_{j} (x - x_{j}) {\hat{W}}_{j} (x, h) = 0, \forall x

These equations enable the explicit evaluation of the correction parameters $α (x)$ and $β (x)$ as:

β (x) = {[\sum_{j} V_{j} (x - x_{j}) \otimes (x - x_{j}) W_{j} (x, h)]}^{- 1} \sum_{j} V_{j} (x_{j} - x) W_{j} (x, h)

α (x) = \frac{1}{\sum_{j} V_{j} W_{j} (x, h) [1 + β (x) • (x - x_{j})]}

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:

{\hat{W}}_{j} (x, h) = W_{j} (x, h) α (x)

that is

α (x) = \frac{1}{\sum_{j} V_{j} W_{j} (x, h)}

\sum_{j} V_{j} {\hat{W}}_{j} (x, h) = 1, \forall x

Note: SPH corrections generally insure a better representation even if the particles are not organized into a hexagonal compact net, especially close to the integration domain frontiers. SPH corrections also allow the smoothing length

h

to values different to the net size

Δ x

to be set.

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.