Abstract
This article presents an integrative, theoretically rigorous, and methodologically detailed exposition on the design, analysis, and numerical implementation of advanced control strategies for continuous casting processes, emphasizing model predictive control for nonlinear parabolic partial differential equation (PDE) systems, lattice Boltzmann method (LBM) based fluid–thermal simulation, and high-performance computing (HPC) implementations on graphics processing units (GPUs). By synthesizing methodological advances from state-of-the-art control theory applied to unsteady PDEs (Yu et al., 2023; Wang et al., 2016; Yu et al., 2018) with kinetic-based fluid modelling and thermodynamics captured by lattice Boltzmann frameworks (d’Humières et al., 2002; He et al., 1998; Lallemand & Luo, 2003), and the practical acceleration strategies using CUDA-enabled GPU computing (Micikevicius, 2009; NVIDIA, 2010; Mudigere, 2009), the paper articulates a comprehensive pipeline: from mathematical problem formulation and discretization strategy, through controller design and stability considerations, to efficient implementation patterns that respect memory access, parallelism, and numerical accuracy on modern heterogeneous architectures. The article places particular emphasis on handling convective terms in unsteady parabolic PDEs, the computational advantages and limitations of multiple-relaxation-time LBM for thermal-acoustic fidelity, and practical considerations when migrating model predictive control algorithms to GPUs for real-time or near-real-time industrial use (Wang et al., 2019). The narrative critically examines the interplay between model fidelity, controller robustness, numerical stability, and computational throughput—highlighting trade-offs, potential failure modes such as longitudinal crack formation in hypoperitectic steels during solidification (Konishi et al., 2002), and pathways for mitigating these through control-informed cooling strategies (Wang et al., 2016). The synthesis culminates in an extended methodological blueprint suitable for researchers and practitioners seeking to develop publication-ready, production-grade simulation-control systems for continuous casting and analogous thermofluid processes.