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Shigeki MatsutaniNational institution of technology, Sasebo college
June 3, 2016
1
Mathematics of Triple-Junction and Singularity Theory:
from CFD of ink-jet printer
2
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..
3
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..
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.
8
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..
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!
12
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).
13
There was a mathematical model of numerical two-phase fluid dynamics!
差分化Discretization
2-phaseEuler Equation
3-phaseEuler Equation
3-phase DifferenceEuler Equation
Applied Math(VOF-method)
Discretization
2-phase DifferenceEuler Equation
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-phaseEuler Equation
1-phase DifferenceEuler Equation
1-phaseAction Functional
2-phaseEuler Equation
2-phase DifferenceEuler Equation
2-phaseAction Functional
3-phaseEuler Equation
3-phase DifferenceEuler Equation
3-phaseAction Functional
Variation
Variation
Applied Math(VOF-method)
Singularity Theory
AnalyticGeometry
差分化
Applied Math(phase-field theory)Arnold, Ebin-Marsden 1970
Diffeomorphism Group+Momentum Map DiscretizationVariation
Discretization
Discretization
1-phaseEuler Equation
1-phase DifferenceEuler Equation
1-phaseAction Functional
2-phaseEuler Equation
2-phase DifferenceEuler Equation
1-phaseAction Functional
3-phaseEuler Equation
3-phase DifferenceEuler Equation
1-phaseAction Functional
Variation
Variation
Applied Math(VOF-method)
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
Applied Math(phase-field theory)Arnold, Ebin-Marsden 1970
Diffeomorphism Group+Momentum Map DiscretizationVariation
Discretization
Discretization
1-phaseEuler Equation
1-phase DifferenceEuler Equation
1-phaseAction Functional
2-phaseEuler Equation
2-phase DifferenceEuler Equation
1-phaseAction Functional
3-phaseEuler Equation
3-phase DifferenceEuler Equation
1-phaseAction Functional
Variation
Variation
Applied Math(VOF-method)
Singularity Theory
AnalyticGeometry
Today’s topic!
18
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..
19
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..
20
Triple-Junction
http://photo-pot.com/?p=23192
21
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!
22
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..
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:
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*# : Surface tension value of * and #.
E = σCASCA + σABSAB + σBCSBC
Interface Energy:
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
26
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!)
E = σCASCA + σABSAB + σBCSBC
= 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
Shape of Droplet with Triple-Junction
EL =σCASCA+(σABーσBC)SAB+λ(VA-V0)
ーー = 0,ーー = 0,ーー = 0δEL
δrδEL
δθδEL
δλ
σBC = σAB + σCAcos θ
Shape of Droplet with Triple-Junction
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
Shape of Droplet with Triple-Junction
σBC = σAB + σCAcos θ
σCAσBC
σAB
σBC > σAB + σCA
θ is not defined! → Wet
Shape of Droplet with Triple-Junction
σBC = σAB + σCAcos θ
θ
σCA
σBC
σAB
σBC > σAB
θσCA
σBC
σAB
σBC < σAB
Shape of Droplet with Triple-JunctionσAB + σCA > σBC
(σBC > σAB - σCA )
33
http://nagare-furoshiki.com/ja/repel.html
超撥水
Shape of Droplet with Triple-Junction
super-water-repellent
34
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..
35
Numer. model of Two-phase field
Problem
To represent two-phase field approximately using a function over a region
36
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.
37
38
Interface =Zero crossing
Intermediate region
Phase field method
Level set methodBasic Approachs!
ε
38
ξ=0
ξ=1
ξ
39
Interface =Zero crossing
Intermediate region
Phase field method
Level set methodBasic Approachs!
ε
39
ξ=0
ξ=1
ξ
We employed the phase field method
40
Intermediate region
Phase field method
Phase field method
ε
40
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
41
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.
43
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)
46
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
50
We have a constant mean curvature surface approximately for constant p1 and p2.
variational method
Approximation of Laplace equation
51
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..
Phase field model with triple-junction
To express the cone singularity in terms of phase-field method
GasLiquid
Solid
53
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!
Phase field model with triple-junction
54
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!
Phase field model with triple-junction
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!
55
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)=
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
sM1
M2
M3
(connected)
{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
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)
66
We evaluate the energy and consider it weakly up toε!
1. x∈Ω,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
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
Evaluate the surface energy!
E(□)= (∑σab(|∇ξa| |∇ξb|)1/2(ξa + ξb))d
3x∫□
a,b
e
Then we have the following estimations:
For ε to 0, the model gives the energy well.
2. x∈supp(ξa)∩supp(ξb)=La∩Lb
s.t. ξa(x)+ξb (x)=1,
|E(supp(ξ1)∩supp(ξ2))ーσ12 A12|<εK
Evaluate the surface energy!
E(□)= (∑σab(|∇ξa| |∇ξb|)1/2(ξa + ξb))d
3x∫□
a,b
e
70
Then we have the following estimations:
For ε to 0, the model gives the energy well.
3. supp(ξ1)∩supp(ξ2)∩supp(ξ3)=L0∩L1∩L2
E(supp(ξ1)∩supp(ξ2)∩supp(ξ3))<εK’
Evaluate the surface energy!
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
2-phaseEuler Equation
3-phaseEuler Equation
3-phase DifferenceEuler Equation
Applied Math(VOF-method)
Discretization
2-phase DifferenceEuler Equation
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-phaseEuler Equation
1-phase DifferenceEuler Equation
1-phaseAction Functional
2-phaseEuler Equation
2-phase DifferenceEuler Equation
2-phaseAction Functional
3-phaseEuler Equation
3-phase DifferenceEuler Equation
3-phaseAction Functional
Variation
Variation
Applied Math(VOF-method)
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
80
Modified Laplace equation:
Modified Laplace equation:
82
83
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
Interface =Zero crossing
Intermediate region
Phase field method
Level set methodBasic Approachs!
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.
Interface =Zero crossing
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
89
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!
91
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
93
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: