# Research Activities

## [Emerging Research Directions]

- Mathematical and Computational Modelling of Electrochemical Systems
- Computational Nonequilibrium Thermodynamics
- Homogenization and Multiscale Methods
**Probing Sharpness of Fundamental Estimates in Hydrodynamics**-- This research effort has opened a "computational window" into some basic questions of the mathematical fluid dynamics. More specifically, we have been interested in the problem of the sharpness of certain fundamental estimates concerning the maximum possible growth, both instantaneously and in finite time, of various quadratic quantities in hydrodynamics such as the enstrophy (the L2 norm of the vorticity) and the palinstrophy (the L2 norm of the vorticity gradient). By "sharpness" we mean the existence of flow fields which may saturate a given analytical bound, in the sense of exhibiting (possibly up to a numerical prefactor) the growth predicted by the estimate. Following an earlier idea of our collaborator, Professor Doering of the University of Michigan, such extreme flows are systematically sought via numerical solution of suitable constrained variational optimization problems. While for the moment we have focused on problems formulated in one and two spatial dimensions (1D and 2D), analogous questions concerning the three-dimensional (3D) Navier-Stokes equation are intrinsically related to the problem of the finite-time singularity formation in this system, an issue recognized by the Clay Mathematics Institute as one of the "Millennium Problems" for the mathematics community. Although the 1D Burgers and 2D Navier-Stokes equations on periodic domains are known to possess global in time regularity, the corresponding upper bounds on the growth of various quantities are obtained with similar techniques as for the 3D Navier-Stokes case, hence the question of their sharpness is in fact quite important for the big picture. We initiated a research program aiming at understanding which of those bounds can be realized by actual flow fields and a number of results concerning the 1D and 2D cases have been reported in Ayala & Protas (2011a, 2011b, 2013a, 2013b). In systematically computing extreme flow events designed to saturate certain key analytical estimates we have thus established a bridge between theory and computation, in the sense that the insights obtained in this way can be, and in fact have already been, used to sharpen some of the analytical estimates (Pelinovsky, 2012). Moreover, in a related study (Farazmand, Kevlahan & Protas, 2011) we used methods of variational optimization to address the problem of realizability of the Kraichnan-Leith-Batchelor theory of the 2D turbulence, resolving in this way an open question of theoretical hydrodynamics. In addition to the aforementioned journal and conference publications, this investigation is also the subject of Diego Ayala's Master's thesis and his upcoming Ph.D. dissertation (to be defended in 2014). Furthermore, among many other research institutions, in March 2012 Dr. Protas was invited to give a seminar on this topic at College de France in the prestigious "Seminaire de Mathematiques Appliquees".**Optimal Identification of Constitutive Relations**-- In this investigation we have developed an efficient computational framework, relying on a variational formulation, which allows one to optimally identify elements of the structure of a governing system (such as, e.g., the dependence of the nonlinear fluxes on the state variables, etc.) based on some available measurements. In contrast to earlier studies, our formulation is developed in the continuous (infinite-dimensional) setting and hence its key ingredients are independent of the particular discretization used and subject to a minimum number of assumptions. Initially begun in response to a question brought up by an industrial partner, this research effort has evolved to acquire a more fundamental character. This inverse problem was framed in terms of the solution of a PDE optimization problem and, although adjoint-based methods are rather well known in such context, for the purpose of the solution of our problem the adjoint-based approach had to be significantly generalized. This was a consequence of the fact that the sensitivities (gradients) had to be computed with respect to the structure of certain terms in the equations, rather than the initial or boundary data as is usually the case. This unusual structure of the problem, in turn, required us to resolve a number of nontrivial questions at the level of numerical analysis and scientific computing. In addition to the journal and conference publications (Bukshtynov, Volkov & Protas, 2011; Bukshtynov & Protas 2012, 2013), this work was also the subject of Dr. Bukshtynov Ph.D. dissertation for which in 2013 he was awarded the prestigious Cecil Graham Dissertation Award of the Canadian Applied and Industrial Mathematics Society. Although our framework was initially conceived in the context of systems described by PDEs (where it was used to identify the dependence of the transport coefficients on the state variables such as, e.g., the temperature-dependent viscosity), it has recently found a number of promising extensions to problems described by systems of ordinary differential equations. One such emerging application concerns the problem of optimal identification of inertial manifolds and turbulence closures in Galerkin reduced-order models (Protas, Noack & Morzynski, 2013). As a novel approach to a classical problem, there are also interesting applications of this framework to a class of inverse problems in vortex dynamics.-
**Topics in Vortex Dynamics**-- Although the problems of vortex dynamics are among the most classical in fluid mechanics and have been studied for a long time, Dr. Protas has made a number of contributions in this area. They concern the analytic and computational aspects of the steady Euler flows characterized by finite-area vortex regions. One of the goals of this research program has been the study of the conditions under which continuous families of such vortex equilibria may exist (Gallizio et al, 2010). This investigation, which involved a combination of mathematical analysis and large-scale computations, represents a step towards resolving an outstanding open question in this field, the so-called "accretion conjecture", which stipulates that, in a given flow geometry, finite area vortex equilibria additionally satisfying the Kutta condition exist if and only if there also exist the corresponding point-vortex equilibria. Other research problems we have addressed concern the development of a general computational framework for studying vortex stability (Elcrat & Protas, 2013) and finding optimal vortex states (Protas 2012). The key mathematical challenge recurrent in these research problems was that Euler flows with finite-area vortices are described by systems of partial differential equations of the free-boundary type and, since the solution of each of the aforementioned problems required the calculation of a suitably-defined Jacobian, this had to be accomplished using advanced methods of shape differentiation. These milestones were extensions of Dr. Protas' earlier work on the control of point-vortex systems which was surveyed in an invited review paper (Protas, 2008). -
**Optimal Control of Partial Differential Equations in Fluid Dynamics and Related Fields**-- These contributions were largely made in response to the research problems raised by Dr. Protas' long-standing industrial partner, General Motors of Canada. Although these problems originated in industrial applications, their resolution triggered a number of advances at the level of fundamental science involving optimization, numerical analysis and scientific computing. The key scientific challenge which had to be overcome concerned the numerical solution of PDE-constrained optimization problems for systems defined on variable domains, such as different shape-optimization problems for PDEs (Peng, Niakhai & Protas, 2013; Protas & Liao, 2008). The task became further complicated in the case of multiphysics and multiscale systems such as those arising in phase-change and solidification applications. A novel computational approach to the solution of a class of generalized inverse Stefan problems was developed in Volkov et al. 2009 and Volkov & Protas 2009. -
**Characterization of Flows Past Obstacles in Unbounded Domains**-- This research effort revolves around a number of outstanding open questions concerning the properties of flows past obstacles in unbounded domains. Despite the importance of these problems for many applied fields such as, for example, aerodynamics, several questions remain unresolved which has to do with certain subtleties characterizing the behavior of such flows at large distances from the obstacle. One class of questions concerns efficient techniques for the calculation of hydrodynamic forces in such settings, and we have addressed some of these issues in Protas (2007), whereas an apparent paradox related to the calculation of forces based on the vorticity impulse formula in steady flows was resolved in Protas (2011). The assumption of steadiness leads to a number of complications when studying flows on unbounded domains. Some open questions concerning the behavior of the steady Oseen flows in the inviscid limit, which go as far back as the 1950s, were settled in Gustafsson & Protas (2013). An innovative numerical technique for the solution of steady Navier-Stokes flows in unbounded domains was developed and thoroughly validated in Jonathan Gustafsson's Ph.D. thesis. While standard approaches to this problem typically involve truncation of the infinite domain to a finite "computational box" (which requires the imposition of some artificial boundary conditions), our new method combined a spectrally-accurate technique defined on the original unbounded domain with explicit control of the solution's behavior at infinity. - Ionut Danaila (Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, France)
- Charles Doering (Departments of Physics, Mathematics, and Complex Systems, University of Michigan, Ann Arbor, USA)
- Alan Elcrat (Department of Mathematics and Statistics, Wichita State University, USA)
- Angelo Iollo (Institut de Mathématiques, Université Bordeaux I, France)
- Nicholas Kevlahan (Department of Mathematics and Statistics, McMaster University, Canada)
- Bernd Noack (Institut PPRIME, Université de Poitiers, France)
- Luca Zannetti (Aerospace Engineering Department, Politecnico di Torino, Italy)
- Early Researcher Award
- Natural Sciences and Engineering Research Council of Canada (Discovery, Collaborative Research and Development, and Research Tools & Instruments grants)
- Automotive Partnership Canada
- AUTO21
- Ontario Centres of Excellence -- Centre for Materials and Manufacturing
- McMaster Centre for Automotive Materials and Manufacturing
- SHARCNET (Graduate Fellowship)
- Foreign Affairs and International Trade Canada (Going Global Innovation Program)
- General Motors of Canada

## [Main Research Themes (past & present)]

## [Current Research Collaborations]

## [Research Funding]

**Generous support provided by these institutions for our research is gratefully acknowledged.**