108 research outputs found

    Data-driven exact model order reduction for computational multiscale methods to predict high-cycle fatigue-damage in short-fiber reinforced plastics

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    Motiviert durch die Entwicklung energieeffizienterer Maschinen und Transportmittel hat der Leichtbau in den letzten Jahren enorm an Wichtigkeit gewonnen. Eine wichtige Klasse der Leichtbaumaterialien sind die faserverstärkten Kunststoffe. In der vorliegenden Arbeit liegt der Fokus auf der Entwicklung und Bereitstellung von Materialmodellen zur Vorhersage des Ermüdungsverhaltens kurzglasfaserverstärkter Thermoplaste. Diese Materialien unterscheiden sich dabei durch ihre Aufschmelzbarkeit und ihrer damit einhergehenden besseren Recyclebarkeit von thermosetbasierten Materialien. Außerdem erlauben die Kurzglasfasern im Gegensatz zu Langfasern eine einfache und zeiteffiziente Herstellung komplexer Komponenten. Ermüdung ist ein wichtiger Versagensmechanismus in solchen Komponenten, insbesondere für Bauteile z.B. in Fahrzeugen, die vibrationsartigen Belastungen ausgesetzt sind. Durch die inherente Anisotropie des Materials sind die experimentelle Charakterisierung und Vorhersage dieses Versagensmechanismus jedoch äußerst zeitintensiv und stellen somit eine wesentliche Herausforderung im Entwicklungsprozess und für die breitere Anwendung solcher Bauteile dar. Daher ist die Entwicklung komplementärer simulativer Methoden von großem Interesse. Im Rahmen dieser Arbeit werden Methoden zur Vorhersage der Ermüdungsschädigung kurzglasfaserverstärkter Werkstoffe im Rahmen einer Multiskalenmethode entwickelt. Die in der Arbeit betrachteten Multiskalenmodelle bieten die Möglichkeit, allein anhand der experimentellen Charakterisierungen der Materialparameter der Konstituenten, d.h. Faser und Matrix, komplexe anisotrope Effekte des Verbundmaterials vorherzusagen. Der experimentelle Aufwand kann dadurch enorm reduziert werden. Dazu werden zunächst Materialmodelle für die Konstituenten des Komposits entwickelt. Mithilfe FFT-basierter rechnergestützter Homogenisierung wird daraus das Materialverhalten des Komposits für verschiedene Mikrostrukturen und Lastfälle vorhergesagt. Die vorberechneten Lastfälle auf Mikrostrukturebene werden mit datengetriebenen Methoden auf die Makroskala übertragen. Das ermöglicht eine effiziente Berechnung von Bauteilen in wenigen Stunden, wohingegen eine entsprechende Berechnung mit geometrischer Auflösung aller einzelnen Fasern der Mikrostruktur auf heutigen Computern viele Jahre dauern würden. Für die Matrix werden unterschiedliche Schädigungsmodelle untersucht. Ihre Vor- und Nachteile werden analysiert. Die Mikrostruktursimulationen geben einen Einblick in den Einfluss verschiedener statistischer Parameter wie Faserlängen und Faservolumengehalt auf das Kompositverhalten. Ein neues Modellordnungsreduktionsverfahren wird entwickelt und zur Simulation des Ermüdungsschädigungsverhaltens auf Bauteilebene angewandt. Weiter werden Modellerweiterungen zur Berücksichtigung des R-Wert-Verhältnisses und viskoelastischer Effekte in der Evolution der Ermüdungsschädigung entwickelt und mit experimentellen Ergebnissen validiert. Das entstandene Simulationsframework erlaubt nach Vorrechnungen auf einer geringen Menge von Mikrostrukturen und Lastfällen eine effiziente Makrosimulation eines Bauteils vorzunehmen. Dabei können Effekte wie Viskoelastizität und R-Wert-Abhängigkeit je nach gewünschter Modellierungstiefe berücksichtigt oder vernachlässigt werden, um immer das effizientste Modell, das alle relevanten Effekte abbildet, nutzen zu können

    Análisis de sustentabilidad: caso de estudio sistemas agroproductivos manejados por Afroecuatorianos, en la zona norte de Ecuador

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    In the northern zone provinces of Ecuador, Carchi, Imbabura, Esmeraldas and Sucumbíos, there are Afro-Ecuadorian ethnicity producers who are dedicated to agricultural production, with conventional agricultural large scale methods such as monoculture, an intensive use and mismanagement of supplies and pesticides, with poor agricultural practices, they cause the ecosystems deterioration and wastage and the environmental contamination, being necessary to determine the sustainability indices in the agro-productive systems to the establishment of measures aimed at the protection of ecosystems, looking to improve the productivity levels in their territory. This research aimed to determine the sustainability of Afro-Ecuadorian agro-productive systems in the northern zone; The information gathering was performed in field visits through surveys in the communities of Mira Canton, Carchi Province, in Ibarra Canton, Imbabura Province, in San Lorenzo Canton, Esmeraldas Province and in Lago Agrio Canton, Sucumbíos Province. Through the SAFA software using, a sustainability analysis determined in four dimensions was established: good governance, environmental integrity, economic resilience and social well-being, creating a sustainability polygon, showing a limited level of sustainability in all dimensions with an average of 11.72% in the North Zone, regarding to Carchi Province an average of 12.97% is established, followed by Esmeraldas with 12.28%, Imbabura with 11.32% and Sucumbíos with 10.33%. The sustainability levels determine that “these agro-productive systems are not sustainable enough in economic, social, environmental and governance aspects”; due to they show a limited average of 11.04%, requiring a specific intervention plan for each province where the governmental agencies and/or non-governmental organizations intervene in order to improve the sustainability of the agro-productive systems in the Afro-Ecuadorian communities of the northern area of the country.En las provincias de la Zona Norte de Ecuador; Carchi, Imbabura, Esmeraldas y Sucumbíos, existen pobladores de la etnia afroecuatoriana que se dedican a la producción agropecuaria, con el uso a gran escala de prácticas agrícolas convencionales como el monocultivo, el uso intensivo y mal manejo de insumos y plaguicidas, y junto a las malas prácticas agrícolas ocasionan el deterioro y desgaste de los ecosistemas y la contaminación ambiental, siendo necesaria la determinación de los índices de sostenibilidad en los sistemas agroproductivos para establecer medidas encaminadas a la protección de los ecosistemas, con la intención de mejorar los niveles de productividad en su territorio. Esta investigación tuvo como objetivo determinar la sostenibilidad de los sistemas agroproductivos de los afroecuatorianos en la zona norte; el levantamiento de información fue realizado en visitas de campo a través de encuestas en las comunidades del Cantón Mira de la Provincia de Carchi, en el Cantón Ibarra de la provincia de Imbabura, en el Cantón San Lorenzo de la provincia de Esmeraldas y en el Cantón Lago Agrio en la provincia de Sucumbíos. Mediante el uso del software SAFA se estableció un análisis de sostenibilidad determinado en cuatro dimensiones: buena gobernanza, integridad ambiental, resiliencia económica y bienestar social, generándose un polígono de sostenibilidad. Asimismo, se identificó un nivel de sostenibilidad limitado en todas las dimensiones, con un promedio de 11,72% en la Zona Norte. En lo concerniente a la provincia de Carchi se establece un promedio de 12,97% seguido por Esmeraldas con un 12,28%, Imbabura con un 11,32% y Sucumbíos con un 10,33%. Los niveles de sostenibilidad determinan que los sistemas agroproductivos no son sustentables en aspectos económicos, sociales, ambientales y de gobernanza; ya que presentan un promedio limitado de 11,04%, siendo necesario un plan de intervención específico para cada provincia en el cual participen organismos del estado y/o no gubernamentales con la finalidad de mejorar la sostenibilidad de los sistemas agroproductivos en las comunidades afroecuatorianas de la zona norte del país

    Fast boundary element methods for the simulation of wave phenomena

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    This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) für ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nützliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten führt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren Einträge die Integration singulärer Kernfunktionen über Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln für die schwierigsten Fälle. Dies ermöglicht die genaue Berechnung der Matrixeinträge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation präsentiert, der die Matrizen komprimiert und die Komplexität der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der räumlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die überlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

    Nonconforming Virtual Element basis functions for space-time Discontinuous Galerkin schemes on unstructured Voronoi meshes

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    We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a nonconforming Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an L2 projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrarily to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as typically employed in DG discretizations. To improve the condition number of the resulting mass matrix, an orthogonalization of the full basis is proposed. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the virtual basis in space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof-dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-time Galerkin finite element predictor to be evaluated. The novel methods are referred to as VEM-DG schemes, and they are arbitrarily high order accurate in space and time. The new VEM-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier-Stokes equations. Numerical results are verified with respect to literature reference solutions and compared in terms of accuracy and computational efficiency to those obtained using a standard modal DG scheme with Taylor basis functions. An analysis of the condition number of the mass and space-time stiffness matrix is also forwarded

    Lattice Boltzmann Methods for Partial Differential Equations

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    Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions

    A new family of semi-implicit Finite Volume / Virtual Element methods for incompressible flows on unstructured meshes

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    We introduce a new family of high order accurate semi-implicit schemes for the solution of non-linear hyperbolic partial differential equations on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is achieved using the semi-implicit IMEX Runge-Kutta schemes, and the novel schemes are proven to be asymptotic preserving and well-balanced. As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier-Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids

    Uncertainty quantification and numerical methods in charged particle radiation therapy

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    Radiation therapy is applied in approximately 50% of all cancer treatments. To eliminate the tumor without damaging organs in the vicinity, optimized treatment plans are determined. This requires the calculation of three-dimensional dose distributions in a heterogeneous volume with a spatial resolution of 2-3mm. Current planning techniques use multiple beams with optimized directions and energies to achieve the best possible dose distribution. Each dose calculation however requires the discretization of the six-dimensional phase space of the linear Boltzmann transport equation describing complex particle dynamics. Despite the complexity of the problem, dose calculation errors of less than 2% are clinically recommended and computation times cannot exceed a few minutes. Additionally, the treatment reality often differs from the computed plan due to various uncertainties, for example in patient positioning, the acquired CT image or the delineation of tumor and organs at risk. Therefore, it is essential to include uncertainties in the planning process to determine a robust treatment plan. This entails a realistic mathematical model of uncertainties, quantification of their effect on the dose distribution using appropriate propagation methods as well as a robust or probabilistic optimization of treatment parameters to account for these effects. Fast and accurate calculations of the dose distribution including predictions of uncertainties in the computed dose are thus crucial for the determination of robust treatment plans in radiation therapy. Monte Carlo methods are often used to solve transport problems, especially for applications that require high accuracy. In these cases, common non-intrusive uncertainty propagation strategies that involve repeated simulations of the problem at different points in the parameter space quickly become infeasible due to their long run-times. Quicker deterministic dose calculation methods allow for better incorporation of uncertainties, but often use strong simplifications or admit non-physical solutions and therefore cannot provide the required accuracy. This work is concerned with finding efficient mathematical solutions for three aspects of (robust) radiation therapy planning: 1. Efficient particle transport and dose calculations, 2. uncertainty modeling and propagation for radiation therapy, and 3. robust optimization of the treatment set-up

    Applications of Special Functions in High Order Finite Element Methods

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    In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

    High-Order Numerical Integration on Domains Bounded by Intersecting Level Sets

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    We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on hypercubes to the curved domains of the integrals. This enables the numerical integration of a wide range of integrands since integration on hypercubes is a well known problem. The mappings are constructed by treating the isocontours of the level sets as graphs of height functions. Numerical experiments with smooth integrands indicate a high-order of convergence for transformed Gauss quadrature rules on domains defined by polynomial, rational, and trigonometric level sets. We show that the approach we have used can be combined readily with adaptive quadrature methods. Moreover, we apply the approach to numerically integrate on difficult geometries without requiring a low-order fallback method

    Computational nonlinear vibration analysis for distributed geometrical nonlinearities in structural dynamics

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    The demand to reduce the impact of aviation on the environment is leading jet engine manu- facturers to increase the fuel and propulsion efficiency of the engines. This in turn is pushing materials to their physical limits by undergoing increasingly higher thermo-mechanical loads. In this regime, blades and other engine components are subjected to large deforma- tions generating nonlinearities that activate new failure mechanisms not dealt with before. Therefore, vibration analysis is essential to develop new methodologies for the accurate prediction of components’ behaviour. This research focuses on investigating the effect of the distributed geometric nonlinearities and rotational speed on the dynamic behaviour of three-dimensional structures. The Green-Lagrange strain measures are employed in this research to express the nonlinear relationship between the displacement and the strain. The nonlinear algorithms used for the numerical simulations are developed based on the Finite Element Method combined with the Harmonic Balance method. The complex geometries are discretised by using the geometric exact three-dimensional solid elements. The forced response functions and the backbone curves for the steady-state response of the nonlinear system can be computed. The research aims to develop and validate methodologies for the identification and control of undesired vibration modes which will inform new design choices. Finite element modelling of the blades generally involves an immense number of degree-of-freedoms, which could be infeasible to compute. The reduced order modelling (ROM) techniques are crucial for achieving an accurate prediction of the nonlinear behaviour in an efficient way. Detailed computation strategies for the intrusive ROM methods are delivered. ROM techniques based on the linear and nonlinear mapping between the full model and the reduced basis are presented. The capabilities and limitations of both methods are assessed. The projection method based on the linear eigenmodes only has a slow converge to the full system. On the other hand, the quadratic manifold method with the static modal derivatives involved in the reduced coordinates provides a fast convergence.Open Acces
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