Ce travail concerne le développement d’un outil numérique pour résoudre des problèmes de plaques et de coques mécaniques et piézo-électriques multicouches. En d’autres termes, nous développons des éléments finis basés sur le modèle de Reissner-Mindlin pour l’analyse des problèmes mécaniques et piézoélectriques de plaques et de coques multicouches. Cet outil doit être le moins coûteux en termes de degrés de liberté, simple à utiliser pour modéliser le problème, sans aucune pathologie numérique classique et satisfaisant :vitesse de convergence et efficacité.Dans le domaine des éléments finis, il existe une théorie appelée La « théorie des déformations du cisaillement du premier ordre »(FSDT), utilisée pour résoudre des problèmes purement mécanique prenant en compte l’effet des contraintes en cisaillement transverse. Dans cette théorie, cinq degrés de liberté (trois translations et deux rotations), sont à chercher.Prenant en compte l’hypothèse d’une approximation du déplacement au niveau de la surface moyenne, en utilisant la loi constitutive 2D.Dans ce travail, nous présentons l’extension piézoélectrique de cette théorie pour des éléments finis quadratiques à huit nœuds pour résoudre des problèmes de plaques et de coques multicouches. En effet, la nouveauté dans ce travail est celle de considérer une loi constitutive 3D afin de calculer non seulement le déplacement transversal au niveau du plan moyen de chaque couche mais aussi les déplacements transversaux en surface supérieure et en surface inférieure de chaque couche. Cette approximation de l’épaisseur fait intervenir deux nouveaux degrés de liberté, qui seront très importants dans l’étude des plaques et des coques épaisses et semi épaisses.Le déplacement mécanique est approximé en utilisant une approche Equivalent Single Layer(ESL) et le potentiel électrique est , quant à lui, approximé par une approche LayerWise (LW).Cette évolution est proposée afin d’acquérir un bon compromis entre le minimum de degrés de liberté et le maximum d’efficacité. D’une part, l’approximation par élément fini pour le potentiel électrique, respectant les coordonnées de l’épaisseur est représenté par une variation linéaire au niveau de chaque couche. D’autres part ce potentiel est constant par élément en chaque interface, ce qui réduit le nombre d’inconnus pour la recherche du potentiel électrique résultant. Plusieurs tests numériques sont présentés dans le but d’évaluer les éléments mécanique et piézoélectrique de plaque PQ8P7 et PQ8P7PZ et de coque CQ8P8 et CQ8P8PZ ainsi que leurs capacités de résoudre les problèmes physiques auxquels ils sont dédiés, en comparant nos résultats aux solutions de référence. / The aim of this work is to develop a computational tool for a multilayred piezolelectic plates and shells: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, there is atheory, called "First-order-shear deformation theories (FSDT) used for the mechanical part in much reserches, taking in account the transverse shear stress effects. There are only five unknows generalized displacement, taking the hypothesis of the midsurface displacement approximation using a 2D constitutive law. In This work we present the piezo electric extension of the FSDT eight nodes plate/shell finite element using seven unknows genralized displacement. Two new mechanical unknowns are added to approximate the thickness in the top and the bottom of the /shell, considering a 3D constitutive law, wich is very interesting study especialy for the thick plate/shell modelisation. The mechanical displacement is approximated using the equivalent single layer approch (ESL) and the electric potential is approximated using the layerwise approch (LW).An evolution is proposed in order to access the best compromise between minimum number of degrees of freedom and maximum efficiency. On one hand, a finite element approximation for the electric potential with respect to the thickness coordinate are presented a linear variation in each layer. On the other hand, the in-plane variation is constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post processing level, the transverse shear stress are deduced using the equilibrium equations.Numerous tests are presented in order to evoluate that capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions.

Ingeniero Civil Mecánico / En la práctica de la ingeniería el método de elementos finitos o MEF se utiliza para realizar cálculos cuando resulta difícil o ineficiente obtener una solución analítica. En los últimos años se ha popularizado el uso de los métodos sin malla ya que estos entregan soluciones más exactas además de ser menos sensibles a las distorsiones de los elementos. Esto último se debe a que los métodos sin malla utilizan un vecindario de nodos para construir la aproximación de las variables, eliminando la necesidad de conexiones elementales (mallado) entre los nodos. La placa de Reissner-Mindlin (RM) se utiliza para el análisis de esfuerzos y deformaciones en placas gruesas. Este modelo supone que la deformación a lo largo del espesor varía en forma lineal además supone que la componente σzz del esfuerzo es nula.Sin embargo, la placa RM presenta el problema de bloqueo por corte, donde la rigidez de la placa aumenta drásticamente a medida que se disminuye el espesor. El objetivo de este trabajo es solucionar el bloqueo por corte mediante la proyección nodal, la cual ya ha sido utilizada para solucionar el bloqueo volumétrico en elasticidad incompresible (análogo al bloqueo por corte). Para este propósito se formula la proyección nodal para el problema de la placa RM. La proyección nodal se formula a partir de la formulación mixta eliminando los grados de libertad de corte S a partir de la discretización de los desplazamientos adyacentes disminuyendo el número de ecuaciones por resolver. Se implementa el esquema para la proyección nodal sin malla con funciones de base de máxima entropía utilizando el software Matlab. Se realizaron experimentos numéricos sobre problemas cuya solución analítica es conocida y se evaluó la convergencia y sensibilidad a distorsiones para la proyección nodal y métodos tradicionales. La proyección nodal converge óptimamente para la norma L2 del error y para la semi-norma H1 del error. La proyección nodal es menos sensible a distorsiones geométricas en la malla base que métodos MEF tradicionales. Se concluye que la proyección nodal soluciona el problema de bloqueo por corte utilizando métodos sin malla. Se consigue una mejor convergencia con respecto a los métodos tradicionales además de disminuir la sensibilidad a distorsiones geométricas.

Ce travail est consacré à l’étude d’estimateurs d'erreur a posteriori de type flux équilibrés et résiduels pour la résolution des équations de Reissner-Mindlin par la méthode des éléments finis. Le mémoire débute par l'introduction du problème aux limites et de son analyse de convergence a priori par la méthode des éléments finis. Nous construisons alors pour une discrétisation conforme un estimateur a posteriori de type flux équilibrés fiable, efficace et robuste en l'épaisseur de la plaque t. Nous obtenons finalement une constante multiplicative égale à 1 pour la fiabilité. Des tests numériques illustrent nos résultats pour différents maillages. Puis nous abordons le cas d’une discrétisation non-conforme, où nous proposons un estimateur a posteriori de type résiduel, utilisant une régularisation de la solution discrète. Des tests numériques illustrent également nos résultats. La suite du travail reprend la discrétisation conforme en construisant un estimateur a posteriori défini à partir de la résolution de problèmes localisés sur les patchs de la triangulation, menant à un choix plus consistant avec le problème aux limites. Le dernier chapitre est consacré à l'estimation a posteriori pour le problème aux valeurs propres de Reissner-Mindlin. L’estimateur obtenu est fiable et efficace pour la norme de l'erreur entre les vecteurs propres, permettant également de majorer l’erreur commise entre les valeurs propres. Des tests numériques illustrent nos résultats. / This work is devoted to the study of equilibrated fluxes and residual a posteriori error estimators for the finite element resolution of the Reissner-Mindlin system. This report begins by the introduction of the boundary value problem and of its a priori convergence analysis in the finite element method context. Then, an equilibrated fluxes a posteriori estimator is built for a conform discretization, which is proven to be reliable, efficient and robust on the plate thickness t. We finally obtain a multiplicative constant equal to 1 for the reliability. Numerical tests illustrate our results on different meshes. Then, we address the non-conforming discretization case, where a residual a posteriori estimator is proposed using a regularisation of the discrete solution. Numerical tests also illustrate our results. Next we come back to the conform discretization by building an a posteriori estimator defined from localised problems resolution on stars, leading to a consistent choice with the boundary value problem. The last chapter is devoted to an a posteriori estimation for the Reissner-Mindlin eigenvalues problem. The obtained estimator is reliable and efficient for the error norm between the eigenvectors, also allowing to evaluate the error between the eigenvalues. Numerical tests illustrate our results.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

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

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.

No description available.

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.