An Efficient Boundary Integral Method for Stiff Fluid Interface Problems

AN EFFICIENT

BOUNDARY INTEGRAL METHOD

FOR

STIFF FLUID INTERFACE PROBLEMS
Oleksiy Varfolomiyev

advisor Michael Siegel

co-advisor Michael Booty

NJIT, 29 April 2015

MOTIVATION
Problem: Study the evolution of interface between two
immiscible, inviscid, incompressible, irrotational fluids of
different constant density in three dimensions

Solution: To formulate and investigate a boundary integral
method for the solution of the stiff internal waves/
Rayleigh-Taylor problem with surface tension and
elasticity

⇢1
⇢2
S

OUTLINE
• Mathematical Model

• The Boundary Integral Method

• The Small-scale Decomposition

• Invariants of the Motion

• Symmetry

• Linear Stability Analysis

• Numerical Method

• Simulation Results

MODEL DESCRIPTION
• Gravity

• Surface Tension

• Elastic Bending Stress
Fluids are driven by

GOVERNING EQUATIONS
Interface parameterization
X(~↵, t) = (x(~↵, t), y(~↵, t), z(~↵, t)), ~↵ = (↵, )
Bernoulli’s Equation for Fluids Velocities Potentials
@ i
@t
r i · Xt +
1
2
|r i|2
+
pi
⇢i
+ gz = 0 in Di
Interface Evolution Equation
Xt = V1ˆt1
+ V2ˆt2
+ Uˆn
r i = ViVelocity potential

BOUNDARY CONDITIONS
Kinematic Boundary Condition
Laplace-Young Boundary Condition
Far-ﬁeld Velocity
V1 · ˆn = V2 · ˆn = U on S
p1 p2 = (1 + 2) on S
V ! 0 as z ! ±1
[p] = ¯Eb4s 2 ¯Eb3
+ 2  + 2 ¯Ebg
Pressure Jump across elastic Interface

Fluid Velocity on S given by the Birkhoﬀ-Rott Integral
W(X) = PV
Z 1
1
Z 1
1
j0
⇥ rXG(X, X0
) d↵0
d 0
+ V1ˆn,
=
1
4⇡
PV
Z 1
1
Z 1
1
j0
⇥
(X X0
)
|X X0|3
d↵0
d 0
+ V1ˆn
Generalized Isothermal Parameterization
E = X↵ · X↵
F = X↵ · X
G = X · X
L = X↵↵ · ˆn
M = X↵ · ˆn
N = X · ˆn
G(↵, , t) = (t)E(↵, , t), F(↵, , t) = 0
µ = 1 2 j = µ↵X µ X↵W = (V1 + V2)/2
Ambrose,
Siegel ‘12
Caﬂish,
Li ‘05

Xt = V1ˆt1
+ V2ˆt2
+ (W · ˆn)ˆn
Interface Evolution Equation
µt + A t =
µ↵
p
E
V 1
(W · ˆt1
) +
µ
p
E
V 2
(W · ˆt2
)
+A
"
2(W · ˆn)2
+ 2V1(W · ˆt1
) + 2V2(W · ˆt2
)
µ2
↵
4E
µ2
4 E
|W|2
#
B 2Gz
Vortex Sheet Density Evolution Equation
= 1 + 2  =
L + N
E
A =
⇢1 ⇢2
⇢1 + ⇢2
G =
AgL
v2
c
B =
2
L(⇢1 + ⇢2)v2
c

ELASTIC INTERFACE
Pressure Jump across the Interface
[p] = ¯Eb4s 2 ¯Eb3
+ 2  + 2 ¯Ebg
g =
LN M2
E2
¯Eb
Gaussian curvature
Bending modulus
Surface Laplacian
Plotnikov,
Toland ‘11
4s =
1
p
EG F2
@
@
E @
@ F @
@↵
p
EG F2
!
+
1
p
EG F2
@
@↵
G @
@↵ F @
@
p
EG F2
!

µt + A t =
µ↵
p
E
V 1
(W · ˆt1
) +
µ
p
E
V 2
(W · ˆt2
)
+A
"
2(W · ˆn)2
+ 2V1(W · ˆt1
) + 2V2(W · ˆt2
)
µ2
↵
4E
µ2
4 E
|W|2
#
+Eb(4s 23
+ 2g) + B + 2Agz
-Equation for the elastic interfaceµ
Tangential Velocities are chosen to preserve parameterization
✓
V1
p
E
◆
↵
✓
V2
p
E
◆
=
tE + 2U ( L N) ( E G)
2 E
✓
V2
p
E
◆
↵
+
✓
V1
p
E
◆
=
2UM F
p
E
( E G)t = 0
Ft = 0
( E G) FRelaxation Terms

Normal Velocity Decomposition
U(X) ⌘ W · ˆn = Us(X) + Usub(X)
Us(X)
Usub(X)
- high order terms dominant at a small scale
- lower order terms
W(X) =
1
4⇡
PV
Z 1
1
Z 1
1
j0
⇥
X0
↵(↵ ↵0
) + X0
( 0
)
E0 3
2
h
(↵ ↵0)
2
+ (t) ( 0)
2
i3
2
d↵0
d 0
+
1
4⇡
PV
Z 1
1
Z 1
1
j0
⇥
8
><
>:
X X0
|X X0|3
X0
↵(↵ ↵0
) + X0
( 0
)
E0 3
2
h
(↵ ↵0)
2
+ (t) ( 0)
2
i3
2
9
>=
>;
d↵d 0
= Ws(X) + Wsub(X)
Taylor expansion of the kernel about X’
Ambrose,
Masmoudi ‘09

Us =
p
4⇡
PV
Z 1
1
Z 1
1
ˆn(µ↵(↵ ↵0
) + µ0
( 0
))
E0 1
2
h
(↵ ↵0)
2
+ (t) ( 0)
2
i3
2
d↵0
d 0
· ˆn
Leading order part of the Normal Velocity
H1f(↵, ) =
1
2⇡
PV
Z 1
1
Z 1
1
f(↵0
, 0
)(↵ ↵0
)
h
(↵ ↵0)
2
+ (t) ( 0)
2
i3
2
d↵0
d 0
,
H2f(↵, ) =
1
2⇡
PV
Z 1
1
Z 1
1
f(↵0
, 0
)( 0
)
h
(↵ ↵0)
2
+ (t) ( 0)
2
i3
2
d↵0
d 0
Riesz Transforms
Us = Ws · ˆn =
1
2

H1
✓
µ↵ˆn
E
1
2
◆
+ H2
✓
µ ˆn
E
1
2
◆
· ˆn
bH1f(k) = i
k1
k
ˆfk, bH2f(k) = i
k2
k
ˆfk
Symbols of the Riesz Transforms
k =
q
k2
1 + k2
2

Xt = Usˆn + V1sˆt1
+ V2sˆt2
+ (U Us)ˆn + (V1 V1s)ˆt1
+ (V2 V2s)ˆt2
The ﬁrst three terms in the RHS give the leading order
behavior at small scales, treated implicitly in the
proposed numerical method. These implicit nonlocal
terms evaluated efﬁciently using the FFT method.
The last three terms in the RHS are of lower order,
treated explicitly.
Evolution Equation Decomposition

Surface Energy
Es =
Z
S
Z
dS =
Z 1
0
Z 1
0
|X↵ ⇥ X | d↵d
Ek = E1
k + E2
k =
1
4
(⇢1 + ⇢2)
Z 1
0
Z 1
0
(µ + A )U |X↵ ⇥ X | d↵d
Kinetic Energy
= 1 + 2
E1
k =
1
2
⇢1
Z 2⇡
0
Z 2⇡
0
Z z(x,y)
0
|r 1|2
dxdydz
µ = 1 2 1 = ( + µ)/2 2 = ( µ)/2

X( ↵, ) = ( x(↵, ), y(↵, ), z(↵, )),
µ( ↵, ) = µ(↵, )
If initial condition possess the symmetry
then symmetry is preserved with the evolution of S
Remark
z has purely real Fourier components
x, y, mu have purely imaginary components
Thus all the Fourier components arrays
have half the usual size!
This accelerates the solution algorithm

Perform a small perturbation of the ﬂat interface

with zero mean vortex sheet strength
X = (2⇡↵ + x0
, 2⇡ + y0
, z0
), with |x0
|, |y0
|, |z0
| ⌧ 1,
µ = µ0
, with |µ| ⌧ 1
Interface velocity at the leading order
W0
=
1
8⇡2
PV
Z 1
1
Z 1
1
µ0
↵(↵ ↵0
) + µ0
( 0
)
[(↵ ↵0)2 + (t)( 0)2]
3/2
d↵0
d 0 ˆk
Normal Velocity
U =
1
4⇡
(H1 (µ↵) + H2 (µ ))

Evolution Equations
dµ0
dt
= B 2Gz0
X0
t = (W0
· ˆk)ˆk ⇡ U0 ˆk
Linear System of ODE’s for the Fourier Components
d
dt
✓
ˆzk
ˆµk
◆
=
✓
0 k
2
B
2 k2
2G 0
◆ ✓
ˆzk
ˆµk
◆
LINEARIZED EQUATIONS
f(ˆ↵) =
X
ˆk
ˆf(ˆk) eiˆk·ˆ↵ ˆf(ˆk) =
Z 1
0
Z 1
0
e 2⇡iˆk·ˆ↵
f(ˆ↵) dˆ↵ k =
q
k2
1 + k2
2

Dispersion Relation k =
B
4 2
k3 G
k
ˆzk(t) =

k
4
p
k
ˆµk(0) +
1
2
ˆzk(0) e
p
kt
+

1
2
ˆzk(0)
k
4
p
k
ˆµk(0) e
p
kt
= ˆzk(0) cosh(
p
kt) +
k
2
p
k
ˆµk(0) sinh(
p
kt)
ˆµk(t) =

1
2
ˆµk(0) +
p
k
k
ˆzk(0) e
p
kt
+

1
2
ˆµk(0)
p
k
k
ˆzk(0) e
p
kt
= ˆµk(0) cosh(
p
kt) + 2
p
k
k
ˆzk(0) sinh(
p
kt)
ˆzk(t) = ˆzk(0) cos(
p
kt) +
k
2
p
k
ˆµk(0) sin(
p
kt),
ˆµk(t) = ˆµk(0) cos(
p
kt)
2
p
k
k
ˆzk(0) sin(
p
kt)
k < 0 Waves Solution
Rayleigh-Taylor Instabilityk 0
Linearized Problem Solution
ˆz00
k (t) =
k
2
ˆµ0
k =
k
2
✓
B
2
k2
2G
◆
ˆzk(t) = k ˆzk(t)

✓
@
@t
◆2
⇠
✓
@
@↵
◆3
⇠
✓
@
@
◆3
✓
@
@t
◆
⇠
✓
@
@↵
◆3/2
⇠
✓
@
@
◆3/2
4t ⇠ (Emh)3/2
, Em = min
(↵, )
E(↵, )
Relation between Time and Space Derivatives
Symbolically
Time step Stability Constraint
ˆz00
k (t) =
k
2
ˆµ0
k =
k
2
✓
B
2
k2
2G
◆
ˆzk(t) = k ˆzk(t)

Dispersion Relation k =
Eb
4
k5 B
4
k3
Gk
ˆz00
k (t) =
k
2
ˆµ0
k =
k
2
✓
Eb
2
k4 B
2
k2
2G
◆
ˆzk(t) = k ˆzk(t)
Elastic Waves Linearized Problem
Relation between Time and Space Derivatives
✓
@
@t
◆2
⇠
✓
@
@↵
◆5
⇠
✓
@
@
◆5 ✓
@
@t
◆
⇠
✓
@
@↵
◆5/2
⇠
✓
@
@
◆5/2
Time step Stability Constraint
4t ⇠ h5/2

Xn+1
= Xn
+ 4t Vn
1 · ˆt1 + Vn
2 · ˆt2 + Un
· ˆn
µn+1
= µn
+ 4t

µn
↵
p
En
V n
1 (Wn
· ˆtn
1) +
µn
p
nEn
V n
2 (Wn
· ˆtn
2)
4t (Bn
+ 2Agzn
)
(t) =
R 1
0
R 1
0
G(↵, , t) d↵d
R 1
0
R 1
0
E(↵, , t) d↵d
EXPLICIT DISCRETIZATION
fn
= fn
ij = f(↵i, j, tn
), i, j = 1, ..., N; n = 0, 1, ...
Boussinesq Approximation

Xn+1
4t V n+1
1s · ˆtn
1 + V n+1
2s · ˆtn
2 + Un+1
s · ˆnn
= Xn
+ 4t
⇥
(Un
Un
s ) · ˆnn
+ (V n
1 V n
1s)ˆtn
1 + (V n
2 V n
2s)ˆtn
2
⇤
µn+1
+ A n+1
+ 4t Bn+1
+ 2Gzn+1
= µn
+ A n
+ 4t
(
µn
↵
p
En
V n
1 (Wn
· ˆtn
1) +
µn
p
En
V n
2 (Wn
· ˆtn
2)
+A
"
2(Wn
· ˆnn
)2
+ 2V n
1 (Wn
· ˆtn
1) + 2V n
2 (Wn
· ˆtn
2)
(µn
↵)2
4En
(µn
)2
4 En
|Wn
|2
# )
IMPLICIT DISCRETIZATION
Arbitrary Atwood Number
n+1
=
Ln+1
+ Nn+1
En
, Ln+1
= Xn+1
↵↵ · ˆnn
, Nn+1
= Xn+1
· ˆnn

Xn+1
+ R(Xn+1
, µn+1
p ) = f(Xn
, µn
p )
µn+1
p + Q(Xn+1
) = h(Xn
, µn
p )
µp := µ + A for convenience µ0
p ⌘ µ0
⌘ 0
µn+1
p = µn+1
+ A n+1
and Xn+1
GMRES solves for
µn+1
+ A2 ˜Kµn+1
= µn+1
p
= 2 ˜Kµ =
1
2⇡
Z Z
µ0
(X X0
) · ˆn
|X X0
|3
|X0
↵ ⇥ X0
| d↵0
d 0
GMRES solves for µn+1
Init guess
Lin System for the Fourier components
˜µn+1
:= µn+1
p A (Xn+1
, µn
)

= 2
Z p
E W · ˆt1 d↵ + R( )
= 2
Z p
G W · ˆt2 d + Q(↵)
R( ) are only modes of 2
Z p
G W · ˆt2 d
Q(↵) are ↵ only modes of 2
Z p
E W · ˆt1 d↵
Integrals are efﬁciently computed using
the Fourier transform technique
Computation of the Velocity Potential

Fast Computation of the Birkhoﬀ-Rott Integral
In a doubly-periodic domain integral over
the fundamental root cell C
G(X, X0
) =
X
m2Z2
˜G(X X0
2 ˜m⇡), ˜m = (m, n)
˜G(X X0
2 ˜m⇡) =
1
[(x x0 2m⇡)2 + (y y0 2n⇡)2 + (z z0)2]1/2
G(X, X0
) =
X
˜m2Z2
0
✓
1
2
˜G(X X0
2 ˜m⇡) +
1
2
˜G(X X0
+ 2 ˜m⇡)
1
2⇡|m|
◆
Sum of periodic ext. of the free space Green’s function
For conditional convergence add reﬂection and const
W =
1
4⇡
PV
Z
C
Z
(µ↵X µ X↵)0
⇥ rXG(X, X0
) d↵0
d 0
Beale ‘04

The Ewald summation technique converts the
slowly convergent sum of algebraic functions into a
rapidly convergent sum of transcendental functions
G(X, X0
) =
1
4⇡
X
m2Z2
˜Rmn(z z0
) cos m(x x0
) cos n(y y0
)
2erfc(m
⇠ )
m
!
+
1
2
X
m2Z2
0
✓
erfc(
p
s⇠)
p
s
erfc(⇡⇠m)
⇡m
◆
+
⇠
p
⇡
˜Rmn(z z0
) =
1
p
m2 + n2
"
e
p
m2+n2(z z0
)
erfc
p
m2 + n2
⇠
+
z z0
2
⇠
!
+e
p
m2+n2(z z0
)
erfc
p
m2 + n2
⇠
z z0
2
⇠
! #
.
s = [(x x0
)/2 m⇡]2
+ [(y y0
)/2 n⇡]2
+ [(z z0
)/2]2
G = Gb + GaEwald Sum⇠ -decay rate parameter

• Integral over the Reciprocal sum is computed with the
trapezoidal rule method spectrally accurate for the periodic
functions
• Integral over the Real space sum is computed with the
method of Haroldsen and Meiron ’98 chosen to give third
order accuracy in space
• Balancing the workload for Reciprocal and Real Space sum
we get overall operation countO(N3/2
)
FAST EWALD SUMMATION

max of num. solution and solution of the linearized problem
x(↵, ) = 2⇡↵
y(↵, ) = 2⇡
z(↵, ) = A0 cos(2⇡↵) cos(2⇡ )
Initial condition
xlin(↵, , t) = 2⇡↵
ylin(↵, , t) = 2⇡
zlin(↵, , t) =
X
k
bzk(0) exp[ (k)t + ik · ~↵]
Linearized problem solution
Numerical Solution Validation
A0 = 0.1
A0 = 0.5

Invariants of motion check
Relative error of the total energy
for varying the time step
A = 0.9, Ag = 1, B = 0.01,
N = 64, 4t = 0.001

Agreement with Baker
Growing modes solution
Explicit and Implicit method
solution match check
Baker, Caﬂisch, Siegel ‘93

• Gravity
• Surface tension
• Gravity & surface tension
Internal Waves

vs

Growing Solution

Stability (Explicit method)
A = 0, Ag = 10, B = 1, Eb = 0.1, t = 0.04

The largest stable time step
t  CE3/2
m /(BN3/2
)
Stability

(Surface Tension Case)

Stability (Hydroelastic Problem)
Aliasing Filter
⇢N (k, l) = exp
(
20
✓
k
N/2
◆10
+
✓
l
N/2
◆10
!)
The largest stable time step
4t ⇠ h5/2

High resolution with aliasing ﬁlter
A = 0, Ag = 5, B = 0, Eb = 0.1, t = 1.625, N = 128, 4t = 0.0025

• Developed method is effective at removing the stiffness
introduced by the surface tension and elasticity
• High order time-step stability constraint for explicit
methods is eliminated by the use of small-scale
decomposition method with semi-implicit discretization
• Method is computationally efﬁcient requiring work
comparable to explicit method
• Presented algorithm can be made arbitrary order accurate
CONCLUSION

G = Gb + Ga
-Reciprocal sum containing far-ﬁeld contributions
-Real space sum containing local contributions
Gb
Ga
Integral over the Reciprocal sum
Ib
(X) =
Z
⇧
f(↵, )rGb
(X, X0
) d↵d
Integral over the Real space sum
Ia
(X) =
Z
⇧
f(↵, )rGa
(X, X0
) d↵d
Ewald Sum

An Efficient Boundary Integral Method for Stiff Fluid Interface Problems

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