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The present paper describes the development of a novel and comprehensive computational framework to simulate solidification problems in materials processing, specifically casting processes. Heat transfer, solidification and fluid flow due to natural convection are modeled. Empirical relations are used to estimate the microstructure parameters and mechanical properties. The fractional step algorithm is modified to deal with the numerical aspects of solidification by suitably altering the coefficients in the discretized equation to simulate selectively only in the liquid and mushy zones. This brings significant computational speed up as the simulation proceeds. Complex domains are represented by unstructured hexahedral elements. The algebraic multigrid method, blended with a Krylov subspace solver is used to accelerate convergence. State of the art uncertainty quantification technique is included in the framework to incorporate the effects of stochastic variations in the input parameters. Rigorous validation is presented using published 1 Corresponding Author

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This paper analyzes the effects of input uncertainties on the outputs of a three dimensional natural convection problem in a differentially heated cubical enclosure. Two different cases are considered for parameter uncertainty propagation and global sensitivity analysis. In case A, stochastic variation is introduced in the two non-dimensional parameters (Rayleigh and Prandtl numbers) with an assumption that the boundary temperature is uniform. Being a two dimensional stochastic problem, the polynomial chaos expansion (PCE) method is used as a surrogate model. Case B deals with non-uniform stochasticity in the boundary temperature. Instead of the traditional Gaussian process model with the Karhunen-Loève expansion, a novel approach is successfully implemented to model uncertainty in the boundary condition.The boundary is divided into multiple domains and the temperature imposed on each domain is assumed to be an independent and identically distributed (i.i.d) random variable. Deep neural networks are trained with the boundary 1 Corresponding Author temperatures as inputs and Nusselt number, internal temperature or velocities as outputs. The number of domains which is essentially the stochastic dimension is 4, 8, 16 or 32.Rigorous training and testing process shows that the neural network is able to approximate the outputs to a reasonable accuracy. For a high stochastic dimension such as 32, it is computationally expensive to fit the PCE. This paper demonstrates a novel way of using the deep neural network as a surrogate modeling method for uncertainty quantification with the number of simulations much fewer than that required for fitting the PCE, thus, saving the computational cost.

Die casting is a type of metal casting in which a liquid metal is solidified in a reusable die. In such a complex process, measuring and controlling the process parameters are difficult. Conventional deterministic simulations are insufficient to completely estimate the effect of stochastic variation in the process parameters on product quality. In this research, a framework to simulate the effect of stochastic variation together with verification, validation, and uncertainty quantification (UQ) is proposed. This framework includes high-speed numerical simulations of solidification, microstructure, and mechanical properties prediction models along with experimental inputs for calibration and validation. Both experimental data and stochastic variation in process parameters with numerical modeling are employed, thus enhancing the utility of traditional numerical simulations used in die casting to have a better prediction of product quality. Although the framework is being developed and applied to die casting, it can be generalized to any manufacturing process or other engineering problems as well.

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In this paper, we demonstrate the combination of machine learning and three dimensional numerical simulations for multi-objective optimization of low pressure die casting. The cooling of molten metal inside the mold is achieved typically by passing water through the cooling lines in the die. Depending on the cooling line location, coolant flow rate and die geometry, nonuniform temperatures are imposed on the molten metal at the mold wall. This boundary condition along with the initial molten metal temperature affect the product quality quantified in terms of micro-structure parameters and yield strength. A finite volume based numerical solver is used to determine the temperature-time history and correlate the inputs to outputs. The objective of this research is to develop and demonstrate a procedure to obtain the initial and wall temperatures so as to optimize the product quality. The nondominated sorting genetic algorithm (NSGA-II) is used for multi-objective optimization in this work. The number of function evaluations required for 1 Corresponding Author

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