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DEVELOPMENT OF SEMI-STOCHASTIC ALGORITHM FOR OPTIMIZING
ALLOY COMPOSITION OF HIGH-TEMPERATURE AUSTENITIC
STAINLESS STEELS (H-SERIES) FOR DESIRED MECHANICAL PROPERTIES
George S. DulikravichMAIDO Institute, Mech. & Aero. Eng. Dept., Univ. of Texas at Arlington
Igor N. EgorovIOSO Technology Center, Moscow, Russia
Vinod K. Sikka and G. MuralidharanOak Ridge National Laboratory, Tennessee
Funded by DoE- Idaho Office and Army Research Office
Ultimate ObjectivesUse and adapt an advanced semi-stochastic
algorithm for constrained multi-objective optimization and combine it with
experimental testing and verification to determine optimum concentrations of alloying elements in heat-resistant and corrosion-resistant H-Series austenitic
stainless steel alloys that will simultaneously maximize a number of alloy’s mechanical
and corrosion properties.
The proposed algorithm also requires a minimized number of
alloy samples that need to be produced and experimentally tested thus minimizing the overall cost of
automatically designing high-strength and corrosion-resistant H-
Series austenitic alloys.
Why this approach?Because the existing theoretical
models for prediction and possible optimization of physical
properties are extremely complex, are not general, and
are still not reliable.
Why optimization?Because brute-force
experimentation would require an enormous matrix of
experimental samples and data that would be too time-
consuming and too costly.
Constrained optimization algorithms
• Gradient Search (DFP, SQP)
• Genetic Algorithms
• Simulated Annealing
• Simplex (Nelder-Mead)
• Differential Evolution Algorithm
• Self-adaptive Response Surface (IOSO) & NNA
Why semi-stochastic optimization?
Because gradient-based optimization is incapable of solving such multi-extremal multi-objective constrained
problems.
The self-adapting response surface formulation used in this optimizer
allows for incorporation of realistic non-smooth variations of
experimentally obtained data and allows for accurate interpolation of
such data.
The main benefits of this algorithm are its outstanding reliability in avoiding local minimums, its computational speed, and a
significantly reduced number of required experimentally evaluated alloy samples as compared to more traditional optimizers like genetic
algorithms.
Parallel Computer of a “Beowulf” type• Based on commodity hardware and public domain software• 16 dual Pentium II 400 MHz and 11 dual Pentium 500 MHz based
PC’s• Total of 54 processors and 10.75 GB of main memory • 100 Megabits/second switched Ethernet using MPI and Linux • Compressible NSE solved at 1.55 Gflop/sec with a LU SSOR solver
on a 100x100x100 structured grid on 32 processors (like a Cray-C90)
How does this apply to alloys?Although of general applicability, the IOSO will be demonstrated on the optimization of the chemical composition of H-Series stainless steels based on Fe-Cr-Ni ternary.
How does it work?1. Start with as large set of
reliable experimental data for the same general class of
arbitrary alloys as you can find anywhere. Response surfaces are then created that are based
on these experimental data
How is additional data created?Artificial neural networks (ANN)
were used for creating the response surfaces. We also used radial-basis functions that were
modified for the specifics of this optimization research.
Summary of the technical approachEvery iteration of multi-objective
optimization consists of:1. Constructing and training the ANN1 for a given set of experimental points.
2. Using ANN1 to create additional data points. Thus, ANN1 is doing what is usually done by complex constitutive
models and expensive experimentation.
3. Determining a subset of experimental points that are the closest to P1 points in
the space of design variables.4. Training the ANN2 so that it gives the
best predictions when applied to the obtained subset of experimental points .
5. Carrying out multi-objective optimization using ANN2 and obtaining
the pre-defined number of Pareto-optimal solutions P2.
Design variablesAs the independent design variables for this problem we considered the
percentage of following components:
C, Mn, Si, Ni, Cr, N. Ranges of their variation were set in
accordance with lower and upper bounds of the available set of
experimental data.
Multiple simultaneous objectivesThe main objective was maximizing the strength of the H-series steel after 100 hours under the temperature of 1800 F. Additional three objectives were to simultaneously minimize the percentages of Mn, Ni, Cr. Thus, the multi-objective optimization problem had 6 independent design variables and 4 simultaneous objectives. We defined the desirable number of Pareto optimal solutions as 10 points.
Accuracy of neural network ANN1
Accuracy of neural network ANN2
An Example of Stochastic Multi-Objective
Constrained Optimization of a Large Experimental Dataset
Sumultaneously:1. Maximize PSI
2. Maximize HOURS3. Maximize TEMP
Critical issuesExisting publicly available experimental data base is
practically non-existent. It needs to be expanded as much as
possible and well documented in order to minimize the number of
future experiments needed.
Fig. 1. Distribution of percentage of sulfur (S)
in database alloys.
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
1 16 31 46 61 76 91 106 121 136 151 166
Multi-objective optimization based on a 158 point experimental dataset
Fig.2. Results of Problem No.1 solution in
objectives’ space.
2000 4000 6000 8000 10000
PSI
2000
4000
6000
8000
10000H
OU
RS
Fig.3. Interdependence of optimization
objectives for Pareto set.
2000 4000 6000 8000 10000
PSI
2000
4000
6000
8000
10000
HO
UR
S
Fig. 4. Sets of Pareto optimal solutions of
problems 2-6.
2000 4000 6000 8000 10000
P S I
0
4000
8000
12000
HO
UR
S
2 - T>=1600
3 - T>=1800
4 - T>=1900
5 - T>=2000
6 - T>=2050
0 0.1 0.2 0.3 0.4 0.5 0.6
C ,%
0
2
4
6
0 0.004 0.008 0.012 0.016
S ,%
0
2
4
6
0.005 0.01 0.015 0.02 0.025 0.03 0.035
P ,%
0
2
4
6
15 20 25 30 35 40
C r,%
0
2
4
6
0 - EXPER IM EN TAL D ATA R AN G E
1 - 3-C R ITER IA O PTIM IZATIO N
2 - T>=1600
3 - T>=1800
4 - T>=1900
5 - T>=2000
6 - T>=2050
1 0 2 0 3 0 4 0 5 0 6 0
N i , %
0
2
4
6
0 . 4 0 . 8 1 . 2 1 . 6 2M n , %
0
2
4
6
0 0 . 5 1 1 . 5 2 2 . 5
S i , %
0
2
4
6
0 0 . 0 4 0 . 0 8 0 . 1 2 0 . 1 6
C u , %
0
2
4
6
0 - E X P E R I M E N T A L D A T A R A N G E
1 - 3 - C R I T E R I A O P T I M I Z A T I O N
2 - T > = 1 6 0 0
3 - T > = 1 8 0 0
4 - T > = 1 9 0 0
5 - T > = 2 0 0 0
6 - T > = 2 0 5 0
0 0.04 0.08 0.12 0.16
M o,%
0
2
4
6
0 0.04 0.08 0.12Pb,%
0
2
4
6
0 0.1 0.2 0.3 0.4
C o,%
0
2
4
6
0 0.4 0.8 1.2 1.6
C b,%
0
2
4
6
0 - EXPER IM EN TAL D ATA R AN G E
1 - 3-C R ITER IA O PTIM IZATIO N
2 - T>=1600
3 - T>=1800
4 - T>=1900
5 - T>=2000
6 - T>=2050
0 0.1 0 .2 0 .3 0 .4 0 .5
W ,%
0
2
4
6
0 0.002 0.004 0.006S n,%
0
2
4
6
0 0.02 0.04 0.06 0.08
A l,%
0
2
4
6
0 0.004 0.008 0.012 0.016
Zn,%
0
2
4
6
0 - E X P E R IM E N TA L D A TA R A N G E
1 - 3-C R ITE R IA O P TIM IZA TIO N
2 - T>=1600
3 - T>=1800
4 - T>=1900
5 - T>=2000
6 - T>=2050
GoalsGoalsThe final outcome of the project
will be the ability of H-Series stainless steel producers and users
to predict either the alloy compositions for desired
properties or properties of given alloy compositions.
Potential payoffSuch capability will have economic
benefit of using the correct alloy compositions and large energy
savings through process improvement by the use of
optimized alloys.
CommercializationAfter the first year, a ready-to-use commercialized version of
the single-property alloy-composition optimization
software will be licensed to U.S. industry and government
laboratories.
Future plans1. Create larger experimental data sets
from the available publications2. Incorporate more design variables
(chemical elements)in the multi-objective optimization
3. Add additional objectives (tensile stress, corrosion resistance, cost of the
material) to the set of multiple simultaneous objectives.
