PSI - Issue 52

Made Wiragunarsa et al. / Procedia Structural Integrity 52 (2024) 583–593 Wiragunarsa et al. / Structural Integrity Procedia 00 (2023) 000–000

587

5

The term D is the JST stabilisation, as reported by Lee et al. [2016], which is added to improve the stability of the numerical scheme, where

D = D 2 ( p ) + D 4 ( p )

(9)

N j = 1

2 W

(2) C

V j ( p j − p i ) ∇

j ( x i )

D 2 ( p ) = κ

(10)

p h min

N j = 1

2 p

2 p

(4) C

3 min

2 W

D 4 ( p ) = − κ

j ( x i )

p h

V j ( ∇

i ) ∇

(11)

j − ∇

Parameters κ (2) and κ (4) are user-defined, h 2 is the Laplacian operator. The second SPH discretisation is applied for the rate of change of the deformation gradient, represented by Equation 12, which V j and ρ j is the volume and mass density. In this research, the B-spline smoothing function is used to calculate the kernel. min is the minimum particle spacing within the domain, and ∇

N j = 1

V j ρ j

d F i dt

( p j − p i ) ⊗ ˜ ∇ W j ( x i )

(12)

=

3.2. Temporal discretisation

The discrete SPH equations which are derived above give a system of ordinary di ff erential equations in the form

d U i dt

= ℜ i ( U i , t )

(13)

the ℜ i ( U i , t ) represents the right-hand side of the SPH spatial discretization with respect to particle i . By using the explicit two-stage TVD Runge-Kutta time integration, the time integration process from t n to t n + 1 will become

n i +∆ t ℜ i ( U n

n )

U ∗ i = U

i , t

(14)

n + 1 )

U ∗∗ i = U ∗ i +∆ t ℜ i ( U ∗ i , t

(15)