Stochastic multi-fidelity surrogate modeling of dendritic crystal growth

Abstract

In this work, we propose a novel framework coupling state-of-the-art multi-fidelity Gaussian Process modeling techniques with input-space warping for a cost-efficient construction of a stochastic surrogate model. During model generation, we achieve high computational efficiency by combining a large number of cheap estimates (low-fidelity model) with only a few, computationally expensive, high-fidelity measurements. We base the fidelity hierarchy on coarse-grid approximations of high-fidelity numerical simulations and show its successful application within the proposed framework. Utilizing coarse-grid approximations for multi-fidelity modeling is attractive for many practical applications, since it often allows for multi-fidelity data generation with a single simulator. As benchmark, we apply this framework to generate a surrogate model for crystal growth velocities in directional dendritic solidification. The derivation of a relation between this tip velocity and process parameters, such as undercooling, thermal diffusivity, capillarity, and capillary anisotropy, has been in the focus of research for decades due to its important role on microstructure evolution during solidification. It defines the thermo-mechanical properties of the solidified part and influences its behavior in subsequent manufacturing steps.As data generator, we use the open-source simulation framework ALPACA, applying a conservative sharp-interface level-set model. We assess the accuracy of the multi-fidelity tip velocity model by using cross-validation techniques. Compared to single-fidelity models purely based on high-fidelity data, our approach improves prediction accuracy significantly but only requires a little cost overhead for data generation. The stochastic nature of the resulting surrogate model allows for quantifying the uncertainty associated with predictions. This motivates the application of the model in Bayesian-optimization algorithms for inverse problems. Also, it may serve as input for microstructure simulations which rely on accurate relations between local solidification velocities and process parameters such as undercooling to predict grain-scale crystalline structures and which need material-dependent model calibration.