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SPH Formulation (2)
• Basic idea: represent a continuum by a (large) set of
particles, whose motion is governed by the conservation
laws of continuum mechanics
• The method is based upon the following identity for a
function f :
f ( r ) = ∫ f( r ′) ⋅δ
( r −r ′)
dV′
r = position vector δ = Dirac's distribution
f = scalar or vectorial field ( ρ, v,...)
• One tries to approximate the former integral by a
“regularization”, i.e. to replace Dirac’s function by a
kernel function W( r, h)
, where h is the characteristic
length
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SPH Formulation (3)
• One may thus define an approximation of the
field f ( r f ( r )
) as:
f ( r ) ≈ f( r ) = f( r ) W( r −r ′, h)
dV′
The kernel must satisfy the following properties:
h→0
∫
• As the regularization length tends to 0, it tends to Dirac’s
distribution:
lim W( r − r′ , h) = δ ( r −r′
)
• It is normalized:
∫ W ( r
, h ) dV′ = 1
• We use a cubic kernel (Monaghan’s W4)
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