Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics

Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years. DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method. However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).

Locating the source of volcanic tremor at stromboli volcano, italy

We have developed a method for locating sources of volcanic tremor and applied it to a dataset recorded on Stromboli volcano before and after the onset of the February 27th 2007 effusive eruption. Volcanic tremor has attracted considerable attention by seismologists because of its potential value as a tool for forecasting eruptions and for better understanding the physical processes that occur inside active volcanoes. Commonly used methods to locate volcanic tremor sources are: 1) array techniques, 2) semblance based methods, 3) calculation of wave field amplitude. We have choosen the third approach, using a quantitative modeling of the seismic wavefield. For this purpose, we have calculated the Green Functions (GF) in the frequency domain with the Finite Element Method (FEM). We have used this method because it is well suited to solve elliptic problems, as the elastodynamics in the Fourier domain. The volcanic tremor source is located by determining the source function over a regular grid of points. The best fit point is choosen as the tremor source location. The source inversion is performed in the frequency domain, using only the wavefield amplitudes. We illustrate the method and its validation over a synthetic dataset. We show some preliminary results on the Stromboli dataset, evidencing temporal variations of the volcanic tremor sources.

Design, development, and analysis of hybrid metal-composite tube for industrial application

In the aerospace, automotive, printing, and sports industries, the development of hybrid Carbon Fiber Reinforced Polymer (CFRP)-metal components is becoming increasingly important. The coupling of metal with CFRP in axial symmetric components results in reduced production costs and increased mechanical properties such as bending, torsional stiffness, mass reduction, damping, and critical speed compared to the single material-built ones. In this thesis, thanks to a novel methodology involving a rubbery/viscoelastic interface layer, several hybrid aluminum-CFRP prototype tubes were produced. Besides, an innovative system for the cure of the CFRP part has been studied, analyzed, tested, and developed in the company that financed these research activities (Reglass SRL, Minerbio BO, Italy). The residual thermal stresses and strains have been investigated with numerical models based on the Finite Element Method (FEM) and compared with experimental tests. Thanks to numerical models, it was also possible to reduce residual thermal stresses by optimizing the lamination sequence of CFRP and determining the influence of the system parameters. A novel software and methodology for evaluating mechanical and damping properties of specimens and tubes made in CFRP were also developed. Moreover, to increase the component's damping properties, rubber nanofibers have been produced and interposed throughout the lamination of specimens. The promising results indicated that the nanofibrous mat could improve the material damping factor over 77% and be adopted in CFRP components with a negligible increment of weight or losing mechanical properties.

Acoustic backscatter of small pelagic fish species in the Adriatic Sea

The aim of the present study is to apply a broad range of techniques to increase the knowledge of acoustic properties of Sprattus sprattus, Scomber colias and Trachurus mediterraneus in the Adriatic Sea. A novel study using tethered live fish but not involving hooks and anesthetic was tested on T. mediterraneus and S. colias through several ex situ experiments using a split-beam scientific echosounder operating at 38, 120, and 200 kHz. The mean TS was estimated for 29 live specimens, resulting in a conversion factor b20 value of -71.4 dB re 1 m2 and -71.6 dB re 1 m2 respectively which is ~3 dB lower than the current one in use in the Mediterranean Sea. Successively, two monospecific trawl hauls were analyzed through the application of in situ approach for the computation of TS values of S. sprattus which led to six b20 values for sprat (range, -68.8 dB re 1 m2 to -65.6 dB re 1 m2), all higher than the current known value of -71.7 dB re 1 m2. The high difference up to 4.2 dB compared to the current value translates in a significant decrease of absolute sprat biomass along the time series un to 20%. Finally, 149 specimens of the three species were collected for backscattering model application(i.e. Kirchhoff-ray mode model (KRM) and Finite Element Method (FEM)) from digital images of the fish body and swimbladder obtained from Computer Tomography (CT) and X-Ray scans. The values resulting from the application of KRM and FEM are in agreement with empirical results. In general terms the present work proposes the acoustic backscatter characterization of S. colias, S. sprattus and T. mediterraneus in the Mediterranean Sea.

Active control of sound radiated by vibrating surfaces on a McLaren supercar

Sound radiators based on forced vibrations of plates are becoming widely employed, mainly for active sound enhancement and noise cancelling systems, both in music and automotive environment. Active sound enhancement solutions based on electromagnetic shakers hence find increasing interest. Mostly diffused applications deal with active noise control (ANC) and active vibration control systems for improving the acoustic experience inside or outside the vehicle. This requires investigating vibrational and, consequently, vibro-acoustic characteristics of vehicles. Therefore, simulation and processing methods capable of reducing the calculation time and providing high-accuracy results, are strongly demanded. In this work, an ideal case study on rectangular plates in fully clamped conditions preceded a real case analysis on vehicle panels. The sound radiation generated by a vibrating flat or shallow surface can be calculated by means of Rayleigh’s integral. The analytical solution of the problem is here calculated implementing the equations in MATLAB. Then, the results are compared with a numerical model developed in COMSOL Multiphysics, employing Finite Element Method (FEM). A very good matching between analytical and numerical solutions is shown, thus the cross validation of the two methods is achieved. The shift to the real case study, on a McLaren super car, led to the development of a mixed analytical-numerical method. Optimum results were obtained with mini shakers excitement, showing good matching of the recorded SPL with the calculated one over all the selected frequency band. In addition, a set of directivity measurements of the hood were realized, to start studying the spatiality of sound, which is fundamental to active noise control systems.

Finite element models of coseismic deformation due to the 2009 L'Aquila (Italy) and 2008 Wenchuan(China) earthquakes

The topic of my Ph.D. thesis is the finite element modeling of coseismic deformation imaged by DInSAR and GPS data. I developed a method to calculate synthetic Green functions with finite element models (FEMs) and then use linear inversion methods to determine the slip distribution on the fault plane. The method is applied to the 2009 L’Aquila Earthquake (Italy) and to the 2008 Wenchuan earthquake (China). I focus on the influence of rheological features of the earth's crust by implementing seismic tomographic data and the influence of topography by implementing Digital Elevation Models (DEM) layers on the FEMs. Results for the L’Aquila earthquake highlight the non-negligible influence of the medium structure: homogeneous and heterogeneous models show discrepancies up to 20% in the fault slip distribution values. Furthermore, in the heterogeneous models a new area of slip appears above the hypocenter. Regarding the 2008 Wenchuan earthquake, the very steep topographic relief of Longmen Shan Range is implemented in my FE model. A large number of DEM layers corresponding to East China is used to achieve the complete coverage of the FE model. My objective was to explore the influence of the topography on the retrieved coseismic slip distribution. The inversion results reveals significant differences between the flat and topographic model. Thus, the flat models frequently adopted are inappropriate to represent the earth surface topographic features and especially in the case of the 2008 Wenchuan earthquake.

Fluid-structure interaction by a coupled lattice Boltzmann-finite element approach

In this thesis, a strategy to model the behavior of fluids and their interaction with deformable bodies is proposed. The fluid domain is modeled by using the lattice Boltzmann method, thus analyzing the fluid dynamics by a mesoscopic point of view. It has been proved that the solution provided by this method is equivalent to solve the Navier-Stokes equations for an incompressible flow with a second-order accuracy. Slender elastic structures idealized through beam finite elements are used. Large displacements are accounted for by using the corotational formulation. Structural dynamics is computed by using the Time Discontinuous Galerkin method. Therefore, two different solution procedures are used, one for the fluid domain and the other for the structural part, respectively. These two solvers need to communicate and to transfer each other several information, i.e. stresses, velocities, displacements. In order to guarantee a continuous, effective, and mutual exchange of information, a coupling strategy, consisting of three different algorithms, has been developed and numerically tested. In particular, the effectiveness of the three algorithms is shown in terms of interface energy artificially produced by the approximate fulfilling of compatibility and equilibrium conditions at the fluid-structure interface. The proposed coupled approach is used in order to solve different fluid-structure interaction problems, i.e. cantilever beams immersed in a viscous fluid, the impact of the hull of the ship on the marine free-surface, blood flow in a deformable vessels, and even flapping wings simulating the take-off of a butterfly. The good results achieved in each application highlight the effectiveness of the proposed methodology and of the C++ developed software to successfully approach several two-dimensional fluid-structure interaction problems.

A Trans-dimensional inversion algorithm to model deformation sources with unconstrained shape in finite element domains

Ground deformation provides valuable insights on subsurface processes with pattens reflecting the characteristics of the source at depth. In active volcanic sites displacements can be observed in unrest phases; therefore, a correct interpretation is essential to assess the hazard potential. Inverse modeling is employed to obtain quantitative estimates of parameters describing the source. However, despite the robustness of the available approaches, a realistic imaging of these reservoirs is still challenging. While analytical models return quick but simplistic results, assuming an isotropic and elastic crust, more sophisticated numerical models, accounting for the effects of topographic loads, crust inelasticity and structural discontinuities, require much higher computational effort and information about the crust rheology may be challenging to infer. All these approaches are based on a-priori source shape constraints, influencing the solution reliability. In this thesis, we present a new approach aimed at overcoming the aforementioned limitations, modeling sources free of a-priori shape constraints with the advantages of FEM simulations, but with a cost-efficient procedure. The source is represented as an assembly of elementary units, consisting in cubic elements of a regular FE mesh loaded with a unitary stress tensors. The surface response due to each of the six stress tensor components is computed and linearly combined to obtain the total displacement field. In this way, the source can assume potentially any shape. Our tests prove the equivalence of the deformation fields due to our assembly and that of corresponding cavities with uniform boundary pressure. Our ability to simulate pressurized cavities in a continuum domain permits to pre-compute surface responses, avoiding remeshing. A Bayesian trans-dimensional inversion algorithm implementing this strategy is developed. 3D Voronoi cells are used to sample the model domain, selecting the elementary units contributing to the source solution and those remaining inactive as part of the crust.

Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.