Improvements to isogeometric blended shells are introduced which blend traditional Reissner-Mindlin shells, and Kirchhoff-Love shells, with an exact interpolation of the shell director increment. A gradient extraction operator is introduced which allows derivatives of basis functions to be exactly expressed as a linear combination of the basis functions themselves. Several benchmarks are investigated and the new blended shell is compared with different shell elements in ABAQUS and NASTRAN. In addition, the effect of different quadrature schemes is included in the comparisons. The new isogeometric blended shell performs comparably in some benchmarks, and even outperforms commercial shell finite elements in some benchmarks. Future improvements to the formulation are discussed.

Tese de mestrado integrado. Engenharia Mecânica. Faculdade de Engenharia. Universidade do Porto. 2012

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnologico / Made available in DSpace on 2016-01-08T19:17:06Z (GMT). No. of bitstreams: 1 99944.pdf: 1541003 bytes, checksum: 1ce956e07d2631dcb9abbc0ea25132e0 (MD5) Previous issue date: 1994 / Formulação e análise de problemas de contato em placas utilizando o Método dos Elementos Finitos. A teoria de placas semi-espessas ou teoria de Mindlin-Reissner é utilizada, e a versão lagrangiana atualizada do princípio dos trabalhos virtuais é adotada para o tratamento da não-linearidade advinda do fato da estrutura poder sofrer grandes deflexões. A desigualdade variacional, característica do problema de contato, é resolvida fazendo-se uso do método da penalidade.

Es werden die vorgestellten Fehlerindikatoren in Bezug auf die Netzsteuerung anhand von drei Beispielen analysiert. Im weiteren werden auch die einzelen MITC-Elemente und ihre Besonderheiten bei dieser Analyse der Netzsteuerung mit berücksichtigt. Als Abschluss werden einige spezielle Fehlerindikatoren vorgestellt, die für die weitere Entwicklung einige interessante Eigenschaften aufzeigen. Im zweiten Teil geht es um die Auswertung mit dem speziellen Ziel der Findung einer optimalen Netzsteuerung. Dabei wird auf die Besonderheiten der Elemente eingegangen sowie auf die Plattendicke und auf ihre Wirkung bei den Fehlerindikatoren. Mit diesen Erkenntnissen wird ein spezieller Fehlerindikator vorgestellt, der die Vorteile aller Fehlerindikatoren aus Teil I vereint.

Within this thesis, thin walled shell structures are discussed with modern element formulationsin the context of the Isogeometric Analysis (IGA). IGA was designed to achieve a directinterface from CAD to analysis. According to the concept of IGA, Non-Uniform RationalB-Splines (NURBS) are used as shape functions in the design and the analysis. Dependingon the polynomial order, NURBS can come along with a high order continuity. Therefore,the curvature of a shell surface can be described directly by the shape function derivativeswhich is not possible within the classical Finite Element Analysis (FEA) using linear meshes.This description of the curvature gives rise to the application of the Kirchho-Love shellformulation, which describes the curvature stiness with the dierentiation of the spatialdegrees of freedom. Based upon this, the formulation can be enhanced with further kinematicalexpressions as the shear dierence vector, which leads to a 5-parameter Reissner-Mindlinformulation. This kinematic formulation is intrinsically free from transverse shear lockingdue to the split into Kirchho-Love and additional shear contributions. The formulation canbe further extended to a 7-parameter three-dimensional shell element, which considers volumetriceects in the thickness direction. Two additional parameters are engaged to describethe related thickness changes under load and to enable the use of three-dimensional materiallaws. In general, three-dimensional shell elements suer from curvature thickness and Poisson'sthickness locking. However, these locking phenomena are intrinsically avoided by thehierarchic application of the shear dierence vector and the 7th parameter respectively. The3-parameter Kirchho-Love, the 5-parameter Reissner-Mindlin and the 7-parameter 3D shellelement build a hierarchic family of model-adaptive shells.This hierarchic family of shell elements is presented and discussed in the scope of this thesis.The concept and the properties of the single elements are elaborated and the dierences arediscussed. Geometrically linear and non-linear benchmark examples are simulated. Convergencestudies are performed and the results are validated against analytical solutionsand solutions from literature, taking into account deections and internal forces. Furthermore,the dierent locking phenomena which occur in analyses with shell formulations areexamined. Several test cases are designed to ensure a validated implementation of the hierarchicshell elements. The element formulations and further pre- and postprocessing featuresare implemented and validated within the open-source software environment Kratos Multi-physics.

This thesis is about the development of a new numerical method for the analysis of composite shells. The present work is based on Reissner Mixed Variational Theorem (RMVT), the Sublaminate Generalized Unified Formulation (S-GUF), and the Ritz approximation. The present work investigates a more efficient way to compute transverse stresses (sigma_xz, sigma_yz, sigma_zz) based upon RMVT, allowing assigning their order of continuity a priori. This is a great advantage compared to a conventional displacement-based approach. In order to enable computing of both global and local responses (depending on the user’sneeds) the S-GUF framework was adopted. The Generalized Unified Formulation (GUF) enables the implementation of approximations with virtually unlimited algebraic order within a single code, and the order could also vary for different variables. In addition to the GUF, the concept of Sublaminate was utilized, allowing for sub-sectioning of the domain in the thickness direction into sublaminates, and it is then possible to apply different formulations in each of these sub-domains. The curvature of the shells is strictly defined by their radius-to- thickness ratio. The flexibility of S-GUF is helpful in the sense that curvature is only introduced and treated if needed by the particular case at hand. The governing equations obtained applying S-GUF to RMVT were solved in a weak formulation using the Ritz approximation. This choice was made to save computational time, which is one of the main benefits of the presented method. Validation of the code was made by comparing results from the present formulation with solutions available in the literature. Good to excellent agreement was found for several benchmark cases, supporting that the formulation is valid and provides reliable solutions.Finally, numerical and analytical considerations about the developed method were made: its numerical stability, how to tune its parameters, and which models result more correct from an analytical standpoint. / Denna avhandling handlar om utvecklingen av en ny numerisk metod för analys av kompositskal.Det aktuella arbetet är baserat på Reissner Mixed Variational Theorem (RMVT), Sublaminate Generalized Unified Formulation (S-GUF) och Ritz-approximationen.Arbetet går ut på att ta fram ett mer effektivt sätt att beräkna spänningar ut ur planet (sigma_xz, sigma_yz, aigma_zz) och utnyttjar RMVT, vilket möjliggör lokal hantering av kontinuitet i varierande ordning, definierad a priori. Detta innebär en stor fördel jämfört med en konventionell förskjutningsbaserad metod. För att möjliggöra beräkning av både global och lokal lösning, beroende påanvändarens behov, antogs S-GUF-ramverket. Den generaliserade enhetliga formuleringen (GUF) gördet möjligt att inom samma formulering implementera approximationer med i princip obegränsad algebraisk ordning, vilka dåocksåkan skilja mellan olika variabler. Förutom GUF används även konceptet Sublaminate som gör det möjligt att dela upp domänen i tjockleksriktningen i underregioner (sublaminate), och det är dåmöjligt att tillämpa olika formuleringar i var och en av dessa subdomäner. Krökningen hos ett skal definiers strikt av förhållandet mellan dess radie och tjocklek. Flexibiliteten hos S-GUF är fördelaktig dåkrökning endast hanteras för de specifika fall där det förekommer. De ekvationer som erhålls genom att applicera S-GUF på RMVT löses påsvag formmed användning av Ritz approximation. Detta val gjordes för att möjliggöra en snabbare beräkningstid, vilket är en av fördelarna med denna metod. Genom att jämföra de resultat med lösningar tillgängliga i litteraturen var det möjligt att validera resultaten och därmed även själva formuleringen. God till utmärkt överensstämmelse påvisades för ett antal olika standardfall vilket styrker att metoden fungerar och attdess lösningar är pålitliga. Slutligen ritades numeriska och analytiska överväganden om metoden här utvecklad, såsom dess numeriska stabilitet, hur man ställer in dess parametrar och vilka modeller somär mer korrekta ur en analytisk synvinkel.

Este trabalho fornece uma contribuição à análise estrutural não-linear de pavimentos de edifícios de concreto armado com o emprego do Método dos Elementos Finitos. A deformação por esforço cortante é considerada, portanto as teorias de Timoshenko e de Reissner-Mindlin são empregadas nas formulações dos elementos de barra e de placa, respectivamente. As posições dos elementos de barra e de placa são independentes e, portanto, podem ser definidas em diferentes planos. Em conseqüência do exposto, o efeito de membrana deve ser necessariamente considerado na modelagem do pavimento. Para completar o modelo mecânico, as não-linearidades físicas descrevem o comportamento do concreto e do aço. A deterioração do concreto no cisalhamento é também considerada através de um modelo simplificado que é proposto para a modelagem do cisalhamento em condições de serviço. / This work gives a contribution to the non-linear structural analysis of reinforced concrete buildings floors using the Finite Element Method. The shear strain components are taken into account by adopting the Timoshenko\'s beam theory together with and the Reissner-Mindlin\'s theory for plate bending. Bar and plate element position are independent and therefore can be defined at different planes. As several level are considered when defining the structure membrane effects are necessary considered. In order to complete the mechanical model, physical non-linearities are also assumed to describe concrete and steel behaviours. The deterioration of the concrete material in shear is also taken account. For this purpose, a simplified model is adopted to compute approximately the damaged shear component in the steel direction.

No description available.

Este trabalho fornece uma contribuição à análise estrutural não-linear de pavimentos de edifícios de concreto armado com o emprego do Método dos Elementos Finitos. A deformação por esforço cortante é considerada, portanto as teorias de Timoshenko e de Reissner-Mindlin são empregadas nas formulações dos elementos de barra e de placa, respectivamente. As posições dos elementos de barra e de placa são independentes e, portanto, podem ser definidas em diferentes planos. Em conseqüência do exposto, o efeito de membrana deve ser necessariamente considerado na modelagem do pavimento. Para completar o modelo mecânico, as não-linearidades físicas descrevem o comportamento do concreto e do aço. A deterioração do concreto no cisalhamento é também considerada através de um modelo simplificado que é proposto para a modelagem do cisalhamento em condições de serviço. / This work gives a contribution to the non-linear structural analysis of reinforced concrete buildings floors using the Finite Element Method. The shear strain components are taken into account by adopting the Timoshenko\'s beam theory together with and the Reissner-Mindlin\'s theory for plate bending. Bar and plate element position are independent and therefore can be defined at different planes. As several level are considered when defining the structure membrane effects are necessary considered. In order to complete the mechanical model, physical non-linearities are also assumed to describe concrete and steel behaviours. The deterioration of the concrete material in shear is also taken account. For this purpose, a simplified model is adopted to compute approximately the damaged shear component in the steel direction.

Cette thèse présente des travaux effectués dans le cadre de l'optimisation de forme de pièces mécaniques, sous critère dynamique, par approche isogéométrique. Pour réaliser une telle optimisation nous mettons en place dans un premier temps les éléments coque au travers des formulations Kirchhoff-Love puis Reissner-Minlin. Nous présentons une méthode permettant d'atteindre les vecteurs normaux aux fibres dans ces formulations au travers de l'utilisation d'une grille mixte de fonctions de base interpolantes, traditionnellement utilisées en éléments finis, et de fonction non interpolantes issues de la description isogéométrique des coques. Par la suite, nous détaillons une méthode pour le couplage de patch puis nous mettons en place la méthode de synthèse modale classique dans le cadre de structures en dynamique décrites par des éléments isogéometriques. Ce travail établit une base pour l'optimisation de forme sous critères dynamique de telles structures. Enfin, nous développons une méthode d'optimisation de forme basée sur le calcul du gradient de la fonction objectif envisagée. La sensibilité de conception est extraite de l'analyse de sensibilité au niveau même du maillage du modèle, qui est obtenue par l'analyse discrète de sensibilité. Des exemples d'application permettent de montrer la pertinence et l'exactitude des approches proposées. / Isogeometric method is a promising method in bridging the gap between the computer aided design and computer aided analysis. No information is lost when transferring the design model to the analysis model. It is a great advantage over the traditional finite element method, where the analysis model is only an approximation of the design model. It is advantageous for structural optimization, the optimal structure obtained will be a design model. In this thesis, the research is focused on the fast three dimensional free shape optimization with isogeometric shell elements. The related research, the development of isogeometric shell elements, the patch coupling in isogeometric analysis, the modal synthesis with isogeometric elements are also studied. We proposed a series of mixed grid Reissner-Minlin shell formulations. It adopts both the interpolatory basis functions, which are from the traditional FEM, and the non-interpolatory basis functions, which are from IGA, to approximate the unknown elds. It gives a natural way to define the fiber vectors in IGA Reissner-Mindlin shell formulations, where the non-interpolatory nature of IGA basis functions causes complexity. It is also advantageous for applying the rotational boundary conditions. A modified reduce quadrature scheme was also proposed to improve the quadrature eficiency, at the same time, relieve the locking in the shell formulations. We gave a method for patch coupling in isogeometric analysis. It is used to connect the adjacent patches. The classical modal synthesis method, the fixed interface Craig-Bampton method, is also used as well as the isogeometric Kirchhoff-Love shell elements. The key problem is also the connection between adjacent patches. The modal synthesis method can largely reduce the time costs in analysis concerning structural dynamics. This part of work lays a foundation for the fast shape optimization of built-up structures, where the design variables are only relevant to certain substructures. We developed a fast shape optimization framework for three dimensional thin wall structure design. The thin wall structure is modelled with isogeometric Kirchhoff-Love shell elements. The analytical sensitivity analysis is the key focus, since the gradient base optimization is normally more fast. There are two models in most optimization problem, the design model and the analysis model. The design variables are defined in the design model, however the analytical sensitivity is normally obtained from the analysis model. Although it is possible to use the same model in analysis and design under isogeomeric framework, it might give either a highly distorted optimum structure or a unreliable structural response. We developed a sensitivity mapping scheme to resolve this problem. The design sensitivity is extracted from the analysis model mesh level sensitivity, which is obtained by the discrete analytical sensitivity analysis. It provides exibility for the design variable definition. The correctness of structure response is also ensured. The modal synthesis method is also used to further improve the optimization eficiency for the built-up structure optimization concerning structural dynamics criteria.