Andrea La Spina

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler-Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations.

A semi-analytic method is proposed to compute the produced plasma and the emitted Cherenkov radiation from hypervelocity impacts on soda–lime glass for various projectiles and impact velocities. First, the Taylor–von Neumann–Sedov blast wave model, coupled with the system of nonlinear Saha equations for multispecies, strongly coupled plasma, is adopted to estimate the hydrodynamic profiles and the ionization state of the target material in the early stage of the impact. Second, the Frank–Tamm formula is considered to investigate the onset of the Cherenkov radiation and to compute the emitted energy. The present approach predicts a linear dependence of the produced total electric charge on the projectile density and a quadratic dependence on the projectile velocity, whereas the emitted Cherenkov radiation scales quadratically with the produced charge if the onset conditions are met.

This work proposes novel hybridizable discontinuous Galerkin (HDG) methods, both in the time and in the frequency domain, to accurately compute the Cherenkov radiation emitted by a charged particle travelling in a uniform medium at superluminal speed. The adopted formulations enrich existing HDG approaches for the solution of Maxwell’s equations by including perfectly matched layers (PMLs) to effectively absorb the outgoing waves and Floquet-periodic boundary conditions to connect the boundaries of the computational domain in the direction of the moving charge. A wave propagation problem with smooth solution is used to show the optimal convergence of the HDG variables and the superconvergence of the postprocessed electric field and a second example examines the role of the PML parameters on the absorption of the electromagnetic field. A series of numerical experiments both in 3D and 2D-axisymmetric components show the capability of the proposed methods to faithfully reproduce Cherenkovian effects in different conditions and their high accuracy is confirmed by comparing the numerical results with the Frank–Tamm formula.

This work proposes a hybridizable discontinuous Galerkin (HDG) method for the solution of magnetohydrodynamic (MHD) problems with weakly compressible flows. A novel fluid formulation that adopts the velocity and the pressure as primal variables is first derived and its superior properties, compared to alternative density–momentum-based approaches, are demonstrated on a simple benchmark. The coupled MHD formulation exhibits superconvergence properties for both the fluid velocity and the magnetic induction, a feature not present in any HDG formulation published in this field. An alternative MHD formulation, adopting a fluid-type solver for the solution of the magnetic subproblem, is also considered and its advantages and disadvantages are discussed. The convergence properties of the proposed formulations for the single physics and for the coupled problem are examined on an extensive set of numerical examples in both two and three dimensions, on structured and unstructured meshes and at low and high Hartmann numbers.

The interest in the simulation of fluid-structure interaction (FSI) phenomena has increased significantly over the years. Despite the constant growth in available computing resources, the demand for more robust and efficient computational methods does not cease.
This thesis proposes novel schemes for the solution of FSI problems with weakly compressible flows. Special attention is devoted to the spatial discretization of the fluid problem by means of the hybridizable discontinuous GALERKIN (HDG) method and to the coupling of the fluid field with the structural one, discretized by means of the continuous GALERKIN (CG) method.

A scheme for the solution of fluid–structure interaction (FSI) problems with weakly compressible flows is proposed in this work. A novel hybridizable discontinuous Galerkin (HDG) method is derived for the discretization of the fluid equations, while the standard continuous Galerkin (CG) approach is adopted for the structural problem. The chosen HDG solver combines robustness of discontinuous Galerkin (DG) approaches in advection-dominated flows with higher-order accuracy and efficient implementations. Two coupling strategies are examined in this contribution, namely a partitioned Dirichlet–Neumann scheme in the context of hybrid HDG–CG discretizations and a monolithic approach based on Nitsche’s method, exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions. Numerical experiments show optimal convergence of the HDG and CG primal and mixed variables and superconvergence of the postprocessed fluid velocity. The robustness and the efficiency of the proposed weakly compressible formulation, in comparison to a fully incompressible one, are also highlighted on a selection of two and three dimensional FSI benchmark problems.

A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche's method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.

In the present study, a weakly compressible formulation of the Navier-Stokes equations is developed and examined for the solution of fluid-structure interaction (FSI) problems. Newtonian viscous fluids under isothermal conditions are considered, and the Murnaghan-Tait equation of state is employed for the evaluation of mass density changes with pressure. A pressure-based approach is adopted to handle the low Mach number regime, ie, the pressure is chosen as primary variable, and the divergence-free condition of the velocity field for incompressible flows is replaced by the continuity equation for compressible flows. The approach is then embedded into a partitioned FSI solver based on a Dirichlet-Neumann coupling scheme. It is analytically demonstrated how this formulation alleviates the constraints of the instability condition of the artificial added mass effect, due to the reduction of the maximal eigenvalue of the so-called added mass operator. The numerical performance is examined on a selection of benchmark problems. In comparison to a fully incompressible solver, a significant reduction of the coupling iterations and the computational time and a notable increase in the relaxation parameter evaluated according to Aitken's Δ^2 method are observed.

The mechanical behaviour of microstructured materials is attracting considerable research interest, from both the academic and the industrial communities.
The phenomenon of wave propagation through these media and its implications are in particularly explored.
The pertinent mathematical techniques are first discussed and several test cases of increasing complexity are illustrated.
The dispersion relations, the phase and the group velocities are derived both in a second and in a micromorphic approach and the main advantages of these models are analysed.
The peculiar phenomenon of localisation is also investigated with regards to damaged materials which exhibit a degradation of their mechanical properties. A simple FEM code, implemented for the study of the wave propagation in a one-dimensional domain, is presented together with an accurate validation. Some simulations are then run with this software and the outputs are compared with the analytical results previously obtained.