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Shocks through layered materials with SPH
Iason Zisis CASAday, 6 Nov 2013
I.Zisis dr. ir. B. vd Linden prof. dr. ir. B.Koren
Hypervelocity impacts
• Small-sized meteorites and space debris��• Impact speed higher than sound speed through target�
ü Velocity: 10 km/s�
ü Time scale: 10 μs�
ü Pressure: 100 GPa�
ü Temperature: 1,000 K �
2
Gallileo, European Space Agency
3
Phenomenology
• Shock compression�• Expansion with phase changes�• Fluid-like behavior�• Severe fragmentation�
• Stresses�
credits European Space Agency
Constitutive models Equation of State
�↵� = �P �↵� + s↵�
P = P (⇢, e)
SPH – Aluminum 7km/s impact
4
Inhomogeneous materials
• Shields from laminated materials�
�• Delamination away from impact point�
• Homogenization�+ existing SPH context�
- only averaged delamination effects�
• Multiphase shock problem�+ reflection-transmission effects�
+ delamination effects localized�
- not thoroughly studied!�
5
O(cm�mm)
How to derive SPH schemes ?
�• Traditional SPH schemes come from function approximations on
moving points��
�
• Particles also define a particle system�
• Variational principles ?�
6
hf(x)i ⇡Z
⌦f(x
o
)W (|x� x
o
|, h)d⌦
Variational SPH framework
• Medium’s density from particles [Monaghan 2005, Price 2008]�
�
• Density evolution equation�
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⇢i =X
j
mjWij(hi)
d⇢idt
=X
j
mjdWij
dt
hi = ⌘⇣mi
⇢i
⌘ 1d
Variational SPH framework
• Lagrange variational principle�
• Particle motion equation�
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de = Tds+ P⇢2 d⇢
d⌦j := Vj =mj
⇢j
d
dt
⇣ @L@vi
⌘� @L
@xi= 0
depends on density estimate
@L@xi
=dejd⇢j
���s
⇣@⇢j@xi
⌘
L =
Z
⌦
⇣12⇢v2 � ⇢e(⇢, s)
⌘d⌦
L =X
j
mj
⇣12v2j � e(⇢j , sj)
⌘
More SPH schemes
• Other density estimates�
• Differential formulations are better for bounded domains��
• Derive conservative SPH schemes…�
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⇢i =X
j
⇢jWij(hi)Vj⇢i = mi
X
j
Wij(hi)
Conservative SPH schemes
10
d⇢idt
=1
⌦i
X
j
mj(vi � vj)rWij(hi)
d⇢idt
=⇢i⌦i
X
j
(vi � vj)rWij(hi)Vj
d⇢idt
=mi
⌦i
X
j
(vi � vj)rWij(hi)
dvi
dt= �
X
j
mj
⇣ Pi
⌦i⇢2irWij(hi) +
Pj
⌦j⇢2jrWij(hj)
⌘
dvi
dt= � 1
⇢i
X
j
⇣Pi
⌦irWij(hi) +
Pj
⌦jrWij(hj)
⌘Vj
dvi
dt= � 1
mi
X
j
⇣ Pi
⇢i⌦i�irWij(hi) +
Pj
⇢j⌦j�jrWij(hj)
⌘
• Mass based�
• Volume based� • Inverse volume based�
Lessons learned
• SPH schemes come from Lagrangian mechanics and can be fully conservative�
�• SPH momentum equation is not a free choice… depends on
density estimate equation �����
“…give me a density estimate and � I will move the SPH particles!” �
11
Still some details left…
• Which scheme to choose for shocks through layered materials ?�…test against exact solutions of Riemann problems�
• Volume based scheme performed the best�
Ø Ideal processes need numerical dissipation�
Ø New artificial mass flux term devised�
12
d⇢idt
=⇢i⌦i
X
j
(vi � vj)rWij(hi)Vj
dvi
dt= � 1
⇢i
X
j
⇣Pi
⌦irWij(hi) +
Pj
⌦jrWij(hj)
⌘Vj +⇧ij
+�ij
Inhomogeneous shock-bar
�• 1D isothermal impact into discontinuous material�
���
�
�13
Ca(x) =⇢
o
(x)v2impact
K(x)
Impact point (t = 0.1)
14
point at initial configuration
Shock on interface (t = 0.2)
15
point at initial configuration
momentum conservation�at machine precision…�
Shock tube (t=0.2)
16
Dissipative processes
• No general framework for natural dissipation in SPH�
• Towards a formulation with natural dissipation�
• GENERIC�
17
e = e(�(k), s) ! de = Tds+ �(k)d⌧ (k)
L =X
j
mj
⇣12v2 � e(⇢j , sj)
⌘
CASAsph platform
18
��
• General SPH solver �with all available schemes�
Thank you…
19