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AN EFFICIENT BOUNDARY INTEGRAL METHOD
FOR STIFF FLUID INTERFACE PROBLEMS
Oleksiy Varfolomiyev
advisor Michael Siegelco-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
MATHEMATICAL MODEL
MODEL DESCRIPTION
• Gravity
• Surface Tension
• Elastic Bending Stress
Fluids are driven by
GOVERNING EQUATIONS
Interface parameterizationX(~↵, 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 EquationXt = V1t
1 + V2t2 + U n
r�i = ViVelocity potential
BOUNDARY CONDITIONS
Kinematic Boundary Condition
Laplace-Young Boundary Condition
Far-field Velocity
V1 · n = V2 · n = U on S
p1 � p2 = �(1 + 2) on S
V �! 0 as z ! ±1
[p] = �Eb4s� 2Eb3 + 2�+ 2Ebg
Pressure Jump across elastic Interface
Fluid Velocity on S given by the Birkhoff-Rott Integral
W(X) = PV
Z 1
�1
Z 1
�1j0 ⇥rXG(X,X0) d↵0d�0 +V1n,
=1
4⇡PV
Z 1
�1
Z 1
�1j0 ⇥ (X�X0)
|X�X0|3 d↵0d�0 +V1n
Generalized Isothermal ParameterizationE = X↵ ·X↵
F = X↵ ·X�
G = X� ·X�
L = X↵↵ · nM = X↵� · nN = X�� · n
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 = V1t1 + V2t
2 + (W · n)n
Interface Evolution Equation
µt +A�t =µ↵pE
�V 1 � (W · t1)
�+
µ�p�E
�V 2 � (W · t2)
�
+A
"2(W · n)2 + 2V1(W · t1) + 2V2(W · t2)� µ2
↵
4E�
µ2�
4�E� |W|2
#
�B� 2Gz
Vortex Sheet Density Evolution Equation
� = �1 + �2 = ��L+N
�EA =
⇢1 � ⇢2⇢1 + ⇢2
G =AgL
v2cB =
2�
L(⇢1 + ⇢2)v2c
ELASTIC INTERFACEPressure Jump across the Interface
[p] = �Eb4s� 2Eb3 + 2�+ 2Ebg
g =LN �M2
E2
Eb
Gaussian curvature
Bending modulus
Surface Laplacian
Plotnikov, Toland ‘11
4s =1p
EG� F 2
@
@�
E @
@� � F @@↵p
EG� F 2
!+
1pEG� F 2
@
@↵
G @
@↵ � F @@�p
EG� F 2
!
µt +A�t =µ↵pE
�V 1 � (W · t1)
�+
µ�p�E
�V 2 � (W · t2)
�
+A
"2(W · n)2 + 2V1(W · t1) + 2V2(W · t2)� µ2
↵
4E�
µ2�
4�E� |W|2
#
+Eb(4s� 23 + 2g) + B+ 2Agz
-Equation for the elastic interfaceµ
Tangential Velocities are chosen to preserve parameterization✓
V1pE
◆
↵
�✓
V2p�E
◆
�
=��tE + 2U (�L�N)� �(�E �G)
2�E✓
V2pE
◆
↵
+
✓V1p�E
◆
�
=2UM � �Fp
�E
(�E �G)t = 0
Ft = 0
��(�E �G) ��FRelaxation Terms
THE BOUNDARY INTEGRAL METHOD
Normal Velocity Decomposition
U(X) ⌘ W · n = 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
�1j0 ⇥
X0↵(↵� ↵0) +X0
�(� � �0)
E0 32h(↵� ↵0)2 + �(t) (� � �0)2
i 32
d↵0d�0
+1
4⇡PV
Z 1
�1
Z 1
�1j0⇥
8><
>:X�X0
|X�X0|3 �X0
↵(↵� ↵0) +X0�(� � �0)
E0 32h(↵� ↵0)2 + �(t) (� � �0)2
i 32
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
n(µ↵(↵� ↵0) + µ0�(� � �0))
E0 12h(↵� ↵0)2 + �(t) (� � �0)2
i 32
d↵0d�0 · n
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
i 32
d↵0d�0,
H2f(↵,�) =1
2⇡PV
Z 1
�1
Z 1
�1
f(↵0,�0)(� � �0)h(↵� ↵0)2 + �(t) (� � �0)2
i 32
d↵0d�0
Riesz Transforms
Us = Ws · n = �1
2
H1
✓µ↵n
E12
◆+H2
✓µ�n
E12
◆�· n
bH1f(k) = �ik1k�
fk, bH2f(k) = �ik2�k�
fk
Symbols of the Riesz Transforms
k� =q�k21 + k22
Xt = Usn+ V1st1 + V2st
2
+ (U � Us)n+ (V1 � V1s)t1 + (V2 � V2s)t
2
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
INVARIANTS OF THE MOTION
Conservation of Mass / Mean Height of the Interface
z =1
4⇡2
Z 2⇡
0
Z 2⇡
0z dxdy =
1
4⇡2
Z 1
0
Z 1
0z |x↵y� � y↵x� | d↵d�
Potential Energy
E
p
= (⇢1 � ⇢2)g
Z 2⇡
0
Z 2⇡
0
Zz(x,y)
0z dxdydz
= (⇢1 � ⇢2)g
Z 2⇡
0
Z 2⇡
0
z
2
2dxdy
= (⇢1 � ⇢2)g
Z 1
0
Z 1
0
z
2
2|J | d↵d�
Jacobian |J | = |x↵y� � y↵x� |
Surface Energy
Es = �
Z
S
ZdS = �
Z 1
0
Z 1
0|X↵ ⇥X� | d↵d�
Ek = E1k + E2
k =1
4(⇢1 + ⇢2)
Z 1
0
Z 1
0(µ+A�)U |X↵ ⇥X� | d↵d�
Kinetic Energy
� = �1 + �2
E
1k
=1
2⇢1
Z 2⇡
0
Z 2⇡
0
Zz(x,y)
0|r�1|2 dxdydz
µ = �1 � �2 �1 = (�+ µ)/2 �2 = (�� µ)/2
Elastic Bending Energy
Esb =
p�
2
Z 1
0
Z 1
0Eb 2 |E| d↵d�
E
tot
= A
Z 1
0
Z 1
0z
2|x↵
y
�
� y
↵
x
�
| d↵d�
+
p�
2
Z 1
0
Z 1
0(µ+A�)U |E| d↵d�
+B
p�
Z 1
0
Z 1
0|E| d↵d�
+
p�
2
Z 1
0
Z 1
0E
b
2 |E| d↵d�
Total Energy
Plotnikov, Toland ‘11
SYMMETRY
X(�↵,��) = (�x(↵,�),�y(↵,�), z(↵,�)),
µ(�↵,��) = µ(↵,�)
If initial condition possess the symmetry
then symmetry is preserved with the evolution of S
Remarkz 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
LINEAR STABILITY ANALYSIS
Perform a small perturbation of the flat interface with zero mean vortex sheet strength
X = (2⇡↵+ x
0, 2⇡� + y
0, z
0), with |x0|, |y0|, |z0| ⌧ 1,
µ = µ
0, with |µ| ⌧ 1
Interface velocity at the leading order
W0 = � 1
8⇡2PV
Z 1
�1
Z 1
�1
µ0↵(↵� ↵0) + µ0
�(� � �0)
[(↵� ↵0)2 + �(t)(� � �0)2]3/2d↵0d�0 k
Normal Velocity
U = � 1
4⇡(H1 (µ↵) +H2 (µ�))
Evolution Equations
dµ0
dt= �B� 2Gz0
X0t = (W0 · k)k ⇡ U 0 k
Linear System of ODE’s for the Fourier Componentsd
dt
✓zkµk
◆=
✓0 k�
2�� B
2�k2� � 2G 0
◆✓zkµk
◆
LINEARIZED EQUATIONS
f(↵) =X
k
f(k) eik·↵ f(k) =
Z 1
0
Z 1
0e�2⇡ik·↵f(↵) d↵ k� =
q�k21 + k22
Dispersion Relation �k = � B4�2
k3� � G�k�
zk(t) =
k�
4�p�k
µk(0) +1
2
zk(0)
�ep�kt
+
1
2
zk(0)�k�
4�p�k
µk(0)
�ep��kt
= zk(0) cosh(p�kt) +
k�2�
p�k
µk(0) sinh(p�kt)
µk(t) =
1
2
µk(0) +�p�k
k�zk(0)
�ep�kt
+
1
2
µk(0)��p�k
k�zk(0)
�e�
p�kt
= µk(0) cosh(p�kt) + 2
�p�k
k�zk(0) sinh(
p�kt)
zk(t) = zk(0) cos(p��kt) +
k�2
p��k
µk(0) sin(p��kt),
µk(t) = µk(0) cos(p��kt)�
2
p��k
k�zk(0) sin(
p��kt)
�k < 0 Waves Solution
Rayleigh-Taylor Instability�k � 0
Linearized Problem Solution
z00k (t) =k�2�
µ0k =
k�2�
✓� B2�
k2� � 2G◆zk(t) = �kzk(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
z00k (t) =k�2�
µ0k =
k�2�
✓� B2�
k2� � 2G◆zk(t) = �kzk(t)
Dispersion Relation �k = �Eb
4k5 � B
4k3 � Gk
z00k (t) =k
2µ0k =
k
2
✓�Eb
2k4 � B
2k2 � 2G
◆zk(t) = �kzk(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
NUMERICAL METHOD
Xn+1 = Xn +4t�Vn
1 · t1 +Vn2 · t2 +Un · n
�
µn+1 = µn +4t
µn↵pEn
�V n1 � (Wn · tn1)
�+
µn�p
�nEn
�V n2 � (Wn · tn2)
��
�4t (Bn + 2Agzn)
�(t) =
R 10
R 10 G(↵,�, t) d↵d�
R 10
R 10 E(↵,�, t) d↵d�
EXPLICIT DISCRETIZATIONfn = fn
ij = f(↵i,�j , tn), i, j = 1, ..., N ;n = 0, 1, ...
Boussinesq Approximation
Xn+1 �4t�V n+11s · tn1 + V n+1
2s · tn2 + Un+1s · nn
�
= Xn +4t⇥(Un �Un
s ) · nn + (V n1 � V n
1s)tn1 + (V n
2 � V n2s)t
n2
⇤
µn+1 +A�n+1 +4t�Bn+1 + 2Gzn+1
�
= µn +A�n +4t
(µn↵pEn
�V n1 � (Wn · tn1)
�+
µn�p
�En
�V n2 � (Wn · tn2)
�
+A
"2(Wn · nn)2 + 2V n
1 (Wn · tn1) + 2V n2 (Wn · tn2)�
(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
↵↵ · nn, Nn+1 = Xn+1�� · nn
Xn+1 +R(Xn+1, µn+1p ) = f(Xn, µn
p )
µn+1p +Q(Xn+1) = h(Xn, µn
p )
µp := µ+A� for convenience µ0p ⌘ µ0 ⌘ 0
µn+1p = µn+1 +A�n+1 and Xn+1GMRES solves for
µn+1 +A2Kµn+1 = µn+1p
� = 2Kµ =1
2⇡
Z Zµ0(X�X0) · n
|X�X0|3|X0
↵ ⇥X0� | d↵0d�0
GMRES solves for µn+1Init guess
Lin System for the Fourier components
µn+1 := µn+1p �A�(Xn+1, µn)
� = 2
Z pE�W · t1
�d↵+R(�)
� = 2
Z pG�W · t2
�d� +Q(↵)
R(�) are � � only modes of 2
Z pG�W · t2
�d�
Q(↵) are ↵� only modes of 2
Z pE�W · t1
�d↵
Integrals are efficiently computed using the Fourier transform technique
Computation of the Velocity Potential
Fast Computation of the Birkhoff-Rott IntegralIn a doubly-periodic domain integral over
the fundamental root cell C
G(X,X0) =X
m2Z2
G(X�X0 � 2m⇡), m = (m,n)
G(X�X0 � 2m⇡) =1
[(x� x
0 � 2m⇡)2 + (y � y
0 � 2n⇡)2 + (z � z
0)2]1/2
G(X,X0) =X
m2Z2
0✓1
2G(X�X0 � 2m⇡) +
1
2G(X�X0 + 2m⇡)� 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↵0d�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 � z
0) cosm(x� x
0) cosn(y � y
0)�
2erfc(
m⇠ )
m
!
+
1
2
X
m2Z2
0✓erfc(
ps⇠)p
s
� erfc(⇡⇠m)
⇡m
◆+
⇠p⇡
Rmn(z � z0) =1p
m2 + n2
"epm2+n2(z�z0)erfc
pm2 + n2
⇠+
z � z0
2⇠
!
+e�pm2+n2(z�z0)erfc
pm2 + n2
⇠� z � z0
2⇠
!#.
s = [(x� x
0)/2�m⇡]2 + [(y � y
0)/2� n⇡]2 + [(z � z
0)/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
NUMERICAL SIMULATION
max of num. solution and solution of the linearized problem
x(↵,�) = 2⇡↵
y(↵,�) = 2⇡�
z(↵,�) = A0 cos(2⇡↵) cos(2⇡�)
Initial conditionxlin(↵,�, t) = 2⇡↵
ylin(↵,�, t) = 2⇡�
zlin(↵,�, t) =
X
k
bzk(0) exp[�(k)t+ ik · ~↵]
Linearized problem solutionNumerical 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 CE3/2m /(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