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Industrial mathematics and practical asymptotics


Before explaining the work plan, it is worth describing the concepts that unify it. Throughout, the basic starting point is the mathematical modelling of complex multiphysical phenomena occurring in industrial processes or in nature, with a view to providing a quantitative description of mechanisms or operation. This is done through the formulation of systems of partial differential equations (PDEs) describing the conservation of mass, heat and momentum, coupled to thermodynamic equilibrium conditions. In general, the derived models can be expected to be three-dimensional and time dependent, although prudent use of asymptotic methods can be expected to identify disparate length and time scales, leading to reduced models that do not sacrifice any of the physics that was present in the original problem, yet are much cheaper to compute numerically. This approach is often termed practical asymptotic, whereby the governing equations are non dimensionalized and systematically simplified to obtain a formulation whose results should agree also quantitatively with those of the original model. For this project, this approach is particularly important, since the complexity of the systems of PDEs that arise renders conventional numerical models, based on 3D computational fluid dynamics (CFD), ineffective, because of the length of computation times and the fact that parameter studies are necessary over a wide range of operating conditions, material properties and geometry dimensions. The work plan acknowledges, however, that some problems will not be as amenable as others to an asymptotic approach, and that it is therefore necessary to combine asymptotic and numerical methods. For this purpose, the commercially available finite-element software Comsol Multiphysics will be used; the applicant already has around fifteen years of experience using it, and has implemented a large number of models with it, even for problems that do not form the basis of this project. (AU)

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Scientific publications (9)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ASSUNCAO, M.; VYNNYCKY, M.; MITCHELL, S. L.. n small-time similarity-solution behaviour in the solidification shrinkage of binary alloy. EUROPEAN JOURNAL OF APPLIED MATHEMATICS, v. 32, n. 2, p. 199-225, . (18/07643-8, 16/12678-0)
VYNNYCKY, MICHAEL. Applied Mathematical Modelling of Continuous Casting Processes: A Review. METALS, v. 8, n. 11, . (18/07643-8)
REDDY, G. M. M.; NANDA, P.; VYNNYCKY, M.; CUMINATO, J. A.. An adaptive boundary algorithm for the reconstruction of boundary and initial data using the method of fundamental solutions for the inverse Cauchy-Stefan problem. COMPUTATIONAL & APPLIED MATHEMATICS, v. 40, n. 3, . (16/19648-9, 18/07643-8)
DEVINE, K. M.; VYNNYCKY, M.; MITCHELL, S. L.; O'BRIEN, S. B. G.. Analysis of a model for the formation of fold-type oscillation marks in the continuous casting of steel. IMA JOURNAL OF APPLIED MATHEMATICS, v. 85, n. 3, p. 385-420, . (18/07643-8)
REDDY, GUJJI MURALI MOHAN; SEITENFUSS, ALAN B.; MEDEIROS, DEBORA DE OLIVEIRA; MEACCI, LUCA; ASSUNCAO, MILTON; VYNNYCKY, MICHAEL. A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D. ALGORITHMS, v. 13, n. 10, . (16/19648-9, 18/07643-8, 17/11428-2)
VYNNYCKY, M.; LACAZE, J.. On the modelling of joint formation in dissolutive brazing processes. JOURNAL OF ENGINEERING MATHEMATICS, v. 116, n. 1, p. 73-99, . (18/07643-8)
VYNNYCKY, MICHAEL; MCKEE, SEAN; MEERE, MARTIN; MCCORMICK, CHRISTOPHER; MCGINTY, SEAN. Asymptotic analysis of drug dissolution in two layers having widely differing diffusivities. IMA JOURNAL OF APPLIED MATHEMATICS, v. 84, n. 3, p. 533-554, . (18/07643-8)
MCKEE, S.; VYNNYCKY, M.; CUMINATO, J. A.. An elementary diffusion problem, Laplace transforms and novel mathematical identities. Journal of Computational and Applied Mathematics, v. 353, p. 113-119, . (18/07643-8)

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