This talk presents an overview of computational methods frequently used for viscoplastic flow and inverse problems, highlighting their evolution from industrial applications to environmental modeling. Initially, we demonstrated that while regularisation techniques are commonly employed in simulations of duct flows involving multiple viscoplastic fluids, they become inadequate when addressing critical yield limits or no-flow conditions. Instead, we showed the augmented Lagrangian method to be superior for accurately capturing these challenging scenarios, as exemplified by multi-layer lubrication and exchange flows. Building upon this foundation, we later developed a novel optimisation approach employing the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). This new technique overcame the computational limitations of traditional augmented Lagrangian (ADMM) methods, achieving linear convergence rates and eliminating the heuristic penalty parameter selection. Extensive numerical experiments demonstrated that this dual-based algorithm significantly improved accuracy and computational efficiency in identifying yield surfaces and unyielded regions. Most recently, we extended these computational optimisation methods to the challenging inverse problem of recovering ice thickness and basal slip conditions in glacier modelling. By employing an augmented Lagrangian optimisation framework, we successfully inferred dual parameters from surface measurements alone. Through this journey, from fundamental yield-limit analyses in ducts, through accelerated optimisation algorithms, to environmental applications in cryosphere dynamics, this talk underscores the continuous evolution and broad applicability of advanced computational methods in fluid mechanics.
Dr Miguel Moyers-Gonzalez is an Associate Professor in the School of Mathematics and Statistics of University of Canterbury, NZ. Previously he was a Lecturer in the Department of Mathematical Sciences at Durham University, UK. He spent two years in the Laboratory of Applied Mathemathics in University of Montreal as a CRM Postdoctoral Fellow. He did his postgraduate studies in the Department of Mathematics and the Institute of Applied Mathematics in UBC, Canada. His Doctoral Thesis was awarded the 2005 Cecil Graham Doctoral Dissertation Award from the Canadian Applied and Industrial Mathematics Society (CAIMS). He was Guest Editor for the Special Volume for the Seventh Viscoplastic Fluids Workshop: From Theory to Applications (VPF7) in the Journal of Non-Newtonian Fluid Mechanics. Miguel’s research field is the mathematical and computational modeling of problems in Continuum Mechanics.