Constraint Satisfaction Approach to the Design of Multi-Component, Multi-Phase Alloys

2014 ◽  
Author(s):  
Edgar Galvan ◽  
Richard J. Malak ◽  
Sean Gibbons ◽  
Raymundo Arroyave
Keyword(s):  
Phase Stability ◽  
Constraint Satisfaction ◽  
Materials Science ◽  
Constraint Satisfaction Problem ◽  
Error Rates ◽  
Optimization Techniques ◽  
Point Mapping ◽  
All Solutions ◽  
Highly Nonlinear ◽  
Thermodynamic Conditions

The development of new materials must start with an understanding of their phase stability. Researchers have used the CALPHAD method to develop self-consistent databases encoding the thermodynamics of phases. In this forward approach, thermo dynamic conditions (processing conditions such as composition, temperature, pressure, etc.) are mapped to equilibrium states. In this research, we are instead interested in the inverse problem of mapping a set of desired phase constitutions to the set of thermodynamic conditions that give rise to them. Recently, search and optimization techniques have been used to determine thermodynamic conditions that yield a particular phase stability state (point-to-point mapping). In this research, we focus on a more general problem: mapping of specific regions in multi-dimensional phase constitution spaces to ranges in values of thermodynamic conditions(set-to-set mapping). In the context of search theory, we are interested in finding all solutions to a Continuous Constraint Satisfaction Problem (CCSP). The problem is typically multi-dimensional, highly nonlinear, and, importantly, contains non-isolated solutions (the solution is ranges of values rather than finite points). Numerical methods for finding all solutions to CCSPs typically rely on branch-and-prune methods, which interleave branching with pruning steps. These methods are mainly designed to address CCSPs with isolated solutions and would be inefficient if applied to the problem at hand. In this work, we describe a novel algorithm to search the thermodynamic phase field for the full set of thermodynamic conditions that result in user-specified phase constitutions. The approach combines techniques from computational materials science, evolutionary computation, and machine learning to approximate the non-isolated solution set to the CCSP. We investigate the performance of the algorithm on an Fe-Ti binary alloy system using ThermoCalc with TCFE7 database. For this system, the algorithm is able to generate solutions with low error rates.

scholarly journals A Constraint Satisfaction Algorithm for the Generalized Inverse Phase Stability Problem

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Edgar Galvan ◽  
Richard J. Malak ◽  
Sean Gibbons ◽  
Raymundo Arroyave
Keyword(s):  
Phase Stability ◽  
Constraint Satisfaction ◽  
Stability Problem ◽  
High Performance ◽  
Generalized Inverse ◽  
Constraint Satisfaction Problem ◽  
Alloy System ◽  
Phase Constitution ◽  
Thermodynamic Conditions ◽  
Binary Alloy System

Researchers have used the (calculation of phase diagram) CALPHAD method to solve the forward phase stability problem of mapping from specific thermodynamic conditions (material composition, temperature, pressure, etc.) to the associated phase constitution. Recently, optimization has been used to solve the inverse problem: mapping specific phase constitutions to the thermodynamic conditions that give rise to them. These pointwise results, however, are of limited value since they do not provide information about the forces driving the point to equilibrium. In this paper, we investigate the problem of mapping a desirable region in the phase constitution space to corresponding regions in the space of thermodynamic conditions. We term this problem the generalized inverse phase stability problem (GIPSP) and model the problem as a continuous constraint satisfaction problem (CCSP). In this paper, we propose a new CCSP algorithm tailored for the GIPSP. We investigate the performance of the algorithm on Fe–Ti binary alloy system using ThermoCalc with the TCFE7 database against a related algorithm. The algorithm is able to generate solutions for this problem with high performance.

scholarly journals Permutation groups with small orbit growth

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manuel Bodirsky ◽  
Bertalan Bodor
Keyword(s):  
Automorphism Group ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Permutation Groups ◽  
First Order ◽  
Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

scholarly journals Optimal control of the Sabatier process in microchannel reactors

2021 ◽  
Vol 128 (1) ◽  
Author(s):  
Sebastian Blauth ◽  
Christian Leithäuser ◽  
René Pinnau
Keyword(s):  
Flow Rate ◽  
Wall Temperature ◽  
Optimization Problems ◽  
Product Yield ◽  
Identification Problem ◽  
State Constraint ◽  
Optimization Techniques ◽  
Reacting Flow ◽  
Microchannel Reactor ◽  
Highly Nonlinear

AbstractWe consider the optimization of a chemical microchannel reactor by means of PDE-constrained optimization techniques, using the example of the Sabatier reaction. To model the chemically reacting flow in the microchannels, we introduce a three- and a one-dimensional model. As these are given by strongly coupled and highly nonlinear systems of partial differential equations (PDEs), we present our software package cashocs which implements the adjoint approach and facilitates the numerical solution of the subsequent optimization problems. We solve a parameter identification problem numerically to determine necessary kinetic parameters for the models from experimental data given in the literature. The obtained results show excellent agreement to the measurements. Finally, we present two optimization problems for optimizing the reactor’s product yield. First, we use a tracking-type cost functional to maximize the reactant conversion, keep the flow rate of the reactor fixed, and use its wall temperature as optimization variable. Second, we consider the wall temperature and the inlet gas velocity as optimization variables, use an objective functional for maximizing the flow rate in the reactor, and ensure the quality of the product by means of a state constraint. The results obtained from solving these problems numerically show great potential for improving the design of the microreactor.

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