Projection-based hyper-reduced order modeling of stress and reaction fields, and application of static condensation for multibody problems

Computational Mechanics' problems are often solved using the Finite Element Method (FEM). The resulting systems of equations may lead to large data and therefore, the solution requires high memory and time to be computed. This situation can be surpassed by applying Reduced Order Modeling (ROM) techniques, allowing the user to capture the system's dominant effects to build a high-fidelity reduced model that gives the possibility to predict and analyze the behaviour of a complex model using low computational resources within a micro time-step. This paper aims to enrich the already implemented Kratos' Rom Application with a reconstruction of the reaction and 2nd Piola Kirkchhoff stress fields. The applied methodology is a projection-based strategy using the Proper Orthogonal Decomposition together with a Gappy Data reconstruction technique. The gappy data comes from building a hyper-reduced order model (HROM). A surrogate model application using static condensation and HROM techniques is proposed to show the possibility of solving multibody systems interfacing Kratos' ROM framework with Mathworks control capabilities in a fast and accurate way. The validation of the applied methodology is given by 3D complex models

Espintensor de Lanczos de tipo O, N, III y conexiones con los campos de Liénard y Wiechert y la ecuación de movimiento de una partícula cargada

Se obtienen los espintensores de Lanczos de tipo O, N, Y III en clasificación Petrov. Al encontrarse el resultado en forma general para tres tipos Petrov se proponen métodos para encontrar todos los demás tipos y se rescata la idea de utilizar el superpotencial de Lanczos corno una herramienta matemática práctica, en particular para los problemas de la electrodinámica clásica. El superpotencial de Weert para la par te acatada del campo de Liénard-Wiechert es deducido. Se deja abierta la posibilidad de estudiar el problema de la ecuación de movimiento de una partícula cargada utilizando los resultados anteriores

C*-Correspondences, Hilbert Bimodules, and their L^p Versions

This dissertation initiates the study of $L^p$-modules, which are modules over $L^p$-operator algebras inspired by Hilbert modules over C*-algebras. The primary motivation for studying $L^p$-modules is to explore the possibility of defining $L^p$ analogues of Cuntz-Pimsner algebras. The first part of this thesis consists of investigating representations of C*-correspondences on pairs of Hilbert spaces. This generalizes the concept of representations of Hilbert bimodules introduced by R. Exel in \cite{Exel1993}. We present applications of representing a correspondence on a pair of Hilbert spaces (\Hi_0, \Hi_1), such as obtaining induced representations of both \Li_A(\X) and \mathcal{K}_A(\X) on \Hi_1, and giving necessary and sufficient conditions on an $(A,B)$ C*-correspondences to admit a Hilbert $A$-$B$-bimodule structure. The second part is concerned with the theory of $L^p$-modules. Here we present a thorough treatment of $L^p$-modules, including morphisms between them and techniques for constructing new $L^p$-modules. We then useour results on representations for C*-correspondences to motivate and develop the theory of $L^p$-correspondences, their representations, the $L^p$-operator algebras they generate, and present evidence that well-known $L^p$-operator algebras can be constructed from $L^p$-correspondences via $L^p$-Fock representations. Due to the technicality that comes with dealing with direct sums of $L^p$-correspondences and interior tensor products, we only focus on two particular examples for which a Fock space construction can be carried out. The first example deals with the $L^p$-module $(\ell_d^p, \ell_d^q)$, for which we exhibit a covariant $L^p$-Fock representation that yields an $L^p$-operator algebra isometrically isomorphic to $\mathcal{O}_d^p$, the $L^p$-analogue of the Cuntz-algebra $\mathcal{O}_d$ introduced by N.C. Phillips in \cite{ncp2012AC}. The second example involves a nondegenerate $L^p$-operator algebra $A$ with a bicontractive approximate identity together with an isometric automorphism \varphi_A \in \op{Aut}(A). In this case, we also present an algebra associated to a covariant $L^p$-Fock representation, but due to the current lack of knowledge of universality of the $L^p$-Fock representation, we only show that there is a contractive map from the crossed product $F^p(\Z, A, \varphi_A)$ to this algebra. This dissertation includes unpublished material

CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We present the Continuous Empirical Cubature Method (CECM), a novel algorithm for empirically devising efficient integration rules. The CECM aims to improve existing cubature methods by producing rules that are close to the optimal, featuring far less points than the number of functions to integrate. The CECM consists on a two-stage strategy. First, a point selection strategy is applied for obtaining an initial approximation to the cubature rule, featuring as many points as functions to integrate. The second stage consists in a sparsification strategy in which, alongside the indexes and corresponding weights, the spatial coordinates of the points are also considered as design variables. The positions of the initially selected points are changed to render their associated weights to zero, and in this way, the minimum number of points is achieved. Although originally conceived within the framework of hyper-reduced order models (HROMs), we present the method's formulation in terms of generic vector-valued functions, thereby accentuating its versatility across various problem domains. To demonstrate the extensive applicability of the method, we conduct numerical validations using univariate and multivariate Lagrange polynomials. In these cases, we show the method's capacity to retrieve the optimal Gaussian rule. We also asses the method for an arbitrary exponential-sinusoidal function in a 3D domain, and finally consider an example of the application of the method to the hyperreduction of a multiscale finite element model, showcasing notable computational performance gains. A secondary contribution of the current paper is the Sequential Randomized SVD (SRSVD) approach for computing the Singular Value Decomposition (SVD) in a column-partitioned format. The SRSVD is particularly advantageous when matrix sizes approach memory limitations

Digital deposition of yttria patterns on titanium sheets

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1997.Includes bibliographical references (leaf 77).The ability to apply Yttria patterns on titanium sheets is required on a secondary manufacturing operation. The freedom to create 2 dimensional patterns as well as 3 dimensional ones by using Three Dimensional Printing allows for the application of Yttria patterns onto these sheets without the use of any type of screens. Two methods for creating these patterns were identified and studied. The first approach consists of selectively printing binder on top of spread layer of powder. A second layer of powder is spread while the binder is still wet. The binder will then dry fixing the Yttria powder to the sheet on the selected positions. Sheets were printed using different particle sizes. Results show that patterns can be applied with good edge definition and uniform thickness using powder ranging from -53[mu]m down to -20[mu]m. The sheets were tested successfully in the secondary operation. And alternative approach consists of mixing the binder and the ceramic powder to create a slurry. The slurry will then be selectively printed onto the sheet using Three Dimensional Printing. A formulation for a -20 [mu]m Yttria powder slurry was developed using Poly Acrylic Acid as dispersant and Polyethylene Glycol as binder. Slurries with 20 volume fraction and less were dispersed and jetted through a 102 [mu]m nozzle. These slurries adhered well to the titanium sheet as they were printed. The formulation was tested successfully in the secondary operation. Following the High-Risk Approach patterns without deflection were printed. For a 102 [mu]m nozzle the best flow rate was determined, as well as the optimal line spacing. The best procedure to print a certain area was to print a first round of lines, dry them and then print lines in between the first set. The average roughness of the layer printed was 14 [mu]m. In order to allow future printing of slurries with a nozzle size of 102 [mu]m and deflection, the design of a print head to accommodate these slurries was also investigated.by Gabriel Fernandez.S.M

Second order equation of motion for electromagnetic radiation back-reaction

We take the viewpoint that the physically acceptable solutions of the Lorentz--Dirac equation for radiation back-reaction are actually determined by a second order equation of motion, the self-force being given as a function of spacetime location and velocity. We propose three different methods to obtain this self-force function. For two example systems, we determine the second order equation of motion exactly in the nonrelativistic regime via each of these three methods, the three methods leading to the same result. We reveal that, for both systems considered, back-reaction induces a damping proportional to velocity and, in addition, it decreases the effect of the external force.Comment: 13 page

Los sistemas de información geográfica (SIG) como herramienta de análisis de la información contenida en el cartilario de Teobaldo I, rey de Navarra (documento 316 del cartulario 1): Tudela y su espacio periurbano

Póster presentado al II Encuentro Internacional de Medievalistas organizado por el Departamento de Geografía e Historia de la Universidad Pública de Navarra, con la colaboración de la Sociedad de Estudios Históricos de Navarra. Pamplona, 21-22 de noviembre de 2013.Este trabajo parte de las investigaciones desarrolladas en la tesis de Mercedes Goñi sobre los usos del suelo en el espacio periurbano de Tudela en época bajomedieval, y de las tareas desarrolladas en el proyecto de investigación dirigido desde la Universidad Pública de Navarra “Espacios de la memoria. Los Cartularios regios de Navarra: construcción y expresión del poder”. En este sentido, una de las preocupaciones principales tanto del proyecto como de la tesis en curso, es la plasmación espacial y cartográfica de los datos contenidos en la documentación medieval navarra, y más en concreto en los cartularios regios del Archivo General de Navarra. En este caso, partimos de los datos contenidos en el primer cartulario de los reyes de Navarra (C1 del Archivo General de Navarra), para presentar a grandes rasgos las posibilidades que ofrece un análisis cartográfico de la documentación medieval, utilizando una cartografía precisa, georreferenciada (Sistemas de Información Geográfica) y un análisis profundo de la toponimia histórica en relación con la actual. Ha de entenderse –en cualquier caso- que la delimitación de los espacios que aquí se realiza supone una hipótesis de trabajo y no una certeza documentalmente probada. En todo caso, creemos que con esta metodología podemos ofrecer nuevas formas de análisis y conocimiento de la realidad social y económica de la Navarra medieval, y una nueva manera de interpretar e interrogar a unas fuentes tan utilizadas como son los cartularios reales

Hyper-reduction for Petrov-Galerkin reduced order models

Projection-based Reduced Order Models minimize the discrete residual of a "full order model" (FOM) while constraining the unknowns to a reduced dimension space. For problems with symmetric positive definite (SPD) Jacobians, this is optimally achieved by projecting the full order residual onto the approximation basis (Galerkin Projection). This is sub-optimal for non-SPD Jacobians as it only minimizes the projection of the residual, not the residual itself. An alternative is to directly minimize the 2-norm of the residual, achievable using QR factorization or the method of the normal equations (LSPG). The first approach involves constructing and factorizing a large matrix, while LSPG avoids this but requires constructing a product element by element, necessitating a complementary mesh and adding complexity to the hyper-reduction process. This work proposes an alternative based on Petrov-Galerkin minimization. We choose a left basis for a least-squares minimization on a reduced problem, ensuring the discrete full order residual is minimized. This is applicable to both SPD and non-SPD Jacobians, allowing element-by-element assembly, avoiding the use of a complementary mesh, and simplifying finite element implementation. The technique is suitable for hyper-reduction using the Empirical Cubature Method and is applicable in nonlinear reduction procedures