Adams, Ethan

Bomarito, Geoffrey

Esham, Kathryn

Townsend, Tyler

English

NASA Technical Reports Server

Formulation of Plasticity Models through 
Symbolic Regression
Geoffrey Bomarito
NASA Langley Research Center
Tyler Townsend
University of Central Florida
Kathryn Esham
The Ohio State University
Ethan Adams
Brigham Young University Idaho
https://ntrs.nasa.gov/search.jsp?R=20200002537 2020-05-24T04:50:42+00:00Z
Motivation for homogenization
WCCM 2018 27/23/2018
metallic alloy
Φ = 𝑓(𝜎)
homogenization
Homogenization methods
Human developed/derived models
 Pros:
 Can be physically based
 Transferability
 Compact and quick to evaluate
 Cons:
 Can take decades in 
development
Typical machine learning models
 Pros:
 Rapid development (training)
 More input → more accurate
 Cons:
 Not transferable
 Not insightful (black box)
 Evaluation is relatively expensive
Response = 𝑓(𝑙𝑜𝑎𝑑𝑖𝑛𝑔)
Can we have the best of both worlds?
7/23/2018 WCCM 2018 3
Φ = 𝑓(𝜎)
Human developed homogenization models
4WCCM 20187/23/2018
 Choose functional form
 Fit parameters (in red)
 Model misfit identified
 Abuse parameters
 Add more physics to 
functional form 
Φ =
 𝜎
𝜎𝑦
2
+ 2𝑞1𝑓
∗𝑐𝑜𝑠ℎ
3
2
𝑞2
𝜎ℎ
𝜎𝑦
− 1 + 𝑞1𝑓
∗ 2
Example: GTN porous plasticity model
Φ =
 𝜎
𝜎𝑦
2
+ 2𝑞1(𝜎)𝑓
∗𝑐𝑜𝑠ℎ
3
2
𝑞2(𝜎)
𝜎ℎ
𝜎𝑦
− 1 + 𝑞1(𝜎)𝑓
∗ 2
Time consuming step!
Symbolic regression and homogenization
5WCCM 20187/23/2018
 Find best fit functional forms and 
parameters simultaneously!
1. Decide what data to use
2. Define fitness to data
3. (Decide how much data to use)
 Attribute physics to portions of equations
Φ = 𝑓(𝜎)
𝒚 = 𝟏𝟎𝒆−𝒙 𝐬𝐢𝐧 𝟐𝒙 + 𝒙
y
x
0 1 2 3
0
4 5
1
2
3
4
5
6
Verification problem: von-Mises plasticity
6WCCM 20187/23/2018
𝜎1
𝜎2
𝜎𝑦𝑖𝑒𝑙𝑑
von Mises
Φ = 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 + 𝜎3 − 𝜎1
2 − 2𝜎𝑦𝑖𝑒𝑙𝑑
2 = 0
RVE:
 Single element model
 von-Mises plasticity
Verification:
 Can we recover 𝜙 from 
looking at response data?
Symbolic regression problem definition
7WCCM 20187/23/2018
1. Decide what data to use
2. Define fitness to data
 proportional loading
 Data for each loading case:
 Principle stresses: σ𝑖
 Principle strains: 𝜖𝑖
 Equivalent plastic strain:  𝜖𝑝
𝐿𝑜𝑎𝑑𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜: [0.675 1.0 0.8250] 𝐿𝑜𝑎𝑑𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜: [0.278 0.238 1.0]
elastic
plastic
Symbolic regression problem definition
8WCCM 20187/23/2018
1. Decide what data to use
2. Define fitness to data
 Φ = 𝑓 𝜎 = 0 (on yield surface)
 Implicit regression
 𝐸 =  
𝑑𝑓 𝜎
𝑑𝜎
:
𝑑𝜎
𝑑𝑡
𝑑𝑓 𝜎
𝑑𝜎
:
𝑑𝜎
𝑑𝑡
→ 0
 𝚽 𝝈 = constant for each loading case
Solving the symbolic regression problem
9WCCM 20187/23/2018
 Using genetic programming
 (Genetic algorithms of computer programs)
 Equations evolve untill they fit the data
 In-house code: bingo
NODE
operators
TERMINAL
constants,
variables
-
*
X X
X
*
1.5
*
𝑦 = 1.5𝑥2 − 𝑥3
Mutation
Crossover
-
*
X X
X
*
1.5
*
-
*
X X
X
*
1.5
*
-
X
**
X X
+
*
X X
X
*
1.5
*
+
*
X 2
X
*
1.5
*
Early results
10WCCM 20187/23/2018
 Looking for yield surface: Φ(𝜎) = 0
 Φ 𝜎 = constant   for each loading case
 Issue: all loading cases are parallel!
 Solution: more complex loading cases
𝜎1 − 𝜎3 + 𝜎2 + 𝜎1 = 𝑐𝑜𝑛𝑠𝑡𝑠𝑎𝑛𝑡
x
y Φ 𝑥, 𝑦 = y = constant
SR
Data: 𝜎1, 𝜎2, 𝜎3,  𝜖
𝑝
Early results
11WCCM 20187/23/2018
 Looking for yield surface: Φ(𝜎) = 0
 Φ 𝜎 = constant   for each loading case
 Issue: all loading cases are parallel!
 Solution: more complex loading cases
x
y Φ 𝑥, 𝑦 = y = constant
Stage 1
𝐿𝑜𝑎𝑑𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜: [0.675 1.0 0.825]
Stage 2 
𝐿𝑜𝑎𝑑𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜: [1.0 0.825 0.675]
𝜎1 − 𝜎3 + 𝜎2 + 𝜎1 = 𝑐𝑜𝑛𝑠𝑡𝑠𝑎𝑛𝑡
Yield surface from 2 stage loading data
12WCCM 20187/23/2018
𝜎3 − 𝜎2 𝜎3 − 𝜎2 − 𝜎2 − 𝜎1 𝜎3 − 𝜎1
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Computation time: 32s on 40 processors
Data: 𝜎1, 𝜎2, 𝜎3,  𝜖
𝑝
SR
Φ = 𝜎3 − 𝜎1
2 + 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Adding hardening
13WCCM 20187/23/2018
No hardening
𝛔𝐲𝐢𝐞𝐥𝐝
𝐩𝐥𝐚𝐬𝐭𝐢𝐜 𝐬𝐭𝐫𝐚𝐢𝐧
Hardening
𝛔𝐲𝐢𝐞𝐥𝐝
𝐩𝐥𝐚𝐬𝐭𝐢𝐜 𝐬𝐭𝐫𝐚𝐢𝐧
Φ = 𝜎3 − 𝜎1
2 + 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Φ = 𝜎3 − 𝜎1
2 + 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 − 𝑐1  𝜖
𝑝 − 𝑐2  𝜖
𝑝 2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Hardening yield surface from 2 stage loading data
14WCCM 20187/23/2018
 𝜎1 − 𝜎3 + 𝜎3 − 𝜎2 𝜎3 − 𝜎2 + 𝜎1 +
47602 + 𝜎1 + 𝜎3 − 𝜎1 − 𝜎3 (𝜎3 + 𝜎3 −
Φ = 𝜎3 − 𝜎1
2 + 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 − 𝑐1  𝜖
𝑝 − 𝑐2  𝜖
𝑝 2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Φ = 𝜎3 − 𝜎1
2 + 𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎3
2 − 19797  𝜖𝑝 − 980185  𝜖𝑝 2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Yield surface now depends on a state variable!
Now it needs a state evolution equation
Data: 𝜎1, 𝜎2, 𝜎3,  𝜖
𝑝
SR
Computation time:  1.5h on 160 processors
Hardening yield surface from 2 stage loading data
15WCCM 20187/23/2018
  𝜖𝑝 =
2
9
 𝜖1 −  𝜖2 2 +  𝜖2 −  𝜖3 2 +  𝜖3 −  𝜖1 2
Data:  𝜖1,  𝜖2,  𝜖3,   𝜖
𝑝
Assuming incremental elastic strains are small
Goal: 
 𝜖1 +  𝜖2 +  𝜖3 = constant
 𝜖1 +  𝜖2 +  𝜖3 = 0 deviatoric plastic strains 
Goal:   𝜖𝑝 =
4
3
 𝜖1
2 +  𝜖1  𝜖2 +  𝜖2
2
 𝜖1
2 +  𝜖1  𝜖2 +  𝜖2
2 −   𝜖𝑝 2 = constant
SR
SR
Data:  𝜖1,  𝜖2,  𝜖3,   𝜖
𝑝
Quick Recap
16WCCM 20187/23/2018
 Verification of von-Mises plasticity
 Non-hardening yield surface
 Hardening yield surface
 State evolution
 Moving Forward:
 How much data is required?
Seconds (on single processor) hours (on multiple processors)
How much data is needed?
17WCCM 20187/23/2018
 Step size
 More dense data = 
 more computation time
 more accurate derivative calculations
 Density needed will depend on 
 complexity of loading scenarios
 Complexity of yield surface
16128
e
rr
o
r 
in
 d
e
ri
v
a
ti
v
e
s
step size
0 5 10 15 20
10-9
10-8
10-7
10-6
10-5
10-4
How much data is needed?
18WCCM 20187/23/2018
 Number of loading scenarios
 No real trend 
(except very low values)
 Minimum case found:
2 4 8
Summary
19WCCM 20187/23/2018
 Set up framework for SR formulation of 
plasticity models
 Implicit symbolic regression of yield surface
 Use non-proportional loading
 Von-Mises verification problem
 Surprisingly little data needed 
 will depend on complexity of yield surface
Φ = 𝑓(𝜎)
homogenization
R
V
E
D
a
ta
S
y
m
b
o
lic
 
R
e
g
re
s
s
io
n
 application to real materials
 adaptive data generation
Future work
20WCCM 20187/23/2018
 bingo (soon to be open source)
 python & c++
 Features:
 Coevolution of fitness predictors
 Island parallelization scheme
 Acyclic graph representation
 Constant optimization
 Age-fitness Pareto selection
performance
robustness
thank you!
geoffrey.f.bomarito@nasa.gov


Formulation of Plasticity Models through Symbolic Regression

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