Summary of this 1 page or a little more . and someone who can do it perfectly not just simple words summary a very good one

Regular Article Cuckoo searching optimal composition of multicomponent alloys by CrossMark molecular simulations Aayush Sharma a, Rahul Singh a, Peter K. Liaw b, Ganesh Balasubramanian a,c, a Department of Mechanical Engineering, lowa State University, Ames, LA 50011, USA b Department of Materials Science E Engineering, University of Tennessee, Knoxville, TN 37996 , USA c Microelectronics Research Center, lowa State University, Ames, IA 50011, USA A R T I C L E I N F O A B S T R A C T Article history: A robust computational design framework that couples the metaheuristic cuckoo search technique with classical Received 1 October 2016 molecular dynamics simulations is employed to optimize the composition of multicomponent alloys for inbeds to predict the influence of atomic concentration of one constituent element (design variable) on the ultiAvailable online 10 January 2017 mate tensile strength (objective function) of the material. The design solutions that correlate the elemental atomCuckoo search Molecular dynamics for discovery of novel multicomponent alloys. ( 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. High-entropy alloy Ultimate tensile strength Multicomponent alloy High-entropy alloys (HEAs), a subset of multicomponent alloys, are computations, and offers recommendations in agreement with typically concentrated solid solutions composed of five or more princi- established experimental results. pal elements, each occupying between 5 and 35 at.\%[1-3]. The different Metaheuristic algorithms enable a nature-inspired generalized optiHEA compositions that have been explored over the last decade show mization scheme to rapidly derive approximate solutions for intractable unique microanostructures and adjustable mechanical properties or gradient free problems [18-20]. Genetic Algorithm (GA) and Particle [4-7]. While the theories explaining the strengthening mechanisms Swarm Optimization (PSO) are the most popular evolutionary algoare still in the early stages of development, few design strategies have rithms with several applications in manufacturing, quality control, probeen proposed for optimizing elemental concentrations to achieve duction, and design [21-29]. However, the effectiveness of a recent targeted phase and material properties [8-10]. Some of these ap- technique called cuckoo search (CS) for multi-modal design applicaproaches include combinatorial material synthesis, numerical schemes tions [30-33], and its superiority in benchmark comparisons [34,35] using ab-initio and molecular simulations, thermodynamics based cal- against PSO and GA makes it an intelligent choice for designing multiculation of phase diagrams, finite element modeling, and the Taguchi element HEAs. CS is a search method that imitates obligate brood paramethod [11-17]. The conventional methods, which involve characteri- sitism of some female cuckoo species specializing in mimicking the zation of microstructures followed by dynamic measurement of the ma- color and pattern of few chosen host birds. The parasitic cuckoo often terial properties, for discovering novel multicomponent alloys are time chooses a nest where the host has just laid its own eggs so that when and resource intensive. The advantage of a computational framework, the first cuckoo chick hatches, it evicts the host eggs out of the nest to on the other hand, lies in the vastness of the parameter space available increase its own food share. Specifically, from an optimization standfor examination, not possible otherwise using only thermodynamic point, CS (i) guarantees global convergence, (ii) has local and global phase analyses or high throughput synthesis. Thus, to design novel mul- search capabilities controlled via a switching parameter ( pa ), and (iii) ticomponent alloys with optimal material compositions for desired mi- uses Levy flights rather than standard random walks to scan the design crostructures and properties, we present a simulation-driven paradigm space more efficiently than the simple Gaussian process [32,36]. that integrates a metaheuristic optimization technique with atomistic We integrate the CS mathematical framework with an atomistic simulation tool, molecular dynamics (MD) in this case, to optimize the elemental composition for a set of model quinary alloy with targeted Corresponding author at: 2092 Black Engineering, Ames, IA 50011, USA. properties. Since analyses using phase diagrams require extensive therE-mail address: bganeshiastate.edu (G. Balasubramanian). modynamic data and computationally expensive ab-initio calculations are limited to systems of a few hundred atoms only, MD simulations are function like Ackley and 6-hump Camel back, revealed 0.2 to be an apchosen as the scientific probe. We construct, verify, and implement a propriate choice for most scenarios at different values of n (5 to 100 ). CS-MD coupled computational design framework and demonstrate its This approach ensured local search to consume around 1/5 th of the performance to tailor the quinary AlCrCoFeNi alloy composition for in- total time, meriting suitable exploration of the global design space withcreasing the tensile strength. in reasonable computing times. Thus, for all simulations of the multiThe CS optimization procedure is based on the algorithm proposed component alloys under CS-MD framework, we use a value of pa equal by Yang and Deb [34]. The CS implementation considers that each egg 0.2. An efficient heuristic optimization framework ensures that the sysin a nest represents a solution governed by the following three idealized tem is not trapped within any local optimum in the design/objective rules: function landscape. CS optimization includes both the local-random 1. At a time, each cuckoo lays one egg and dumps it in a randomly-cho- walk and the global-explorative random walk with Lvy flights. Thus, sen nest. a guaranteed global convergence is an intrinsic feature and a unique ad- 2. Only the best nest with highest quality eggs is carried over to the vantage of the CS framework [30-33]. We believe that a comparison of computational times between CS and just the local search would sug- 3. The probability that the host bird discovers the cuckoo egg is pa ( 0 , gest that since the local search explorations could be stuck in local opti1) for a fixed number of available host nests. If/When discovered, the mum, they will consume more computational resources/time (although host bird can either get rid of the cuckoo egg or build a completely we did not run independent simulations to verify this). In relation to this, CS would certainly be more efficient in exploring desired maxiCuckoo search possesses the advantage of a balanced combination ma/ minima at both local and global search gradients [34,35]. of both the local-random walk and the global-explorative random The modified CS algorithm implemented here is shown in Fig. 1a, walk. The switching parameter, pa, controls the selection between which is adapted from a generalized description of the standard CS these two walks. A local random walk is represented as x1t+1= technique [34]. The CS-MD framework, illustrated in Fig. 1b, involves xit+sH(pa)(xjtxkt)=0 where, = random number, optimizing user-declared design variables, such as the elemental conH(u)= Heaviside function, s= step-size, xit&xkt= two different solu- centration in the alloy for the desired property, the ultimate tensile tions selected randomly by random permutation, = entry wise prod- strength in this case. Each cycle (generation) of CS involves comparing uct. A global random walk or Levy Flights is represented as different solutions (nests) to retain the best candidate while replacing xit+1=xit+L(s,), where L(s,)=31()sin(28)11,(ss00) and >0 all unfavorable solutions with newer alternatives predicted via the globsults. For every cycle, only the alloy with the elemental concentration to the original algorithm [34]. that yields the maximum ultimate tensile strength is retained amongst The convergence rate in CS optimization has been often found to be all the available solutions, referred as the best nest/solution ( g ). least dependent on the choice of key parameters like number of nests The CS-MD framework is tested for robustness by varying input (n) and switching parameter (pa)[32,36]. Our initial trials on test concentrations from 2.5% to 97.5%, in increments of 2.5%. A single MD-output file was used as the 'output' for all concentrations. The algorithm. Here, the same colored histograms represent the variation code provided the correct elemental concentrations (input) and cal- in the design parameter (atomic concentration) values with each iteraculated the maximum strength as equivalent for all cases tested. tion of the CS cycle. The fluctuations in the histograms during the differThe implementation of multiple input parameters in this framework ent iterations indicate that the algorithm is employing different design is a natural extension of this predictive scheme. values to arrive at a global optimum and achieve the desired objective All the atomistic simulations of the binary, ternary subsets, and the function of the increased strength in the alloy. While the local search quinary alloy are performed with LAMMPS [37]. In a 2.02.0 consumes about 1/5 th of the total search time, the rest is required for 2.0nm FCC lattice, elements (Al-Cr-Co-Fe-Ni) were randomly arranged the global search when using a switching parameter of pa=0.2. In to form the alloy system of 32,000 atoms, having periodic boundary each iteration, the complete exploration of the design variable by the conditions imposed in all directions. Energy minimization is carried different cuckoo nests analyzes the favorable Al elemental \% (atomic) out, using the conjugate gradient algorithm with energy and force that increases the strength of the binary AlyFe alloy. tolerance set to 1015 units. First initialization at 2200K under an iso- The CS-MD results (Fig. 2a and b) suggest that as Al \% reduces the thermal-isobaric (NPT) ensemble at a pressure of 0MPa for 90 picosec- strength of the binary (Al-Fe) alloy increases from 4000MPa for 20% onds (ps) to melt the alloy using equilibrium MD simulations. This step Al to a final value of 5000MPa for 9%Al in 100 objective function evalis followed by rapid quenching of the alloy under the NPT ensemble at uations. Each objective function evaluation involves a complete MD 0 MPa with a cooling rate of 3.8K/ps to reach 300K. We employ the computational analysis, where for a particular composition, a Nos-Hoover thermostat and barostat, each with a coupling time of nanoscopic structure of the alloy is simulated by quenching from 1ps. Next, the structure is allowed to equilibrate for another 90ps. A 2200K to 300K, followed by the uniform tensile deformation in the time step of 0.001ps is maintained throughout all our MD simulations. 100 direction. The low Fe solubility in Al promotes several stable The quenched alloy is, then, further equilibrated under the NPT and and metastable phases that often lead to the formation of a hard and NVT (canonical) ensembles successively. The pressure and temperature brittle intermetallic with the reduced formability in rapidly-quenched constraints each with the coupling time of 1ps are imposed by the Al alloys [41-43]. The predictions of the CS-MD approach for the high Nos-Hoover thermostat and barostat, for a total time of 90ps, followed strain deformation of AlxFe alloys shows reasonable agreement with by the NVT ensemble, for further 90 ps. Finally, the entire system is sim- the literature, where an increase in Fe % promotes higher tensile ulated in the absence of thermodynamic constraints for further 90ps strength in quenched alloys [44-48]. under the NVE (microcanonical) ensemble to ensure that we obtain The ternary Fe-Ni-Cr alloy with the design variable as the atomic \% of an equilibrated structure. Next, tensile loading of the alloy is performed Fe is next employed as a test bed for the material-optimization method. independently at 300K. The simulation cell is deformed in the x-direc- Fig. 2c shows the design-space extent when the different cuckoos extion of 100 with a strain-rate of 1010s1, for the engineering strain plore possible nests/solutions over the different cycles. The variations of 0.9%, while lateral boundaries are controlled using the NPT equations of the design parameter and objective function with each cycle of CS of motion to zero pressure. We employ the 12-6 Lennard-Jones poten- are shown in Fig. 2d that shows an increase in the strength of the Fetial with the functional details described in our earlier work [38]. The NiCr alloy with the increase in the Fe content. Within 200 objective different parameters for the force field, as employed in previous MD function evaluations, the algorithm is able to predict the Fe content simulations, are also available as the supplementary information [39, (at.\%) that effectively raises the strength in the ternary alloy by 4%. 40]. Each alloy examined under the CS framework underwent the struc- An increase in Fe content in alloys (binary and ternary) promotes the ture preparation (melting, quick quenching and equilibration), followed formation of intermetallic phases that enhance strength (tensile) in by high strain deformation, as described elsewhere in details [38]. the modified alloy [42,44,45,4951]. The predictive capability of the CS-MD optimization framework for We extend our optimization analysis to quinary alloy compositions multicomponent alloys is first verified for a binary Al-Fe alloy, followed constituted of the Al-Cr-Co-Fe-Ni elements. The Al content is the design by ternary and quinary combinations. The selected optimization param- variable, and the objective is to examine trends for the design vable eters for the binary, ternary, and quinary alloys are listed in Table 1 to- that could result in a higher strength of the quinary alloy. Predictions gether with the upper and lower limits of the design variable for each from the design framework in 120 objective function evaluations across material. For the binary case, the Al elemental \% (atomic) is chosen to 6 cycles of CS runs suggest that the strength (UTS) of the quinary alloy be the design variable, while the objective function is to increase the ul- can increase by 35%, Fig. 2 fillustrates the evolution of UTS as a function timate tensile strength (UTS) for the alloy. UTS is the maximum stress of Al at.\% over the computed cycles. We find that reducing Al concentrathat a material can sustain within specific strain limits, which in this in- tion from 10% to 1% at the expense of increasing Fe fraction in our vestigation is 0 to 90% strain at a specified strain rate of 1010s1. The quinary alloy, contributes to increasing the strength of the multicompochoice of the extremely-high strain rate, which is difficult to realize in nent alloy system. The trends observed from our design framework are typical experiments, is required to derive demonstrative predictions in good agreement with experimental measurements [52], that show an from MD simulations within reasonable wall times. The design space increase in UTS from 620MPa to 635MPa at 298K for a decrease in Al\% shown in Fig. 2a, c, and e has the elemental \% (atomic) of the design var- from 6.4 to 2.2 at.\% for the Al CrCoFeNi HEA. We conjecture that Al proiable in the z-axis varying with the number of iterations and number of motes the clustering transition [38], and as its concentration reduces nests presented along the y and the x directions, respectively. The num- while that of Fe increases, the strength of the multicomponent alloy inbers of nests denote the number of solutions considered. The total num- creases. Our predictions, which contradict the increase of strength and ber of evaluations of the objective function is the product of the number hardness with increasing Al content in single phase solid solutions of nests and number of iterations. The predictive landscape in Fig. 2a, c (near equiatomic HEAs) [5], suggest that for the optimized composiand e represent the exploratory walks performed by the different tions intermetallic, multi-phase or possibly amorphous structures are nest/solutions ( =10 for binary, 20 for ternary and quinary) of the CS promoted in multicomponent alloys. It is also important to clarify that Table 1 Optimization parameters used for the different alloys. (a) (b) (c) Fig. 2. Design-space-exploration map [panels - (a), (c), and (e)], and design variable (concentration) and objective function (UTS) variation with each cycle of CS [panels - (b), (d), and (e)], for different multicomponent alloys - binary Al-Fe, ternary Fe-Ni-Cr, and quinary Al-Cr-Co-Fe-Ni - examined in the CS-MD optimization framework. the different multi-component alloy systems considered in the study algorithm is sufficiently robust to be not confined by local optimum are defect-free [53,54], and we report values (UTS) that are in the solutions and predict the global maxima/minima. We employ the realm of ideal-strength [53,55-57]. However, in actual materials dislo- technique to examine the variation of mechanical strength under cations, twinning, stacking faults, and microstructural defects, to name high strain-rate deformation in binary, ternary, and quinary multia few, significantly contribute to strength characteristics. component alloys. The results of the computational scheme reveal In summary, we develop and implement a combined cuckoo the correlation between the concentration of a single element (design search and molecular dynamics based design framework to optimize variable) and ultimate tensile strength (objective function) that are the composition of multicomponent alloys for the desired structural qualitatively in agreement with earlier experimental measurements. property, such as high tensile strength. The metaheuristic simulation The proposed technique accelerates the selection and composition of elements for a multicomponent alloy system with desired structures and properties, overcoming the limitations of trial-and-error strategies for exploring the vast materials landscape for such complex structures