Mathematics of Triple-Junction and Singularity Theory: from CFD …smatsutani.jp/CFD_Singularity20160602.pdf · 2019-06-19 · Shigeki Matsutani. National institution of technology,

Shigeki MatsutaniNational institution of technology, Sasebo college

June 3, 2016

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Mathematics of Triple-Junction and Singularity Theory:

from CFD of ink-jet printer

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１．Background２．Motivation:

From Ink-jet printer３．Math of Triple-Junction and Singularity

３-1. Problem３-２. Physics of Triple-Junction３-３. Numer. model of Two-phase field３-４. Numer. Model of Triple-Junc..

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1990 2000 2010 2016

Canon: industrial Math Analysis

Abelian Func. TheoryEmma Previato

Quantized ElasticaJohn McKay

Integrablesystem

Today’s topic

Study on Singular strata in Jacobi variety

Caustics problem

JMSJ 2007,2014

Math is of use!

for other math fields:e.g., singularity theory is of use for abelian function theory.

for acutal life:e.g., singularity theory is ofuse for ink-jet printer.

Both are different!

Math is of use for actual life!e.g., for an ink-jet printer.

Leonardo da Vinci said “Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics.”

In XXI century, we have so many cases that mathematical fields have the strong power for industrial studies.No one predicts what mathematical subject is of use, in this sense, after decades because no one predicts what technology is of use for actual life, basically.

Math is of use for actual life!e.g., for an ink-jet printer.

Leonardo da Vinci said “Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics.”

It is probabilistic whether some mathematical fields are of use or not. It cannot be predicted!⇒ Every math field has the possibility but

it is important that we should not be tooconscious whether our theory becomes ofuse or not. We study it to be established!

⇒ If a math field is well-established, it will be of use, in this sense, when the time comes.

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Purpose of study: to evaluate micro fluid dynamics in an ink-jet printer numerically.

Numerical computation:emission of ink droplets

Emission device of ink droplets

Several x 10 um

heater

Vapor bubble

Model (M-Nakano-Shinjo 2011)

Matsutani, Nakano, Shinjo,Math Phys Anal Geom (2011) 14:237–278

heaterEmission device of ink droplets

Triple-Junction is Singular!Triple-phase：Solid, Liquid, Gas!

125

Purpose of study: to evaluate micro fluid dynamics in an ink-jet printer numerically.

The singularity of Triple-Junction isthe cone C{0,1} or C[0,1]!

C{0,1} ×(0,1) ⊂ C[0,1] × (0,1)

Today’s topic is the singularity of Triple-Junction!

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Mathematical Problem

To construct a mathematical model in order to evaluate fluid dynamics with triple-phase field and triple junctions numerically!

We can assume that these fields are liquids (incompressible fluids).

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There was a mathematical model of numerical two-phase fluid dynamics!

差分化Discretization

2-phaseEuler Equation


3-phase DifferenceEuler Equation

Applied Math（VOF-method）

Discretization


By revising the mathematical model of two-phase fluid dynamics to obtain that of three-phase field!

差分化

Applied Math（phase-field theory）Arnold, Ebin-Marsden 1970

Diffeomorphism Group+Momentum Map DiscretizationVariation

Discretization

Discretization



1-phaseAction Functional







Variation

Variation


Singularity Theory

AnalyticGeometry

差分化



Discretization

Discretization










Variation

Variation


Singularity Theory

AnalyticGeometry

In order to obtain the three-phase Euler equation, we used primitive facts in

1. Diffeomorphism Group2. Momentum Map3. Analytic Geometry4. Singularity Theory.

差分化



Discretization

Discretization










Variation

Variation


Singularity Theory

AnalyticGeometry

Today’s topic!

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Triple-Junction

http://photo-pot.com/?p=23192

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http://nagare-furoshiki.com/ja/repel.html

超撥水

super-water-repellent

Reduced Mathematical Problem

To construct a mathematical model in order to evaluate the triple junctions numerically!

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Shape of Droplet is determined so that it minimizes the interface energy under the preservation of volume.(For a small droplet, the gravity is neglected!)

A:Liquid

B : Solid

C:Gas

Shape of Droplet with Triple-Junction

SCA : Area of Interface between C and ASAB : Area of Interface between A and BSBC : Area of Interface between B and Cσ*# :positive real number dep. on * and #.

E = σCASCA ＋ σABSAB ＋ σBCSBC

Interface Energy：


A:Liquid

B : Solid

C:Gas


SCA : Area of Interface between C and ASAB : Area of Interface between A and BSBC : Area of Interface between B and Cσ*# : Surface tension value of * and #.


Interface Energy：


A:Liquid

B : Solid

C:Gas


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p1-p２=σ・κ(x)

Laplace equation for surface tension

Shape of interface (surface tension)

κ= div n for the normal unit vector field nκ／２ is the mean curvature,

p1,p２:pressureσ:surface tension value

The constant mean curvature surface is realized.

σBC = σAB ＋ σCAcos θ

θ

σCA

σBC

σAB

A:Liquid

B : Solid

C:Gas

Shape of Droplet with Triple-Junctionθ： Contact angle

part of sphere

(The pressures in matters are naturally defined!)


= σCASCA ＋ σABSAB ＋ σBC (Const.ーSAB ）

= σCASCA ＋(σABーσBC)SAB ＋Const.

SAB = πr2 sin2θSCA = 2πr2 (1ーcosθ)

VA = ーπr3 (2+cosθ)(1ーcosθ)243

A:Liquid

B : Solid

C:Gas


EL =σCASCA＋(σABーσBC)SAB＋λ(VA-V0)

ーー = 0，ーー = 0，ーー = 0δEL

δrδEL

δθδEL

δλ



A:Liquid

B : Solid

C:Gas

EL = πr2[2σCA (1ーz)+ (σABーσBC) (1ーz2)ーrλ((2+z)(1ーz)2/3ーV0)]

z= cosθ

πr[4σCA (1ーz)+ 2(σABーσBC) (1ーz2)

ーrλ(2+z)(1ーz)2]=0

πr2 [ー2σCA ー2(σABーσBC) z

+rλ(1+z)(1ーz)]=0

ーー = 0：δEL

δr

ーー = 0：δEL

δz

σCA z +(σABーσBC)=0



σCAσBC

σAB

σBC ＞ σAB ＋ σCA

θ is not defined! → Wet



θ

σCA

σBC

σAB

σBC > σAB

θσCA

σBC

σAB

σBC < σAB

Shape of Droplet with Triple-JunctionσAB ＋ σCA ＞ σBC

(σBC ＞ σAB - σCA )

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http://nagare-furoshiki.com/ja/repel.html

超撥水


super-water-repellent

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Numer. model of Two-phase field

Problem

To represent two-phase field approximately using a function over a region

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Why function？We consider the lattice in a concerned region.

Why function？

By using functions over lattice points, cells, and edges, we represent the matters.

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Interface =Zero crossing

Intermediate region

Phase field method

Level set methodBasic Approachs!

ε

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ξ=0

ξ=1

ξ

39


Intermediate region

Phase field method


ε

39

ξ=0

ξ=1

ξ

We employed the phase field method

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Intermediate region

Phase field method

Phase field method

ε

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ξ=0

ξ=1

ξ

By making the widthεof the intermediate regions over several lattice units, the points whose value is 0.5 represent the interface!

Intermediate region

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The phase field method for two-phase field requires a smooth partition of unityover a continuous domain in R3

ξ(x)＋ξｃ(x)=1

ξ=0

ξ=1

Phase field method

1. We consider the phase field with intermediate region ofε or a smooth partition of unity over a continuous domain in R3

2. We compute the interface energy approximately.3. By variational method, we derive an differential equation.

Approach

The difference equation of the shape, numerical model

Discritization(Restriction to a lattice points)

It is not difficult to obtain a difference equation of the shape from the differential equation.

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Ω：3 dim Domain Ω⊂R3

Ｂ：a connected compact set in ΩIntegral Representation of Area A of ∂Ｂ

Ｂ

Ω{ 1 x∈Bo

0 x∈Bcθ(x)=

Characteristic Function

A = |∇θ(x)| d3x

θ is differentiable as a hyperfunction.

∫d3x := dx1dx2dx3

Ω

44q

s1s2

A = |∇θ(x)| d3x∫= ∂ｑθ(q)dq d2s

q(x)=0

∫( )∫= dθ d2s∫( )∫ = d2s∫

∂B∂B

Ω

Ω

ξθ

Category of the hyperfunctions

Category of the smooth functions

(up to ε)

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Ω{ 1 x∈Mo

0 x∈Lcξ(x)=

Aξ = |∇ξ(x)| d3x

ξ: smooth, monotonic increasing for normal direction

∫

ε/2=dist(p,B), p∈M, Lc

M

LＢ

｜Aξ ーA | < ε・KAξ

Integral approximation of the area of ∂Ｂ in terms of ξ-function

Filtration of closed sets M ⊂ Ｂ ⊂ L s.t.

KAξ>0

(connected)

Ω

q

s1s2

Aξ = |∇ξ(x)| d3x∫= ∂ｑξ(q)dq d2s

q(x)=0

∫( )∫< |１＋ε／min Ra(s)| d

2s∫∂B

<(1+ ε／Rinf) d2s∫∂B

a=1,2

Rinf :inf. of the principal curvature radius Ra(s)

Ω

Ω

Assumeε<< Rinf :inf of the principal curvature radius

NG case:

κ／２： approximation of mean curvature because mean curvature is given as

div n/2for the normal unit vector field n

Integral approximation of the area of ∂Ｂ in terms of ξ-function

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We have a constant mean curvature surface approximately for constant p1 and p2.

variational method

Approximation of Laplace equation

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Phase field model with triple-junction

To express the cone singularity in terms of phase-field method

GasLiquid

Solid

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Idea：1. Express these regions in terms of smooth

partition of unity.2. Compute the integral approximation of the area

of the interfaces!3. We derive the Laplace equation approximately.

Problems：・To estimate the effect from the triple-junction on the integral since it is singular.

・To show that the model is not ill-posed!


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These problems are of the category of Ｃ∞-functions since we consider smooth partition of unity!

But it is difficult to estimate it and we use the idea of the singularity theory!


Problems：・To estimate the effect from the triple-junction on the integral since it is singular.

・To show that the model is not ill-posed!

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Stratification:

Ｖ1 ⊂ Ｖ2⊂ Ｖ3

There is a natural stratification (滑層分割) and we consider “differentiable” submanifolds.

We express the triple-phases in terms of closed sets whose union corresponds to the domain and intersection corresponds to interface!

B1∩B２ ⊂∂Ba, a=1,2

B1∩B２∩B3 ⊂∂(B1∩B２),

B1B2

B3

Ba : closed set

Ω＝B1∪B２∪B3

Stratification:

B1B2

B3

B1∩B２ ⊂∂Ba, a=1,2

B1∩B２∩B3 ⊂∂(B1∩B２),

B1∩B２∩B3 ⊂ B1∩B２ ⊂B1,

Stratification:

B1B2

B3

{1 x∈Ma

o

0 x∈Lacξa (x)=

Ma⊂Ｂa⊂La ε=dist(p, Ba) p∈ Ma, Lac

ξ1(x)＋ξ2 (x)＋ξ3 (x)=1

ξasmooth, mono.-increasing for normal dir.

Ma，La：closed sets

Partition of Unity

(connected)

{1 x∈Ma

o

0 x∈Lacξa (x)=


ξ1(x)＋ξ2 (x)＋ξ3 (x)=1



Partition of Unity

ｓM1

M2

M3

(connected)

{1 x∈Ma

o

0 x∈Lacξa (x)=


ξ1(x)＋ξ2 (x)＋ξ3 (x)=1



Partition of Unity

ｓL1

L2

L3

ε (connected)

F(ξ)＝ (∑σab（|∇ξa| |∇ξb|)1/2(ξa + ξb)-p(x))d3x∫

Ωa,b

p(x)＝ ∑ｐaξa（ｘ）

Integral Approximation of the area in terms of the Partition of Unity

Aab ＝（|∇ξa| |∇ξb|)1/2(ξa + ξb) d3x∫

Ω

Surface Free Energy

ξ

We derive the equation using the stratified structure of this system!

1. x∈Ω，s.t. ξa(x)=1,

2. x∈supp(ξa)∩supp(ξb)=La∩Lb

s.t. ξa(x)+ξb (x)=1,

3. supp(ξ1)∩supp(ξ2)∩supp(ξ3)=L0∩L1∩L2

ｓM1

M2

M3

B1B2

B3

We estimate the accuracy of this model each part:

1. x∈Ω，ξa(x)=1

ーーF[ξ] =0δδξ

０＝０

Results of variation method

for each region:

ｓM1

M2

M3

65We show that this equation does not have bad effects on the model!

2. x∈supp(ξa)∩supp(ξb) s.t. ξa(x)+ξb (x)=1

→ reproduces the interface of two-phase field

3. x∈ supp(ξ1)∩supp(ξ2)∩supp(ξ3)

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We evaluate the energy and consider it weakly up toε!

1. x∈Ω，ξa(x)=1,




ｓM1

M2

M3

Evaluate the surface energy!

E(□)＝ (∑σab（|∇ξa| |∇ξb|)1/2(ξa + ξb))d

3x∫□

a,b

e

Then we have the following estimations:

1. x∈Ω，ξa(x)=1,E({x∈Ω，ξa(x)=1})=0


E(□)＝ (∑σab（|∇ξa| |∇ξb|)1/2(ξa + ξb))d

3x∫□

a,b

e


For ε to 0, the model gives the energy well.



｜Ｅ(supp(ξ1)∩supp(ξ2))ーσ12 A12|＜εK


E(□)＝ (∑σab（|∇ξa| |∇ξb|)1/2(ξa + ξb))d

3x∫□

a,b

e

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For ε to 0, the model gives the energy well.


Ｅ(supp(ξ1)∩supp(ξ2)∩supp(ξ3))＜εK’


E(□)＝ (∑σab（|∇ξa| |∇ξb|)1/2(ξa + ξb))d

3x∫□

a,b

By applying the idea of the stratification to this model, we can estimate the accuracy of this model.

We conclude that this model approximates the triple-junction well up to ε!

ｓM1

M2

M3

B1B2

B3

(c)

30[degree]

(a)

45[degree]

(b)

60[degree]

90[degree]

(d)

120[degree]

(e)

Using the model (via the difference Euler equation with viscous force term), the surface tension values reproduce the expected contact angle !

Matsutani, Nakano, Shinjo,Math Phys Anal Geom (2011) 14:237–278

73

Reduced Mathematical Problem

To construct a mathematical model in order to evaluate the triple junctions numerically!

To construct a mathematical model in order to evaluate fluid dynamics with triple-phase and triple junctions numerically!

Mathematical Problem

差分化Discretization





Discretization


By revising the mathematical model of two-phase fluid dynamics to obtain that of three-phase field!

差分化



Discretization

Discretization










Variation

Variation


Singularity Theory

AnalyticGeometry

t=1.0[µsec] t=2.0[µsec]

t=4.0[µsec] t=5.0[µsec]t=3.0[µsec]

30[degree]

time

t=0.0[µsec]

Meniscus MotionMatsutani, Nakano, Shinjo,Math Phys Anal Geom (2011) 14:237–278

We can compute the three-phase fluid dynamics with triple-junction numerically!

77

We can compute the three-phase fluid dynamics with triple-junction numerically!

Numerical computation:emission of ink droplets

Emission device of ink droplets

Several x 10 um

heater

Vapor bubble

78

Thank you for your attention!

79

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Modified Laplace equation：

Modified Laplace equation：

82

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phase field法による2相の流体方程式（オイラー方程式）：

変分原理

84


Intermediate region

Phase field method


ε

84

ξ=0

ξ=1

ξ

Level Set method

Zero points of signed distant function express the interface!⇔ The ridge appears inside!

Its singularity causes difficulty.


Level set method

86Intermediate region

Phase field method

Phase field method

遷移領域

ε

86

ξ=0

ξ=1

ξ

By making the widthεof the intermediate regions over several lattice units, the points whose value is 0.5 represent the interface!

87

We employed the phase field method

The phase field method for two-phase requires a smooth partition of unity overa continuous domain in R3

ξ(x)＋ξｃ(x)=1 ξ=0

ξ=1

88

Deviation from thoday’s topic

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Singularity of light ray＝Caustics

Michel Berry

Catastrophe theory and Caustics1970’s.

http://blog.sigfpe.com/2005/08/caustics-and-catastrophes.html

Orbits from an electron emission device for an display cause caustics. (1998）Though it has a bad effect on phosphor (蛍光体), it cannot be avoided due to the stability. We could concentrate ourselves on another countermeasure.

experiment ｖｓ theory

Singularity theory is of use for many situations in industrial studies besides inkjet printer!

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q

s1s2

Aξ = |∇ξ(x)| d3x∫= ∂ｑξ(q)dq d2s

q(x)=0

∫B( )∫

< |１＋ε／min Ra(s)| d2s∫

B<(1+ ε／Rinf) d2s∫

∂B

a=1,2

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1, ｒ＜Rーε/2,0, r＞R+ε/2,(r-R)/ε, otherwise

ξ(ｒ)=

Aξ = |∇ξ(x)| d3x∫

= 4π(R+q)2/ε dq∫= 4π/ε [R2q+Rq2+q3/3]ε/2

ーε/2

ε/2

ーε/2

= 4πR2+πε2/3

Example: ball with R (but not smooth ξ)

Assumeε<< Rinf :inf of the principal curvature radius

NG case:

Documents

Mathematics of Triple-Junction and Singularity Theory: from CFD …smatsutani.jp/CFD_Singularity20160602.pdf · 2019-06-19 · Shigeki Matsutani. National institution of technology,