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.