Higher-Order Localization Relationships Using the MKS Approach

Tony Fast and Surya R. Kalidindi

Department of Materials Science and Engineering

Drexel University

Philadelphia,PA

Material Hierarchy

McDowell DL. A perspective on trends in multiscale plasticity. International Journal of Plasticity 2010;26:1280.c

LOCALIZATION ~ TOP-DOWN

HOMOGENIZATION ~ BOTTOM-UP

Localization is important to track microstructure evolution at lower length scales in concurrent multiscale modeling

Concurrent Multi-scaling

1. Homogenize coarse scale2. Isolate region where homogenization fails 3. Implement lower scale numerical simulations4. Isolate region where homogenization fails 5. Implement fine scale numerical simulations

…..

Liu W, McVeigh C. Computational Mechanics 2008;42:147

Quasicontinuum, Bridging Scale Method

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$

$

$

$

$

$

Develop efficient, effective localization models for DNS to reduce cost

• Many Inputs and Many Outputs• Repetitive simulation is demanding

• Simulation produces a lot of data, but what information is determined about the system?

microstructure signal local response

Stress, strain, evolution

DATA VS. KNOWLEDGE

How can information about new structures be extracted?What knowledge is gained?

• DSP representation of local structure-local response• Localization relationship and its influence coefficients

• Extended from Kroner’s Green’s function

• Influence coefficients capture the combined point effects of the MS configuration on the local response

microstructure signal local response influence coefficients

DATA VS. KNOWLEDGE

𝑝𝑠=∑𝑡=0

𝑆−1

∑𝑖=1

𝐼

𝑎❑𝑖𝑡𝑚

𝑖𝑠+𝑡❑

microstructure signal local response influence coefficients

DATA VS. KNOWLEDGE

𝑝𝑠=∑𝑡=0

𝑆−1

∑𝑖=1

𝐼

𝑎❑𝑖𝑡𝑚

𝑖𝑠+𝑡❑

In the Materials Knowledge System, influence coefficients capture knowledge (physics) of the system as a convolution filter with the microstructure. They provide a database to efficiently

extract, store, and recall local microstructure-processing-property linkages.

Facilitates exploration of the local response of other MS configurations

Microstructure is inherently discrete because of probe size and resolution limits of physical model or characterization method

Microstructure

Position (s)

FIRST-ORDER FILTERDecompose MS into salient features

{𝑠1 ,𝑠2 ,𝑠3 }→𝑠

is the digital MS signal

𝑚1𝑠❑

𝑚2𝑠❑

DISCRETE MICROSTUCTURE TO DIGITAL SIGNAL

Spinodal decomposition of binary alloy

– Phase field modeling

Elastic and Thermo-Elastic Response of Dual Phase Composites

– Finite Element Analysis

Materials Knowledge SystemsEffective Localization Models

Evolution Field

Accuracy

5.12

1 E

E

𝑝𝑠=∑𝑡=0

𝑆−1

∑𝑖=1

𝐼

𝑎❑𝑖𝑡𝑚

𝑖𝑠+𝑡❑

Landi, G., S. R. Niezgoda, et al. (2009). Acta Materialia 58(7): 2716-2725T. Fast, S. R. Niezgoda, S. R. Kalidindi, Acta Materialia 59, 699 (2011)..

Establishing Knowledge Databases

• Coefficients calibrated to validated direct numerical simulations

• Calibration using OLSF facilitated by decoupling spatial components using Fourier transform

– Drastic reduction in complexity and parallelizable

• Limited to weakly nonlinear systems linkages assumed to be linear• Further extension relies on nonlinear system identification methods

𝔍 (𝑝𝑠 )𝑘=∑h=1

𝐻

𝐴❑h𝑘𝑀

h𝑘❑𝔍𝑝𝑠=∑

𝑡=0

𝑆−1

∑h=1

𝐻

𝑎❑h𝑡𝑚

h𝑠+𝑡❑

First Order Filter

<s+1> Higher-Order Filter <s+2> Higher-Order Filter

Higher-Order Microstructure Signals

Number of HO signals grows rapidly

Higher-Order Coefficients (Convolution Filters)

Strain

• IC relating to Higher-Order Signals are Volterra kernels that capture strong nonlinear interactions

𝑝𝑠=∑𝑡=0

𝑆−1

∑𝑖=1

𝐼

𝑎❑𝑖𝑡1~𝑚𝑖

𝑠+𝑡 1❑

H

h

H

h

S

t

S

t

htts

htts

hts

hhhttts

N N

N

N

N

Nmmmap

1 1

1

0

1

01 1

1

2

21

1

1

21

21

HO

𝔍 (𝑝𝑠 )𝑘=∑h=1

𝐻

𝐴❑h𝑘𝑀

h𝑘❑𝔍

• Contrast (nonlinearity) – Young’s modulus ratio• Uniaxial 1-1 strain• Random distribution of phases in microstructure

Finite Element Simulation of Dual Phase Composite

2

1

E

E

FEMε=5e-4

• Redundant signals translate to linear relationships in spectral domain

• Assumptions of HOIC– Nearest vectors (neighbors) contribute most– Higher-Order coefficients capture nonlinearity best

Protocols for Establishing Higher-Order Terms

Case Combination of Coefficients Selected1 First Order Coefficients2 Second Order Coefficients up to first neighbors3 Second Order Coefficients up to second neighbors

…

7 Second Order Coefficients up to sixth neighbors8 Seventh Order Coefficients up to first neighbors

9Seventh Order Coefficients up to first neighbors plus Second Order Coefficients from second neighbors to sixth neighbors

nsm

nm1nm3

nm5

nm6

nm2

• Different HOICs calibrated from 400 FEM simulations• Training (calibration) set vs. Validation set describes how well knowledge is

captured

• Improvement in accuracy decreases with distant neighbors• Combining coefficients improves accuracy and precision while maintaining

tractability

52

1 E

E10

2

1 E

E

Stronger Contrast Knowledge Systems

Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order

Accurate Strain Localization

• HOIC of increasing order captures local information better

52

1 E

E10

2

1 E

E

• Drastic improvement of linkages of FOIC• Accuracy has a strong dependence on nonlinearity

Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order

Accurate Strain Distribution

Characteristics of distribution are improved particularly in the tails with HOIC

52

1 E

E10

2

1 E

E

Case 1: First Order Case 2 – 7: Second Order Case 8-9: Seventh Order

Extension to larger domainsdrastic time savings

FEM required 45 min on supercomputerMKS required 15 seconds on a desktop computer

MKS – NlogN(N)

Case 9: Seventh-Order to First Neighborhood and Second-Order to Sixth Neighborhoods

153 influences coefficients have finite memory and decay to zero at larger distances

Conclusions

• Higher-order coefficients are crucial in developing effective localization models for strongly nonlinearity systems

• Systematic selection of higher-order neighborhoods facilitated the development of this work– These concepts hinge off the finite-memory of the physical

interactions

• MKS provides drastic time savings over DNS particularly in the extension to larger spatial domains