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Nature inspired algorithms for the solution of inverse heat transfer problems applied to distinct unsteady heat flux orientations in cylindrical castings


One of the most important parameters in the foundry design of castings is the heat transfer coefficient at the casting/mold surface (h), which significantly affects the solidification cooling rate and defines the as-cast microstructure and, consequently, its properties. The inverse heat transfer problem (IHTP) is a needed task to be solved in many casting processes to evaluate this coefficient. In this work, four different nature inspired algorithms [genetic algorithm (GA), evolutionary strategy (ES), artificial immune system (AIS) and particle swarm algorithm (PSO)] have been integrated to a numerical model of solidification to solve the IHTP problem with a view to evaluating the transient profile of h during solidification of three different cylindrical casting setups (outward, inward and upward). The aim is to reduce the number of iterations to achieve successful results by the association between metaheuristics and different sets of h values. Therefore, an objective function was modeled to match the temperatures predicted by the numerical model to those of experimental measurements. Sets of 10, 20, 50, 100 and 200 h values used to compose its transient profile have been evaluated by the four metaheuristics approaches aiming to characterize each casting situation.

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This study was financed by: FAPESP (São Paulo Research Foundation: grant 2009/52197-7) and CNPq—National Council for Scientific and Technological Development and IFSP—Federal Institute of Education, Science and Technology of São Paulo (Grant PRP-IFSP 226/2016) for their financial support.

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Appendix A: Nature Inspired Algorithms

Among the several types of nature inspired algorithms, four have been chosen to be compared due to their particular characteristics. The Genetic Algorithm (GA) metaheuristic has mechanisms of variability (mutation and recombination) similar to the Evolutionary Strategy (ES). While GA starts with a single population that interacts itself over generations, ES interacts with different populations which can provide greater variability. On the other hand, the Particle Swarm optimization (PSO) is the starting point of metaheuristics inspired by a swarm of particles, in which the optimization considers the spatial relationships of each candidate solution (particle) in space over time, e.g., local, and relative velocities of the particles. Artificial Immune System (AIS) deals with both global and local searches, since its suppression function delimits a region of interest representing the zone at which the antibody survives and therefore more antibodies can be randomly created by clonal expansion (De Castro, 2008; Silva-Santos et al., 2015; Silva-Santos, Goncalves, et al., 2018). This, consequently, brings another difference in relation to the previous metaheuristics due to the variation in the size of the population along the optimization.

The nature inspired algorithms are herein designated by a bi-dimensional matrix (Eq. 20), where each line is associated with a candidate solution that is differently named in each heuristic, such as antibody in AIS, chromosome in GA, parent or offspring in ES and particle in the PSO algorithm. Therefore, according to Eq. (20), each of the n matrix lines is associated with one candidate solution and the m columns of them are the attributes necessary to be searched in the optimization procedure described in previous studies (Silva-Santos et al., 2015; Silva-Santos et al., 2018). Furthermore, each of the m attributes of the n individuals of the population can vary in the range from \(VM \in_{{\left[ {1:m} \right]}}\) to \(VMax_{{\left[ {1:m} \right]}}\). This hn,m attribute constraint is associated with the number of heat transfer coefficients h, where here m columns are executed with 10, 20, 50, 100 and 200 values to designate different random input values of h.

$$ Population = \left[ {\begin{array}{*{20}c} {h_{1,1} } & {h_{1,2} } & {h_{1,3} } & \ldots & {h_{1,m} } \\ {h_{2,1} } & {h_{2,2} } & {h_{2,3} } & \ldots & {h_{2,m} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {h_{n,1} } & {h_{n,2} } & {h_{n,3} } & \ldots & {h_{n,m} } \\ \end{array} } \right] \vee \left\{ {VMin_{m} \le h_{n,m} \le VMax_{m} } \right\} $$

Each candidate solution, i.e., each line of the matrix (Eq. 20), is randomly generated by Eq. (21), where its probability density function is given by a uniform distribution function given by Eq. (22), where u is the result from the randomly distribution limited by d and g designated as the normalized values between the lowest (0) and highest (1) value interval limits, respectively. In addition, each hn is a data vector with m attributes and associated with each Population matrix to identify each candidate solutions, where their attributes values are set in the predefined range between \(VMin_{{\left[ {1:m} \right]}}\) = 400 and \(VMax_{{\left[ {1:m} \right]}}\) = 10,000 (Silva-Santos, Goulart, et al., 2018).

$$ h_{{n,\left[ {1:m} \right]}} = rand\left( {} \right)_{{\left[ {1:m} \right]}} \cdot \left( {VMax_{{\left[ {1:m} \right]}} - VMin_{{\left[ {1:m} \right]}} } \right) + VMin_{{\left[ {1:m} \right]}} $$
$$ u = \frac{1}{d - g} \vee \left[ {d,g} \right) \cup 0 $$

Considering the development of the metaheuristics in Scilab toolbox, according to its manual, the rand function seed is by default initiated in zero (Scilab, 2020).

The number of candidate solutions is fixed in GA and PSO, but in ES it is cyclically resized along each iteration by the parents and offspring populations considered. In AIS, the number of antibodies depends on the convergence of results achieved along the iterations. These population resize in AIS and ES algorithms is essentially associated with their search space solution. In AIS, the antibody population is initiated with 30% of the predefined n population size and it can be cloned (expanded) until n to maximize the search exploration. When the AIS is converging, the number of antibodies is randomly reduced until 10% of m according to the suppression criteria defined by antibodies local proximity defined by \(\sigma_{local}\). The σlocal is a parameter to determine a local search space region to identify best solutions in this region, which will be further detailed.

Nevertheless, the ES population resize is iteratively controlled, where each generation starts with µpar that generates another novel λoff offspring’s, filling out the Population matrix with n lines (n = µpar + λoff). Afterwards, only µpar best individuals in the Population is selected to be reused in the next generation, returning the matrix size to µpar lines. Therefore, λoff is associated with the number of offspring population, as µpar defines the number of individuals in the parents´ population. Furthermore, considering the numerical accuracy necessary to define the heat transfer coefficient (h) vector, its random values were defined as double float precision.

Other different computational approaches to characterize each algorithm is associated with randomly variational operators that are designated as recombination and mutation in GA and ES, by their evolutive based structure (Fig. 13) (De Castro, 2008; Hruschka et al., 2009; Salto & Alba, 2009). In GA and ES a pair of candidate solutions are required to generate a new one, but in ES, only parents are considered to generate a new candidate solution designated by chromosome in GA and offspring in ES (Coello Coello, 2017; Kazimipour et al., 2014). This recombination is herein randomly performed by the vector breaking point to consider the m attributes from each individual. As example, to generate a certain individual m/2 attributes were considered from two different previous solution candidates (Silva et al., 2014). On the other hand, in ES and GA, only the mutation operator is similar in both algorithms, and they are designated randomly by the Cauchy values in the range of \(VMin_{{\left[ {1:m} \right]}}\) and \(VMax_{{\left[ {1:m} \right]}}\). Therefore, these operators are responsible for generating new and different candidates.

Fig. 13
figure 13

General evolutionary algorithm (ES and GA) dataflow representation

The AIS and PSO algorithms are different from the evolutive algorithms (GA and ES). The PSO algorithm was initially proposed to mimetize the bird flock flying behavior and, since them, variations have been developed to overcome some limitations, such as the local minima search (Ali et al., 2020; García-Nieto, et al., 2007; Khouadjia et al., 2012; Miranda et al., 2018). This mimetization procedure concerns solving multidimensional space problems by positioning each bird in the space according to its previous experience or randomly generated, associating them with their neighbors’ positions. Therefore, each element is considered as a particle described by a velocity vector without mass and volume (Holden & Freitas, 2008; Khare & Rangnekar, 2013).

The main vector velocity calculation is described by Eq. (23), where Vglobal is associated with the global particle velocity in each iteration, C1 and C2 are constants to provide the stochastic acceleration terms by the particle weighting representation for Pbest (particle best value) and Gbest (global best positions), respectively (Michalis et al., 2017; Tian & Shi, 2018).

$$ Vglobal_{{i,j}} = Vglobal_{{i,j}} + C_{1} \cdot rand() \cdot \left( {P_{{best\;i,j}} - Vlocal_{{i,j}} } \right) + C_{2} \cdot rand() \cdot \left( {P_{{best\;i,G_{{best}} }} \; - \;Vlocal_{{i,j}} } \right) $$

The global and local velocity parameters of the particles, C1 and C2 respectively, aims to support particles evolution along the iterations in global optimization perspective and to improve local search-based solution in each iteration.

Finally, the AIS is a different category of nature inspired algorithm to mathematically mimetize some general relations between antibodies and antigenic relations (Caetano et al., 2019; De Castro & Von Zuben, 2020; Diakonikolas et al., 2017; Silva & Dasgupta, 2016; Vidal et al., 2018; Zhang, 2019).

Two main optimization procedures in this category are detached from Evolutionary Algorithms (GA and ES) and swarm intelligence (PSO) algorithms. At first, such as already mentioned, there is a global search when every antibody in the population is considered to return best solution and a local search exploitation determined by σlocal to select best candidate solution in this region. The second original difference is the population resize along the generations.

Therefore, the matrix number of lines of Eq. (20) is dynamic along the iterations in order to abstract the self-adaptivity occurred in the vertebrate immune system according the antibodies and antigenic matching to suppression as an immune response (De Castro & Von Zuben, 2020).

Herein, an opt-Ainet algorithm-based version was developed to attempt these antibodies and antigenic dynamic relations by the suppression mechanism in certain regions delimited by the input parameter σ. When σlocal < σ, a local search is considered to identify the best individual in the referred suppression region Eq. (24) and each σlocal is calculated by Euclidean distances measurements in Eq. (25) (Caetano et al., 2019; Diakonikolas et al., 2017; Silva & Dasgupta, 2016):

$$ Supression_{Region} = \left\{ {\left( {\sigma_{Local} ,\sigma } \right) \vee \sigma_{Local} \le \sigma } \right\} $$
$$ \sigma _{{Local}} = \sqrt {\left( {h_{{i1,j1}} - h_{{i2,j2}} } \right)^{2} } \left| {0 \le \left( {j_{1} \ne j_{2} } \right) \le n_{{Parameters}} } \right|0 \le \left( {i_{1} \ne i_{2} } \right) \le n_{{Antibodies}}$$

where the σlocal is the local parameter determination for each j parameter of pi1 or pi2 antibodies, henceforth, j1 and j2 are different attributes of each antibody.

Appendix B: Optimization Convergence

Considering the inverse heat transfer complexity to determine the h coefficients in different inward, outward and upward flow directions and computational limitations already mentioned, these conditions converge in a large runtime machine to execute each optimization trial. Consequently, usual convergence benchmarking analysis with multiple executions to consider the average iterative values is not viable under the problem circumstances. Henceforth, considering the results with different h coefficient vector values for each heat transfer flow direction is possible to prospect a convergence pattern, as described next.

At first, Fig. 14 presents the inward heat transfer flow with 10, 20, 50, 100 and 200 h values with each of the natural inspired algorithms. The ES algorithm usually returned best results with lower numbers of h values (10, 20 and 50). However, with 100 and 200 every metaheuristic achieved similar values with the same 70 iterations. The PSO executions with lowest number of h values random initial population low searched value persist along many iterations, possibly motivated by small population variation. On the other hands, the PSO in h with 100 and 200 values is close to a linear convergence along the iterations.

Fig. 14
figure 14

Inward heat transfer flow optimization convergence curves with a 10, b 20, c 50, d 100 and (200) h values

These convergence observations in inward heat transfer flow are replicated to outward system. Furthermore, the convergence curves with different h also presents similar patterns along the iterative process. Therefore, with lowest number of h values the optimization reached acceptable results, such as presented in Fig. 15 for 10 (a), 20 (b), 50 (c), 100 (d) and 200 (d) values.

Fig. 15
figure 15

Outward heat transfer flow optimization convergence curves with a 10, b 20, c 50, d 100 and (200) h values

Finally, the upward heat transfer flow with largest runtime machine to be processed by the numerical model herein presented required by them. However, the metaheuristics convergence follows similar characteristics than previous inward and outward results. Therefore, the ES algorithm returned best results with lowest numbers of h values, as shown in Figs 16(a) and (b) for 10 and 20 h values, respectively. Furthermore, the almost linear descendent convergence along the iterative process is similar in 100 and 200 h values, respectively presented in Fig. 16(d) and (e).

Fig. 16
figure 16

Upward heat transfer flow optimization convergence curves with a 10, b 20, c 50, d 100 and (200) h values

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da Silva-Santos, C.H., Goulart, P.R., Bertelli, F. et al. Nature inspired algorithms for the solution of inverse heat transfer problems applied to distinct unsteady heat flux orientations in cylindrical castings. J Intell Manuf (2022).

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  • Solidification
  • Inverse heat transfer problem (IHTP)
  • Numerical modeling
  • Nature inspired algorithms
  • Metaheuristic optimization