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