The main topic of the thesis is the synthesis of control laws for interacting agent-based dynamics and their mean-field limit. In particular, after a general introduction, in the second chapter a linearization-based approach is used for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into a nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics. The performance and robustness of the proposed methodology is assessed through different numerical experiments in collective dynamics. In the other chapters of the thesis I present some related projects, robustness of systems with uncertainties, a proximal gradient approach for sparse control and an application in crowd evacuation dynamics.

IIn this work we present a new class of well-balanced, arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid. Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. Note that the same space-time DG scheme on a collocated grid would lead to ten non-zero blocks per element in 2D and seventeen non-zero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh. This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations.

In this work we develop a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. Specifically, we adopt either the continuous Galerkin (CG) predictor, which does not allow discontinuities in time and is suitable for smooth time evolutions, or the discontinuous Galerkin (DG) predictor which can handle stiff source terms that might produce jumps in the local space-time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space-time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The node solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration is connected with the new one at the future time level with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving space-time control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. In our algorithm either a simple and robust Rusanov-type numerical flux or a more sophisticated and less dissipative Osher-type numerical flux is employed. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Furthermore we focus on the following issues to improve the algorithm efficiency: the time evolution, the numerical flux computation across element boundaries and the high order WENO reconstruction procedure. First, a time-accurate local time stepping (LTS) algorithm for unstructured triangular meshes is derived and presented, where each element can run at its own optimal time step, given by a local CFL stability condition. Then, we propose a new and efficient quadrature-free formulation for the flux computation, in which the space-time boundaries of each element are split into simplex sub-elements. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element, hence allowing the flux integrals to be evaluated on the space-time reference control volume once and for all analytically during a preprocessing step. Finally, we consider the very new a posteriori MOOD paradigm, recently proposed for the Eulerian framework, to overcome the expensive WENO approach on moving meshes. The MOOD technique requires the use of only one central reconstruction stencil because the limiting procedure is carried out a posteriori instead of a priori, as done in the WENO formulation.

Model order reduction is a technique to reduce computational times of parameterized PDEs while maintaining good accuracy of the approximated solution. Reduced basis methods (RB) are the most common algorithms for reducing the complexity of parameterized PDEs and nowadays they are widely applied and very actively researched in numerous fields. We propose two ideas to further enhance model reduction: the Fundamental Order Reduction Method (FOR) and offline error estimators for RB methods. The FOR method uses nonlinear combinations of the solutions to build the reduced model and use simple affine evaluations to execute the online stage. On the other hand, offline estimators are a class of estimators that move a-posteriori operations to the offline stage, reducing in this way the load of computations in the online stage. We apply these two ideas to an EEG equation which is useful for detecting the position where an epilepsy seizure begins inside the brain. We present two known ways to solve this equation: direct approach and subtraction approach, and show theoretical and numerical results of the application of the RB and FOR methods. We prove that is not feasible to apply model reduction in the direct approach but show that it is possible in the subtraction approach. Afterwards we solve the inverse problem associated with the EEG equation using a combination of the FOR method and neural networks.

Recent advances in medical science regarding the role of the venous system in the development of neurological conditions has renewed the attention of researchers in this district of the cardiovascular system. The main goal of this thesis is to perform a theoretical study of Chronic CerebroSpinal Venous Insufficiency (CCSVI), a venous pathology that has been associated to Multiple Sclerosis. CCSVI is a condition in which main cerebral venous drainage pathways are obstructed. Its impact in cerebral hemodynamics and its connection to Multiple Sclerosis is subject of current debate in the medical community. In order to perform a credible study of the haemodynamical aspects of CCSVI, a sufficiently accurate mathematical model of the problem under investigation must be used. The venous system has not received the same attention as the arterial counterpart by the medical community. As a consequence, the mathematical modeling and numerical simulation of the venous system lies far behind that of the arterial system. The venous system is a low-pressure system, formed by very thin-walled vessels, if compared to arteries, that are likely to collapse under the action of gravitational or external forces. These properties set special requirements on the mathematical models and numerical schemes to be used. In this thesis we present a closed-loop multi-scale mathematical model of the cardiovascular system, where medium to large arteries and veins are represented as one-dimensional (1D) vessels, whereas the heart, the pulmonary circulation, capillary beds and intracranial pressure are modeled as lumped parameter models. A characteristic feature of our closed-loop model is the detailed description of head and neck veins. Due to the large inter-subject variability of the venous system, we perform a patient-specific characterization of major veins of the head and neck using MRI data collected in collaboration with the Magnetic Resonance Research Facility of the Wayne State University, Detroit (USA). Computational results are carefully validated using published data for the arterial system and most regions of the venous system. For head and neck veins validation is carried out through a detailed comparison of simulation results against patient-specific Phase-Contrast MRI flow quantification data. Regarding the development of novel numerical schemes, we construct high-order accurate, robust and efficient numerical schemes for 1D blood flow in elastic and viscoelastic vessels, as well as a solver for vessel networks. The solver is validated in the context of an in vitro network of vessels for which experimental and numerical results are available. After validation of both, the mathematical model and the numerical methodology, we use our theoretical tool to study the influence of different CCSVI patterns on cerebral hemodynamics. CCSVI patterns are defined by the medical literature as combinations of venous obstructions at different locations. Here we used two strategies. First, we take a venous configuration corresponding to a healthy control and explore the effect of different CCSVI patterns by modifying this network. Then, we characterize our venous network with the geometry of a real CCSVI patient and compare results with the ones obtained for the healthy control. The presented model provides a powerful tool to study still unresolved aspects of cerebral blood flow physiology, as well as several venous pathologies. Furthermore, it constitutes an ideal platform for improving currently used algorithms and for integrating fundamental physiological processes, such as detailed hemodynamics, regulatory mechanisms and transport of substances.

In the present work we solve systems of partial differential equations (PDE) for hyperbolic conservation laws using semi-implicit numerical methods on staggered meshes applied both to the class of finite volume (FV) and both to the family of high order Discontinuous Galerkin (DG) finite elements schemes. In particular, we want to show that these new semi-implicit schemes can be applied in several fields of applied sciences, such as in geophysical flows and compressible fluids in compliant tubes. Inside this thesis we distinguish two big parts. First, we consider staggered semi-implicit schemes for compressible viscous fluids flowing in elastic pipes. This topic is very important in several practical applications of civil, environmental, industrial and biomedical engineering. Here, we analyse the accuracy and the computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric rigid and elastic pipes. We consider two families of differential models that can be used to predict the pressure and velocity distribution along the tube. One is the so called 2Dxr PDE model which is derived from the full compressible Navier-Stokes equations under the assumptions of a hydrostatic pressure and an axially symmetric geometry. The second family is a simple 1D non-conservative PDE system based on the cross-sectionally averaged version of the Navier-Stokes equations in cylindrical coordinates. In this last case, the use of a simple steady friction model is not enough to simulate the wall friction phenomena in highly transient regime. As a consequence the wall friction model has to be frequency dependent and, following previous studies present in the literature, we consider the classes of convolution integral (CI) models and instantaneous acceleration (IA) models. We carry out a rather complete analysis of the previously-mentioned methods for the simulation of flows characterized by fast transient regime in rigid and compliant tubes. The numerical results show that the convolution integral models are clearly better than instantaneous acceleration models concerning accuracy. Moreover, for CI models, instead of computing the convolution integrals, which is very time- and memoryconsuming, we express these methods via a set of additional ODEs for appropriate auxiliary variables. This trick improves the computational efficiency of these methods substantially, since it avoids the direct computation of the convolution integral. In addition, semi-implicit finite volume methods are significantly superior to classical explicit finite volume schemes in terms of computational efficiency, however, providing the same level of accuracy. We then proceed by extending the finite volume discretization of the 1D and 2Dxr PDE models to arbitrary high-order of accuracy in space introducing a new SIDG scheme on staggered meshes. Both models include the effects of the viscosity and of the wall motion. The nonlinear convective terms are discretized explicitly by using a classical RKDG scheme of arbitrary high-order of accuracy in space and third order of accuracy in time. The continuity equation is integrated over the elements that belong to the main grid, while the momentum equation is integrated over the control volumes of the edgebased staggered dual grid. Inserting the discrete momentum equation into the discrete continuity leads to a mildly nonlinear algebraic system for the degrees of freedom of the pressure, which is solved by using the (nested) Newton method of Brugnano, Casulli and Zanolli. We use the -method in order to get second order of accuracy in time for the implicit part of the scheme. In addition, the schemes have to obey only a mild CFL condition based on the fluid velocity and not based on the sound speed; consequently these schemes work also in the low Mach number regime and even in the incompressible limit of the Navier-Stokes equations. This is a very important property, which is the so-called asymptotic preserving (AP) property of the scheme. We carry out several numerical tests in order to validate this novel family of numerical methods against available exact solutions and experimental data. We also report numerical convergence tables in order to show that the new schemes indeed achieve high order of accuracy in space. In the second part of the thesis, we present a new class of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin schemes on staggered Cartesian grids for the solution of the 1D and 2D shallow water equations (SWE) and of the Euler equations both expressed in conservative form. Here, the starting point is the unlimited arbitrary high order accurate staggered SIDG scheme proposed by Dumbser and Casulli (2013). For this metho d, the mass conservation equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. According to the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. Furthermore, the staggered SIDG is also extended to the Euler equations of compressible gasdynamics. Here, the governing PDE are rewritten using a flux vector splitting technique. The convective terms are updated using an explicit Runge-Kutta DG integrator. Then, the discrete momentum equation, which is integrated again on the dual grid, is coupled with the discrete energy equation that is discretized on the control volumes of the main grid. The pressure is efficiently obtained solving a linear system combined with an iterative Picard iteration procedure.

Recent advances in medical science regarding the interaction and functional role of fluid compartments in the central nervous system have attracted the attention of many researchers across various disciplines. Neurotoxins are constantly cleared from the brain parenchyma through the intramural periarterial drainage system, glymphatic system and meningeal lymphatic system. Impairment of these systems can potentially contribute to the onset of neurological disorders. The goal of this thesis is to contribute to the understanding of brain fluid dynamics and to the role of vascular pathologies in the context of neurological disorders. To achieve this goal, we designed the first multi-scale, closed-loop mathematical model of the murine fluid system, incorporating: heart dynamics, major arteries and veins, microcirculation, pulmonary circulation, venous valves, cerebrospinal fluid (CSF), brain interstitial fluid (ISF), Starling resistors, Monro-Kellie hypothesis, brain lymphatic drainage and the modern concept of CSF/ISF drainage and absorption based on the {\em Bulat-Klarica-Orešković} hypothesis. The mathematical model relies on one-dimensional Partial Differential Equations (PDEs) for blood vessels and on Ordinary Differential Equations (ODEs) for lumped parameter models. The systems of PDEs and ODEs are solved through a high-order finite volume ADER method and through an implicit Euler method. The computational results are validated against literature values and magnetic resonance flow measurements. Furthermore, the model is validated against {\em in-vivo} intracranial pressure waveforms acquired in healthy mice and in mice with impairment of the intracranial venous outflow. Through a systematic use of our computational model in healthy and pathological cases, we provide a complete and holistic neurovascular view of the main murine fluid dynamics. We propose a hypothesis on the working principles of the glymphatic system, opening a new door towards a comprehensive understanding of the mechanisms which link vascular and neurological disorders. In particular, we show how impairment of the cerebral venous outflow might potentially lead to accumulation of solutes in the parenchyma, by altering CSF and ISF dynamics. This thesis also concerns the development of a high-order ADER-type numerical method for systems of hyperbolic balance laws in networks, based on a new implicit solver for the junction-generalized Riemann problem. The resulting ADER scheme can deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting of networks of one-dimensional sub-domains. Also, we design a novel one-dimensional mathematical model for collecting lymphatics coupled with a Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions. The resulting mathematical model gives each lymphangion the autonomous capability to trigger action potentials based on local fluid-dynamical factors.

This PhD thesis presents a novel second order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume scheme for nonlinear hyperbolic systems, written both in conservative and non-conservative form, whose peculiarity is the nonconforming motion of interfaces. Moreover it has been coupled together with specifically designed path-conservative well balanced (WB) techniques and angular momentum preserving (AMC) strategies. The obtained result is a method able to preserve many of the physical properties of the system: besides being conservative for mass, momentum and total energy, also any known steady equilibrium of the studied system can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and minimal dissipation on moving contact discontinuities even for very long computational times. The core of our ALE scheme is the use of a space-time conservation formulation in the construction of the final Finite Volume scheme: the governing PDE system is rewritten at the aid of the space-time divergence operator and then a fully discrete one-step discretization is obtained by integrating over a set of closed space-time control volumes. In order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods, we adopt a nonconforming treatment of sliding interfaces, which requires the dynamical insertion or deletion of nodes and edges, and produces hanging nodes and space-time faces shared between more than two cells. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements, the mesh quality remains high and, as a direct consequence, also the time step remains almost constant in time, even for highly sheared vortex flows. Moreover, due to the space-time conservation formulation, the geometric conservation law (GCL) is automatically satisfied by construction, even on moving nonconforming meshes. Our nonconforming ALE scheme is especially well suited for modeling in polar coordinates vortical flows affected by strong differential rotation: in particular, the novel combination with the well balancing make it possible to obtain great results for challenging astronomical phenomena as the rotating Keplerian disk. Indeed, we have formulated a new HLL-type and a novel Osher-type flux that are both able to guarantee the well balancing in a gas cloud rotating around a central object, maintaining up to machine precision the equilibrium between pressure gradient, centrifugal force and gravity force that characterizes the stationary solutions of the Euler equations with gravity. To the best knowledge of the author this work is original for various reasons: it is the first time that the little dissipative Osher scheme is modified in order to be well balanced for non trivial equilibria, and it is the first time that WB is coupled with ALE for the Euler equations with gravity; moreover the use of a well balanced Osher scheme joint with the Lagrangian framework allows, for the first time within a Finite Volume method, to maintain exactly even moving equilibria. In addition, the introduced techniques demonstrate a wide range of applicability from steady vortex flows in shallow water equations to complex free surface flows in two-phase models. In the last case, studied on fixed Cartesian grids, the new well balanced methods have been implemented in parallel exploiting a GPU-based platform and reaching the very high efficiency of ten million of volumes processed per seconds. Finally, in the case of vortical flows we propose a preliminary analysis on how to increase the accuracy of the method by exploiting the redundant conservation law that can be written for the angular momentum, as proposed in Després et al. JCP 2015. Indeed, an easy manipulation of the Euler equations allows to write its additional conservation law: clearly it does not add any supplementary information from the analytical point of view, but from a numerical point of view it provides extra information in particular in the case of rotating systems. We present both a master-slave approach, to deduce a posteriori a more precise approximation of the velocity, and some coupled approaches to investigate how the entire process can take advantage from considering directly the angular momentum during the computation within a strong coupling with other variables. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of the new methods for both smooth and discontinuous problems, close and far away from the equilibrium, in one and two space dimensions. Many of the presented results show a great enhancement with respect to the state of the art.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnologico / Made available in DSpace on 2012-10-16T09:27:39Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T19:48:46Z : No. of bitstreams: 1 99783.pdf: 2839707 bytes, checksum: d2985fb59a12e6dcc16d383fbff0dd07 (MD5) / Desenvolvimento de uma metodologia numérica adequada para a solução de problemas aerodinâmicos a qualquer velocidade e extensão desta para a solução de alguns problemas aeroelásticos, empregando-se os métodos de volumes finitos e elementos finitos. Estuda técnicas para acelerar a convergência empregadas na solução de escoamento compreensíveis e/ou incompreensíveis como passos de tempo local, média dos resíduos e ?multigrip?. Resultados numéricos são apresentados para um cilindro, uma placa, o aerofólio NACA 0012, um painel e um cilindro hemisférico.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnologico / Made available in DSpace on 2012-10-16T22:38:07Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T17:27:10Z : No. of bitstreams: 1 88963.pdf: 3847761 bytes, checksum: 74479c384d134ff79db869c636edbf92 (MD5) / Neste trabalho é desenvolvida uma metodologia numérica, em volumes finitos, no sistema de coordenadas generalizadas, utilizando malhas adaptativas coincidentes com a direção do vetor velocidade, visando minimizar os erros de interpolação decorrentes do fator do vetor velocidade ser, em geral, oblíquo às linhas de malha. É proposta uma função de interpolação para a avaliação das variáveis na face do volume de controle, com base nas equações de conservação, permitindo que as influências dos efeitos convectivos, difusivos, de pressão e existência de termos fontes sejam consideradas. Esta função de interpolação é denominada neste trabalho de WUDS-E, por ser uma extensão do esquema WUDS. É um fato bastante conhecido as dificuldades de convergência enfrentadas quando diferenças centrais são empregadas para a função de interpolação sem a adição de termos dissipativos. A alternativa já em uso há décadas e o emprego de esquemas "upwind" e híbridos, evitando a origem de coeficientes negativos que, dependendo do método de solução do sistema linear empregado, podem ser desastrosos. Estes métodos, entretanto,em suas formas mais simplificadas, isto é, unidimensionais, resultam em considerável introdução de difusão numérica quando as linhas coordenadas não estão alinhadas com o vetor velocidade.