Mathematical Physics - Volume II - Numerical Methods

5.3 Conservation Equations

167

5.3 Conservation Equations

The conservation equations in Lagrangian framework are given by: D ρ D t = − ρ ∇ · v ,

(5.8)

= ∇ ·

σ ρ +

σ ρ 2 ·

D v D t

D v D t

1 ρ 1 ρ

∇ · σ or

∇ ρ ,

(5.9)

=

σ · v

D E D t

D E D t

1 ρ 2

σ : ∇ v or

σ : ∇ ( ρ v ) −

∇ ρ

(5.10)

ρ 2 ·

=

=

D D t is the material time derivative and v = ˙ x .

Where

Equations (5.9) and (5.10) are the forms proposed by Monaghan [47]. Kernel in terpolation allows the derivation of the basic SPH form of these conservation equations as: D ρ ( x ) D t = Z Ω ρ ( x ′ ) ∇ · v ( x ′ ) W ( | x − x ′ | , h ) d Ω , (5.11) D v ( x ) D t = Z Ω ∇ · σ ( x ′ ) ρ ( x ′ ) ! W ( | x ′ − x | , h ) d Ω + + Z Ω σ ( x ′ ) [ ρ ( x ′ )] 2 · ∇ ρ ( x ′ ) W ( | x ′ − x | , h ) d Ω , (5.12) D E ( x ) D t = Z Ω σ ( x ′ ) [ ρ ( x ′ )] 2 : ∇ ( ρ ( x ′ ) v ( x ′ )) W ( | x ′ − x | , h ) d Ω − − Z Ω σ ( x ′ ) · v ( x ′ ) [ ρ ( x ′ )] 2 · ∇ ρ ( x ′ ) W ( | x − x ′ | , h ) d Ω . (5.13) All of the above equations contain integrals of the form: Z Ω W ( | x − x ′ | , h ) f ( x ′ ) ∂ g ( x ′ ) ∂ x ′ d x ′ . (5.14) Using a Taylor series expansion at point x ′ = x , it follows: Z Ω W ( | x − x ′ | , h ) f ( x ′ ) ∂ g ( x ′ ) ∂ x ′ d x ′ = = Z Ω f ( x ) ∂ g ( x ) ∂ x +( x − x ′ ) d d x f ( x ) ∂ g ( x ) ∂ x + · · · W ( | x − x ′ | , h ) d x ′ , (5.15)