kesco非線形解析セミナー 2018,7,5 先進材料の成 …...1...

1

先進材料の成形に起因するばらつき・不確かさを考慮した確率的非線形マルチスケール解析の

実用化に向けて

慶應義塾大学理工学部機械工学科高野直樹

[email protected]

KESCO非線形解析セミナー 2018,7,5

Towards practical stochastic nonlinear multiscale

analysis for advanced materials

Considering variability and/or uncertainty due to

manufacturing process

事後配布用抜粋版

Target material & manufacturing process

Composite Materials

複合材料Additive Manufacturing

3D積層造形

1.4

1.5

1.6

1.7

1.8

56 58 60 62 64 66

Fiber volume fraction of specimen, (%)

Thic

knes

s of

spec

imen

, t

(mm

)

A

B

C D

E

macro

fV

Experimental fact for GFRP laminate by HL

- Glass plain woven fabric + Unsaturated polyester

- Hand layup of 4-layered laminate

- 5 students (inter-individual difference)

- 5 specimens per student (intra-individual difference)

Hagiwara, Takano, et al., MEJ (2017)

0

200

400

600

0.00 0.01 0.02 0.03 0.04

Fracture stress

Str

ess

(MP

a)

Strain

Knee-point

A

B

C

D

E

0

0

200

400

600

0.00 0.01 0.02 0.03 0.04

Fracture stress

Str

ess

(MP

a)

Strain

Knee-point

A

B

C

D

E

0

Experimental fact for GFRP laminate by HL


Young’s modulus (G

Pa)

23

25

27

29

31

33

35

37

56 58 60 62 64 66

E

BA

D

C

macro

fV

480

500

520

540

560

580

600

56 58 60 62 64 66


Fra

cture

str

ess

(MP

a)

A B

C

D

E

macro

fV


Kn

ee-p

oin

t st

ress

(M

Pa)

70

90

110

130

56 58 60 62 64 66

A

B

C

D

E

macro

fV

Young’s modulus

Knee-point stress

Fracture stress

Hagiwara, Takano, et al., MEJ (2017)

Aim of stochastic modeling & simulation

■先進材料の開発段階において、成形・造形プロセスパラメータは多数あるため最適なパラメータ選択は困難であるが、製品の性能における「ばらつき」がさらに問題を複雑化している。

■ばらつき（variability）、あるいは不確かさ（uncertainty）を考慮したCAE技術の開発により、解決策を見出したい。

■非線形挙動の中で、ばらつき・不確かさがいかに伝播、拡大するかは難問である。本講演では、複合材料を主たる対象として、損傷進展を考慮した確率的非線形マルチスケールシミュレーション法を紹介する。

Introduction to multiscale simulation

N.Takano et al., Trans.JSME (1997)

What is Multiscale Simulation ? - Virtual Microscope

2


N.Takano et al., Trans.JSME (1997)

Damage propagation


0 0.2 0.4 0.6 0.8 110-5

10-4

10-3

10-2

10-1

100

101

SquareHexagonal

Kozeny-Carman

Fiber Volume Fraction

μK

22 /

r2

Gebart (Square) Gebart (Hexagonal)

Homogenization theory

繊維含有率

Gebartの式（正方配列モデル）

Gebartの式（細密充填モデル）

Kozeny-Carmanの式

正方配列モデル細密充填モデル

均質化法による予測値

Fiber volume fraction

Gebart (A)

Gebart (B)

Kozeny-Carman

N.Takano et al., Compo.Sci.Tech. (2002)

30°

y1

30°

y1

30°

y1

30°

y1

Permeability

UD

Sheared woven

fabric

(A)

(B)


Aramid plain knitted fabric

Polypropylene

N.Takano et al., Int.J.Solids Struct. (2001)

Deep-draw forming of FRTP


DDD ,,AVXHH

XV

D

A

Geometrical parameters

Physical parameters of

constituent materials

fiber volume fraction

morphology

HD homogenized

(macroscopic)

properties

Upscaling

(homogenization/averaging)

Downscaling

(localization)

1mm

,,, DDD aVxHH

Macro Micro

Nonlinear problems

Asymptotic homogenization method

- Developed in late 1970s by applied mathematicians

in France, Russia and USA

- Physical quantities are asymptotically expanded using

the scale ratio between macro- and micro-scales

- Take a limit of for homogenization (averaging)

- Both macro- and microscopic quantities are calculated

(localization)

- Has been applied to various nonlinear problems

- Has been applied to materials with random micro-

architecture such as porous media

- Has been validated in many problems by comparison

with experimentally measured results

0

Takano et al., IJSS(2001, 2003), IJMS (2010, 2016), JBSE(2011)


X : Geometrical parameter

D : Physical parameter

Homogenized macroscopic matrix: HD

DDD ,XHH

Periodicity Scale ratio：i

i

y

x

1x

2x 1y

2y

Y 1y

2y

3y

1x

2x

3x

RVE

Y

1

MTOT

r

r

Y= Y

1 2 3r , , ,...,MTOT

1Y

2Y

3Y

rYMTOTY

yxuxuu ,10

L

Matrix

fiber

Homogenized model Perturbation

＝＋

x

xuyχyxu

01 ,

Asymptotic expansion（漸近展開）

xu0 yxu ,1

Matrix

fiber

3


Y

klT

Y

Tkl dYdY QDBBDBχ1

othersfor 0

for 1

123123332211

klijq

qqqqqq

ij

TklQ

Characteristic displacements（特性変位）

Y

H dYY

χBIDD1

EχBIDσ

Homogenization（均質化）

Localization（局所化）

σEDσ H

YdY

Y

1

E : Macro-strain (given)

Multiscale large deformation analysis

Takano, et al.,

IJSS (2000)大変形後にも周期性の仮定が許容されれば均質化法は適用可能である

Multiscale large deformation analysis Uncertainty or Variability at microscale

HD homogenized

(macroscopic)

properties

- Uncertainty or variability is included

- Parameterization of morphology is difficult

- Sampling w.r.t. geometrical parameters

- Repeated homogenization analyses

Basarudin, Takano et al., Mat.Trans.(2013)

Wen, Takao et al., J.Multiscale Model. (2015)

Macro Micro DDD DX

HH fAVXfff ,,

Probability

density function f

HX

N

j

jXH

jfXff DD

1

XV

D

A

Geometrical parameters

Physical parameters of

constituent materials

fiber volume fraction

morphology

Geometrical random parameters

■一般には、加工誤差による形状・寸法に関するCADデータとの差異、およびばらつき

■繊維強化プラスチック複合材料の事例では、ミクロ構造、形態（morphology）のパラメータ化は難しい

幾何的パラメータの要件

・計測可能であること・パラメータ間の独立性、従属性が定義できること・FEMデータ生成に利用できること

Parameterization: 2D woven GFRP laminate

暫定的なユニットセルモデル

A

ユニットセル内の幾何的情報

積層ずれ

sネスティング

LL ,

matintL

Hagiwara, Takano et al.,

Mech. Eng. J. (2017)

4


micro

f

minor

i

major

i

meso

fiiwarp VLLVLAA ,,,,,

繊維束配置

メゾ繊維含有率

繊維束の形状

ミクロ繊維含有率

縦糸厚さ

,,,,,,,, matintmatsurfmacro

f LsALLLVtXX

マクロな幾何的情報

表面樹脂層

ユニットセル寸法・ネスティング・層間樹脂層・積層ずれ

積層パラメータ

・板厚

・マクロ繊維含有率・ユニットセル幅・ユニットセル厚さ

ユニットセル内の幾何的情報

Hagiwara, Takano et al., Mech. Eng. J. (2017)


micro

f

minor

i

major

i

meso

fiiwarp VLLVLAA ,,,,,

,,,,,,,, matintmatsurfmacro

f LsALLLVtXX

)

2cos()

2cos(

2

1

2

2

1

1

l

Al

As

24

1

,24

11,,

i

micro

ifV

micro

f

micro

f

minor

i

major

i

meso

f

meso

f VdVVV

VLLVV

Hagiwara, Takano et al., Mech. Eng. J. (2017)

0

2

4

6

8

10

12

14

16

18

48 52 56 60 64 68 72 76

n=391 , μ=62.51% , σ=5.11%

Pro

bab

ilit

y(%

) n=391

μ=62.5, σ=5.1

mesofV


Real specimen

Statistical data based models

Hagiwara, Takano et al.,

Mech. Eng. J. (2017)

)

2cos()

2cos(

2

1

2

2

1

1

l

Al

As

Acquisition of statistical database : Future work

ParameterizationStatistical

measurement

RVE modeling &

stochastic simulation

Replacement by process simulation

- Resin flow among fiber bundles

- Resin permeation into fiber bundles

- Deformation by

- Pressure by roller

- Pressure by resin flow

Multiphysics problem including

- Navier-stokes equation

- Darcy’s law

- Large deformation

Use of COMSOL 5.2


0 fluidfluid u

fluidfluidfluidfluidfluid

p uuu2

0 fluid

fluidu

fluidfluidfluid

fluidfluid

fluidp uuκ

uu21

繊維束間流れ

繊維束内流れ

繊維束変形

FSΠ0Π ,

ECS :

T

solidsolid

T

solidsoliduuuuE

2

1

大変形を伴うため，幾何非線形性を考慮する

繊維束を流体で満たされた多孔体と考える

i =1

i =n

j =1

j =n

warpx

weftx

zx

東野,高野ら,日本材料学会学術講演会(2017)


繊維含有率は領域の面積に反比例する

変形前後の面積要素の関係はNansonの公式によって記述される

0

0

ff VS

SV

0

0

0001

0

f

a

T

f VS

Jda

V

nnFF

F：変形勾配S：変形後の面積：変形前の面積0S 0fV ：の初期値fV

：変形前の微小面積要素0da 0n ：変形前の領域の単位法線ベクトル

FdetJ

000

1 Jdada TnnFF

0aFa dJd T

繊維含有率はF （変形勾配）から求められる

この2式より，


5


49

49.5

50

50.5

51

51.5

52

① ② ③ ④ ⑤ 平均

断面②の形状変形断面④の形状変形

メゾ繊維含有率，


[%]

初期値50%


メゾ繊維含有率

A B C

D E F

①②③④⑤

y

zx

A B

D E

C

FL

Z

T


Physical random parameters

First-order perturbation approximation

w.r.t. physical random parameter

QoI

0 0.03

Deterministic

First-order perturbation

DDD fff 10DDDD

0Exp

00

Df Probabilistic density function (PDF)

0Exp DD

i

im mn

mniDmnimniiD fDf

MTOT

1

6

1

6

,,0

in PDαD

i Constituent material number

nm,

othersfor 0

),(),,(),(for 1 mnnmjiijmnP

Component in matrixD

Wen, Takano, et al., Acta Mech (2018)

Physical random parameters

MTOT

i m mn

mniDX

HmniX

HHX ff

jjj

1

6

1

6

,

1

,

0DDαD

Y

000Y

1d

jj XXH

χBIDDY

Y

1,

0

Y,

01

, YY1

ddDj

ijj XmnimniXmnX

Hmni χBDχBIQD

Y

0th and 1st order terms of characteristic displacement

0th and 1st order terms of homogenized stress-strain matrix

Homogenized stress-strain matrix

Y

01

Y

00 YY dd klTTkl

X j

QDBBDBχ

kl

mniimnT

imnikl

mnT

iTkl

Xmni dDdDdj

0

Y

0,

Y

0,

1

Y

01, YYY χBPBQPBBDBχ

Verification

x y

z

unit：μm

8 10

3016

2

x yz

Material #1

Material #2

Material #5

Material #3

Material #4

21 physical random parameters were assumed

0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Proposed method

Monte Carlo simulation

(10,000 samples)

11HD 11

HD(GPa) (GPa)

Pro

bab

ilis

tic

den

sity

Pro

bab

ilis

tic

den

sity

Expected value 49.34 Gpa

Standard deviation 0.326 GPa



0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Proposed method


(10,000 samples)

11HD 11

HD(GPa) (GPa)

Pro

bab

ilis

tic

den

sity

Pro

bab

ilis

tic

den

sity





0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Proposed method


(10,000 samples)

11HD 11

HD(GPa) (GPa)

Pro

bab

ilis

tic

den

sity

Pro

bab

ilis

tic

den

sity





0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

47.8 48.3 48.8 49.3 49.8 50.3 50.847.8 48.3 48.8 49.3 49.8 50.3 50.8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Proposed method


(10,000 samples)

11HD 11

HD(GPa) (GPa)

Pro

bab

ilis

tic

den

sity

Pro

bab

ilis

tic

den

sity





Wen, Takano, et al., Acta Mech (2018)

11HD

Physical and geometrical random parameters

Physical parameter

if αD

Geometrical parameter

jXf

ほ

H

X jf D

Stochastic

homogenization

Micro-strain

EχBEχBIα

NMAT

i m mn

mnimni ff1

6

1

6

,

1

,

0

αf

EMacro-strain

N

j

H

Xj

H

jfXff

1

DD

EχBEχBIα

MTOT

i m mn

mniDXmniXX ffjjj

1

6

1

6

,1

,0

HX

N

j

jXH

jfXff DD

1

submitted paper is in review

Damage propagation analysis

1 2

3

174,320 elements

246,085 nodes

Particle 100Coating 1

Matrix 10

Young’s modulus (GPa)

TEE 00000 31

002.031 E

- Damage occurs only in the coating layer

- Damage criterion using effective strain of coating layer

- Young’s modulus of damaged coating element was

reduced to GPa 410

Assumptiongs:

journal paper is

in preparation

6

Damage propagation analysis

Macroscopic strain E31

Dam

aged

vo

lum

e fr

acti

on

to

co

atin

g v

olu

me

(-)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.05 0.1 0.15 0.2 0.25210

1

2

3

4

5

Exp ( ̅ )Exp ( ̅ ) + 1×S.D. ( ̅ )Exp ( ̅ ) + 2×S.D. ( ̅ )

0.045

0.047

0.049

0.051

0.053

0.055

0.04 0.0420.041 10-2

1

0.196

0.198

0.2

0.202

0.204

0.206

0.0790.080.0810.08 10-2

2

3

0.28

0.282

0.284

0.286

0.288

0.29

0.119 0.1210.12 10-2

0.375

0.377

0.379

0.381

0.383

0.385

0.159 0.1610.16 10-2

4

0.416

0.418

0.42

0.422

0.424

0.426

0.19 0.210.2 10-2

5

journal paper is

in preparation

Merit of stochastic simulation : summary

Stiffness /

Str

ength

Heat

insula

tio

n

Stiffness / Strength Heat insulation

,H H PD D X D

Porosity

ratio P

No design solution

no controlled morphology

Target property


, ,H H P AD D X D

,H H PD D X D

Design solution

Stiffness /

Str

ength

Heat

insula

tio

n

Porosity

ratio P

Improvement of

morphology

Target property

Target porosity

Experiment


95%95%

Stiffness /

Str

ength

Heat

insula

tio

nPorosity

ratio PDesign solution

Target property

n

j

HXj

H APfXffj

1

,, DDDImprovement of

morphology


95%95%

Stiffness /

Str

ength

Heat

insula

tio

n

Porosity

ratio PDesign solution

Target property

n

j

HXj

updateH APfXffj

1

,, DDD

Posterior probability

Target porosity

Wen, Takano, et al., J.Multiscale Model. (2015)

Application to metal 3D printed component

高野ほか, 日本材料学会学術講演会(2018)

☑造形パラメータが非常に多く、造形受託企業の経験やノウハウに基づき選択されているのが現状

☑未知の形状だと造形の成否が事前にわからない

☑造形受託企業内部において技術伝承の見通しが立っていない

☑発注者（医者）と造形受託企業の双方にとって品質保証の手順、手法論が定まっていない

☑ CAEの有効利用には、まだ道のりが遠い

委託造形する際の問題点

7


高野ほか, 日本材料学会学術講演会(2018)

研究室 CADデータ

造形受託企業 A B C D

研究室各種の計測、数値シミュレーション

Maraging steel Aluminum alloyCompany

A

B

C

D

3D printer

EOS INT M290

3D Systems ProX300

EOS INT M280

EOS INT M290

dcL 56

c

c

d

cross-section

of strut

x

yz

)mm :unit(

26

4

1

L

d

c

本日はマルエージング鋼についてのみ発表


Takao, et al., IJMS (2017)

サポート構造

(c) Type 3 determined by

company D

(d) Type 4 determined by

discussion with

company D

(e) Type 5 determined by

discussion with

company D

(a) Type 1 determined by

company A

(b) Type 2 determined by

company B and C

(c) Type 3 determined by

company D

(d) Type 4 determined by

discussion with

company D

(e) Type 5 determined by

discussion with

company D

(a) Type 1 determined by

company A

(b) Type 2 determined by

company B and C

Type 1 Type 2 Type 3

各社が経験とノウハウに基づき設定したら、3社3様になった

Type 1は積層高さが低いので、時間短縮と低コスト化がはかれる（社長の方針）

Geometrical uncertainty

非常に顕著な造形不良がみられた

モデル化して統計的計測

ak

bk

qk

0

0.1

0.2

0.3

0.4

1.5 1.8 2.1 2.4 2.7

Fre

qu

ency

(%

)

ak (mm)

Number of struts 2,160

Average 2.11 mm

Standard deviation 0.35 mm

an

bn ah

bh

kink

notch hole Int.J.Mech.Sci. Vol.134, pp.347-356 (2017)

マイクロCT

Physical uncertainty

0.96

0.97

0.98

0.99

1

1.01

1.02

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04

JN

JY

HN

ON

OY

Normalized volume by CAD model

No

rmal

ized

den

sity

by

EOS

dat

a sh

eet

*

*

*

* *

0.96

0.97

0.98

0.99

1

1.01

1.02

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04

JN

JY

HN

ON

OY

J (Without heat treatment)

J (With heat treatment)

H (Without heat treatment)

O (Without heat treatment)

O (With heat treatment)

A (Without heat treatment)

A (With heat treatment)

C (Without heat treatment)

0.96

0.97

0.98

0.99

1

1.01

1.02

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04

JN

JY

HN

ON

OY





O (With heat treatment)

0.96

0.97

0.98

0.99

1

1.01

1.02

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04

JN

JY

HN

ON

OY





O (With heat treatment)C (With heat treatment)

B (Without heat treatment)3%

mnimnimnimni fDD ,

20

,,, VarVar mnimnimnimni fDD ,

20

,,, VarVar

重量（電子天秤）と体積（アルキメデス法）から算出したみかけの密度

一次漸近展開近似

QoI

0 0.03

Deterministic

First-order perturbation

Int.J.Mech.Sci. Vol.134, pp.347-356 (2017)


0.00

0.50

1.00

C社 A社 B社Company D Company B Company A

0.0

1.0

0.5

Normalized homogenized Young’s modulus

Numerical analysis Experiment

Int.J.Mech.Sci. Vol.134, pp.347-356 (2017)

☑定性的には特徴をとらえている

☑造形前に可能性として予測できる

☑造形経験の統計データに基づくCAE活用の新しい可能性

Stochastic simulation

(N=81)

Experiment

(N=1)

Summary

Process

parameters

Mechanical

propertiesMicrostructure

(RVE models)

Stochastic

analysis

Parameterization Statistical

measurement

FEM model

generation

Feedback information

referred

- Physical

- Geometrical

Future work: Replacement by

process simulation

(multiphysics simulation)

Posterior probability

n

j

HXj

H

jfXff

1

DD

kesco非線形解析セミナー 2018,7,5 先進材料の成 …...1...

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