A.E. Gheribi, C. Audet, S. Le Digabel, E. Bélisle and A. D. Pelton

Transcription

1 Calculating optimal conditions for alloy and process design using thermodynamic and properties databases, the FactSage software and the Mesh Adaptive Direct Search (MADS) algorithm. A.E. Gheribi, C. Audet, S. Le Digabel, E. Bélisle and A. D. Pelton During alloy and process design, it is often desired to identify regions of design or process variables for which certain calculated functions have optimal values under various constraints, for example: compositions of minimum liquidus temperature in an N-component alloy; compositions where the amount of precipitate in a given phase is maximized or minimized during annealing or rolling; other calculated functions such as densities, vapour pressures and viscosities; or the overall cost. The present work reports on the development of software, linked to the FactSage thermodynamic and properties database system, to perform such calculations. The software uses the Mesh Adaptive Direct Search algorithm (MADS) designed to solve nonsmooth optimization problems for which the objectives and constraints are typically outputs of computer simulations. Numerical results for several examples are presented A. E. Gheribi, research associate E. Bélisle, research associate and A. D. Pelton, Professor emeritus CRCT - Center for Research in Computational Thermochemistry, Department of Chemical Eng., École Polytechnique (Campus de Université de Montréal), Box 6079, Station Downtown, Montréal, Québec, Canada H3C 3A7 C. Audet, Associate professor, and S. Le Digabel, associate professor, GERAD and Department of Mathematics and Industrial Eng., École Polytechnique (Campus de Université de Montréal), Box 6079, Station Downtown, Montréal, Québec, Canada H3C 3A7. "

2 I. Introduction The FactSage thermodynamic computer system consists [1-3] of a suite of program modules and several large evaluated thermodynamic databases. The program modules access the databases to perform chemical equilibrium calculations by means of a general Gibbs energy minimization algorithm. The FactSage databases contain the thermodynamic properties as functions of temperature, pressure and composition for over 5000 pure substances and hundreds of multicomponent solid and liquid solutions of metals, oxides, salts, etc. These solution databases have been developed over the last 35 years by the following approach. First an appropriate mathematical model is developed, based upon the structure of the solution, giving the thermodynamic properties as functions of composition and temperature. Next, all available thermodynamic and phase diagram data from the literature are simultaneously optimized to obtain one set of critically evaluated self-consistent parameters of the models for all phases in 2-component, 3-component and, if data are available, higher-order sub-systems. Finally the models are used to estimate the thermodynamic properties of N-component solutions from the database of parameters for lower-order sub-systems. As well as thermodynamic data, the FactSage databases contain data on densities, lattice parameters and liquid viscosities as functions of temperature, pressure and composition. The FactSage software accesses the databases to calculate phase diagrams and the conditions of phase equilibria in multicomponent systems. FactSage can be used to follow the course of equilibrium or Scheil-Gulliver cooling; predict freezing ranges, segregation of alloying elements and phase formation and the accompanying volume and enthalpy changes; calculate the amounts of various precipitates during subsequent annealing; etc. In principle, one can use the software to screen potential multicomponent alloys, searching for compositions having desired properties and phase assemblages (e.g., minimum liquidus temperature, desired freezing range, desired density range and shrinkage ratio, type and amount of precipitates, etc.). However, the software performs these calculations only at fixed given alloy compositions (or through the calculation of two-dimensional phase diagram sections). Hence, in order to screen a potential N-component alloy for compositions having optimal properties under a given set of constraints one would need #

3 to calculate the properties over a grid of compositions, where the number of grid points varies to the exponent (N-1), and then interpolate between grid points to determine the extrema. For example, a 6-component alloy would require millions of calculations involving several days of computation. The aim of the present work is to apply powerful optimization tools to search automatically for optimal conditions. To this end, we use an algorithm for constrained multi-objective optimization and combine it with the FactSage software to determine optimal compositions (or optimal values of other variables such as annealing or rolling temperature). Several statistical heuristic methods have been used in the literature. These include pure random search, so-called simulated annealing [4], genetic algorithms [5, 6] and neural network methods [7,8]. Non-stochastic methods have been developed such as Scheraga s diffusion method [9] and a class of branch and bound methods due to Floudas [10]. The neural network methods and genetic algorithms can be used to relate the properties of alloys to their chemical composition. However, the reliability and versatility of these heuristic methods are dubious as shown by Bhadeshia and Sourmail [11]. Meza and Martinez [12] studied a molecular conformation problem and concluded that a direct search algorithm was superior to a genetic algorithm or simulated annealing. In the present study, we apply the Mesh Adaptive Direct Search (MADS) algorithm [13] for nonsmooth optimization under general constraints. Direct search methods are designed to solve black-box optimization problems under very general types of constraints, without computing or estimating derivatives, and require no information about the topology of the objective functions. This is essential if one is to optimize extremely complex and non-smooth functions with many local extrema such as liquidus temperatures, or discontinuous functions with many singularities such as the mass of precipitate in a primary phase after annealing. The MADS algorithm compares only function values and thus is easy to implement and use. Moreover, it possesses mechanisms to escape from local solutions. It is also superior to fixed grid enumeration methods in which the grids are generated with fixed steps. Such methods are feasible for small systems but become computationally prohibitive for large systems. The underlying grid used in MADS, $

4 by contrast, is adaptive and automatically adjusts in response to the values of the properties being calculated. Section 2 of the present article describes the MADS algorithm for single and bi-objective optimization. In section 3, three sample problems are presented and numerical results are shown. Section 4 concludes with an analysis of the results. II. Black-box optimization with the MADS algorithm Black-box optimization encompasses a large class of optimization problems. It occurs typically when the functions describing the problem are evaluated by a computer code, a simulation, or an experiment. These evaluations are usually costly, even when they fail to return a value, and so one wishes to use as few function calls as necessary. We write our optimization problems in the form (1) where f is a function from to and is a subset of. Derivatives of the functions defining the problems are generally not explicitly available, and are generally difficult to approximate. We distinguish three types of constraints: (i) Unrelaxable constraints which must not be violated by any trial point. For example, the simulation can only be executed if the variables are positive. This allows constraints that simply return the value feasible or not feasible; (ii) Relaxable constraints which may be violated. The simulation will still execute and a measure of how far the constraint has been violated is returned; (iii) Hidden constraints [14,15] : this is a convenient terminology for excluding the set of points in the feasible region at which the black-box fails to return a value (that is, when the simulation crashes). The present work uses the MADS [13] algorithm for the following reasons: (i) it is backed by a rigorous hierarchical convergence analysis involving the non-smooth calculus [13] ; (ii) it has been tested on many real engineering problems [16, 17, 18, 19]. %

5 MADS is an iterative method in which the black-box functions are evaluated at several trial points located on a discretization of the space of variables, called the mesh. The coarseness of this mesh, at iteration k of the algorithm, is driven by the mesh size parameter. The principle of MADS as a direct search algorithm is schematized in Fig. 1. The algorithm submits trial points to the black-box (in the present case to FactSage). The results of these evaluations are then examined and used to generate new trial points. Each MADS iteration is composed of three steps: the POLL, the SEARCH, and the updates. The SEARCH step is flexible and allows trial points anywhere on the mesh. This liberty is permissible because the convergence analysis does not rely on this step. Specific SEARCH steps can be tailored for specific applications or one may define generic strategies such as Latin- Hypercube (LH) sampling [20] or Variable Neighborhood Search (VNS) [21]. The VNS strategy consists in perturbing the current iterate and conducting a local descent from the perturbed points. This allows escape from local extrema at which the algorithm could become trapped. The POLL step is more rigidly defined and is the key to the convergence analysis. It explores the mesh at trial points neighboring the current incumbent solution. The ancestor of the MADS algorithm, called coordinate search, constructed the POLL points by varying one and only one coordinate of the incumbent solution by a certain step size in the positive and negative directions. This led to a POLL set containing exactly 2n points, where n is the dimension of the space of variables. The MADS algorithm explores the space of variables more thoroughly. The numerical results presented in the present article are generated by the strategy detailed in [22], and form an orthogonal set of directions which grows asymptotically dense in the space of variables. Iteration k of the MADS algorithm terminates at the update step, and there are two possible outcomes. If either the SEARCH or POLL step produced a better solution than x k, then x k+1 is set to that better solution, and the parameter that dictates the mesh coarseness is increased (for example, it is doubled). This implies that the distance from the next POLL points to x k+1 will be greater than the distance from the previous POLL points to x k. The other possible outcome is that neither the SEARCH nor the POLL succeeded in improving x k. In that case, x k+1 is set equal to x k and the mesh size parameter is decreased (for example, divided by two). The consequence is that the next POLL points will be closer to x k+1 = x k than the previous ones. &

6 A drastic way to handle a constraint consists in rejecting any trial point that violates it. This is called the extreme barrier approach, and it was used by early versions of MADS [13]. However, for many optimizations finding a feasible point is a large part of solving the problem, and so a more subtle way to handle the relaxable constraints was developed [23] in which a constraint violation function h : is used to measure the aggregate amount by which the relaxable constraints are violated. A threshold on h is imposed, and trial points whose constraint violation exceeds the threshold are rejected. As the algorithm unfolds, the threshold is reduced, but not necessarily at each step. This approach is called the progressive barrier approach. In many real-world problems, decisions depend on multiple and conflicting criteria. There is often no unique solution that is simultaneously optimal for all criteria, Multi-objective optimization aims at identifying the best trade-offs between the criteria. The solution to such a problem is the set of best trade-off points selected according to an order relation such as the Pareto dominance relation. The vector is said to dominate the vector if all objective values evaluated at u are superior to or equal to those evaluated at v with at least one strict inequality. The set of vectors in that are not dominated by any other member of is called the Pareto set or Pareto front [24]. BIMADS [25] is an algorithm that generates an approximation of the Pareto front when there are two criteria (i.e. two objective functions). It solves a series of constrained single-objective formulations (which are not obtained by simply adding weights to the objectives) using MADS. The series of formulations is constructed so as to attempt a uniform coverage of the Pareto front, even in the case where the Pareto front is nonconvex or disjoint. An illustration of a Pareto front is shown in Fig. 2 for the minimization of two normalized objective functions P1 and P2. The points represent the objective function values (P1, P2) associated with a set of trial points. The squares are said to be dominated while the circles are Pareto points: For each dominated point B, there exists at least one Pareto point A such that and, with at least one of these inequalities being strict. The choice of which point on the Pareto front is optimal must then be made by the user, based upon the relative importance of the properties P 1 and P 2. As illustrated in Fig. 2, a Pareto front may be continuous over some ranges of P 1 and P 2, and discontinuous over others. '

7 III. Numerical results for three sets of sample problems A. Calculation of minimum liquidus temperature of a salt system Systems with low melting temperatures have numerous applications in process design. In this example we calculate unconstrained and constrained minimum liquidus temperatures in the 9- component LiCl-NaCl-KCl-RbCl-CsCl-MgCl 2 -CaCl 2 -SrCl 2 -BaCl 2 system. These are singleobjective minimizations. Calculations were performed with the FTsalt database of FactSage. The thermodynamic models used in developing the FTsalt database and the agreement between calculated and experimental phase diagram and thermodynamic data have been described in detail [1-3]. The calculated liquidus temperatures are estimated to be accurate to within 15 o C or less. The input to the FactSage software required to perform the calculations is shown in Fig. 3. Step 1: Define the system components. The software then automatically retrieves all relevant data from the database. Step 2: Specify the type of FactSage calculation to be performed. In this example, the user specifies (by placing the letter P beside the liquid phase) that he wishes to calculate the temperature of first precipitation of a solid phase from the liquid, i.e the liquidus temperature. Step 3: Define the properties to be minimized or optimized and the parameters of MADS. A detailed manual for the operation of the software may be found in [1]. Our purpose here is simply to show that the input requires no more than a few entries. A liquidus temperature is a very complex function of composition with many local extrema and singularities. The minimization of such a function provides a stringent test of the software. For (

8 the present 9-component system the absolute minimum temperature (out of 692 local liquidus minima) was found after 375 calls to FactSage by the MADS algorithm, requiring less than two hours computation on a 3.16 GHz machine. In contrast, a simple grid-based brute-force enumeration approach would require over one billion calculations to generate a comparable solution for a system with 9 components. The calculated unconstrained absolute minimum liquidus temperature of 239 o C and the liquid composition at this minimum are shown in Table Ia, while Table Ib shows the calculated amounts of the solid phases in equilibrium with the liquid at the minimum. FactSage, of course, also calculates the compositions of the equilibrium solid phases although these have not been shown in Table Ib. The liquid at the unconstrained minimum contains over 50 % CsCl, which is expensive. A second minimization was thus performed under the constraint: (wt. % CsCl) < 5. RbCl and LiCl are also expensive. Additional constrained minimum calculations were performed under the constraint (wt. % CsCl + wt. % RbCl) < 5 and for the simultaneous constraints (wt. % CsCl + wt. % RbCl) < 5 and (wt. % LiCl) < 5. All results are shown in Tables 1a and 1b. B. Calculation of minimum liquidus temperature of a metallic alloy The FactSage FSstel database was used. Although this database is intended for calculations primarily in Fe-rich alloys, the calculations are also generally reliable even in systems containing no Fe. Details of the development of the database, of the thermodynamic models used, and of the agreement with experimental thermodynamic and phase diagram data are described in [1]. The liquidus calculations are estimated to be accurate to within 25 o C or less. The calculated unconstrainted absolute minimum liquidus point of the 8-component steel Fe-Cr- Co-Mn-W-Mo-C-V as well as minima under several different constraints are shown in Tables 2a and 2b. Note that the solution under the constraint (wt. % Fe) > 50 gives a liquid phase containing no Cr, Co W or V and hence may be compared to the unconstrained minimum for Fe- )

9 -Mn-Mo-C alloys calculated and confirmed experimentally by Gomez-Acebo et al. [26]. The calculated liquidus temperature (1035 o C) compares well with the value (1035 o C) reported by Gomez-Acebo et al., while the calculated (Table IIb) solid phases at the minimum compare well with those reported by these authors (fcc + cementite + hcp + traces of M 7 C 3 ). As shown in Table IIa, no more than 805 calls to FactSage were ever needed, requiring less than 1.5 hours on a 3.16 GHz machine. In a future publication we will present a new algorithm which allows the calculation not only of the absolute liquidus minimum but of all local minima on a liquidus surface. C. Bi-objective optimization - first example In this example, we seek simultaneously to minimize the liquidus temperature and to maximize the heat capacity of the liquid phase at a temperature 15 o C above the liquidus for the system LiCl-NaCl-KCl-Li 2 CO 3 -Na 2 CO 3 -K 2 CO 3. Such a calculation might for example be of interest in designing fluids for heat transfer or storage. In these calculations, the FactSage FTsalt database (discussed in Section 3.1) was used. The calculated Pareto front in the absence of constraints is shown in Fig. 4 and selected Pareto points are listed in Table III. The choice of which composition on the Pareto front is optimal must then be made by the user depending upon the relative importance that the user attaches to the liquidus temperature and the heat capacity. Let us now introduce the constraint that the density of the liquid 15 o C above the liquidus must be less than 1.85 g/cm 3. The calculated constrained Pareto front is plotted in Fig. 4 and selected points on the front are listed in Table III. *

10 D. Bi-objective optimization - second example In this example we seek to maximize the amount of precipitate in the primary phase of an MS50- type tool steel after annealing subsequent to Scheil-Gulliver cooling (in order to maximize certain mechanical properties) while minimizing the cost of the raw materials. The constraints on the composition are shown in Table IV. Raw materials costs were taken from the London Metal Exchange (as of August, 2009) The FactSage FSstel database (discussed in Section 3.2) was used. To calculate the amount of precipitate, two consecutive FactSage calculations are required. In the first calculation, a thermodynamic simulation of rapid Scheil-Gulliver cooling is made. In this calculation it is assumed that diffusion in all solid phases is forbidden and thermodynamic equilibrium occurs only at the solid/liquid interface. Although a Scheil-Gulliver simulation cannot describe microsegregation, for most alloys the results are reasonably close to reality. In the second FactSage calculation, the primary phase at the completion of the Scheil-Guliver cooling is allowed to reach internal thermodynamic equilibrium at an annealing temperature and the amount of resulting precipitate is calculated. The present software permits the two steps to be easily specified by the user, and the two consecutive FactSage calculations are automatically performed each time the MADS algorithm queries FactSage. Pareto fronts calculated at annealing temperatures of 1000 o C and 1100 o C are shown in Fig. 5 and selected Pareto points are listed in Table V. These calculation required 3125 and 3550 calls respectively by the MADS algorithms to FactSage in order to calculate the entire Pareto front, requiring approximately 36 hours on a 3.16 GHz machine. E. Design of AZ91 magnesium alloys with Y and Ce The preceding examples were all chosen to illustrate the use of the software rather than as examples of actual industrial design problems. The present example is more illustrative of how the software can be applied to real current design questions. "+

11 We consider the design of AZ91 magnesium alloys (Mg-Al-Zn-Mn-Si alloys with nominal Al and Zn contents of 9.0 and 1.0 wt. % respectively). The goals are to improve deep drawing properties (which are important in the production of Mg sheet for automobile bodies) and corrosion resistance by additions of yttrium and cerium while minimizing the liquidus temperature and the cost. The calculations were performed with the FactSage FTlite (light metals alloys) database. For Mg alloys this database is the result of extensive modeling and optimization [27]. Calculated liquidus temperatures are estimated to be accurate to within 10 o C. It has been shown [28-31] that the addition of Y significantly improves the ultimate tensile strength and hardness of AZ91 alloys while Ce additions improve corrosion resistance. This is due mainly to precipitation of Al n Y (1 n 3) and Al 11 Ce 3. In the present example we search for alloy compositions which maximize the mass of Al 3 Y and Al 11 Ce 3 precipitates in the primary hcp phase, while minimizing the liquidus temperature and the cost. Let us assume that the objectives of the user in order of priority are: 1- Maximum weight of Al 3 Y precipitate in Mg matrix after annealing at 350 o C (following Scheil-Gulliver cooling) 2- Minimum liquidus temperature 3- Minimum cost of raw materials 4- Maximum weight of Al 11 Ce 3 precipitate in Mg matrix after annealing at 350 o C The constraints on the alloy composition are shown in Table VI along with the cost of the raw materials. The goal is to provide a list of alloy compositions which provide a good compromise among these four objectives. ""

12 1. Addition only of Y As a first example we consider the addition only of Y (no Ce). That is, we consider only the first three objectives. The following strategy is used (although other strategies are also possible). In a first step, we simultaneously maximize the amount of Al 3 Y precipitate while minimizing the liquidus temperature subject to the constraints of Table VI (with 0% Ce). The calculated Pareto front is shown in Fig. 6. Since the mass of Al 3 Y precipitate and the liquidus temperature are the our principal objectives, we require that the optimization of the cost (third priority) be subject to the constraints of lying close to the Pareto front of Fig. 6. Accordingly, we draw the straight line shown in Fig. 6 and require that the cost be minimized subject to the constraint of lying above and to the left of this line. That is: m(al 3 Y)(ppm) > T liq ( o C) (2) Next, we simultaneously maximize the mass of Al 3 Y precipitate and minimize the cost subject to the constraint of Eq. (2). The calculated Pareto front is shown in Fig. 7 and selected points from the Pareto front are shown in Table VII. 2. Addition of Y + Ce As a second example, we consider all four objectives. The following (non-unique) strategy was used. In a first step, the mass of Al 3 Y precipitate and liquidus temperature were simultaneously optimized as in the previous example (exept that now Ce contents of < 1 % are allowed). The calculated Pareto front is shown in Fig. 8. The line shown in Fig. 8 was then drawn as an acceptable compromise for subsequent optimizations: m(al 3 Y)(ppm) > T liq ( o C) (3) "#

13 Next, the mass of Al 3 Y precipitate and cost were optimized subject to the constraint of Eq. (3). The calculated Pareto front is shown in Fig. 9 along with the acceptable compromise line: m(al 3 Y)(ppm) > C($/tonne) (4) Finally, we maximize the mass of both Al 3 Y amd Al 11 Ce 3 precipitates under the two constraints (2) and (4) as well the constraints of Table VI. The resultant calculated Pareto front is shown in Fig. 10 and selected points are shown in Table VIII. 4. Conclusions The FactSage thermodynamic computer system [1-3] consists of several large evaluated databases and program modules which access the databases to perform chemical equilibrium calculations by means of a general Gibbs energy minimization algorithm. The databases contain evaluated thermodynamic data for thousands of compounds and hundreds of multicomponent solutions of metals, salts, oxides, etc. The databases contain model parameters obtained by coupled thermodynamic/phase diagram optimization. For example, the FSstel database for steels contains 28 components and 87 solution phases; the FTsalt database for salts contains 28 components and 72 solution phases; the FToxid database for oxides contains 21 components and 51 solution phases; etc. With the software one can calculate phase diagram sections or phase fraction charts of multicomponent, follow the course of equilibrium or Scheil-Gulliver cooling and calculate the amount of precipitation during subsequent annealing, calculate enthalpy effects, vapour pressures, etc. As well as thermodynamic data, the databases contain data for densities, lattice parameters and viscosities. The FactSage system can be used to screen potential systems, searching for compositions having a desired set of properties and phase constitution under a given set of constraints. For instance, one could search for alloys within a given composition range, with a liquidus temperature below a given value, with a desired freezing range, with a maximum or minimum amount of precipitates after annealing, with a density, shrinkage ratio or viscosity within a given range, etc. "$

14 However, to perform such searches by simply performing tens of thousands of calculations over a grid of compositions is prohibitively time-consuming for any but the simplest systems. The present work has extended the capability of FactSage by coupling the calculations with automatic optimization search software. To this end we have used the MADS (Mesh Adaptive Direct Search) algorithm [13]. The direct search method solves black-box optimization problems under very general constraints, without computing or estimating derivatives and requires no information about the topology of the functions being optimized. This is essential for the optimization of complex, non-smooth functions such as liquidus temperatures or amounts of precipitates after annealing. The MADS software has been integrated into the FactSage system through a new program module called FactOptimal [1]. In the present article, the use of the software is illustrated through several examples: calculating the minimum liquidus temperatures of a 9-component chloride system and of a 5-component metal alloy; simultaneously (bi-objective optimization) minimizing the liquidus temperature and maximizing the heat capacity (at 15 o C above the liquidus) of a 6- component salt system; maximizing the amount of precipitate in the primary phase of an MS50- type tool steel following annealing subsequent to Scheil-Gulliver cooling, while simultaneously minimizing the cost of the raw materials; designing a 7-component AZ91 Mg-alloy containing Ce and Y in order to maximize the amount of Al 3 Y and Al 11 Ce 3 precipitates (to improve deep drawing properties and corrosion resistance respectively) while simultaneously minimizing the liquidus temperature and cost of raw materials. In future publications we shall describe extensions to FactOptimal which permit (i) the calculation of all local minima on a liquidus surface, (ii) the calculation of all points on a liquidus surface at which isothermal reactions occur (minima, eutectics, peritectics, saddle points, maxima); (iii) target calculations to search for systems having properties with a given value or within a range of values such calculations will permit the optimization not only of composition but also of such variables as annealing or rolling temperatures. Work is also underway to couple FactOptimal with empirical equations or databases relating the amounts and compositions of phases with properties such as creep resistance, M S temperature, yield strength, castability, etc., the goal being a true system of Integrated Computational Materials Engineering. "%

15 Acknowledgements This research was supported by funding from the Natural Sciences and Engineering Research Council of Canada (NSERC) Magnesium Strategic Research Network. More information on the Network can be found at Constructive discussions with Prof. Christopher W. Bale, Prof. Patrice Chartrand and Dr. Christian Robelin were much appreciated. "&

16 References 1. The FactSage system: 2. C. W. Bale, P. Chartrand, S. A. Decterov, G. Eriksson, K. Hack, R. Ben Mahfoud, J. Melançon, A. D. Pelton and S. Petersen, CALPHAD (2002) 26, C.W. Bale, E. Bélisle, P. Chartrand, S.A. Decterov, G. Eriksson, K. Hack, I.-H. Jung, Y.-B. Kang, J. Melançon A.D. Pelton, C. Robelin and S. Petersen, CALPHAD (2009) 33, S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Science New Series (1983) 220, D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Kluwer Academic Publishers, Boston, MA (1989) 6. R.S Judson, E.P. Jaeger, A.M. Treasurywala and M.L. Peterson, J. Comp. Chem., (1993) 14, Haykin S., Neural Networks, 2nd Edition, Prentice Hall, (1999) 8. Fausett L., Fundamentals of Neural Networks, Prentice-Hall, (1994) 9. J. Kostrowicki, L. Piela, B. Cherayil and H.A. Scheraga, J. Phys. Chem. (1991) 95, C.D. Maranas and C. A. Floudas J. Chem. Phys. (1994) 100, H. K. Bhadeshia and T. Sourmail, Japan Soc. Promotion of Science, 123rd Committee on Heat-Resisting Materials and Alloys (Tokyo, Japan) (2003) 44, J.C. Meza and M. L. Martinez, J. Comput. Chem. (1994) 15, C. Audet and J. E. Dennis, SIAM J. Optimiz. (2006) 17, T. D. Choi and C. T. Kelley, SIAM J. Optimiz. (2000)10, A. R. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, MPS/SIAM Book Series on Optimization. SIAM, Philadelphia, (2009). 16. C. Audet, V. Béchard, and J. Chaouki, Optimization and Engineering, (2007) 9, C. Audet, J. E. Dennis, and S. Le Digabel, Comput. Optim. Appl., (2010) 46, R. E. Hayes, F. H. Bertrand, C. Audet, and S. T. Kolaczkowski. Can. J. Chem. Eng., (2003) 81, M. Kokkolaras, C. Audet, and J. E. Dennis, Optimiz. Eng., (2001) 2, B. Tang, J. Amer. Statistical Assoc., (1993) 88, C. Audet, V. Béchard, and S. Le Digabel, J. Global Optim., (2008) "'

17 22. M. A. Abramson, C. Audet, J. E. Dennis, and S. Le Digabel, SIAM J. Optimiz., (2009) C. Audet and J. E. Dennis, SIAM J. Optimiz., (2009) 20, P.L. Yu, J. Optimiz. Theory App., (1974) 14, C. Audet, G. Savard, and W. Zghal, SIAM J. Optimiz. (2008) 19, , 26. T. Gomez-Acebo, M. Sarasola and F. Castro, Calphad 27, (2003) 27. Y.-B Kang, C. Aliravci, P.J. Spencer, G. Eriksson, C.D. Fuerst, P. Chartrand and A.D. Pelton, (2009) JOM, 61, W. Shou-Ren, G. Pei-Quan, Y. Li-Ying and W. Yanjun, J. Mater. Eng. Perform., (2009), 18, G-D, TONG H-F, LIU and Y-H, LIU, Trans. Nonferrous Met. Soc. China (2010) 20, L. "í#ek, M. Greger, L.A. Dobrza$ski, I. Ju%i&ka,R. Kocich, L. Pawlica and T. Ta$ski J. Achievements Materials and Manufacturing Eng., (2006) 18, Y.L. Song, Y.H. Liu, S.H. Wang, S.R. Yu and X.Y. Zhu, Mater. Corros., (2007) 58, "(

18 Tables and Figures Table Ia Calculated compositions of absolute minimum liquidus temperature of the LiCl-NaCl- KCl-RbCl-CsCl-MgCl 2 -CaCl 2 -SrCl 2 -BaCl 2 system as well as minima under various constraints on composition (FSC = number of FactSage calculations). Constraint LiCl NaCl KCl RbCl CsCl MgCl 2 CaCl 2 SrCl 2 BaCl 2 T liq ( o C) FSC 1 Unconstrained wt. % CsCl < wt.% CsCl+RbCl < wt.% CsCl+RbCl < 5 and wt.% LiCl < Table Ib Calculated amounts of solid phases and enthalpy of fusion at unconstrained and constrained minima of the liquidus of the LiCl-NaCl-KCl-RbCl-CsCl-MgCl 2 -CaCl 2 -SrCl 2 -BaCl 2 system (SS = solid solution) g Liquid-> 19.84g KCl-CsCl (SS) g LiCl-[MgCl 2 ] (SS) g CsSrCl g LiCs 2 Cl g Cs 2 MgCl 4, H m = kj/kg g Liquid-> 50.50g KCl-RbCl (SS) g LiCl-NaCl-[MgCl 2 ] (SS) g (K,Rb)Sr 2 Cl 5 (SS) g LiRbCl 2, H m = kj/kg g Liquid-> 32.47g NaCl-KCl (SS) g LiCl-[MgCl 2 ] (SS) g KSr 2 Cl g K(Mg, Ca)Cl 3 (SS) g LiKMgCl 4, H m = kj/kg g Liquid-> 31.02g Na 2 MgCl 4 -K 2 MgCl 4 -Li 2 MgCl 4 (SS) g CaCl 2 -[MgCl 2 ] (SS) g BaCl g (K,Cs)CaCl 3 (SS) g LiKMgCl g MgBa 2 Cl 6 (S), H m = kj/kg. ")

19 Table IIa Calculated compositions of absolute minimum of liquidus temperature of the Cr-Co- Mn-W-Mo-C-V system as well as minima under various constraints on composition (FSC = number of FactSage calculations). Constraint Fe Cr Co Mn W Mo C V T liq ( o C) FSC 1 Unconstrained wt.% Fe > wt.% Fe > wt.% Cr > wt.% Cr > 5 and wt.% Fe > wt.% W > wt.% W > 5 and wt.% Co < Table IIb Calculated amounts of solid phases and enthalpy of fusion at unconstrained and constrained minima of the liquidus of the Fe-Cr-Co-Mn-W-Mo-C-V system 1 100g Liquid-> 50.10g FCC g Cementite g HCP, H m =-299 kj/kg g Liquid -> 51.71g FCC g Cementite g HCP, H m =-221 kj/kg g Liquid -> 41.94g FCC g Cementite+26.87g "-carbide, H m = kj/kg g Liquid -> 57.77g FCC g M 7 C g M 6 C g HCP BCC, H m = kj/kg g Liquid -> 48.06g FCC+48.59g Cementite g M 7 C g VC, H m = kj/kg g Liquid -> 57.94g FCC g Cementite g WC, H m = kj/kg g Liquid -> 61.61g FCC g BCC g µ-phase g HCP, H m = kj/kg. "*

20 Table III Selected points from Pareto fronts of Fig. 4. LiCl NaCl KCl Li 2 CO 3 Na 2 CO 3 K 2 CO 3 T liq Cp density ( o C) (J/kg) (g/cm 3 ) Unconstrained constraint: density at T liq +15 o C < 1.85 g/cm #+

21 Table IV Composition of M50 tool steel. Fe Cr Co Mn W Mo C V Min Max #"

22 Table V Selected points from Pareto fronts of Fig. 5. Fe Cr Co Mn W Mo C V $/tonne Total mass of precipitates per 100g of alloy (g) T annealing =1000 o C T annealing =1100 o C ##

23 Table VI Composition limits of AZ91+Y+Ce magnesium alloys and cost of raw material (London Metal Exchange, August 2009) Al Mg Mn Si Zn Y Ce Y+Ce Min Max Cost ($/tonne) #$

24 Table VII Selected points on the Pareto front of Fig. 7. Al Mg Mn Zn Y Si Cost $/tonne Mass of Al 3 Y per 100g of alloy (ppm) T liq ( o C) #%

25 Table VIII Selected points on the Pareto front of Fig. 10. Al Mg Mn Zn Y Si Ce Mass of Al 3 Y per 100g of alloy (ppm) Mass of Al 11 Ce 3 per 100g (ppm) Tliq ( o C) Cost $/tonne #&

26 Figures Fig. 1. : General principle of a direct search method. The algorithm iteratively constructs lists of trial points that are evaluated by the black-box.

27 Fig. 2. Illustration of Pareto concept. Two normalized objective function P 1 and P 2 are minimized.

28 Fig. 3. Input for calculation of the minimum liquidus temperature of the LiCl-NaCl-KCl-RbCl- CsCl-MgCl 2 -CaCl 2 -SrCl 2 -BaCl 2 system. Step 1

29 Step 2

30 Step3

31 Fig. 4. Pareto front describing the minimum liquidus temperature and maximum heat capacity at 15 o C above the liquidus of the LiCl-NaCl-KCl-Li 2 CO 3 -Na 2 CO 3 -K 2 CO 3 system (i) unconstrained, (ii) under constraint on density.

32 Fig. 5. Pareto front describing minimum cost and maximum total mass of precipitate in primary phase after annealing at 1000 o C and 1100 o C following Scheil-Gulliver cooling of M50-type tool steel.

33 Fig. 6. Pareto front which simultaneously maximizes the mass of precipitate Al 3 Y in Mg matrix in AZ91+Y after annealing at 350 o C following Scheil-Gulliver cooling and minimizes the liquidus temperature

34 Fig. 7. Pareto front which simultaneously maximizes the mass of precipitate Al 3 Y in Mg matrix in AZ91+Y after annealing at 350 o C following Scheil-Gulliver cooling and minimizes the cost under the constraint mass(al 3 Y)(ppm) > T liq ( o C) and under the constraints of Table VI (with 0% Ce)

35 Fig. 8. Pareto front which simultaneously maximizes the mass of precipitate Al 3 Y in Mg matrix in AZ91+Y+Ce after annealing at 350 o C following Scheil-Gulliver cooling and minimizes the liquidus temperature

36 Fig. 9. Pareto front which simultaneously maximizes the mass of precipitate Al 3 Y in Mg matrix in AZ91+Y+Ce after annealing at 350 o C following Scheil-Gulliver cooling and minimizes the cost under the constraint mass (Al 3 Y) (ppm) > T liq ( o C)

37 Fig. 10. Pareto front which simultaneously maximizes the masses of precipitates Al 3 Y and Al 11 Ce 3 in Mg matrix in AZ91+Y+Ce after annealing at 350 o C following Scheil-Gulliver cooling under the two constraints mass(al 3 Y) (ppm) > T liq ( o C) and mass (Al 3 Y) (ppm) > C($/tonne)