Current approaches to exploring materials and manufacturing (or processing) design spaces in pursuit of new/improved engineered structural materials continue to rely heavily on extensive experimentation, which typically demand inordinate investments in both time and effort. Although tremendous progress has been made in the development and validation of a wide range of simulation toolsets capturing the multiscale phenomena controlling the material properties and performance characteristics of interest to advanced technologies, their systematic insertion into the materials innovation efforts has encountered several hurdles. The most common of these are related to (i) the lack of a generalized (applicable to a wide variety of materials classes and phenomena) mathematical framework that allows objective extraction and synergistic integration of the high value materials knowledge (defined from the perspective of producing reliable process-structure-property (PSP) linkages) from all available datasets (including a variety of multiscale experiments and simulations), while accounting for the inherent uncertainty associated with each dataset, (ii) the lack of formal approaches that identify objectively where to invest the next effort (could be a new experiment or a new simulation) for maximizing the likelihood of success (i.e., meeting or exceeding the designer-specified combinations of materials properties) at any step of the innovation effort, and (iii) the lack of experimental techniques that are specifically designed to provide the quality and quantity of information needed to calibrate the large number of material parameters present in most multiscale materials models. This talk will describe ongoing efforts in my research group aimed at addressing the gaps identified above.

1. Materials Innovation Driven by Data
and Knowledge Systems
Surya R. Kalidindi
Funding: NIST, AFOSR

2. Materials-Manufacturing Nexus:
Valley of Death
Reduced-order, uncertainty-
quantified, Process-
Structure-Property (PSP)
linkages predicting multiscale
multiphysics material’s
responses are critically
needed for successful
extension of the digital
thread of manufacturing to
fully exploit the materials
design space

3. Materials Innovation Supported by Digital
Knowledge Systems
www.comsol.com
DESIGN &
MANUFACTURING
P-S-P
P-S-P
P-S-P
• Continuously updated
with information from all
sources – experiments
and models
• Quantified uncertainty
allowing objective
decisions

4. Foundational Elements of Materials Knowledge Systems
• Data management systems: proprietary file
formats, metadata capture, accessible analytic
tools, e-collaboration platforms
• Quantification of material structure: statistics,
uncertainty quantification, multiple
length/structure scales, diverse materials classes
• High throughput experimental assays: surrogate
tests for screening, material sample libraries
• Physics-aware machine learning: extrapolation
vs. interpolation, identification of the controlling
physics (i.e., model forms and parameter values)
• Information fusion from disparate sources:
multiscale experiments, multiscale models

5. Structure-Property Linkages
• Property: 𝑃 ∈ ℛ; 𝑝 𝑃 = 𝒩 𝑃 𝑃, 𝜎 𝑝
2
• Microstructure: includes all relevant details
of the material’s (hierarchical) internal
structure; inherently requires a high-
dimensional representation; demands a
suitable parametrization for exploration of
the space; 𝝁 ∈ 𝓜; 𝑝 𝝁 = 𝒩 𝝁 𝝁, 𝜮 𝝁
• Governing (embedded) Physics: includes
prescription of relevant conservation laws,
constitutive equations, parameter values,
and any other physics-based constraints
relevant to the phenomena being studied;
needs a suitable parameterization for
exploration of the space; 𝝋 ∈ 𝚽; 𝑝 𝝋 =
𝒩 𝝋 𝝋, 𝜮 𝝋
     
0εσ
εεCε
CCxεxCxσ



,
T,
0
,00
,

6. n-Point Spatial Correlations
• Spatial correlations capture all of the salient measures of the microstructure
• Efficient codes for computing them are now available through PyMKS code repository
𝑓𝒓
𝑛𝑝
=
1
𝑆 𝒓
1
𝐽
𝒔 𝑗=1
𝐽
(𝑗)
𝑚 𝒔
𝑛 (𝑗)
𝑚 𝒔+𝒓
𝑝
𝑓𝑟
ℎℎ′
=
# 𝑇𝑟𝑖𝑎𝑙𝑠 𝑆𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙
# 𝑇𝑟𝑖𝑎𝑙𝑠 𝐴𝑡𝑡𝑒𝑚𝑝𝑡𝑒𝑑
Solid
Pore
0
1
%50
%0
Complete Set of 𝑓𝑟
ℎℎ′
for
all possible 𝑟
S. R. Kalidindi, “Hierarchical Materials Informatics”, Butterworth Heinemann, 2015.

7. Reduced-Order Representations Using PCA
𝑓𝑟
(𝑗)
≈
𝑘=1
𝑅
𝛼 𝑘
(𝑗)
𝜑 𝑘𝑟 + 𝑓𝑟 𝑅 ≪ 𝑚𝑖𝑛 𝐽 − 1 , 𝑅
Microstructure m = (7.95, 0.46, 0.78)
S. R. Kalidindi, “Hierarchical Materials Informatics”, Butterworth Heinemann, 2015.

8. Polycrystal Microstructures
𝑚 𝑔, 𝒙 ≈
𝐿
𝐿
𝒔
𝑺
𝑀𝒔
𝐿
𝑇𝐿 𝑔 𝜒 𝒔 𝒙
𝑓 𝑔, 𝑔′ 𝒓 ≈
𝐿
𝐿
𝐾
𝐿
𝒕
𝑺
𝐹𝒕
𝐿𝐾
𝑇𝐿 𝑔 𝑇 𝐾 𝑔′ 𝜒𝒕 𝒓
𝑓 𝑔, 𝑔′ 𝒓 =
1
𝑉𝑜𝑙(Ω 𝒓)
Ω 𝒓
𝑚 𝑔, 𝒙 𝑚 𝑔′, 𝒙 + 𝒓 𝑑𝒙
𝐹𝒕
𝐿𝐾
=
1
𝑺 𝒕
𝒔
𝑺 𝒕
𝑴 𝒔
𝐿 𝑴 𝒔+𝒕
𝐾

9. Specification of Governing Physics:
Conventional Approaches
     
ConditionsBoundary
0
,00
,
0εσ
εεCε
CCxεxCxσ



,
T,
Representation of governing physics in the form of partial
differential equations, material constitutive laws, and
boundary conditions is ideal for “forward” numerical
simulations, not for inverse solutions in materials innovation

10. Kroner’s Statistical Continuum Theories
Challenges: principal value problem; selection of reference medium; convergence
𝜺 𝑥 = 𝑰 −
𝑉 𝐻
𝜶 𝑟, 𝑛 𝑚 𝑥 + 𝑟, 𝑛 𝑑𝑛𝑑𝑟
+
𝑉 𝑉 𝐻 𝐻
𝜶 𝑟, 𝑟′, 𝑛, 𝑛′ 𝑚 𝑥 + 𝑟, 𝑛 𝑚 𝑥 + 𝑟 + 𝑟′, 𝑛′ 𝑑𝑛𝑑𝑛′ 𝑑𝑟𝑑𝑟′ − ⋯ 𝜺 𝑥
𝜺 𝑥 = 𝒂 𝑥 𝜺 𝑥
Elastic Localization
n : Local State
𝑚 𝑥, 𝑛 : Microstructure Function
𝜶 𝑟, 𝑛 : Physics capturing kernel; independent of 𝑚 𝑥, 𝑛

11. MKS Framework: Discrete Local States
𝑚 𝑠
ℎ
: local
microstructure
descriptor
𝜶 𝑡
ℎ
: influence of local
microstructure at
microscale
Convolution
𝒑 𝑠: local
response
𝑚 𝑥, 𝑛 = 𝑚 𝑠
ℎ 𝜒ℎ 𝑛 𝜒 𝑠 𝑥 ; 𝜶 𝑟, 𝑛 = 𝜶 𝑡
ℎ
𝜒ℎ 𝑛 𝜒 𝑟 𝑡
𝒑 𝑠 =
ℎ=1
𝐻
𝑡=1
𝑆
𝜶 𝑡
ℎ
𝑚 𝑠+𝑡
ℎ
+
ℎ=1
𝐻
ℎ′=1
𝐻
𝑡=1
𝑆
𝑡′=1
𝑆
𝜶 𝑡𝑡′
ℎℎ′
𝑚 𝑠+𝑡
ℎ
𝑚 𝑠+𝑡′
ℎ′
+ ⋯ 𝒑
Main Idea: Calibrate 𝜶 𝑡
ℎ
on
selected microstructures of the
material system of interest and the
FE predicted response fields for
those microstructures; Use the
same 𝜶 𝑡
ℎ
on new microstructures
of the same material system to
predict their response fields.

12. Efficient Parametrization of Governing
Physics Using Green-function based Kernels
pmmmp h
tts
h h t t
h
ts
hh
tt
h t
h
ts
h
ts 





   ...'
'
' '
'
'
 𝛼 𝑡
ℎ
, 𝛼 𝑡𝑡′
ℎℎ′
, … capture the physics and are independent of
the details of the material structure captured in 𝑚 𝑠
ℎ
 𝝋 = 𝛼 𝑡
ℎ
, 𝛼 𝑡𝑡′
ℎℎ′
, … offers a powerful representation.
 The dependence of 𝛼 𝑡
ℎ
on constitutive parameters and
boundary conditions can be efficiently parametrized using
powerful Fourier representations
 The convolution structure allows cheap computations
using discrete Fourier transforms
 Highly consistent with convolution neural networks for
injection of machine learning approaches

13. MKS Prediction of Composite Stress-Strain
Responses
f2 = 25%
f2 = 50%
f2 = 75%
FEM
MKS
EffectiveStress,a.u.
Strain
CPU Time
MKS: 0.5 s
FEM: up to ~24 hrs
validation RVEs
Latypov and Kalidindi, Journal of Computational Physics, 346, 2017

14. Reduced-order
microstructure
representation
𝐸𝑒𝑓𝑓 𝜎 𝑦
New protocol is 10,000x faster
than traditional protocols in
prediction of 𝜎 𝑦
Structure-Property Linkages: -Ti Polycrystals
Paulson, Priddy, McDowell, Kalidindi, Acta Materialia, 129, 2017

15. Application: Ranking for Fatigue
𝜺 𝑡𝑜𝑡𝑎𝑙 →
0.5% applied strain
amplitude
MKS + Explicit Integration CPFEM
Predict 𝜺 𝑡𝑜𝑡𝑎𝑙
using MKS
localization
Estimate 𝜺 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 (𝐩𝐨𝐬𝐭 − 𝐚𝐧𝐚𝐥𝐲𝐬𝐞𝐬)
Construct distribution of
extreme fatigue indicator
parameters (FIPs)
New protocol is 40X
faster than traditional
protocols for ranking
HCF resistance
𝑭𝑰𝑷 𝑭𝑺 =
∆𝜸 𝒎𝒂𝒙
𝒑
𝟐
𝟏 + 𝒌
𝝈 𝒎𝒂𝒙
𝒏
𝝈 𝒚
Priddy, Paulson, Kalidindi, McDowell, Materials and Design, 154, 2018

16. Images to Properties: Steel Scoops Excised
from High-Temperature Exposed Components
Iskakov et al., Acta Materialia, 144, pp. 758-767, 2018

17. ATOMIC STRUCTURE DATASETS FROM MD:
171 Fe GRAIN BOUNDARIES
125 STGBs with <100>, <110> and <111> tilt axes
• 50 <100> STGBs with different misorientation angles
• 50 <110>
• 25 <111>
19 TWGB
• 10 <100> TWGBs
• 4 <110>
• 5 <111>
27 ATGB

18. DATASET CONTENT: SELF-INTERSTITIAL
ATOM & VACANCY FORMATION ENERGY
AT ATOMIC SITES WITHIN 15A OF GB

19. TWO LOCAL ATOMIC STRUCTURE DESCRIPTORS
Wikipedia
1
Local KDE based PCF
2
Rotationally invariant 3-point stats
(Bispectrum)
 Rotational invariance is achieved by
projecting density function onto 3-
Sphere
 PCF is computed for each
neighborhood using Epanechnikov
Kernel

20. MLP REGRESSION IS USED TO BUILD
STRUCTURE-PROPERTY LINKAGES
Wikipedia
Test MAE = 1.51%
Test RMSE = 4.38 %
12
PCs
12
Perceptrons
8
Perceptrons

21. • 45 x 45 x 45 Microstructure. Each color
represents a distinct crystal lattice orientation
randomly selected from cubic FZ.
• FEM prediction: 3 minutes with 16
processors on a supercomputer
• MKS prediction: 30 seconds with only 1
processor on a standard desktop computer
Stress Fields in Polycrystals
Yabansu and Kalidindi, Acta Materialia, 94, pp. 26–35, 2015

22. For a 43X43X43 RVE
the FEM analysis
required 15 hours on a
one 2.4 GHz AMD
processor node in the
Georgia Tech super
computer cluster, while
the MKS predictions
were obtained were
obtained in 306.5
seconds on the same
resource
Plastic Strain Rates in Two-Phase Composites
Montes De Oca Zapiain et al., Acta Materialia, 2017

23. Learning PSP Linkages Through Fusion
of Disparate Information Sources
Process
Structure
Property
GoverningPhysics
• Two main classes of information
sources: (i) multiscale measurements,
and (ii) physics-based multiscale
simulations.
• In experiments, we probe unknown
physics by measuring suitable inputs
and outputs.
• In simulations, we assume the governing
physics and explore the response
depends on inputs.
• Statistical approaches can play a key role
in the objective fusion of information.

24. Bayesian Update for Governing Physics
• Structure-Property Observations: 𝑃 ∈ 𝑷, 𝝁 ∈ 𝑴 and 𝜎 𝑃
2
, 𝜮 𝝁
• Governing Physics: 𝝋, 𝜮 𝝋
• 𝑝 𝝋 𝑴, 𝑷, 𝜎 𝑃
2
, 𝜮 𝝁 , 𝜮 𝝋 ∝ 𝑝 𝑷 𝑴, 𝝋, 𝜎 𝑃
2
, 𝜮 𝝁 , 𝜮 𝝋 𝑝 𝝋 𝜮 𝝋
• The likelihood 𝑝 𝑷 𝑴, 𝝋, 𝜎 𝑃
2
, 𝜮 𝝁 , 𝜮 𝝋 can be computed using the reduced-
order models already calibrated to the best available computational/modeling
toolsets
Selecting the Next Experiment
• Potential New Observations: 𝑷, 𝑴
• Evaluate 𝑝 𝑃 𝝁, 𝑴, 𝑷, 𝜎 𝑃
2
, 𝜮 𝝁 , 𝜮 𝝋 = 𝑝 𝑃 𝝁, 𝝋, 𝜎 𝑃
2
, 𝜮 𝝁 , 𝜮 𝝋 𝑝 𝝋 𝑴, 𝜮 𝝋 𝑑𝝋
• Identify the 𝝁 producing the highest uncertainty in 𝑃 as the next experiment
to be conducted

25. Application: Indentation Experiments and Simulations
β
α
1µm 500µm
5mm
3Å
Need high-throughput protocols that are capable of probing mechanical
responses of small volumes of material at different hierarchical length scales

26. Spherical nanoindentation stress-strain curves
Kalidindi and Pathak. Acta Mat., 56, pp. 3523-3532, 2008.
Indentation strain
Indentationstress
𝑌𝑖𝑛𝑑
𝜎𝑖𝑛𝑑 = 𝐸𝑖𝑛𝑑 𝜀𝑖𝑛𝑑
Load
Displacement
𝐻𝑖𝑛𝑑
𝜎𝑖𝑛𝑑 =
𝑃
𝜋𝑎2
𝜀𝑖𝑛𝑑 =
4
3𝜋
ℎ
𝑎
𝑆 =
𝑑𝑃
𝑑ℎ 𝑒
= 2𝑎𝐸𝑒𝑓𝑓
𝑃 =
4
3
𝐸𝑒𝑓𝑓 𝑅 𝑒𝑓𝑓
1
2
ℎ 𝑒
3
2
𝑆: elastic unloading stiffness
𝜎𝑖𝑛𝑑: indentation stress
ε𝑖𝑛𝑑: indentation strain
2𝑎
2.4𝑎
ℎ

27. Physics-Based Multiscale Simulations
T**1**
*T**
**
)T}FF{(detFT
I}F{F
2
1
E
]C[ET




1
p*
FFF


𝐅p = 𝐋p 𝐅p
𝐋p =
𝛼
𝛾 𝛼 𝐒 𝐨
𝛼
𝛾 𝛼 = 𝛾o
τ 𝛼
s 𝛼
1/m
sgn τ 𝛼
τ 𝛼 = 𝐓∗ ⋅ 𝐒 𝐨
𝛼

28. Calibrating model to experiments
𝐸𝑒𝑓𝑓 𝑔, 𝐶11, 𝐶12, 𝐶44 =
𝑙=0
∞
𝑚=1
𝑀 𝑙
𝑞,𝑟,𝑠=0
∞
𝐴𝑙
𝑚𝑞𝑟𝑠
𝐾𝑙
𝑚
𝑔 𝑃𝑞 𝐶11 𝑃𝑟 𝐶12 𝑃𝑠 𝐶44

29. s (MPa)
Literature 146.12 to 161
Model
Prediction
155.4 ± 3.5
As-cast Fe3%Si
Orientation
( 1, , 2 )
Experimental#
Yind (GPa)
339.8, 54.4, 46.1 1.13 ± 0.04
103.7, 121.6, 49.9 1.12 ± 0.02
232.5, 53.1, 324.0 1.12 ± 0.16
83.2, 125.4, 30.4 1.10 ± 0.02
3.0, 41.3, 76.4 1.09 ± 0.04
194.7, 79.7, 317 1.07 ± 0.01
50.0, 38.1, 250.1 1.06 ± 0.02
114.2, 85, 173.5 0.85 ± 0.04
170.0, 102.6, 357.9 0.91 ± 0.06
163.6, 78.8, 168 0.93 ± 0.04
259.9, 238.0, 145.8 1.0 ± 0.06
𝑌𝑖𝑛𝑑 𝑠 , Φ, 𝜑2 = 𝑠
𝑙=0
∞
𝑚=1
𝑀 𝑙
𝐴𝑙
𝑚
𝐾𝑙
𝑚
(Φ, 𝜑2)
Calibrating model to experiments

30. • Emerging concepts and toolsets in Data science
and Cyberinfrastructure can be strong enablers
for systematic mining and automated capture
of Materials Knowledge and its dissemination
using broadly accessible “open” platforms
• The fusion framework is foundational to the
development and implementation of
autonomous explorations of the unimaginably
large materials and process design spaces while
synergistically leveraging all available
experimental and simulation data
Summary Statements