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.

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.

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).

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.

Development of musculoskeletal models for the design and the pre-clinical validation of hip resurfacing prosthesis

Background. The surgical treatment of dysfunctional hips is a severe condition for the patient and a costly therapy for the public health. Hip resurfacing techniques seem to hold the promise of various advantages over traditional THR, with particular attention to young and active patients. Although the lesson provided in the past by many branches of engineering is that success in designing competitive products can be achieved only by predicting the possible scenario of failure, to date the understanding of the implant quality is poorly pre-clinically addressed. Thus revision is the only delayed and reliable end point for assessment. The aim of the present work was to model the musculoskeletal system so as to develop a protocol for predicting failure of hip resurfacing prosthesis. Methods. Preliminary studies validated the technique for the generation of subject specific finite element (FE) models of long bones from Computed Thomography data. The proposed protocol consisted in the numerical analysis of the prosthesis biomechanics by deterministic and statistic studies so as to assess the risk of biomechanical failure on the different operative conditions the implant might face in a population of interest during various activities of daily living. Physiological conditions were defined including the variability of the anatomy, bone densitometry, surgery uncertainties and published boundary conditions at the hip. The protocol was tested by analysing a successful design on the market and a new prototype of a resurfacing prosthesis. Results. The intrinsic accuracy of models on bone stress predictions (RMSE < 10%) was aligned to the current state of the art in this field. The accuracy of prediction on the bone-prosthesis contact mechanics was also excellent (< 0.001 mm). The sensitivity of models prediction to uncertainties on modelling parameter was found below 8.4%. The analysis of the successful design resulted in a very good agreement with published retrospective studies. The geometry optimisation of the new prototype lead to a final design with a low risk of failure. The statistical analysis confirmed the minimal risk of the optimised design over the entire population of interest. The performances of the optimised design showed a significant improvement with respect to the first prototype (+35%). Limitations. On the authors opinion the major limitation of this study is on boundary conditions. The muscular forces and the hip joint reaction were derived from the few data available in the literature, which can be considered significant but hardly representative of the entire variability of boundary conditions the implant might face over the patients population. This moved the focus of the research on modelling the musculoskeletal system; the ongoing activity is to develop subject-specific musculoskeletal models of the lower limb from medical images. Conclusions. The developed protocol was able to accurately predict known clinical outcomes when applied to a well-established device and, to support the design optimisation phase providing important information on critical characteristics of the patients when applied to a new prosthesis. The presented approach does have a relevant generality that would allow the extension of the protocol to a large set of orthopaedic scenarios with minor changes. Hence, a failure mode analysis criterion can be considered a suitable tool in developing new orthopaedic devices.

Creep and damage models for the service behaviour of structural members

Deformability is often a crucial to the conception of many civil-engineering structural elements. Also, design is all the more burdensome if both long- and short-term deformability has to be considered. In this thesis, long- and short-term deformability has been studied from the material and the structural modelling point of view. Moreover, two materials have been handled: pultruded composites and concrete. A new finite element model for thin-walled beams has been introduced. As a main assumption, cross-sections rigid are considered rigid in their plane; this hypothesis replaces that of the classical beam theory of plane cross-sections in the deformed state. That also allows reducing the total number of degrees of freedom, and therefore making analysis faster compared with twodimensional finite elements. Longitudinal direction warping is left free, allowing describing phenomena such as the shear lag. The new finite-element model has been first applied to concrete thin-walled beams (such as roof high span girders or bridge girders) subject to instantaneous service loadings. Concrete in his cracked state has been considered through a smeared crack model for beams under bending. At a second stage, the FE-model has been extended to the viscoelastic field and applied to pultruded composite beams under sustained loadings. The generalized Maxwell model has been adopted. As far as materials are concerned, long-term creep tests have been carried out on pultruded specimens. Both tension and shear tests have been executed. Some specimen has been strengthened with carbon fibre plies to reduce short- and long- term deformability. Tests have been done in a climate room and specimens kept 2 years under constant load in time. As for concrete, a model for tertiary creep has been proposed. The basic idea is to couple the UMLV linear creep model with a damage model in order to describe nonlinearity. An effective strain tensor, weighting the total and the elasto-damaged strain tensors, controls damage evolution through the damage loading function. Creep strains are related to the effective stresses (defined by damage models) and so associated to the intact material.

Experimental analysis and FE Modeling of Aluminium alloys Microstructural Evolution in Extrusion

Extrusion is a process used to form long products of constant cross section, from simple billets, with a high variety of shapes. Aluminum alloys are the materials most processed in the extrusion industry due to their deformability and the wide field of applications that range from buildings to aerospace and from design to automotive industries. The diverse applications imply different requirements that can be fulfilled by the wide range of alloys and treatments, that is from critical structural application to high quality surface and aesthetical aspect. Whether one or the other is the critical aspect, they both depend directly from microstructure. The extrusion process is moreover marked by high deformations and complex strain gradients making difficult the control of microstructure evolution that is at present not yet fully achieved. Nevertheless the evolution of Finite Element modeling has reached a maturity and can therefore start to be used as a tool for investigation and prediction of microstructure evolution. This thesis will analyze and model the evolution of microstructure throughout the entire extrusion process for 6XXX series aluminum alloys. Core phase of the work was the development of specific tests to investigate the microstructure evolution and validate the model implemented in a commercial FE code. Along with it two essential activities were carried out for a correct calibration of the model beyond the simple research of contour parameters, thus leading to the understanding and control of both code and process. In this direction activities were also conducted on building critical knowhow on the interpretation of microstructure and extrusion phenomena. It is believed, in fact, that the sole analysis of the microstructure evolution regardless of its relevance in the technological aspects of the process would be of little use for the industry as well as ineffective for the interpretation of the results.

Modelling and analysis of thin-walled beams in the context of the Generalized Beam Theory

In this work, the Generalized Beam Theory (GBT) is used as the main tool to analyze the mechanics of thin-walled beams. After an introduction to the subject and a quick review of some of the most well-known approaches to describe the behaviour of thin-walled beams, a novel formulation of the GBT is presented. This formulation contains the classic shear-deformable GBT available in the literature and contributes an additional description of cross-section warping that is variable along the wall thickness besides along the wall midline. Shear deformation is introduced in such a way that the classical shear strain components of the Timoshenko beam theory are recovered exactly. According to the new kinematics proposed, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition. Later, a procedure for a posteriori reconstruction of all the three-dimensional stress components in the finite element analysis of thin-walled beams using the GBT is presented. The reconstruction is simple and based on the use of three-dimensional equilibrium equations and of the RCP procedure. Finally, once the stress reconstruction procedure is presented, a study of several existing issues on the constitutive relations in the GBT is carried out. Specifically, a constitutive law based on mirroring the kinematic constraints of the GBT model into a specific stress field assumption is proposed. It is shown that this method is equally valid for isotropic and orthotropic beams and coincides with the conventional GBT approach available in the literature. Later on, an analogous procedure is presented for the case of laminated beams. Lastly, as a way to improve an inherently poor description of shear deformability in the GBT, the introduction of shear correction factors is proposed. Throughout this work, numerous examples are provided to determine the validity of all the proposed contributions to the field.