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Constrained optimization algorithms

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Constrained optimization algorithms

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  1. DEVELOPMENT OF SEMI-STOCHASTIC ALGORITHM FOR OPTIMIZINGALLOY COMPOSITION OF HIGH-TEMPERATURE AUSTENITICSTAINLESS STEELS (H-SERIES) FOR DESIRED MECHANICAL PROPERTIESGeorge S. DulikravichMAIDO Institute, Mech. & Aero. Eng. Dept., Univ. of Texas at ArlingtonIgor N. EgorovIOSO Technology Center, Moscow, RussiaVinod K. Sikka and G. MuralidharanOak Ridge National Laboratory, TennesseeFunded by DoE- Idaho Office and Army Research Office

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. Constrained optimization algorithms • Gradient Search (DFP, SQP) • Genetic Algorithms • Simulated Annealing • Simplex (Nelder-Mead) • Differential Evolution Algorithm • Self-adaptive Response Surface (IOSO) & NNA

  7. Why semi-stochastic optimization?Because gradient-based optimization is incapable of solving such multi-extremal multi-objective constrained problems.

  8. 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.

  9. 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.

  10. 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)

  11. 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.

  12. 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

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. Accuracy of neural network ANN1

  19. Accuracy of neural network ANN2

  20. An Example of Stochastic Multi-Objective Constrained Optimization of a Large Experimental DatasetSumultaneously:1. Maximize PSI2. Maximize HOURS3. Maximize TEMP

  21. 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.

  22. Fig. 1. Distribution of percentage of sulfur (S) in database alloys.

  23. Multi-objective optimization based on a 158 point experimental dataset

  24. Fig.2. Results of Problem No.1 solution in objectives’ space.

  25. Fig.3. Interdependence of optimization objectives for Pareto set.

  26. Fig. 4. Sets of Pareto optimal solutions of problems 2-6.

  27. GoalsThe 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.

  28. 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.

  29. 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.

  30. Future plans1. Create larger experimental data sets from the available publications2. Incorporate more design variables (chemical elements)in the multi-objective optimization3. Add additional objectives (tensile stress, corrosion resistance, cost of the material) to the set of multiple simultaneous objectives.

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