Implementing numerical algorithms to optimize the parameters in kampmann–wagner numerical (kwn) precipitation models

ABSTRACT The Kampmann–Wagner Numerical (KWN) model of precipitation is a powerful tool to simulate the precipitation of the second phase considering the nucleation, growth, and coarsening.

Some quantities such as interfacial energy and nucleation site number density are required to accomplish the simulation. Practically, those quantities are hard to measure in the experiment

directly, and the derivation of those quantities through modeling can also be costly. In this work, we hereby adopt the minimization algorithm implemented in the open-source Scipy Python

package to derive that important information in terms of very limited experimental data. The convergence and robustness of different algorithms are discussed. Among those algorithms, the

Nelder–Mead and Powell algorithms are successfully applied to optimize multiple parameters during KWN modeling. This work will shed light on the design of experiments/processes and

facilitate integrated computational materials engineering (ICME). SIMILAR CONTENT BEING VIEWED BY OTHERS MICRO-CONTINUUM APPROACH FOR MINERAL PRECIPITATION Article Open access 10 February

2021 PHASE-FIELD MODEL OF PRECIPITATION PROCESSES WITH COHERENCY LOSS Article Open access 12 March 2021 PROBABILISTIC NUCLEATION GOVERNS TIME, AMOUNT, AND LOCATION OF MINERAL PRECIPITATION

AND GEOMETRY EVOLUTION IN THE POROUS MEDIUM Article Open access 12 August 2021 INTRODUCTION The interest in modeling precipitation has increased dramatically in recent years. Continuing

research effects have been made to develop general and composition-dependent models for the aging effect of alloys to predict the evolution of precipitate fraction, size, number density,

morphology, and solid solution solute level. The reported approaches for this purpose fall into two categories: direct detailed approaches for visualization purposes1,2,3 and physically

based internal state variable approaches4,5. The former is represented by the phase-field method (PFM) or finite difference method and excels in providing detailed descriptions of

nucleation, growth, and coarsening. The latter, represented by the Kampmann–Wagner Numerical (KWN) model6, can handle precipitation involving multi-scale transportation phenomena such as

concurrent nucleation, growth, coarsening of an ensemble of precipitates, precipitation in matrix with an uneven compositional profile, etc. The KWN model is also implemented in some

commercial software such as Thermo-Calc software as the TC-PRISMA module7,8,9 as well as some open-source codes such as Kawin developed by Ury et al.5. To set up a simulation, a suitable

CALPHAD thermodynamic database and mobility database are required. According to classical nucleation theory (CNT), some parameters such as interfacial energy and nucleation site density are

considered crucial quantities which has significant effects on the modeling of precipitation. However, those quantities may be hard to determine precisely due to the complexity of

microstructure, e.g.: the reported interfacial energy of \({\delta }^{{\prime} }\) precipitate in Al–Li alloys varies from 0.007 to 0.11 J/m−2 10 and that of _β_″ precipitate in Al–Mg–Si

alloys varies from 0.08 to 0.25 J/m−2 11,12,13,14. According to Miesenberger et al.15, the large range of measured interfacial energy from different works could be caused by the complexity

of the microstructure. It is reported that both the volume and interface contributions to the Gibbs free energy for nucleation are functions of temperature and size of the nucleus and the

chemical composition of precipitate and matrix. The effects such as diffuse interface, interface energy size effect, and heterogeneous nucleation energy with varying nucleation location

(e.g. dislocation, grain boundary, bulk) play a significant role in the determination of the interfacial energy. In KWN modeling, there are some other important parameters which could also

impact the precipitation behavior such as elastic strain, elastic modulus, etc. Those parameters can be accurately calculated or measured directly, and thus are not good candidates for

optimization. Parameters such as interfacial energy and nucleation site density cannot be measured directly, and in many cases end up acting as tuning or calibration parameters.

Traditionally, the parameters of KWN models are set by numerous trials and analyses by experienced researchers. Human-intuited materials discovery involves modeling, experiments, and

analysis by human researchers. In the data-driven paradigm, high throughput screening and numerical optimization enable automated data generation, storage, and analysis, with increased data

sizes and processing speed. By replacing human intuition in the paradigm, active learning and inverse design in data-driven discovery further enable the iterative optimization of the

parameters toward the best solutions. Meanwhile, fundamental studies using the physics-based approaches can introduce additional insights into optimizing the parameters. Such a framework is

shown in Fig. 1. In this paper, we hereby evaluate multiple optimization algorithms to help to determine those quantities even when experimental data are limited, e.g.: only 3–5 points of

measured mean radius as a function of aging time. We will also show the robustness of this approach and analyze the advantages and disadvantages of different optimization algorithms. RESULTS

The applicability of the optimization algorithms in this study is demonstrated through two common commercial alloys: Al and Ni-based alloys. In this section, we will try to optimize the

multiple parameters in the KWN modeling of Al–Mg–Si aluminum and IN718 superalloys. These examples are good representatives for those categories. AL–MG–SI ALLOYS Al–Mg–Si is an important

category in commercial aluminum alloys (known as 6000 series aluminum). In this class of aluminum alloys, the primary strengthening precipitate is the _β_″ phase. Here we adopted the alloy

AA6005 with composition Al-0.562Mg-0.016Cu-0.607Si-0.054Mn-0.002Fe (weight pct) which is aged at 185 °C16. In this study, we will try to optimize single and multiple parameters in the KWN

model. The thermodynamic and kinetics databases are TCAL8 and MOBAL7, respectively. According to the literature11,12,13,14, the calculated interfacial energy ranges from 0.08 to 0.25 J/m−2.

The rest parameters are set as indicated in the literature17 such as nucleation site density (6.25 × 1028 m−3) and morphology of precipitate (needle-like). The input of strain was taken as

_ϵ_11 = −0.046, _ϵ_22 = −0.046 and _ϵ_33 = 0.0007. The optimization of parameters is categorized into two aspects: (1) optimize interfacial energy; (2) optimize both interfacial energy and

nucleation site density. By default, in the case of homogeneous nucleation, each atom in the whole volume of the matrix phase is a potential nucleation site. However, defects such as grain

boundary, dislocations, and point defects because of quenching or deformation are inevitable. Therefore, while nucleation site density is theoretically a physical value that can be

determined by understanding where a precipitate should be nucleating, it is necessary to consider the nucleation site density as an additional parameter to optimize, at least over a range

that can be assigned some physical meaning. For the optimization of interfacial energy, the applicable range is set as [0.07, 0.26], which is a little wider than the value reported in the

experiment. During the simulation, the nucleation site density is set as 6.25 × 1028 m−3 as indicated in the literature17. Figure 2a shows the optimization of interfacial energy starting at

70 mJ/m2 by Nelder–Mead and Powell algorithms and Fig. 2b shows the variation of mean squared error (MSE) with iterations. Figure 2b shows that the minimization of MSE is much faster by the

Powell method than the Nelder–Mead algorithm. However, the parameter will jump suddenly at iteration No. 17, which contributes to the massive increase of MSE. This phenomenon is caused by

the termination check of the Powell algorithm. According to Vassiliadis and Conejeros18, the termination procedure is computationally expensive since the entire minimization problem has to

be resolved at least twice until the tight convergence criteria are satisfied. In the end, more iterations are taken for the Powell algorithm even if the MSE with Powell decreases much

faster. Besides the interfacial energy, the number of nucleation sites is another variables which should be determined. According to Myhr et al.14 it is noted that the nucleation site

density _N_ is physically reasonable to fall within a range where the classic nucleation theory is normally considered to be valid. Therefore, we would further explore the reasonable site

density with an optimization algorithm. Starting with interfacial energy equal to 0.07 J/m2 and site density equal to 1028 (m−3), the evolution of parameters during Nelder–Mead and Powell

optimization is shown in Fig. 3a, b, respectively. The corresponding MSE is shown in Fig. 3c. The converged results are different for the two algorithms. The Nelder–Mead optimization is

converged at interfacial energy 79.13 mJ/m2 and nucleation site density 3.46 × 1025 m−3 while Powell optimization is converged at interfacial energy 95.62 mJ/m2 and nucleation site density

9.89 × 1027 m−3. Both points can be viewed as a local minimum of MSE, while the MSE of Nelder–Mead optimization is 5.03 × 10−19 and that of Powell optimization is 7.22 × 10−19. The results

are summarized in Table 1. Figure 4a, b shows the simulated radius of _β_″ with optimization of only interfacial energy and both interfacial energy and nucleation site density, respectively.

The fitting from both algorithms shows very good agreement with the experiment by Myhr et al.16. In the case of the optimization of only interfacial energy, both algorithms lead to the same

results. However, the results optimized by two different algorithms are quite different. The decision of which to rely on could be made by more detailed analysis in terms of the observed

microstructure. One possible approach is to check the nucleation site practically. since the nucleation site density has some physical meaning. Within the TC-PRISMA module implemented in

Thermo-Calc software, the heterogeneous nucleation site can be further quantified as follows: if the particle nucleates in bulk, the nucleation site density can be estimated as 8.60 × 1028 

m−3, which corresponds to every possible atomic site in the simulation volume; If the particle nucleates at the grain boundary when the grain size is 100 μm, the site density is 6.53 × 1023 

m−3. The full derivation of heterogeneous nucleation site densities and other potential sites such as dislocations, grain corners, and edges, can be found in the user guide of the TC-PRISMA

module9. Experimentally, the nucleation of _β_″ precipitates is in the bulk, so optimizations of nucleation site density which are more than an order of magnitude away from the bulk value

are likely non-physical in this alloy system. Therefore, the optimized parameters with the Powell algorithm (interfacial energy 95.62 mJ/m2 and nucleation site density 9.89 × 1027 m−3) may

be the better optimization for this alloy system. Taking the optimized parameters from Powell algorithm to simulate the precipitation of two different Al–Mg–Si alloys, we simulated the mean

particle length, number density and volume fraction as a function of aging time. The results are shown in Fig. 5a–c, respectively. Compared to the experimental results measured by Du et

al.19 and Qian et al.20, the simulated particle size and number density show good agreement with the experiment. However, the simulated volume fraction exhibits significant discrepancy with

the experiment, especially when compared to Du’s result. As shown in Fig. 5c, the volume fraction of _β_″ is overestimated at around 104 and 105 s. However, after 106 s, the simulated volume

fraction is less than the experimental data. As characterized by Dutta et al.21 and Edwards et al.22, the sequence of precipitation in Al–Mg–Si alloy is cluster of Si atoms → GP-I zones →

mixture of GP-II- zones _β_″ \(\to \,{\beta }^{{\prime} }\,\to \,\beta\) (Mg2Si). In this process, the nucleation _β_″ depends on the location of GP zones. Therefore, with much less

nucleation site, the nucleation stage may take longer time than the homogeneous nucleation. On the other hand, _β_″ start to transfer to \({\beta }^{{\prime} }\), forming a mixture of _β_″,

\({\beta }^{{\prime} }\) and other precipitates, after 208 h as reported by Du et al.19. However, other precipitates are ignored in the simulation. Therefore, the measured results could be

larger than the simulation since all precipitates are included in the experiment. NI-BASED SUPERALLOY: IN718 The Ni-based superalloy IN718 is widely used for its combination of strength and

weldability. It is strengthened by a combination of \({\gamma }^{{\prime} }\) and _γ_″. The co-precipitation phenomenon of these two precipitates has garnered a lot of attention over the

years. Sundararaman et al. reported the simultaneous nucleation of \({\gamma }^{{\prime} }\) and _γ_″ 23. Theska et al.24,25,26 and Drexler et al.27 characterized the microstructure of

precipitates in terms of particle radius, number density, phase fractions as well as aspect ratios. Besides experimental characterization, computational tools can also be employed to

simulate the co-precipitation of \({\gamma }^{{\prime} }\) and _γ_″, as reported by Zhang et al.28. Yu et al.29 further studied the effects of aging conditions on the microstructure of

precipitates. However, the uncertainty surrounding the interfacial energy of the precipitates presents challenges to using these tools in a predictive way. Therefore, it is beneficial to

accelerate the determination of interfacial energy through calibration with limited experimental measurements such as mean radius or particle length of precipitates. The composition of the

alloys can be found in Table 2. Some of these results and their implications on process variation have been previously published by the authors29. In this paper, we will focus on the

optimization used to calibrate the interfacial energies. The experimental data are derived from Han et al.30, Sundararaman et al.23, and Devaux et al.31. The precipitate phases are \({\gamma

}^{{\prime} }\) and _γ_″. The thermodynamic and kinetics in the simulation are TCNI12 and MOBNI6, respectively. To set up the simulation, some info about the precipitates should first be

reviewed. The morphology of \({\gamma }^{{\prime} }\) is typically spherical or cuboidal, and the morphology of _γ_″ is mostly plate-like (oblate spheroidal). This can be described as

$$\frac{{x}_{1}^{2}}{{l}^{2}}+\frac{{x}_{2}^{2}}{{l}^{2}}+\frac{{x}_{3}^{2}}{{r}^{2}}\,\le \,1(l \,>\, r)$$ (1) where _x_1, _x_2, and _x_3 are the coordinates of a point on the surface,

and _l_ is the horizontal radius of the oblate and _r_ is the thickness of the spheroid. The aspect ratio _α_ = _l_/_r_ > 1. According to Devaux et al.31, the eigenvalue of transformation

strain is _ϵ_11 = 0.0086, _ϵ_22 = 0.0086, and _ϵ_33 = 0.0286. However, the particles may lose coherency after some time. As a simplification, we set a fixed aspect ratio for the _γ_″

precipitates in this work. In this regard, the elastic strain energy in this study will affect the nucleation rate and nuclei size but will not change the aspect ratio of the particle during

the coarsening of precipitates. Additionally, the TC-PRISMA implementation assumes spherical nuclei and will only apply the fixed aspect ratio in the growth stage. According to Zhang et

al.32, the elastic constant is a function of temperature. In this study, they are assumed to be constants over the temperature ranging from 973 to 1023 K as _C_11 = 174 GPa, _C_12 = 92.4 

GPa, and _C_22 = 110 GPa. Figures 6 and 7 show the evolution of interfacial energy with iterations for \({\gamma }^{{\prime} }\) and _γ_″ precipitate, respectively. After iterations by

Nelder–Mead algorithm, the interfacial energy of \({\gamma }^{{\prime} }\) at 973, 998, and 1023 K are 20.4, 24.6_,_ and 17.8 mJ/m2, respectively, while the interfacial energy are 20.4,

24.3_,_ and 17.9 mJ/m2, respectively after iterations by Powell algorithm. The converged interfacial energy is very close to each other and the value lies within the applicable range

summarized by Ardell33. On the other hand, the interfacial energy of _γ_″ at 973, 998, and 1023 K are 17.84, 27.34_,_ and 17.3 mJ/m2, respectively, after optimized by Nelder–Mead algorithm

and that are 17.98, 27.66_,_ and 17.15 mJ/m2 after optimized by Powell algorithm. The optimized value is well below the applicable range summarized by Schleifer et al.34. The simulated

\({\gamma }^{{\prime} }\) and _γ_″ particle sizes are shown in Fig. 8a, b, respectively, which is the optimized interfacial energy. As summarized by Ardell33, interfacial energy between _γ_

and \({\gamma }^{{\prime} }\) is between 10 and 40 mJ/m2. The fitted interfacial energy through both algorithms lies within the reported range. However, according to Schleifer et al.34, the

derived interfacial energy between _γ_ and _γ_″ is between 90 and 200 mJ/m2. The optimized value is well below the lower limit of experiments. One possible reason is the co-precipitation.

Co-precipitation is a characteristic phenomenon in Ni-based superalloys. A lot of researchers investigated this phenomenon through the PFM since PFM has a great advantage of studying

heterogeneous microstructure2,35,36. Sriram et al.2 stated that the driving force for nucleation could be $$\Delta G=\Delta {G}_{v}-{E}^{\rm{self}}-{E}^{\rm{int}}$$ (2) where Δ_G_ is the

chemical driving force for nucleation, _E_self is the self-elastic energy for forming a critical _γ_″ nucleus and _E_int is elastic interaction energy between an existing \({\gamma

}^{{\prime} }\) precipitate and the _γ_″ nuclei. Ji et al.35 estimated _E_int as 0.375 GPa (equivalent to 2.68 × 103 J/mol) at maximum in IN718. Therefore, the effects of elastic interaction

are not negligible. To avoid redundancy, we would only focus on the precipitation of _γ_″ at 1023 K for the multivariate optimization of interfacial energy and elastic interaction energy

additions. Initiating from the interfacial energy 0.09 J/m2 and energy addition −1000 J/mol, the evolution of parameters during Nelder–Mead and Powell optimization is shown in Fig. 9a, b,

respectively. The corresponding MSE is shown in Fig. 9c. Here we take the aspect ratio of _γ_″ as 1.1 since the aspect ratio at the initial stage is close to 1 according to Schleifer et

al.34. The results show that MSE will converge at interfacial energy 95.41 mJ/m2, energy addition −958.2 J/mol and interfacial energy 94.12 mJ/m2, energy addition −894.7 J/mol for

Nelder–Mead and Powell, respectively. The calculated MSEs are 3.29 × 10−18 and 3.38 × 10−18 for Nelder–Mead and Powell algorithms, respectively. The results are summarized in Table 3 and the

simulated mean lengths of _γ_″ precipitates are shown in Fig. 10a correspondingly. The results with optimized parameters from Nelder–Mead and Powell optimization are shown as orange and

green curves, respectively. These results indicate that considering a reasonable energy addition (below the maximum elastic interaction energy reported by Ji et al.35), the simulated _γ_″

precipitation can exhibit a good agreement with interfacial energy which lies within the calculated range of other investigations in the literature34 (90–200 mJ/m2). Likewise, the elastic

interaction from the _γ_″ will also have an effect on the precipitation of \({\gamma }^{{\prime} }\). Therefore, we imposed an energy addition of −500 J/mol on \({\gamma }^{{\prime} }\). The

fitted interfacial energy becomes 84.5 mJ/m2. The simulated mean particle length of _γ_″ precipitate with updated parameters is shown in Fig. 10b (light purple curve). In contrast with the

parameters fitting in Fig. 10a, the agreement becomes relatively poor in Fig. 10b at the early stage of precipitation. The overestimation is caused by treating the heterogeneous

co-precipitation as a mean-field approximation is hypothesized in KWN models. The simulated mean particle sizes, number density as well the volume fraction of both precipitates are shown in

Fig. 11. In comparison with the experimental results by Anderson et al.37 and Ahmadi et al.38, the simulated sizes of precipitates and number density at aging time >2 × 105 s and the

simulated total volume fraction of \({\gamma }^{{\prime} }\) and _γ_″ precipitates exhibit a good agreement. The disagreement at early stage of precipitation could be caused by the

heterogeneous nucleation during the co-precipitation2,24. According to Han et al.30, the ratio of volume fraction between _γ_″ and \({\gamma }^{{\prime} }\) is around 2.5 to 4.0. Under this

condition, the ratio of volume fraction between two precipitates varies from 2.64 to 4.12 after 104 s (Fig. 11c), which shows a good agreement with the experiment. DISCUSSION According to

the Scipy documentation and lecture notes39,40, the optimization algorithms can be classified into three categories: derivative-free, gradient, and trust-region. All of them are used for a

local minimization. To be noted, there is no analytical solution of KWN models. Therefore, it is hard to derive the Jacobian (first-order derivative) or Hessian (second-order derivative),

which is required for gradient and trust-region-based optimization algorithms. Therefore, a derivative-free method will be more appropriate to optimize the KWN models. Therefore, Nelder–Mead

and Powell methods are employed in this work. The categories of algorithm are shown in Table 4. The Nelder–Mead algorithm is a generalization of dichotomy approaches to high-dimensional

spaces. According to Varoquaux et al.40, since Nelder–Mead does not rely on computing gradients, it can work on functions that are not locally smooth such as experimental data, as long as

they display a large-scale bell-shaped behavior. Furthermore, Powell’s method is not too sensitive to local ill-conditioning quadratic functions. As a result, there could be several

“jumps" of MSE during the optimization processes in Fig. 6. As shown in Fig. 12, the surface of MSE with the parameters interfacial energy and site number density is highly

non-quadratic. To overcome this problem, we would suggest to determine the site number density of nucleation in terms of nucleation site observed in the experiment. For example, when the

site number density is set as 6.25 × 1028/m−3 8, the variation of MSE will become a function of interfacial energy is shown in Fig. 13. Under this condition, the interfacial energy can be

optimized as 99.29 mJ/m2. According to Wagner et al.6, the coarsening behavior of precipitates is based on Lifshitz and Slyozov, and Wagnerz (called LSW theory) and the analytical solution

of interfacial energy derived by Ardell and Ozolins41 while the exponent of growth rate is fitted with experimental results. Here we compare the optimized interfacial energy of \({\gamma

}^{{\prime} }\) in Ni–Al binary alloys with the calculation as shown in Table 5. The formalization of theoretical calculation can be found in the subsection of the “Methods” section. The

optimized results in this work vary within 10% of tolerance. The difference could be caused by the heterogeneity of microstructure41. Two different algorithms in this work show promising

results in the optimization of parameters during KWN modeling. However, there are obvious limitations to these algorithms. The biggest one is that the global minimum of MSE cannot be

guaranteed. Therefore, it is always suggested to check the agreement with other experimental data in case the iterations stop at some non-physical local minimum. Alternatively, with some

prior physical knowledge of the material system can be determined or have bounds set on their optimization ranges beforehand. Moreover, in this study, we focused on optimizing the parameters

in KWN models for practical purposes when experimental data is limited, instead of quantifying uncertainties. The uncertainty quantification is suggested as a future work where more

advanced algorithms such as Bayesian optimization methods could be adopted. Even though the Bayesian algorithm is more powerful when dealing with multivariate optimization42, the algorithms

in this work still have their advantage due to its simplified form. In conclusion, we have demonstrated how optimization algorithms can be implemented to help optimize some of the physical

parameters needed for KWN precipitation modeling in this work. The precipitation processes of two different alloys were investigated using this technique. The results indicated that the

Nelder–Mead or the Powell algorithm can be adopted to optimize parameters needed for KWN modeling such as interfacial energy, and nucleation site density. We have shown that with 40–100

iterations, the interfacial energy of a _β_″ precipitate in a multicomponent Al–Mg–Si alloy can be optimized. The optimized value lies within the applicable range determined in other

publications. Additionally, the simulated precipitation process shows very good agreement with the experiment in terms of the mean particle length and particle number density. After

determining the nucleation site from micrographic analysis, the optimized interfacial energy can be accurately determined to be 99.29 mJ/m2. We also used a IN718 superalloy as an example to

optimize the interfacial energy of \({\gamma }^{{\prime} }\) and _γ_″ precipitates at different aging temperatures. The optimized interfacial energy of \({\gamma }^{{\prime} }\) shows very

good agreement with values determined in other publications while that of _γ_″ is well below. This is hypothetically caused by the elastic interaction due to the co-precipitation of

\({\gamma }^{{\prime} }\) and _γ_″. The interfacial energy and energy additions are taken as multivariate parameters for an optimization. The simulated particle size, number density, and

volume fraction of both precipitates agree well with the experiment except for the early stage of aging because the heterogeneous nucleation due to co-precipitation at early stages may not

be captured by this mean-field model. Furthermore, this work gives a possible solution to determine the unknown parameters during KWN precipitation modeling by using numerical optimization

algorithms, enabling usage in a more predictive capacity. As KWN modeling has been implemented in commercial software such as Thermo-Calc, Pandat, and some open-source simulation tools such

as Kawin5, this could help ease the adoption of these tools. These findings can help to facilitate the integrated computational materials engineering (ICME) process and have the potential to

accelerate materials design. METHODS THE KAMPMANN–WAGNER NUMERICAL (KWN) MODEL The KWN model is modified from the particle coarsening theory put forward by Lifshitz and Slyozov43, and

Wagnerz44 (called LSW theory) and has been successfully implemented as a module called TC-PRISMA in Thermo-Calc9 or PanPrecipitation in Pandat software45. At its core, the KWN model is a

mean-field model of precipitation, tracking the evolution of average properties of a bulk volume rather than the localized properties surrounding an individual precipitate. Since particles

are not tracked individually, the particle size distribution (PSD) is modeled as a continuous function where particles are placed in uniformly spaced bins that represent the size of the

particles. At each time step, nucleation, growth, and dissolution behaviors are calculated and the PSD is updated to reflect any changes while still adhering to strict mass balance and

continuity equations. In CNT46,47, the formation of second-phase particles is due to the heterogeneous fluctuations in a metastable solid solution. These fluctuations are assumed to occur

constantly as a result of the thermal instability of the parent phase. The net free energy change of nucleus formation as a function of its radius is shown in $$\Delta G=-\frac{4}{3}\pi

{R}^{3}\Delta {G}_{v}+4\pi {R}^{2}\gamma$$ (3) The critical radius _R_*-radius, at which the volumetric and interfacial energy contributions are in a state of unsteady equilibrium, can be

derived as $${R}^{* }=\frac{2\gamma }{\Delta {G}_{v}}$$ (4) The corresponding energy barrier can be derived as $$\Delta {G}^{* }=\frac{4}{3}\pi \gamma {{R}^{* }}^{2}=\frac{16}{3}\frac{\pi

{\gamma }^{3}}{\Delta {G}_{v}^{2}}$$ (5) The time-dependent nucleation rate _J_(_t_) is given by $$J(t)={J}_{s}exp\left(\frac{-\tau }{t}\right)$$ (6) where _J__s_ is the steady-state

nucleation rate, _τ_ is the incubation time for establishing steady-state nucleation conditions and _t_ is the time. The steady-state nucleation rate _J__s_ is expressed by $${J}_{s}=Z{\beta

}^{* }N\,exp\left(\frac{-\Delta {G}^{* }}{kT}\right)$$ (7) where _Z_ is the Zeldovich factor, _β_* is the rate at which atoms or molecules are attached to the critical nucleus, _N_ is the

number of available nucleation sites per unit volume, −Δ_G_* is the Gibbs energy of a critical nucleus, _k_ is Boltzmann’s constant, and _T_ is the temperature in Kelvin. In the case of

homogeneous nucleation, each atom in the whole volume of the matrix phase is a potential nucleation site. However, quantity _N_ can also be input when the nucleation happens in the defects

such as grain boundary and dislocations. The rest of the quantities can be calculated as follows: The Zeldovich factor (_Z_) is a measure of the probability that supercritical nuclei with a

radius slightly larger than the critical radius have a probability of passing back across the free energy barrier and dissolving in the matrix. It is related to the thermodynamics of the

nucleation process in $$Z=\frac{{V}_{m}^{\beta }}{2\pi {N}_{A}{({R}^{* })}^{2}}\sqrt{\frac{\gamma }{kt}}$$ (8) where _N__A_ is the Avogadro number and _R_* is given by Eq. (4). _β_* reflects

the kinetics of mass transport in the nucleation process and is given by Svoboda et al.48: $${\beta }^{* }=\frac{4\pi {({R}^{* })}^{2}}{{a}^{4}}{\left[\mathop{\sum }\limits_{i =

1}^{k}\frac{({X}_{i}^{\beta /\alpha }-{X}_{i}^{\alpha /\beta })}{{X}_{i}^{\alpha /\beta }{D}_{i}}\right]}^{-1}$$ (9) where _a_ is the lattice parameter, \({X}_{i}^{\beta /\alpha }\) and

\({X}_{i}^{\alpha /\beta }\) are the mole fractions of element _i_ at the interface in the precipitate and matrix, respectively and _D__i_ is the corresponding diffusion coefficient in the

matrix. Theoretically described by Langer and Schwartz49, the KWN model deals with concurrent nucleation, growth, and coarsening. Employing PSD _f_(_R_, _t_) in terms of particle size

(radius) _R_ and time _t_, the approach proceeds by simultaneously solving continuity equation $$\frac{\partial f(R,t)}{\partial t}=-\frac{\partial }{\partial R}[v(R)f(R,t)]+j[R,t]$$ (10)

where _v_(_R_) is the growth rate of a particle of size _r_, and _j_(_R_, _t_) is the distributed nucleation rate. In a pseudo-steady state approximation, the growth rate is simplified into

solving the Laplace equation along the radial direction $$v(R)=\left(\frac{dR}{dt}\right)=\frac{2\gamma {V}_{m}^{p}K}{R}\left(\frac{1}{{R}^{* }}-\frac{1}{R}\right)$$ (11) where _R_ and _R_*

are the radius and critical radius of the precipitate, and _K_ is the kinetic parameter that is related to the solute composition and mobility. Neglecting the cross-diffusion, _K_ can be

expressed as $$K={K}_{\gamma }\cdot {K}_{\rm{shp}}\cdot {K}_{\rm{sphere}}$$ (12) where _K__γ_ is the parameter that takes into account the Gibbs–Thomson effect due to interfacial energy

anisotropy. For a needle shape, \({K}_{\gamma }={\root{3}\of{\alpha }}\). For a plate shape, \({K}_{\gamma }={\root{3}\of{{\alpha }^{2}}}\) where _α_ is the aspect ratio of the ellipsoidal

particle and _K_shp is the parameter that takes into account the non-spherical concentration field around the particle and can be expressed as \({K}_{\rm{shp}}=\frac{2\,{\root{3}\of{{\alpha

}^{2}}e}}{ln(1+e)-ln(1-e)}\) where _e_ is the eccentricity of the ellipsoidal particle and \(e=\sqrt{1-\frac{1}{{\alpha }^{2}}}\). _K_sphere can be expressed as

$$K={K}_{\rm{sphere}}={\left[\mathop{\sum }\limits_{i = 1}^{k}\frac{{({X}_{i}^{\beta /\alpha }-{X}_{i}^{\alpha /\beta })}^{2}{\epsilon }_{i}}{{X}_{i}^{\alpha /\beta }{M}_{i}}\right]}^{-1}$$

(13) \({X}_{i}^{\beta /\alpha }\) and \({X}_{i}^{\alpha /\beta }\) are shown in Fig. 14 (noted as \({X}_{c}^{\gamma /\alpha }\) and \({X}_{c}^{\alpha /\gamma }\), respectively). The

derivation can be found in the guidance of TC-PRISMA9. After the nucleation and growth have been calculated and the particle size distribution updated, it is necessary to perform a mass

balance to update matrix solute concentrations. To conserve the mass, the sum of solute atoms in the matrix and the precipitate must be equal to the initial number of solute atoms. The

initial mole fraction of component i in the matrix phase \({X}_{0i}^{m}\), the new concentration \({X}_{i}^{m}(t)\) can be obtained from the following relationship

$${X}_{0i}^{m}=\left(1-\sum _{p}{\int_{0}^{\infty }}\frac{4\pi {r}_{p}^{3}f({r}_{p})}{3{V}_{m}^{p}}d{r}_{p}\right){X}_{i}+\sum _{p}{\int_{0}^{\infty }}{\int_{0}^{{t}_{j}}}\frac{4\pi

{r}_{p}^{2}f({r}_{p},t)v({r}_{p},t)}{{V}_{m}^{p}}{X}_{i}^{p}({r}_{p},t)dtd{r}_{p}$$ (14) where \({X}_{i}^{p}({r}_{p},t)\) is the mole fraction of element _i_ at the interface in the

precipitate phase _p_ of particle size _r__p_ at time _t_. _f_(_r__p_, _t_), _v_(_r__p_, _t_) and \({V}_{m}^{p}\) are the PSD function, growth rate, and molar volume of the precipitate phase

_p_, respectively. The elastic energy is calculated through Eshelby’s theory50,51 with a reasonable computational cost. The elastic strain energy can be obtained by

$${E}^{el}=-\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}^{T}V$$ (15) where _V_ is the particle volume, \({\epsilon }_{ij}^{T}\) is the transformation strain (eigenstrain) and _σ__i__j_ is the

elastic stress, which can be calculated as $${\sigma }_{ij}={C}_{ijkl}({\epsilon }_{kl}-{\epsilon }_{kl}^{T})$$ (16) where _ϵ__k__l_ is the total strain and _C__i__j__k__l_ is the stiffness

tensor of the material. The total strain can be calculated as $${\epsilon }_{ij}={S}_{ijkl}{\epsilon }_{kl}^{* }$$ (17) where _S__i__j__k__l_ can be calculated by Eshelby tensor _D_ with

$${S}_{ijmn}=-\frac{1}{2}{C}_{lkmn}({D}_{iklj}+{D}_{jkli})$$ (18) The Eshelby tensor can be calculated by $${D}_{ijkl}=-\frac{abc}{4\pi}{\int_{0}^{\pi}}{\int_{0}^{2\pi}}{{\Omega

}}_{ij}{n}_{k}{n}_{l}\frac{sin\theta }{{\beta }^{3}}d\phi d\theta$$ (19) where _a_, _b_, and _c_ are ellipsoid axes. _n__i_(_i_ = 1, 2, 3) are the unit directional vector normal to the

spherical surface and \(\beta =\sqrt{({a}^{2}co{s}^{2}\phi +{b}^{2}si{n}^{2}\phi )si{n}^{2}\theta +{c}^{2}co{s}^{2}\theta }\). Ω_i__j_ is the Green function. The derivation of Ω_i__j_ can be

found in Khachaturyan52. OPTIMIZATION ALGORITHMS The optimization process can be described as follows: * 1. Take initial guess of parameters [\({p}_{1}^{0}\),\({p}_{2}^{0}\), …,

\({p}_{i}^{0}\)] where _i_ is the number of parameters to optimize. * 2. Simulate the precipitation process through the KWN method implemented in the TC-PRISMA module of Thermo-Calc

software. * 3. Quantify the MSE with simulated results through the MSE $${\rm{MSE}}=\frac{{\sum }_{n}{({y}_{n}^{\rm{calc}}-{\hat{y}}_{n})}^{2}}{n}$$ where \({y}_{n}^{\rm{calc}}\) is the

calculated quantity through TC-PRISMA and \({\hat{y}}_{n}\) is the experimental data and _n_ is the number of samples in the experiments. The justification of MSE to optimize parameters is

shown in the later subsection. * 4. Apply the minimization algorithm in Scipy to find $$argmi{n}_{[{p}_{1}^{t},{p}_{2}^{t}...{p}_{i}^{t}]}\left(\frac{{\sum

}_{n}{({y}_{n}^{\rm{calc}}-{\hat{y}}_{n})}^{2}}{n}\right)$$ where [\({p}_{1}^{t}\),\({p}_{2}^{t}\), …, \({p}_{i}^{t}\)] are the updated parameters. * 5. Repeat steps 2–4 until the

convergence within the range of relative tolerance is achieved. * 6. Output the optimized parameters. In step 4, SciPy is an open-source scientific computing library for the Python

programming language. Since version 1.0, numerous optimization algorithms have been developed39. In this study, Scipy version 1.7 is employed. Table 5 shows the collection of optimization

algorithms. Ideally, algorithms to optimize KWN modeling should satisfy the following conditions: (1) allowed to boundary-constrain the minimization algorithm; (2) Jacobian matrix is not

required; (3) stopping criterion is allowed. Boundary-constrained minimization algorithms are preferred since the parameters in the KWN modeling have specific physical meanings. Therefore,

there is always a limit for any of those quantities, e.g.: the coherent interfacial energy ranges from 0.01 to 0.5 J/m2 for most metals. In this regard, only Nelder–Mead, Powell, TNC, and

L-BFGS-B among all the algorithms in Table 5 are satisfied. Besides the bounds of simulation, the relative step size is also crucial. After careful testing, the automatic step size of

Nelder–Mead and Powell algorithms are properly set in the simulation while the step sizes of TNC and L-BFGS-B algorithms should be manually input, which will be time-consuming. Therefore,

Nelder–Mead and Powell algorithms will be employed and analyzed in this work. NELDER–MEAD ALGORITHM Gao et al.53 present the modified Nelder–Mead algorithm. This algorithm aims to solve the

unconstrained optimization problem $$minf({\bf{x}})$$ (20) where \(f:{{\mathbb{R}}}^{n}\to {\mathbb{R}}\) is the called objective function and _n_ is the dimension. A simplex is a geometric

figure in _n_ dimensions that is the convex hull of _n_ + 1 vertices. According to Gao and Han53, the Nelder–Mead method first generates a sequence of simplices to approximate an optimal

point. At each iteration, the vertices _x__j_(_j_ = 1. . . _n_ + 1) of the simplex are ordered according to _f_ as $$f({{\bf{x}}}_{1})\,\le \,f({{\bf{x}}}_{2})\le ...f({{\bf{x}}}_{n+1})$$

(21) where X1 is referred to the best vertex and X_n_+1 is referred to the worst vertex. This algorithm will apply those four possible operations: reflection (_α_), expansion (_β_),

contraction (_γ_), and shrink (_δ_). The notation in the bracket is the scalar parameters of corresponding operations. Therefore, within one iteration, the operations will execute

sequentially as below: * 1. Sort as described in Eq. (19). * 2. Reflection. Compute the reflection point X_r_ from $${{\bf{x}}}_{r}=\bar{{\bf{x}}}+\alpha (\bar{{\bf{x}}}-{{\bf{x}}}_{n+1})$$

(22) Evaluate _f__r_ = _f_(X_r_). If _f_1 ≤ _f__r_ ≤ _f__n_, replace X_n_+1 with X_r_. * 3. Expansion. If _f__r_ < _f_1 then compute the expansion point X_e_ by

$${{\bf{x}}}_{e}=\bar{{\bf{x}}}+\beta ({{\bf{x}}}_{r}-\bar{{\bf{x}}})$$ (23) Evaluate _f__e_ = _f_(X_e_). If _f__e_ < _f__r_, replace X_n_+1 with X_e_, otherwise replace X_n_+1 with X_r_.

* 4. Outside contraction. If _f__n_ ≤ _f__r_ ≤ _f__n_+1, compute the outside contraction point $${{\bf{x}}}_{oc}=\bar{{\bf{x}}}+\gamma ({{\bf{x}}}_{r}-\bar{{\bf{x}}})$$ (24) Evaluate

_f__o__c_ = _f_(X_o__c_). If _f__o__c_ ≤ _f__r_, replace X_n_+1 with X_o__c_, otherwise go to step 6. * 5. Inside contraction. If _f__r_ ≥ _f__n_+1, compute the inside contraction point

_x__i__c_ from $${{\bf{x}}}_{ic}=\bar{{\bf{x}}}-\gamma ({{\bf{x}}}_{r}-\bar{{\bf{x}}})$$ (25) Evaluate _f__i__c_ = _f_(X_i__c_). If _f__i__c_ < _f__n_+1, replace X_n_+1 with X_i__c_;

otherwise, go to step 6. * 6. Shrink. For 2 ≤ _i_ ≤ _n_ + 1, define $${{\bf{x}}}_{i}={{\bf{x}}}_{1}+\delta ({{\bf{x}}}_{i}-{{\bf{x}}}_{1})$$ (26) The process can be shown graphically in Fig.

15a. This process will terminate until the convergence condition is satisfied. POWELL ALGORITHM The Powell method, introduced by M.J.D Powell54, can be viewed as a gradient-free

minimization algorithm in its basic form. It requires repeated line search minimizations, which may be carried out using univariate gradient-free, or gradient-based procedures. The procedure

can be described below: * 1. Initialization: select an accuracy _ϵ_ > 0, and a starting point X(0). Set the initial search directions S(_i_) to be the unit vectors along each coordinate

axis, for _i_ = 1, …, _n_. Set the main iteration counter to _k_ = 0, and the cycle counter _i_ = 1. * 2. Directional univariate minimization (take a 2D problem as an example and the graphic

schematic is shown in Fig. 15b). The process can be explained as follows: * Starting at X(0), perform a 1D optimization along along S(1) to find extremum X(1) * Starting at X(1), perform a

1D optimization along along S(2) to find extremum X(2) * Define S(3) to be in the direction connecting X(0) to X(2) * Starting at X(2), repeat steps 1 to 3 until the convergence condition is

met. * 3. Termination check: a satisfactory termination criterion is generally to stop whenever at any stage of the algorithm the change in the variables is less than the required accuracy

when ∥X(_n_+1) − X(_k_)∥ ≤ _ϵ_. According to Vassiliadis and Conejeros18, Powell gives a more elaborate termination check procedure. It is shown that the termination procedure is expected to

be more reliable, but it is more computationally expensive since the entire minimization problem has to be resolved at least twice until the tight convergence criteria are satisfied.

JUSTIFICATION OF MSE TO OPTIMIZE KWN MODEL In statistics, the MSE measures the average of the squares of the errors. As an estimator of given parameter \(\hat{p}\), the MSE of \(\hat{p}\)

with respect to an unknown parameter _p_ is defined as

$${\rm{MSE}}(p)={E}_{p}[{(\hat{p}-p)}^{2}]={E}_{p}[{(\hat{p}-{E}_{p}[\hat{p}])}^{2}]+{({E}_{p}[\hat{p}]-p)}^{2}=Va{r}_{p}(\hat{p})+Bia{s}_{p}{(\hat{p},p)}^{2}$$ (27) where _E__p_ is the mean

with respect to the parameters _p__i_. According to Eq. (27), it is shown that the minimization of MSE equals to the minimization of variance and bias. The minimization of bias shows how

good the estimator is in estimating the real values and the minimization of variance of parameters \(\hat{p}\) will make sure the \(\hat{p}\) converge to a certain value. The physicality of

the model is not determined by MSE value, but by the KWN framework (e.g.: mean-field assumption and CNT). THEORETICAL CALCULATION OF INTERFACIAL ENERGY IN KWN MODELS According to Ardell33,

within the framework of theories of KWN models, the analytical solution of interfacial energy between _γ_ and \({\gamma }^{{\prime} }\) can be expressed as $$\sigma =\frac{\Delta

{X}_{e}{G}_{{m}^{{\prime}{\prime}}}}{2{V}_{m} < z > }{\left(\frac{k}{\kappa }\right)}^{1/n}$$ (28) where Δ_X__e_ is the equilibrium concentration between _γ_ and \({\gamma }^{{\prime}

}\), \({G}_{{m}^{{\prime}{\prime}}}\) is the curvature of the molar Gibbs free energy at equilibrium concentration of _γ_, _V__m_ is the molar volume of \({\gamma }^{{\prime} }\) phase,

<_z_> = <_R_>/_R_* and <_R_> is the average radius and _R_* is the critical radius of precipitates, _k_ and _κ_ is a rate constant that incorporates the thermodynamic and

kinetic parameters of the alloy system and _n_ is the parameter obtained from fitting the particle size distributions. _n_ can be fitted as 3 by the experimental data. In this regard, the

dependence of _σ_ on temperature _T_ can be calculated as a function of temperature which can be expressed as $$\sigma =76.7\pm 15.5-(0.055\pm 0.017)\cdot T$$ (29) The unit of Eq. (29) is

mJ/m2. The comparison of theoretical calculation and optimized value is summarized in Table 6. DATA AVAILABILITY The authors declare that the data supporting the findings of this study are

available within the article on reasonable request from the corresponding author. CODE AVAILABILITY The source code of Scipy optimizate module in this work can be found in the website:

https://github.com/scipy/scipy/tree/main/scipy/optimize. Please contact the corresponding author for accessing the TC-python code. REFERENCES * Yu, T. et al. H-phase precipitation and its

effects on martensitic transformation in NiTi-Hf high-temperature shape memory alloys. _Acta Mater._ 208, 116651 (2021). Article  CAS  Google Scholar  * Sriram, H. et al. Formation

mechanisms of coprecipitates in Inconel 718 superalloys. _Acta Mater._ 249, 118825 (2023). Article  CAS  Google Scholar  * Yu, T., Anderson, P., Mills, M. & Wang, Y. Simulating

martensitic transformation in NiTi-Hf–effects of alloy composition and aging treatment. _Acta Mater._ 276,120038 (2024). * Du, Q. et al. Modeling over-ageing in Al-Mg-Si alloys by a

multi-phase CALPHAD-coupled Kampmann-Wagner Numerical model. _Acta Mater._ 122, 178–186 (2017). Article  CAS  Google Scholar  * Ury, N. et al. Kawin: an open source Kampmann-Wagner Numerical

(kwn) phase precipitation and coarsening model. _Acta Mater._ 255, 118988 (2023). Article  CAS  Google Scholar  * Wagner, R., Kampmann, R. & Voorhees, P. W. _Homogeneous Second-Phase

Precipitation_ Ch. 5, 309–408 (Wiley, 2013). https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527603978.mst0388. * Wu, K., Chen, Q. & Mason, P. Simulation of precipitation kinetics

with non-spherical particles. _J. Phase Equilib. Diff._ 39, 571–583 (2018). Article  CAS  Google Scholar  * Chen, Q., Wu, K., Sterner, G. & Mason, P. Modeling precipitation kinetics

during heat treatment with CALPHAD-based tools. _J. Mater. Eng. Perform._ 23, 4193–4196 (2014). Article  CAS  Google Scholar  * AB Thermo-Calc. The precipitation module (tc-prisma) user

guide 2024a https://thermocalc.com/support/documentation/ (2024). * Noble, B. & Bray, S. Use of the Gibbs-Thompson relation to obtain the interfacial energy of \({\delta }^{{\prime} }\)

precipitates in Al-Li alloys. _Mater. Sci. Eng. A_. 266, 80–85 (1999). Article  Google Scholar  * Bahrami, A., Miroux, A. & Sietsma, J. An age-hardening model for Al-Mg-Si alloys

considering needle-shaped precipitates. _Metall. Mater. Trans. A_ 43, 4445–4453 (2012). Article  CAS  Google Scholar  * Bardel, D. et al. Coupled precipitation and yield strength modelling

for non-isothermal treatments of a 6061 aluminium alloy. _Acta Mater._ 62, 129–140 (2014). Article  CAS  Google Scholar  * Bardel, D. et al. Cyclic behaviour of a 6061 aluminium alloy:

Coupling precipitation and elastoplastic modelling. _Acta Mater._ 83, 256–268 (2015). Article  CAS  Google Scholar  * Myhr, O. R., Grong, y. & Pedersen, K. O. A combined precipitation,

yield strength, and work hardening model for Al-Mg-Si alloys. _Metall. Mater. Trans. A_ 41, 2276–2289 (2010). Article  Google Scholar  * Miesenberger, B., Kozeschnik, E., Milkereit, B.,

Warczok, P. & Povoden-Karadeniz, E. Computational analysis of heterogeneous nucleation and precipitation in AA6005 al-alloy during continuous cooling DSC experiments. _Materialia_ 25,

101538 (2022). Article  CAS  Google Scholar  * Myhr, O. R., Grong, Ø. & Andersen, S. J. Modelling of the age hardening behaviour of Al-Mg-Si alloys. _Acta Mater._ 49, 65–75 (2001).

Article  CAS  Google Scholar  * Chen, H.-L., Chen, Q. & Engström, A. Development and applications of the TCAL aluminum alloy database. _Calphad_ 62, 154–171 (2018). Article  CAS  Google

Scholar  * Vassiliadis, V. S. & Conejeros, R. _Powell Method_ 2001–2003 (Springer, Boston, MA, USA, 2001). https://doi.org/10.1007/0-306-48332-7_393. * Du, Q., Holmedal, B., Friis, J.

& Marioara, C. D. Precipitation of non-spherical particles in aluminum alloys Part ii: numerical simulation and experimental characterization during aging treatment of an Al-Mg-Si alloy.

_Metall. Mater. Trans. A_ 47, 589–599 (2016). Article  CAS  Google Scholar  * Qian, F., Mørtsell, E. A., Marioara, C. D., Andersen, S. J. & Li, Y. Improving ageing kinetics and

precipitation hardening in an Al-Mg-Si alloy by minor Cd addition. _Materialia_ 4, 33–37 (2018). Article  CAS  Google Scholar  * Dutta, I. & Allen, S. A calorimetric study of

precipitation in commercial aluminium alloy 6061. _J. Mater. Sci. Lett._ 10, 323–326 (1991). Article  CAS  Google Scholar  * Edwards, G., Stiller, K., Dunlop, G. & Couper, M. The

precipitation sequence in Al–Mg–Si alloys. _Acta Mater._ 46, 3893–3904 (1998). Article  CAS  Google Scholar  * Sundararaman, M., Mukhopadhyay, P. & Banerjee, S. Some aspects of the

precipitation of metastable intermetallic phases in INCONEL 718. _Metall. Trans. A_ 23, 2015–2028 (1992). Article  Google Scholar  * Theska, F. et al. On the early stages of precipitation

during direct ageing of alloy 718. _Acta Mater._ 188, 492–503 (2020). Article  CAS  Google Scholar  * Theska, F., Stanojevic, A., Oberwinkler, B., Ringer, S. & Primig, S. On conventional

versus direct ageing of alloy 718. _Acta Mater._ 156, 116–124 (2018). Article  CAS  Google Scholar  * Theska, F., Stanojevic, A., Oberwinkler, B. & Primig, S. Microstructure-property

relationships in directly aged alloy 718 turbine disks. _Mater. Sci. Eng. A_ 776, 138967 (2020). Article  CAS  Google Scholar  * Drexler, A. et al. Experimental and numerical investigations

of the ″ and \({\gamma }^{{\prime} }\) precipitation kinetics in alloy 718 _γ_. _Mater. Sci. Eng. A_. 723, 314–323 (2018). Article  CAS  Google Scholar  * Zhang, F. et al. Simulation of

co-precipitation kinetics of \({\gamma }^{{\prime} }\) and _γ_″ in superalloy 718. In _Proc. 9th International Symposium on Superalloy 718 & Derivatives: Energy, Aerospace, and

Industrial Applications_ 147–161 (Springer, 2018). * Yu, T., Barkar, T., Lancelot, C.-M. & Mason, P. An ICME framework to predict the microstructure and yield strength of Inconel 718 for

different heat treatments. In _Proc. 10th International Symposium on Superalloy 718 and Derivatives_ (eds Ott, E. A. et al.) 415–427 (Springer Nature, Switzerland, Cham, 2023). * Ya-fang

Han, P. D. & Chaturvedi, M. C. Coarsening behaviour of _γ_ ″ and \({\gamma }^{{\prime} }\) -particles in Inconel alloy 718. _Metal Sci_. 16, 555–562 (1982). Article  Google Scholar  *

Devaux, A. et al. Gamma double prime precipitation kinetic in alloy 718. _Mater. Sci. Eng. A_ 486, 117–122 (2008). Article  Google Scholar  * Zhang, R. et al. Temperature-dependent misfit

stress in gamma double prime strengthened Ni-base superalloys. _Metall. Mater. Trans. A_ 51, 1860–1873 (2020). Article  CAS  Google Scholar  * Ardell, A. J. Temperature dependence of the

_γ_/\({\gamma }^{{\prime} }\) interfacial energy in binary Ni–Al alloys. _Metall. Mater. Trans. A_. 52, 5182–5199 (2021). Article  CAS  Google Scholar  * Schleifer, F., Holzinger, M., Lin,

Y.-Y., Glatzel, U. & Fleck, M. Phase-field modeling of _γ_/_γ_″ microstructure formation in Ni-based superalloys with high _γ_″ volume fraction. _Intermetallics_ 120, 106745 (2020).

Article  CAS  Google Scholar  * Ji, Y. et al. Predicting coherency loss of _γ_″ precipitates in IN718 superalloy. _Metall. Mater. Trans. A_ 47, 3235–3247 (2016). Article  CAS  Google Scholar

  * Shi, R. et al. Growth behavior of \({\gamma }^{{\prime} }\)/_γ_ ″ coprecipitates in Ni-base superalloys. _Acta Mater_. 164, 220–236 (2019). Article  CAS  Google Scholar  * Anderson, M.

J. et al. Mean-field modelling of the intermetallic precipitate phases during heat treatment and additive manufacture of Inconel 718. _Acta Mater._ 156, 432–445 (2018). Article  CAS  Google

Scholar  * Ahmadi, M. R. et al. Modeling of precipitation strengthening in Inconel 718 including non-spherical _γ_″ precipitates. _Model. Simul. Mater. Sci. Eng._ 25, 055005 (2017). Article

  Google Scholar  * Virtanen, P. et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. _Nat. Methods_ 17, 261–272 (2020). Article  CAS  PubMed  PubMed Central  Google

Scholar  * Varoquaux, G. et al. Scipy Lecture Notes: Release 2015.2. https://doi.org/10.5281/zenodo.31736 (Zenodo, 2015). * Ardell, A. J. & Ozolins, V. Trans-interface

diffusion-controlled coarsening. _Nat. Mater._ 4, 309–316 (2005). Article  CAS  PubMed  Google Scholar  * Frazier, Peter I. "Bayesian optimization." _Recent advances in

optimization and modeling of contemporary problems_. 255–278 (Informs, 2018). * Lifshitz, I. M. & Slyozov, V. V. The kinetics of precipitation from supersaturated solid solutions. _J.

Phys. Chem. Solids_ 19, 35–50 (1961). Article  Google Scholar  * Wagner, C. Theorie der alterung von niederschlägen durch umlösen (Ostwald-reifung). _Z. Elektrochem. Ber. Bunsenges. Phys.

Chem._ 65, 581–591 (1961). CAS  Google Scholar  * Cao, W. et al. Pandat software with PanEngine, PanOptimizer and PanPrecipitation for multi-component phase diagram calculation and materials

property simulation. _Calphad_ 33, 328–342 (2009). Article  CAS  Google Scholar  * Russell, K. C. Nucleation in solids: the induction and steady state effects. _Adv. Colloid Interface Sci._

13, 205–318 (1980). Article  CAS  Google Scholar  * Kashchiev, D. _Nucleation_ (Elsevier, 2000). * Svoboda, J., Fischer, F., Fratzl, P. & Kozeschnik, E. Modelling of kinetics in

multi-component multi-phase systems with spherical precipitates: I: Theory. _Mater. Sci. Eng. A_ 385, 166–174 (2004). Google Scholar  * Langer, J. S. & Schwartz, A. J. Kinetics of

nucleation in near-critical fluids. _Phys. Rev. A_ 21, 948–958 (1980). Article  CAS  Google Scholar  * Eshelby, J. D. The determination of the elastic field of an ellipsoidal inclusion, and

related problems. In _Proc. Royal Society of London Series A. Mathematical and Physical Sciences_ 241, 376–396 (1957). * Eshelby, J. D. The elastic field outside an ellipsoidal inclusion. In

_Proc. Royal Society of London Series A. Mathematical and Physical Sciences_ 252, 561–569 (1959). * Khachaturyan, A. G. _Theory of Structural Transformations in Solids_ (Courier

Corporation, 2013). * Gao, F. & Han, L. Implementing the Nelder-Mead simplex algorithm with adaptive parameters. _Comput. Optim. Appl._ 51, 259–277 (2012). Article  Google Scholar  *

Powell, M. J. D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. _Comput. J._ 7, 155–162 (1964). Article  Google Scholar  *

Nelder, J. A. & Mead, R. A simplex method for function minimization. _Comput. J._ 7, 308–313 (1965). Article  Google Scholar  * Powell, M. J. _A Direct Search Optimization Method That

Models the Objective and Constraint Functions by Linear Interpolation_ (Springer, 1994). * Powell, M. J. Direct search algorithms for optimization calculations. _Acta Numer._ 7, 287–336

(1998). Article  Google Scholar  * Powell, M. J. A view of algorithms for optimization without derivatives. _Math. Today Bull. Inst. Math. Appl._ 43, 170–174 (2007). Google Scholar  * Polak,

E. & Ribiere, G. Note sur la convergence de méthodes de directions conjuguées. _Rev. française d’informatique Rech. opérationnelle. Série rouge_ 3, 35–43 (1969). Google Scholar  *

Nocedal, J. & Wright, S. J. _Numerical Optimization_ (Springer, 1999). * Byrd, R. H., Lu, P., Nocedal, J. & Zhu, C. A limited memory algorithm for bound constrained optimization.

_SIAM J. Sci. Comput._ 16, 1190–1208 (1995). Article  Google Scholar  * Zhu, C., Byrd, R. H., Lu, P. & Nocedal, J. Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale

bound-constrained optimization. _ACM Trans. Math. Softw._ 23, 550–560 (1997). Article  Google Scholar  * Nash, S. G. Newton-type minimization via the Lanczos method. _SIAM J. Numer. Anal._

21, 770–788 (1984). Article  Google Scholar  * Schittkowski, K. On the convergence of a sequential quadratic programming method with an augmented lagrangian line search function. _Math.

Operat. Stat. Ser. Optim._ 14, 197–216 (1983). Google Scholar  * Schittkowski, K. The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search

function: Part 1: convergence analysis. _Numer. Math._ 38, 83–114 (1982). Article  Google Scholar  * Powell, M. J. A new algorithm for unconstrained optimization. In _Nonlinear Programming_

31–65 (Elsevier, 1970). * Steihaug, T. The conjugate gradient method and trust regions in large scale optimization. _SIAM J. Numer. Anal._ 20, 626–637 (1983). Article  Google Scholar  *

Gould, N. I., Lucidi, S., Roma, M. & Toint, P. L. Solving the trust-region subproblem using the Lanczos method. _SIAM J. Optim._ 9, 504–525 (1999). Article  Google Scholar  * Lenders,

F., Kirches, C. & Potschka, A. trlib: A vector-free implementation of the gltr method for iterative solution of the trust region problem. _Optim. Methods Softw._ 33, 420–449 (2018).

Article  Google Scholar  * Conn, A. R., Gould, N. I. & Toint, P. L. _Trust Region Methods_ (SIAM, 2000). Download references ACKNOWLEDGEMENTS We are grateful to Dr. Qing Chen for helping

revise the manuscript in this work. There is no applicable funding for this work. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Thermo-Calc Software Inc., 4160 Washington Rd, Suite 230,

Canonsburg, PA, 15317, USA Taiwu Yu, Adam Hope & Paul Mason Authors * Taiwu Yu View author publications You can also search for this author inPubMed Google Scholar * Adam Hope View

author publications You can also search for this author inPubMed Google Scholar * Paul Mason View author publications You can also search for this author inPubMed Google Scholar

CONTRIBUTIONS Taiwu Yu: conceptualization, data collection, code writing, data analysis, and paper writing. Adam Hope: conceptualization, data calibration, and manuscript modification. Paul

Mason: conceptualization. CORRESPONDING AUTHOR Correspondence to Adam Hope. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION

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algorithms to optimize the parameters in Kampmann–Wagner Numerical (KWN) precipitation models. _npj Comput Mater_ 10, 235 (2024). https://doi.org/10.1038/s41524-024-01415-2 Download citation

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