On Performance of Data Models and Machine Learning Routines for Simulations of Casting Processes

Various numerical simulation techniques based on Eulerian, Lagrangian, Arbitrary Lagrangian/Eulerian (ALE), Mixed Lagrangian/Eulerian (MiLE), and Multi Material ALE (MMALE) formulations have been proposed for the casting processes [4]. Many of the current CFD solvers for industrial casting processes are based on these rigorous mathematical and analytical methods, which utilize some conventional, dynamic, and hybrid mesh strategies.

However, the dynamic nature and transient generation of billets for continuous and semi-continuous casting processes are among the great difficulties for the simulation of these processes, since the location and the shape of the boundaries/interfaces are dynamic [4]. The numerical grids need to either be generated at the start of the simulations and activated later during the simulation or a dynamic mesh scheme needs to be implemented from the start of simulation [5].

In this research work, a fixed domain strategy has been utilized for the fluid-thermal CFD simulations where the entire mesh for the numerical domain has been created at start of simulation. Since the results of CFD simulations need to be postprocessed and filtered for the database building exercise, all casting scenarios have been carried out using identical mesh and geometric boundaries. However, variations of initial melt temperature, cooling and casting speed for different scenarios were implemented using varying input values.

The solidification phenomena during casting processes have their unique complex attributes where change of phase, grain formation and thermal evolution require special numerical attention. Conventional numerical approaches include simplified conduction-based energy transfer mechanism, explicit conduction-convection scheme, and enthalpy concepts which can be analytical calculated during the casting process. The most popular phase transition analytical equation for industrial casting applications is the Scheil-Gulliver equation [6] where the solid fraction is defined by melt flow, partition coefficient, solidus and liquidus temperatures.

In this research work, the directChillFoam open-source solver [7] has been adopted for our vertical casting application which can be used as an extension of OpenFOAM simulation package [8]. For the phase transformation herein, the continuum mixture formulation based on a single-region incompressible approach is adopted where the conservation of mass, momentum, energy, and species are considered, according to the model originally proposed in [9]. Thermal and solutal buoyancy effects are hence considered by means of additional source terms added to the conservation equations (i.e. Boussinesq formulation). The solver takes advantage of the PIMPLE algorithm, which allows for choice of either transient (i.e. PISO algorithm) or steady-state (i.e. SIMPLE algorithm) calculations [10]. Porosity and slurry flows are considered independently and the transition from one to another happens at the coherence fraction. For the thermal evolution during the casting process, the heat transfer coefficients (HTCs) are provided along the billet’s outer surface.

For the casting case study herein, the validation case in directChillFoam [7], based on the vertical direct-chill casting experiment by Vreeman et al. [11], was taken. A general description of the case setup is publicly available in the official documentation [12]. Only minor changes have been applied to the original case, mainly for allowing the parametrical study required to train/validate the data model with different combinations of initial melt temperature, cooling flow rate, and casting speed.

A wedge-shaped axisymmetric pseudo-2D domain, meshed with a structured grid is adopted. Here the element density along the axial length was slightly increased with respect to the original setup and a linear grading employed to avoid sudden changes in the mesh element size. (see Fig. 1). Material properties correspond to the binary alloy Al-6 wt% Cu. The phase-change obeys to temperature-dependent tabular values of the melt fraction (e.g. obtained from CALPHAD software package) and the solute properties are set to follow the Lever-rule. The HTCs at the mould-melt interface (primary cooling) were locally averaged, depending on the value of the solid fraction. The heat transfer at the billet-water interface (secondary cooling) was introduced as tabular values obtained from the correlation proposed in [13].

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The simulations were run in transient mode for ensuring a correct description of the process physics at the steady process conditions. The runtime was set to 2000 s, more than doubling the period required for reaching quasi-steady state. It should be noted that some conditions show stronger permanent fluctuating behaviour than others due to the chaotic nature of the coupled phenomena involved. For that reason, time-averaged results between over the last 1000 s of the temperature and melt-fraction magnitudes were saved for all cell-centers using ParaView software (i.e. “temporal statistics” filter). These values were then exported in a CSV format, along with their spatial information (i.e. x‑ and z‑coordinates) for allowing further postprocessing.

The increasing use of reduced order and real-time models and their crucial role in modelling and control of material processes for twinning and shadowing frameworks have inspired much research in recent years, especially for dynamic processes like casting. Although these data models are based on databases which are made of existing and live data for manufacturing processes, their level of accuracy for engineering applications has thoroughly been scrutinized by researchers [14, 15]. Some of the most popular data solvers for material processes application is the proper orthogonal decomposition (POD), singular value decomposition (SVD), and proper generalized decomposition (PGD), which are based on mathematical eigen-base schemes [15]. Data solvers based on these techniques can be defined as decomposition solvers (spatial and time) where algebraic approximations of system responses can be created to allow for fast reconstruction of system characteristics. The initial database for these reduced order models (ROM) can be originated from experimental work, numerical simulations, or mined data where the variations of pre-defined parameters are considered to fill the so-called snapshot matrix. For the POD and SVD the factorization of the responses can be written in mathematical form as [14, 15]:where x is the eigenvector of A, R is the full or reduced system responses in the database, the de-composed matrices of U and V^T are orthogonal matrices related to spatial and temporal decomposition and Σ is the eigen value matrix.

Among other popular data solvers are the two-stage kriging and principal component analyses (PCA) solvers where the primary dimensional-reduction task is performed by eliminating less important dimensions while keeping data characteristics by maintaining important data dimensions [16]. Other conventional regression and symbolic regression methods are also among most established techniques for data solvers where variations of input data can be ordered based on their impacts. Symbolic regression (SR) and genetic algorithm symbolic regression (GASR) are among more sophisticated regression techniques where regression analysis can be performed within a multi-dimensional search space [16]. The clustering method is also one of the famous data solvers that work based on data-points grouping and clustering by detecting some common data features. It can be defined as an unsupervised learning scheme to trace patterns in available data for better data insights.

Different data science schemes based on mathematical decomposition, projection and classification techniques have already been developed for fast predictor\corrector models. However, in recent years by advances of data science techniques and ML routines, attention has been given to the new generation of real-time models with very fast response times. To elaborate on the development of these real-time models for industrial applications, the governing equation for a simple case of transient mechanical vibration can be considered as:where [K], [M], [C] and [u] are stiffness, mass, damping and displacement matrices of the vibrating system. If the tempo-spatial characteristic of the vibrating system is decomposed, the frequency domain system characteristic equation can be written as;where \(\omega_{n}\) and \(\dot{u}_{n}\) are the system natural frequencies and Fourier coefficients. Thus, the final governing equation in frequency domain can be written as \(\left[\widehat{K}_{f}\right]\left[u\right]=\widehat{F}[t]\) where \(\left[\widehat{K}_{f}\right]\) and \(\widehat{F}[t]\) are damped dynamic stiffness and force spectral decomposed matrices. However, since the characteristic equation of the system herein is calculated using the system properties (i.e. geometry, material properties, boundaries), the computational time and efforts to solve the equations are substantial. The alternative data driven governing equation for the response of the vibrating system can be considered as [16]:

Where [C] is the estimated system characteristics (using data techniques), [X(x,t)] is the input values (from snapshot matrix cases) and \([Y\left(x,t\right)]\) can be interpreted as desired response at specific values for variables. Here these data techniques are used which are based on the same mathematical decomposition and eigenvalue concepts to form the system characteristic equation. However, data from calculated or measured responses of the system for the range of variables are used to formulate the system matrices and characteristic equations. The snapshot matrices are used to define potential scenarios for database building using variation of process parameters. The calculated/measured data from these scenarios can be utilized to form a response database which holds basic characteristic information about the system. If data eigen solvers like SVD or POD are used to solve system responses for desired parameters, the optimal expansion of the matrix can be written as:

Where U represents the spatial eigenvector decomposition, Σ is the singular values diagonal matrix (e.g., eigenvalues) and V^Trepresent temporal decomposition for transient system responses. For casting processes where the thermal evolution is driving the solidification and grain structure formation, the temperature prediction can be carried out using:where T_k are temperatures at process time t. For these processes, the data solver needs to be accompanied with an appropriate data interpolator for accurate predictions.

The application of real-time models for material processes like casting can be examined to investigate the accuracy and reliability of these techniques for the rapid design and real-time optimizations. For this research work, the concepts and challenges of creating reliable real-time models for casting processes were undertaken and a vigorous framework has been set up to create enough data for database building exercise. The development work involves key steps and milestones to guarantee the accuracy, reliability, and relevance of the models as:

Although, for the full multi-physical and multi-scale aspects of the casting processes, full fluid-thermal-mechanical and coupled macrostructure simulations are required, real-time models are only developed based on fluid-thermal simulations using the CFD solver in this research work.

The combination of an appropriate data solver and a suitable interpolator should be considered for dynamic processes like casting. In the research work herein, different data solvers including Eigen (e.g. POD, SVD), regression, clustering, and FFT solvers along with Kriging, InvD (inverse distance), and radial basis function (RBF) data interpolators have been employed to build efficient real time solvers for casting processes. The combination of eigen solvers with Kriging interpolators has been successfully used for predictions in dynamic and transient processes (i.e., time-dependent predictions). On the other hand, regression solvers with both normalzation and standardization schemes have been used for data with complex patterns.

Data interpolators like Kriging can generally provide accurate predictions even at unobserved points of interest, while it is capable of estimating uncertainty associated with these predictions. Consider grid points in a simulation as N (x, y, z, t, T), where x, y, z are the spatial coordinates and t and T represent the time and calculated temperatures at those points. For Kriging to predict the temperature value at unobserved desired point N_k (x_k, y_k, z_k, t_k, T_k), it can estimate the value as:where \(\lambda_{i}\) are the Kriging weights, \(N(x_{i},y_{i},z_{i},t_{i},T_{i})\) are the calculated temperature values at nearby points at that time and n is the considered number of nearby calculated points.

The industrial applications of data real-time models for dynamic processes like casting have their unique benefits and challenges. From the start, for the data model building exercise, the poor quality of data and their under-representative nature for the multi-dimension search space (i.e. defining variations of process parameters) can lead to errors and inaccuracies. The balance of data density and their spatial distributions/resolutions can also affect the model predictions, especially for processes like casting with high data gradient (e.g. high cooling/heating curves). In addition, the selection of an appropriate data solver and its associated data interpolar can greatly affect the performance of the data model for material processes. The issues of accurate fitting into process initial conditions (e.g. initial melt temperature) along with precise fitting to zones with high cooling/heating curves are among challenging aspects of model building exercises. The right calibration and data training schemes using ML routines are among other challenges for the building of an accurate data model. Finally, the proper validation of the real-time models for normal, near boundary, and extreme cases along with their associated computational time are also among cumbersome tasks for model building scientists.

In the work herein, using efficient sampling techniques, snapshot matrices are defined in a way to represent the basic process scenarios in a balanced way. The combination of process parameter variations representing the spatial size of the search space (i.e. parameters variation limits) have been defined for basic scenarios. Furthermore, several DOEs which represent the normal, near boundary and extreme process scenarios were utilized for strict validations of data models for real world scenarios. The data postprocessing for real-time models can also be performed in 1D, 2D, or 3D manners, where results from DOEs and data models are compared along lines, within plane contours or bodies. For this study, only the 1D and 2D contour results were compared and validated using DOEs scenarios. Figure 2a shows the vertical casting machine with the round shape cast billet, while Fig. 2b shows the rotated 2D temperature contour for a validation scenario where a snake shape path is defined to record temperatures at computational points. The challenge here is that, since the data produced by CFD solver is best optimized for the smallest solver matrices’ sizes, the data order might not be appropriate for the data model. The CFD solver data order might generate data strings with a very high gradient (jumping from end of billet to top of melt pool), which is unfeasible for the data model. In this sense, extra postprocessing is required to prepare and transform the data for appropriate use of data models where more moderate gradients along with less sharp change/discontinuities are observed.

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Digital shadowing of the casting process allows us to evaluate various scenarios (sets of process parameters) and the consequent impact on the quality of e.g. the microstructure. At the same time, the risk of causing a bleed-out or freeze-in can be assessed. While CFD solvers need hours to simulate each scenario, a reduced order model allows for fast screening of various process parameter combinations. Furthermore, during the production the ROM can suggest optimized parameters to workers to ensure their safety and high product quality. These human-machine interactions are currently being designed and developed in the research project ‘opt1mus’ and are shown in Fig. 3. Hence, end users can request advice for a specific alloy/format combination and a desired microstructure quality via a web interface which has been implemented using the “PhysicsWorks” framework by “Solidification” (https://​physicsworks.​io/​). This request triggers the simulation and training of a ROM to suggest suitable parameters before and real time guidance during casting. The model is consequently improved from correlations between the actual process conditions and the measured microstructure to serve improved predictions for the next requests. Figure 3 shows the schematic view of the whole framework.

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