68 research outputs found

    Snapshots location for reduced order models: an approach based on proper orthogonal decomposition and mesh adaptivity techniques.

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    75 p.La solución numérica de flujos de fluidos requiere un coste computacional relativamente grande y hay casos en los que el coste puede llegar a ser inasumible. En muchos de estos casos la mejor opción es utilizar ¿Métodos de Orden Reducido¿ (ROM). Se presenta un trabajo en este ámbito, concretamente con el método Proper Orthogonal Decomposition (POD). La principal novedad desarrollada localiza las zonas del espacio paramétrico de mayor sensibilidad a la hora de alimentar el modelo reducido. Partiendo de una distribución uniforme de los ¿snapshots¿ (casos resueltos del modelo completo), la investigación ha desarrollado un método rápido, eficiente y preciso para estimar un error que puede ser interpretado como la sensibilidad o cantidad de información que aporta cada ¿snapshot¿ al modelo reducido. El trabajo se ha aplicado en dos casos: Primero sobre unas ecuaciones concretas de un flujo ideal con un término fuente específico, y segundo, sobre un caso CFD de un flujo no viscoso incidiendo sobre un perfil alar.bcam:basque center to applied mathematics SCM Grid: sustainability construction & material

    Snapshots location for Reduced Order Models: An approach based on Proper Orthogonal Decomposition and Mesh Adaptivity Techniques

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    Solving Partial Differential Equations (PDE), is a key issue in science and engineering since they are widely used in many real problem modelling including fluid mechanics, acoustics, heat and mass transfer ...etc. Exact solutions for PDEs and Ordinary Differential Equations (ODE) can be obtained in a very few and simplified cases only, numerical approximations are used instead (i.e. Finite Volume Method (FEM), Finite Volume Method (FVM)...). However, and despite the availability of important supercomputing facilities, due to the huge number of degrees of freedom these methods have, they may still suffer from cost-effectiveness performance. They are mainly two contexts where one need to solve PDEs with a lower computational cost: Real-time context and Many-query context. Examples for the former are: parameter-estimation, control, flying simulator..., and for the latter: optimization, multi model/scale simulation... To reduce significantly simulation time (often on the expense of accuracy) Reduced Order Modelling (ROM) techniques are introduced. The main idea is to reduce the initial solving space dimension (as in finite elements method) to a subspace with a significantly reduced dimension and then solve for the projected solutions. Proper Orthogonal Decomposition (POD) is one of the more used ROM strategies. The presented work focuses on this technique, which has two challenging steps: (i) the snapshot location and (ii) the error estimate on the parameter space that drive the process to search new snapshot location. As a consequence of these two steps, POD applied to PDEs is considered as belonging to the well-known Greedy Algorithm family. This thesis brings a mesh adaptivity approach as the process to find the new parameter space locations. This process will be driven by a new error estimate based on Leave One Out Cross Validation (LOOCV) technique. We could say that this error estimate is universal, in the sense that it is not problem dependent. In addition, it is well known POD lack of accuracy when dealing with PDEs which solution contains shocks. Here, a new interpolation approach improves this, for shocked solutions. Finally, we present that the POD reduced basis is optimal just in average, and a new local basis is presented (Sorted Gram-Schmidt (SGS)) to be coupled with POD one. The criteria to decide which basis is better to be used for each new parameter value is defined as well. The whole proposed strategy efficiency is validated against a mathematical (exact) solutions of an incompressible, steady state flow equations, and on CFD solutions of an inviscid flow around a NACA0012 airfoil

    Adiabatic theory for slowly varying Hamiltonian systems with applications to beam dynamics

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    In questo lavoro si presentano due modelli in cui la teoria dell'invarianza adiabatica può essere applicata per ottenere peculiari effetti nel campo della dinamica dei fasci, grazie all'attraversamento di separatrici nello spazio delle fasi causato dal passaggio attraverso determinate risonanze d'un sistema. In particolare, si esporrà un modello bidimensionale con cui è possibile trasferire emittanza tra due direzioni nel piano trasverso: si fornirà una spiegazione del meccanismo per cui tale fenomeno ha luogo e si mostrerà come prevedere i valori finali di emittanza che un sistema raggiunge in tale configurazione. Questi risultati sono confermati da simulazioni numeriche. Simulazioni numeriche e relativi studî parametrici sono anche i risultati che vengono presentati per un altro modello, stavolta unidimensionale: si mostrerà infatti come un eccitatore esterno oscillante dipolare la cui frequenza passi attraverso un multiplo del tune della macchina permetta di catturare le particelle d'un fascio in un certo numero d'isole stabili

    Variational Time Integrators in Computational Solid Mechanics

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    This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton's principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether's theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether's theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does. A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark. In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.</p

    Calculus of Variations

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    Since its invention, the calculus of variations has been a central field of mathematics and physics, providing tools and techniques to study problems in geometry, physics and partial differential equations. On the one hand, steady progress is made on long-standing questions concerning minimal surfaces, curvature flows and related objects. On the other hand, new questions emerge, driven by applications to diverse areas of mathematics and science. The July 2012 Oberwolfach workshop on the Calculus of Variations witnessed the solutions of famous conjectures and the emerging of exciting new lines of research

    Divergence Measures

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    Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures

    Dynamical Systems Theory

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    The quest to ensure perfect dynamical properties and the control of different systems is currently the goal of numerous research all over the world. The aim of this book is to provide the reader with a selection of methods in the field of mathematical modeling, simulation, and control of different dynamical systems. The chapters in this book focus on recent developments and current perspectives in this important and interesting area of mechanical engineering. We hope that readers will be attracted by the topics covered in the content, which are aimed at increasing their academic knowledge with competences related to selected new mathematical theoretical approaches and original numerical tools related to a few problems in dynamical systems theory
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