Giorgio Bornia. Research statement. Primary interests

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1 Giorgio Bornia Research statement 2500 Broadway and Boston Lubbock, TX Primary interests My main research interests lie within the following areas: Optimal control for partial differential equations Fluid dynamics, multi-physics problems Finite element methods, multigrid and domain decomposition methods Scientific computing, design of finite element software I am interested in the study of optimal control problems constrained by partial differential equations both from a theoretical and from a numerical point of view. I believe that a deep insight into the study of optimal control problems requires a complete treatment encompassing theoretical, numerical and computational aspects. First of all, it is necessary to understand the conditions that guarantee well-posedness in an infinite-dimensional framework as well as convergence of numerical discretizations. Then, it is of great importance to study how a solution to these problems can be effectively achieved with state-of-the-art algorithms and computational tools. In this way, a tight link between mathematical formulations and issues arising from applications can be established. In my PhD dissertation I presented a new approach to the boundary optimal control of the incompressible steady MHD equations. Optimal control problems for the MHD equations are of great interest due to a wide range of applications in science and engineering. The approach I proposed is based on the introduction of divergence-free lifting functions for the Dirichlet boundary conditions of the magnetic field. With this approach, boundary control problems can be formulated in terms of extended distributed problems, bringing theoretical and computational advantages. In fact, compatibility conditions involving the boundary data and controls are automatically satisfied and do not need to be explicitly enforced. Also, boundary controls can be sought in natural half-integer Sobolev spaces, without stronger regularity requirements, and a standard distributed control implementation can be achieved. I considered a mathematical formulation of the boundary optimal control problem in terms of the minimization of a velocity-tracking cost functional constrained by the steady incompressible MHD equations. For the state equations, I have shown existence of a solution by using an abstract setting for the analysis of nonlinear mixed problems of the Navier-Stokes type. Afterwards, the existence of a global minimizer for the optimal control problem can be shown via the usual direct method in the calculus of variations. Following the Lagrange multiplier approach, I derived an optimality system, whose solutions are candidate solutions for the optimal control problem. In order to achieve the numerical solution of the optimality system, I considered a classical Taylor-Hood finite element discretization for mixed problems based on the LBB theory. I implemented a gradient-type algorithm with the purpose of diminishing the numerical 1/5

2 oscillations induced by a decoupling of the state, adjoint and control equations. I developed a finite element object-oriented code with multigrid solver to obtain the solution of the optimality system with the proposed approach. To that purpose, I made use of various existing libraries for parallel linear algebra solvers (PETSc), mesh and multigrid operator generation (libmesh) and file input/output (HDF5). The numerical results I achieved for both two- and three-dimensional computations show that a possible minimum for the optimal control problem can be computed in a robust and accurate manner. See [21, 20, 18, 6] for further details. At present, I am studying the extension of the proposed lifting function approach for boundary optimal control problems arising in coupled thermal fluid dynamics. For these problems I am considering algorithms for the numerical solution of the optimality system based on multigrid solvers with domain decomposition smoothers. These strategies rely on a domain decoupling rather than an equation decoupling, as it occurs in classical gradienttype algorithms. It is interesting to study the convergence behaviour of these methods, in particular the robustness with respect to the penalty parameters in the objective functional [3]. Further activities Other research activities of my interest are in the fields of Fluid Structure Interaction; Thermal hydraulics of innovative nuclear reactors; Volume Of Fluid (VOF) methods for two-phase flows. Concerning fluid-structure interaction, I am currently working on a comparison of domain decomposition smoothers in multigrid algorithms for the solution of steady-state incompressible FSI problems [2]. Also, I contributed to study a monolithic projection method for incompressible fluid-structure interaction in [15]. Concerning the thermal hydraulics of innovative nuclear reactors, I worked on the development of a FEM code for a preliminary CFD study of a pool-type lead cooled fast reactor, belonging to the list of Generation IV selected reactor projects (see [16, 26]). The development of this FEM code falls within a joint effort involving various European universities and research institutes towards the establishment of an integrated platform for the simulation of nuclear reactors and facilities (see [28, 25]). The simulation of the turbulent behaviour of liquid metal coolants is at present a great challenge, since standard turbulence models do not match experimental data and heat transfer correlations appropriately. I contributed to study new turbulence models for low Prandtl number fluids (see [25, 27, 17]). In the field of two-phase flows, I studied the numerical properties of the height function method for the calculation of normal vector and curvature of an interface line between two different fluid phases (see [7, 22]). Publications Articles in journals [1] B. Athukorallage, G. Bornia, T. Paragoda, and M. Toda. Willmore-type energies and Willmore-type surfaces in space forms. JP Journal of Geometry and Topology, in press. 2/5

3 [2] E. Aulisa, S. Bna, and G. Bornia. A comparison of multigrid smoothers for stationary incompressible Fluid-Structure Interaction problems in preparation. [3] E. Aulisa, G. Bornia, and S. Manservisi. Boundary Control Problems in Convective Heat Transfer with Lifting Function Approach and Multigrid Vanka-Type Solvers. Communications in Computational Physics, 18: , [4] G. Bornia and S. Manservisi. A multigrid approach for a domain decomposition solver over non-matching grids. International Journal of Numerical Analysis and Modeling, submitted. [5] G. Ke, E. Aulisa, G. Bornia, and V. Howle. Preconditioning techniques for Picard, Newton and mixed Picard-Newton methods for the Bénard convection problem in preparation. [6] G. Bornia, M. D. Gunzburger, and S. Manservisi. A distributed control approach for the boundary optimal control of the steady MHD equations. Communications In Computational Physics, 14(3): , [7] G. Bornia, A. Cervone, S. Manservisi, R. Scardovelli, and S. Zaleski. On the properties and limitations of the height function method in two-dimensional Cartesian geometry. Journal of Computational Physics, 230(4): , Articles in conference proceedings [8] B. Athukorallage, E. Aulisa, G. Bornia, T. Paragoda, and M. Toda. New advances in the study of Generalized Willmore surfaces and flow. In Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, page 11, [9] E. Aulisa, S. Bnà, and G. Bornia. Multigrid Solver with Domain Decomposition Smoothing for Steady-State Incompressible FSI Problems. In ECCOMAS Thematic Conference - 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, COMPDYN 2015, May 2015 Crete Island, Greece, volume 1, pages Institute of Structural Analysis and Antiseismic Research School of Civil Engineering National Technical University of Athens (NTUA), [10] G. Bornia. On a class of constrained boundary control problems treated with a lifting function approach. In AIP Conference Proceedings of the American Institute of Physics, volume 1558, page 4, [11] G. Bornia and M. Toda. Preface - Symposium on Geometric Methods for Integrable Systems and PDE with Applications to Engineering, Biology and Medicine. In AIP Conference Proceedings of the American Institute of Physics, page 4, [12] S. Bnà, S. Manservisi, and G. Bornia. A Penalty-Projection Algorithm for Incompressible Fluid-Structure Interaction. In Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering, pages 1 19, Wien, Austria, September 10 14, [13] G. Bornia and S. Manservisi. Coupled boundary optimal control problems in thermal fluid dynamics with lifting function approach and Vanka-type solvers. In Proceedings of the 3/5

4 6th European Congress on Computational Methods in Applied Sciences and Engineering, pages 1 19, Wien, Austria, September 10 14, [14] S. Bnà, G. Bornia, D. Cerroni, S. Manservisi, F. Menghini, and R. Scardovelli. Heat transfer numerical simulations with the four parameter κ- ω - κ t - ω t model for low-prandtl number liquid metals. In Proceedings of the XXX UIT Heat Transfer Conference, Bologna, Italy, June 25 27, [15] S. Bnà, G. Bornia, and S. Manservisi. A monolithic FEM multigrid penalty-projection solver for incompressible fluid-structure interaction. In Proceedings of the First ECCOMAS Young Investigators Conference ECCOMAS YIC 2012, pages 1 14, Aveiro, Portugal, April 24 27, [16] S. Bnà, F. Bassenghi, G. Bornia, S. Manservisi, and R. Scardovelli. Thermo-hydraulic analysis of a LFR Generation IV reactor with a porous medium approach. In Proceedings of the XXIX UIT Heat Transfer Congress, Turin, Italy, June 20 22, [17] C. Carraria Martinotti, F. Bassenghi, S. Bnà, G. Bornia, S. Manservisi, and R. Scardovelli. Effects of buoyancy on mixed turbulent heat transfer to heavy liquid metals in vertical annuli. In Proceedings of the XXIX UIT Heat Transfer Congress, Turin, Italy, June 20 22, [18] G. Bornia and S. Manservisi. Three-dimensional computations for boundary optimal control problems in incompressible Magnetohydrodynamics. In Proceedings of the ECCO- MAS Thematic Conference on CFD & Optimization ECCOMAS CFD&OPT 2011, pages 1 19, Antalya, Turkey, May 23 25, [19] F. Bassenghi, G. Bornia, A. Cervone, S. Manservisi, and R. Scardovelli. A comparison between a pressure projection method and a fully coupled multigrid FEM Navier-Stokes solver. In Proceedings of the XXVIII UIT Heat Transfer Congress, pages , Brescia, Italy, June 21 23, [20] F. Bassenghi, G. Bornia, A. Cervone, S. Manservisi, and R. Scardovelli. Extended boundary approach for optimal control of incompressible steady MHD equations. In Proceedings of the XXVIII UIT Heat Transfer Congress, pages , Brescia, Italy, June 21 23, [21] G. Bornia, A. Cervone, and S. Manservisi. Optimal control for incompressible steady MHD flows via constrained extended boundary approach. In Proceedings of the V European Conference on Computational Fluid Dynamics (ECCOMAS-CFD 2010), pages 1 19, Lisbon, Portugal, June 14 17, [22] G. Bornia, A. Cervone, S. Manservisi, and R. Scardovelli. A study of the approximation of an interface line with the height function method. In Proceedings of the XXVII UIT Heat Transfer Congress, pages , Reggio Emilia, Italy, June 22 24, Reports [23] F. Bassenghi, G. Bornia, S. Manservisi, R. Scardovelli, F. Donato, C. Lombardo, and M. Polidori. Optimization and validation of the CFD modules in the NURISP platform. CERSE-UNIBO Report RdS 1350, University of Bologna, /5

5 [24] G. Bornia, D. Cerroni, S. Manservisi, M. Polidori, and F. Donato. FEM-LCORE code: parallelization, turbulence models and code integration. CERSE-UNIBO Report RdS 1351, University of Bologna, [25] F. Bassenghi, G. Bornia, L. Deon, S. Manservisi, P. Meloni, and M. Polidori. Implementation and validation of the NURISP platform. CERSE-UNIBO Report RL 1308, University of Bologna, [26] G. Bornia, M. Finelli, S. Manservisi, V. Mikhin, M. Polidori, and K. Voukelatou. Development and validation of FEM-LCORE code for the thermal hydraulics of open cores. CERSE-UNIBO Report RL 1307, University of Bologna, [27] G. Bornia, C. Carraria Martinotti, S. Manservisi, and R. Scardovelli. Validation of heat transfer correlations on single rod data. THINS Project Report - EC FP7, University of Bologna, [28] F. Bassenghi, G. Bornia, A. Cervone, and S. Manservisi. FISSICU platform on CRESCO- ENEA Grid for thermal-hydraulic nuclear engineering. CERSE-UNIBO Report RL 1302, University of Bologna, /5