Corrected SPH Approximation of a Function

Corrected SPH formulation 1 2 has been introduced in order to satisfy the so-called consistency conditions:(1)
Ω W ( y x , h ) = 1 , x
(2)
Ω ( y x ) W ( y x , h ) = 0 , 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)

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:(4)
j V j W ^ j ( x , h ) = 1 , x
(5)
j V j ( x x j ) W ^ j ( x , h ) = 0 , x
These equations enable the explicit evaluation of the correction parameters α ( x ) and β ( x ) as:(6)
β ( x ) = [ j V j ( x x j ) ( x x j ) W j ( x , h ) ] 1 j V j ( x j x ) W j ( x , h )
(7)
α ( x ) = 1 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:(8)
W ^ j ( x , h ) = W j ( x , h ) α ( x )
that is(9)
α ( x ) = 1 j V j W j ( x , h )
(10)
j V j W ^ j ( x , h ) = 1 , x
注: 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ to values different to the net size Δ x to be set.
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.