B-Spline based uncertainty quantification for stochastic analysis

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided

O método dos elementos finitos generalizados aplicado com elementos triangulares na análise dinâmica de estruturas em estado plano

Orientador: Prof. Dr. Marcos ArndtDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa : Curitiba, 31/10/2022Inclui referênciasResumo: A análise dinâmica das estruturas é um tema atualmente relevante, haja vista que o avanço tecnológico tem permitido a execução de estruturas cada vez mais esbeltas, que estão diretamente suscetíveis aos efeitos dinâmicos. Os problemas de vibrações livre de estruturas em estado plano podem ser aproximados pelo Método dos Elementos Finitos Generalizados (MEFG), sendo que o foco do presente estudo e a apresentação de duas propostas de formulação de elementos mestres triangulares enriquecidos com funções de forma trigonométrica, pois a natureza oscilatória destas funções é caraterística das soluções destes tipos de problemas. A primeira proposta (chamada de MEFG-D) consiste na degeneração de um elemento quadrilateral já enriquecido e a segunda (chamada de MEFG-2) é feita através a proposição de novas funções de enriquecimento tomadas com base em metodologias de geração de espaço enriquecido para os elementos de barras. Relativo a análise modal de chapas em estado plano verifica-se que a segunda metodologia é mais vantajosa em relação a primeira, e quando comparada com os resultados obtidos pelo Método do Elementos Finitos Tradicional, o MEFG-2 produz melhores aproximações para o campo de frequências naturais situadas no início do espectro de frequências.Abstract: The dynamic analysis of structures is a relevant topic, given that advanced technology has allowed the execution of slender structures, which are more suitable to the dynamic effects. The in-plane vibration of structures can be approximated through Generalized Finite Element Method (GFEM). The purpose of this work is to present two triangular finite elements enriched with trigonometric functions, considering that oscillatory characteristic of this functions are common within these problems. The first methodology (it is called GFEM-D) is made from an enriched quadrilateral element. The second proposal (called GFEM-2) is done by generating new enrichment functions, based on the methodology of building enrichment functions for bar elements. To the in-plane vibration of plates, it is found that the GFEM-2 its better than GFEM-D. When compared to the traditional FEM, the GFEM-2 has a better approximation to the low natural frequency field. In this way, the method proves to be suitable for use in these types of problems

Stokesin yhtälöiden homogenisaatio rei'itetyssä alueessa

We consider homogenisation of the Stokes equations in a domain perforated with small holes, representing a porous medium. We propose an approach called the energy decomposition method, where the problem is divided into two components in different subspaces. One of these can be understood as representing the perturbation due to the microscale inhomogeneities. By finding an approximate solution to this subproblem, we are able to represent the effect of the inhomogeneities on the total problem. The method is simple to implement and does not presuppose a periodic distribution of the holes, making it a promising starting point for stochastic homogenisation.Tutkimus käsittelee Stokesin yhtälöiden homogenisaatiota alueessa, jossa on pieniä reikiä eli esteitä, joiden läpi virtaus ei pääse etenemään. Tilanne vastaa virtausta huokoisen aineen läpi. Energiahajotelmaksi kutsutussa menetelmässä Stokesin yhtälöitä vastaava minimointitehtävä jaetaan kahteen aliavaruuteen, joista toinen edustaa mikroskooppisten esteiden ratkaisuun aiheuttamaa häiriötä. Kun tämä alitehtävä ratkaistaan likimääräisesti, häiriö pystytään esittämään ratkaisematta alkuperäistä ongelmaa suoraan. Menetelmän toteutus on yksinkertainen, eikä se edellytä väliaineelta periodista rakennetta. Energiahajotelma vaikuttaakin lupaavalta lähtökohdalta stokastiselle homogenisaatiolle

Jubilea a zprávy

summary:Čižmár, Ján: Za profesorom Medekom. Adámek, Jiří: Rozloučení s Janem Reitermanem. Havela, L.; Sechovský, V.: Kolokvium o systémech s 5f elektrony. Nývlt, Miroslav: Mezinárodní konference o materiálovém výzkumu. Kováčik, V.; Sechovský, V.: Mezinárodní konference o systémech se silně korelovanými eletrony - SCES '92. Drábek, Pavel: 1st World Congress of Nonlinear Analysts. Lukáč, Pavel: Evropská vědecká konference: Základní aspekty dislokačních interakcí. Marek, Ivo: Zpráva o 4. mezinárodním sympoziu o numerické analýze (ISNA '92). Nožička, František: Zpráva o konferenci: Matematická optimalizace - teorie a aplikace v ekonomii a technice

Overview and evolution of CFD methods

Bakalářská práce je zaměřena na zmapování a charakteristiku CFD metod v jejich historickém kontextu. Porovná přednosti a nevýhody jednotlivých typů a zmíní oblasti, ve kterých byly dané metody využity.The bachelor thesis is focused on mapping and characterization of CFD methods in their historical context. It compares advantages and disadvantages of different types and mention the areas in which have been used these methods.

Relocation of the 1962 earthquake swarm in West Bohemia

Since the swarm at the turn of 1985/86 the current activity of seismic swarms in western Bohemia takes place mainly in the area of Nový Kostel at the eastern edge of the Cheb basin. However, the location of previous activities, where only more remote seismic stations were in operation, is not exactly known. This also applies to the swarm of 1962, which this bachelor thesis is focused on. Seismic bulletin data are available from eight seismic stations at a suitable distance for sufficient localization quality for this swarm. The theoretical part of the thesis presents research results of fundamental existing localization methods and their properties. In the practical part, the location of the earthquake epicenter of swarm 1962 is presented.Súčasná aktivita seizmických rojov v Západných Čechách sa odohráva predovšetkým v oblasti obce Nový Kostel na východnom okraji Chebskej panve. To platí od roja na prelome rokov 1985/86. Poloha predošlých aktivít, kedy boli v prevádzke len vzdialenejšie seizmické stanice, avšak nie je presne známa. To sa taktiež týka roju z roku 1962, ktorým sa táto bakalárska práca zaoberá a ku ktorému sú dostupné seizmické dáta z ôsmich staníc vo vhodnej vzdialenosti pre postačujúcu kvalitu lokalizácie. Teoretická časť práce predstavuje výsledky rešerše základných existujúcich metód lokalizácie a ich vlastnosti. V praktickej časti je mriežkovým prehľadávaním určené ohnisko zemetrasného roju 1962.Ústav hydrogeologie, inž. geologie a užité geofyzikyInstitute of Hydrogeology, Engineering Geology and Applied GeophysicsFaculty of SciencePřírodovědecká fakult

A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear Parabolic Problems

This Chapter aims to investigate the error estimation of numerical approximation to a class of semilinear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the finite element method for which the meshes are allowed to change in time. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto 2003, enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order in L∞H1 for Lipschitz and non-Lipschitz nonlinearities. In this Chapter, some challenges will be addressed to deal with nonlinear term by employing a continuation argument

A goal-oriented finite element method and its extension to pgd reduced-order modeling

RÉSUMÉ: Nous proposons une méthode éléments finis formulée pour des quantités d’intérêt. L’objectif est d’accroître la précision des solutions numériques pour ces quantités, choisies par l’utilisateur, sans pour autant perdre en précision globale. Les approches traditionnelles visant à contrôler l’erreur en quantité d’intérêt utilisent habituellement la solution d’un problème adjoint pour: (i) estimer l’erreur en quantité d’intérêt; et (ii) savoir comment adapter la discrétisation afin d’obtenir un espace éléments finis capable de mieux représenter les quantités d’intérêt de la solution. Ces approches s’inscrivent donc dans un procédé itératif de prédictions-corrections. Nous proposons d’utiliser cette même solution adjointe conjointement avec un probleme primal modifié, tel que sa solution soit ajustée à une valeur plus précise de la quantité d’intérêt. Ainsi, nous résolvons dans un espace qui est déjà adapté à la quantité d’intérêt. L’originalité de la présente approche consiste à utiliser la solution du problème adjoint non pas en tant que substitut de la solution exacte/référence pour l’estimation d’erreur et l’adaptation, mais en extrayant de celle-ci des valeurs des quantités d’intérêt extrêmement précises. Ces valeurs sont ensuite utilisées dans une minimisation sous contrainte de l’énergie (problème primal contraint) afin d’obtenir une solution plus précise en quantité d’intérêt. Ensuite, nous étendons cette approche en quantité d’intérêt à un contexte de réduction de modèles en utilisant la PGD. Ces méthodes reposent généralement sur des représentations spectrales, et sont de plus en plus utilisées pour simuler des problèmes en haute dimension. En ne considérant que les principaux modes propres de la solution, ces méthodes déjouent la malédiction de la dimensionnalité et rendent possibles des simulations auparavant inenvisageables.----------ABSTRACT: We present a finite element formulation of boundary-value problems that aims at constructing approximations specifically tailored for the estimation of quantities of interest of the solution, hence the name goal-oriented finite element method. The main idea is to formulate the problem as a constrained minimization problem that includes refined information in the goal functionals, so that the resulting model is capable of delivering enhanced predictions of the quantities of interest. This paradigm constitutes a departure from classical goal-oriented approaches in which one computes first the finite element solution and subsequently adapts the mesh via a greedy approach, by controlling error estimates measured in terms of quantities of interest using a posteriori dual-based error estimates. The formulation is then extended to the so-called Proper Generalized Decomposition method, an instance of model order reduction methods, with the aim of constructing reduced-order models tailored for the approximation of quantities of interest. Model order reduction methods aim at circumventing the curse of dimensionality arising from the high number of parameters of a given problem, by uncovering and/or exploiting lower dimensional structures present in the model or in the solution. Numerical examples are disseminated throughout the dissertation. They appear at the end of each of the three main chapters and Chapter 5 consists of an application example, namely a parametrized electrostatic cracked composite material

Variable Kinematic Finite Element Formulations Applied to Multi-layered Structures and Multi-field Problems

L'abstract è presente nell'allegato / the abstract is in the attachmen