This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed
in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Delr
llil aXICAL ENGINEERING
SAND2020-10247C
THE UNIVERSITY OF UTAH
Reduced-physics model with symbolic
regression using Bingo for fatigue life
prediction of welds in shell structures.
Keven Carlson, Dr. Jacob Hochhalter, Dr. John Emery
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned
subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
(A.
Departmentof
MECHANICAL ENGINEERING
TET THE UNIVERSITY OF UTAH
• Problem Definition and Overview
• Finite Element Model
•Stress Intensity Factor Computation
•Symbolic Regression
• Results
• Conclusion
Departmentof
MECHANICAL ENGINEERING
117THE UNIVERSITY OF UTAH
Problem Definition
• Stress Intensity Factors (SIFs) drive crack growth.
• SIFs are accurately calculated in 3-D continuum element models
• Computationally expensive
• Shell element models are computationally cheaper, but cannot model
3-D crack growth.
• "Line-weld" elements are used in shell element models to represent
welds.
• We seek a low-fidelity crack growth model that maps stresses in shell
elements to SIFs using Symbolic Regression (SR).
FA_
Departmentof
MECHANICAL ENGINEERING
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FRANC3D
111
1
irfrliq'k
1. FRANC3D used to insert and grow
cracks into model mesh.
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1114101
2. Reads displacements and
material properties from Abaqus
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3. Uses M-integral to compute SIFs
NAPIr
4. SIF results along crack front
determine crack kink angle
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through 80% of thickness
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Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
Symbolic Regression
• Machine Learning: Computers
using data to create predictive
models
• Genetic Programming:
Evolution of computer programs
based on fitness
(?)
• Symbolic Regression: Searches
space of mathematical
functions to fit equation to
inputted data
2.2
(11
+ 7* cos( )
(1 )1)-
'LT
Departrnentof
MECHANICAL ENGINEERING
THE UNIVERSITY OF UTAH
Simple Example
• Data artificially generated from
equation f(x) = 3 * sin(x) + 2
• Allowed equation operators: +,
sin, cos
x, =,
• Trained bingo until error tolerance <
le-4, stack size = 40, population size =
100, max generations = 50
• Output equation:
f(x) = 3 * sin(x)+ 2
Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
Pa reto Front
• Experimental data
• Possible noise or small error
6
— Complexity: 12
— Complexity: 6
• Overfitting
• Model fits equation to noise
5
4-
• Not generalization
• Performs poorly on test data
3>,
• Illustrated when noise is
randomly added to data
• Tradeoff between complexity
and interpretability
2-
0
4
6
8
10
12
Departmentof
MECHANICAL ENGINEERING
117THE UNIVERSITY OF UTAH
Bingo
• NASA-produced, open-source
symbolic regression software
using evolutionary algorithms
• Parallel island evolution strategy
can be used with MPI
• Island evolution strategy allows
for periodic migration of models
between islands
.
.
island A
island D
Island B
■
•
A
island C
Dynamic Island Model Base on Spectral Clustering in Genetic
Algorithm
(Qinxue Meng et al. 2018)
ti
Departmentof
Overview
MECHANICAL ENGINEERING
117THE UNIVERSITY OF UTAH
BC combination
00-
Define BCs so
displacements
in both models
is similar
BC
Max allowable
displacements
Sobol'
sensitivity
analysis on BCs
effect on max
principal stress
BC combinations
Crack insertion
and growth in
continuum
element model
using
FRANC3D
SIFs and crack size data
Symbolic
regression
using Bingo
Shell element
model is
simulated using
determined
boundary
conditions
Interprettable,
closed form
solution to
describe data
1
Line weld stresses
Line weld stress mapping is left for
future work
FA_
Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
• Problem Definition and Overview
• Finite Element Model
•Stress Intensity Factor Computation
•Symbolic Regression
• Results
• Conclusion
Departmentof
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117 THE UNIVERSITY OF UTAH
Finite Element Model Geometry
• C-Channel:
• Welded to a plate
• Flange length: 50.8 mm in
y-direction
• Constant thickness of
3.048 mm
• Plate:
• 254 mm in z-direction
• 203.2 mm in x-direction
• 3.048mm in y-direction
• Weld Thickness: 3.048mm
• Idealized as one solid
material with no heataffected zone from weld
Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
• Problem Definition and Overview
• Finite Element Model
•Stress Intensity Factor Computation
•Symbolic Regression
• Results
• Conclusion
Departmentof
MECHANICAL ENGINEERING
117THE UNIVERSITY OF UTAH
Training Data Generation
• 180 different BC combinations
• Number of discrete displacements applied to surfaces dependent on
sensitivity
• 0.05 mm initial crack size
• Much lower than what non destructive evaluation would detect
• Crack inserted perpendicular to MPS at location on weld surface with
greatest MPS
• Crack grown to 80% thickness of weld or about 2.4 mm at crack steps
of 0.24 mm
• Smaller crack step sizes show minimal difference on output
• Important values used in symbolic regression model: SIF, MPS, crack
geometry, initial crack orientation, crack growth path
Departmentof
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117 THE UNIVERSITY OF UTAH
Note:
• Some cracks grow downward
into the plate
• Not likely behavior in practice
because of the weld boundary
• The crack growth path
• Plane fit to points of all crack
fronts after initial crack insertion
• Thickness
• Defined as the distance from crack
insertion to where the crack would
break through the surface
Downward growing crack.
Departmentof
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117 THE UNIVERSITY OF UTAH
Different Crack Growth Directions
•
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Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
• Problem Definition and Overview
• Finite Element Model
•Stress Intensity Factor Computation
•Symbolic Regression
• Results
• Conclusion
Departmentof
MECHANICAL ENGINEERING
117THE UNIVERSITY OF UTAH
•
K1
CT 7-\/1.
Bingo Model For SIFs
f(2c )2)t Ili)112)
• a: crack depth
• c: half crack width
• t: thickness relative to crack growth path
• a: maximum principal stress at point of crack insertion
• 1/1 = (x)y)z)
• 112 = (x2)y2)z2)
• ill: normal unit vector of crack at insertion
• /12: normal unit vector of plane fit through all crack front points
Departmentof
MECHANICAL ENGINEERING
1111 THE UNIVERSITY OF UTAH
Normal vectors
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Moor Atibm,
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Departmentof
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UTAH
• Problem Definition and Overview
• Finite Element Model
•Stress Intensity Factor Computation
•Symbolic Regression
• Results
• Conclusion
Departmentof
MECHANICAL ENGINEERING
1111 THE UNIVERSITY OF UTAH
Symbolic Regression
• Model from Pareto front with
operations that made sense
O 11 -
• i.e. some models had equations
with components of the form x'
• Of the lower complexity models,
performed best on test dataset
O 10 -
O 09 -
• Mean absolute error: 0.0543
O 07 -
O 05 00
2.5
l
5.0
7.5
10.0
12.5
Complexity
15.0
'
17.5
'
20.0
1
Gradient Boosting
• Boosting
• Using an ensemble of"weak" learners that together create a strong one.
• Gradient Boosting
• Each "weak" learner is found from the error of the previous learner
• Prediction Ds found with: El= 4=1 1;1,( ❑ where 1;1„( ❑ is each function
found through symbolic regression using the results from
I;11(Q=
:11;1/(Q.
Subtract First Model
.
.
•
46
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0.6
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—0.3
0.0
0.2
0.4
alt
0.6
0.8
0.0
0.2
0.4
a/t
0.6
0.8
• Becomes more correlated with a/t
)1)
c
Departmentof
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Second and Third Models
• Second Model
• Trained on residual error off1 on train dataset.
• Similar selection method as first model
• Fitness: 0.0482
a
P — IYIt
f2 = —0.1657
• Third Model
0.1657(p(p *ppyr.25 + 0.176
• Trained on residual error off1 +f2
• Fitness: 0.0466
f3
y (z2 0.933)
z(z 1)± 1
-\/
11)).
MECHANICAL ENGINEERING
THE UNIVERSITY OF UTAH
i/,4/\
*afbr
Check For Convergence
• Fourth model did not improve
overall model.
0.056 -
• Test error is below training error,
may be a result of fewer
datapoints and thus fewer
chances of large outliers.
• 80% train, 20% test randomly
split
Mean Absolute Error
'LT
Departrnentof
0.054 -
0.052
0.050 -
0.048 -
Test Error
Train Error
2
3
Number of Functions
)).
Departmentof
Removing Outliers
MECHANICAL ENGINEERING
117 THE UNIVERSITY OF UiAH
f1 + f2
fl
120 -
100 -
80 -
60 -
40 -
20 -
fl =
25
50
75
100
125
25
50
75
100
1.5 -k 0.141) + 0 1 CC)
Oh
Departmentof
MECHANICAL ENGINEERING
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• Problem Definition and Overview
•Symbolic Regression
• Finite Element Model
•Sensitivity Analysis
• Fracture Analysis
• Results
• Conclusion
Departmentof
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Future Implications
a 2
Fs =[M1 + M2
(
t
• Raju-Newman Semi-Elliptical
Surface Problem
• Gradient Boosting with SR can
automate this process
For a/c
a 4
M3
(
doh
t
1
= 1.13 — 0.09ea—)
M2= — 0.54 +
0.89
0.2 +
()
M3 =0.5
a)
24
1
+ 14(1 —
0.65 +
c
a)
2
g --= 1 +[0.1 + 0.35(— (1 — sin 4-17
t
-
Departmentof
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Conclusion
• Symbolic regression can give interpretable results of how each input
affects the output
• Models for SIFs given MPS and crack geometry found to be much
easier for SR than line-weld stresses to MPS
• Gradient boosting can be used with SR to improve the overall model
when more complex equations are expected
• Use gradient boosting with SR to revisit classical fracture mechanics
SIF solutions
• Simpler geometry could be beneficial for finding physically
interpretable results
)).
Departmentof
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Thank You
• Acknowledgements:
• Sandia National Laboratories
• Dr. John Emery
• Dr. Geoffrey Bomarito
• Jonas Merrell
• University of Utah Center for High Performance Computing
fi
Sandia
National
laboratories
This work was funded in part by Sandia National Laboratories. Sandia National Laboratories is a
multimission laboratory managed and operated by National Technology & Engineering Solutions of
Sandia, L.L.C., a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department
of Energy's National Nuclear Security Administration under contract DE-NA0003525.