kesco非線形解析セミナー 2018,7,5 先進材料の成 …...1...
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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
naoki@mech.keio.ac.jp
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
Fiber volume fraction of specimen, (%)
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
Fiber volume fraction of specimen, (%)
Fra
cture
str
ess
(MP
a)
A B
C
D
E
macro
fV
Fiber volume fraction of specimen, (%)
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
Introduction to multiscale simulation
N.Takano et al., Trans.JSME (1997)
Damage propagation
Introduction to multiscale simulation
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)
Introduction to multiscale simulation
Aramid plain knitted fabric
Polypropylene
N.Takano et al., Int.J.Solids Struct. (2001)
Deep-draw forming of FRTP
Introduction to multiscale simulation
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)
Asymptotic homogenization method
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
Asymptotic homogenization method
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
Parameterization: 2D woven GFRP laminate
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)
Parameterization: 2D woven GFRP laminate
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
Parameterization: 2D woven GFRP laminate
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
Acquisition of statistical database : Future work
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)
Acquisition of statistical database : Future work
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,
東野,高野ら,日本材料学会学術講演会(2017)
5
Acquisition of statistical database : Future work
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,
ミクロ繊維含有率
[%]
初期値50%
ミクロ繊維含有率
メゾ繊維含有率
A B C
D E F
①②③④⑤
y
zx
A B
D E
C
FL
Z
T
東野,高野ら,日本材料学会学術講演会(2017)
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
Expected value 49.33 Gpa
Standard deviation 0.322 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
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
Expected value 49.33 Gpa
Standard deviation 0.322 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
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
Expected value 49.33 Gpa
Standard deviation 0.322 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
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
Expected value 49.33 Gpa
Standard deviation 0.322 GPa
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
Merit of stochastic simulation : summary
, ,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
Merit of stochastic simulation : summary
95%95%
Stiffness /
Str
ength
Heat
insula
tio
nPorosity
ratio PDesign solution
Target property
n
j
HXj
H APfXffj
1
,, DDDImprovement of
morphology
Merit of stochastic simulation : summary
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
Application to metal 3D printed component
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
Application to metal 3D printed component
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
J (Without heat treatment)
J (With heat treatment)
H (Without heat treatment)
O (Without heat treatment)
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
J (Without heat treatment)
J (With heat treatment)
H (Without heat treatment)
O (Without heat treatment)
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)
Application to metal 3D printed component
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
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