Relevant bibliographies by topics / Monolithic finite element formulation

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  2. Dissertations / Theses
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Academic literature on the topic 'Monolithic finite element formulation'

Author: Grafiati
Published: 6 September 2023

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Journal articles on the topic "Monolithic finite element formulation"

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Gupta, Adhip, and C. S. Jog. "A Monolithic Finite Element Formulation for Magnetohydrodynamics Involving a Compressible Fluid." Fluids 7, no. 1 (January 7, 2022): 27. http://dx.doi.org/10.3390/fluids7010027.

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This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) involving a compressible fluid based on a continuous velocity–pressure formulation. The entire formulation is within a nodal finite element framework, and is directly in terms of physical variables. The exact linearization of the variational formulation ensures a quadratic rate of convergence in the vicinity of the solution. Both steady-state and transient formulations are presented for two- and three-dimensional flows. Several benchmark problems are presented, and comparisons are carried out against analytical solutions, experimental data, or against other numerical schemes for MHD. We show a good coarse-mesh accuracy and robustness of the proposed strategy, even at high Hartmann numbers.
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Antunes, A. R. E., P. R. M. Lyra, R. B. Willmersdorf, and S. M. A. Bastos. "An implicit monolithic formulation based on finite element formulation for incompressible Navier–Stokes equations." Journal of the Brazilian Society of Mechanical Sciences and Engineering 37, no. 1 (March 18, 2014): 199–210. http://dx.doi.org/10.1007/s40430-014-0155-x.

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Sun, WaiChing. "A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain." International Journal for Numerical Methods in Engineering 103, no. 11 (April 30, 2015): 798–839. http://dx.doi.org/10.1002/nme.4910.

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Kutlu, Akif. "Mixed finite element formulation for bending of laminated beams using the refined zigzag theory." Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 235, no. 7 (July 2021): 1712–22. http://dx.doi.org/10.1177/14644207211018839.

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This study presents a mixed finite element formulation for the stress analysis of laminated composite beams based on the refined zigzag theory. The Hellinger–Reissner variational principle is employed to obtain the first variation of the functional that is expressed in terms of displacements and stress resultants. Due to C0 continuity requirements of the formulation, linear shape functions are adopted to discretize the straight beam domain with two-noded finite elements. The proposed formulation is shear locking free from nature since it introduces displacement and stress resultant terms as independent field variables. A monolithic solution of the global finite element equations is preferred, hence the stress resultants are directly obtained from the solution of these equations. The in-plane strain measures of the beam are obtained directly at the nodes over the compliance matrix and stress resultants by avoiding error-prone spatial derivatives. Following, transverse shear stresses are calculated from the equilibrium equations at the post-processing level. This simple but effective finite element formulation is first verified and tested for convergence behavior. The robustness of the approach is shown through some examples and its accuracy in predicting the displacement and stress components is revealed.
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Панасюк, Леонид, Leonid Panasyuk, Галина Кравченко, Galina Kravchenko, Елена Труфанова, Elena Trufanova, Инал Тарба, Inal Tarba, Лаша Цвейба, and Lasha Cveyba. "FINITE ELEMENT MODELLING OF INTERACTION BUILDING FRAME AND SLAB-PILE FOUNDATION." Construction and Architecture 7, no. 1 (April 19, 2019): 34–38. http://dx.doi.org/10.29039/article_5c646f16bffb38.56532696.

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The article deals with the simulation of joint work of slab grillage and monolithic frame of the building by finite element method. The finite-element model is developed in the spatial formulation according to the complex scheme "upper structure-base plate-pile Foundation". The pile field was modeled by pliable rods with stiffness corresponding to the average draft of the pile field. Static and dynamic calculations are performed in the ING+software package. The results of the stress-strain state of the building frame elements demonstrate the correctness of this approach to take into account the compliance of the base.
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Lozovskiy, Alexander, Maxim A. Olshanskii, and Yuri V. Vassilevski. "A finite element scheme for the numerical solution of the Navier–Stokes/Biot coupled problem." Russian Journal of Numerical Analysis and Mathematical Modelling 37, no. 3 (June 1, 2022): 159–74. http://dx.doi.org/10.1515/rnam-2022-0014.

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Abstract A finite element method for a monolithic quasi-Lagrangian formulation of a fluid–porous structure interaction problem with a corrected balance of stresses on the fluid–structure interface is considered. Deformations of the elastic medium are not necessarily small and are modelled using Saint Venant–Kirchhoff (SVK) constitutive relation. The stability of the method is proved in a form of energy bound for the finite element solution.
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Chen, Xiangxiang, Xudong Chen, Andrew Chan, Yingyao Cheng, and Hongfan Wang. "A FDEM Parametric Investigation on the Impact Fracture of Monolithic Glass." Buildings 12, no. 3 (February 25, 2022): 271. http://dx.doi.org/10.3390/buildings12030271.

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Due to the brittleness, monolithic glass may fracture under impact, resulting in catastrophic sequences. The combined finite-discrete element method, i.e., FDEM, is employed to investigate both the oblique and the perpendicular impact failures of monolithic glass parametrically, particularly the soda-lime glass. Using FDEM, glass is discretised into discrete elements where a finite element formulation is incorporated, leading to accurate evaluation of the contact forces and structural deformation. Following the basic theories of the FDEM, a cohesive Mode I fracture model of glass is briefly introduced. Numerical examples are given for the verification of the employed fracture model and the applicability of the FDEM, and comparisons have been made against the computational and experimental results in the literature. The investigated parameters include the impact velocity, the impact angle, the material properties of glass, etc. The obtained results not only revealed the impact fracture mechanism of soda-lime glass but also provided guidance for its design and manufacturing.
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Grabmaier, Sebastian, Matthias Jüttner, and Wolfgang Rucker. "Coupling of finite element method and integral formulation for vector Helmholtz equation." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 4 (July 2, 2018): 1405–17. http://dx.doi.org/10.1108/compel-08-2017-0346.

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Purpose Considering the vector Helmholtz equation in three dimensions, this paper aims to present a novel approach for coupling the finite element method and a boundary integral formulation. It is demonstrated that the method is well-suited for many realistic three-dimensional problems in high-frequency engineering. Design/methodology/approach The formulation is based on partial solutions fulfilling the global boundary conditions and the iterative interaction between them. In comparison to other coupling formulation, this approach avoids the typical singularity in the integral kernels. The approach applies ideas from domain decomposition techniques and is implemented for a parallel calculation. Findings Using confirming elements for the trace space and default techniques to realize the infinite domain, no additional loss in accuracy is introduced compared to a monolithic finite element method approach. Furthermore, the degree of coupling between the finite element method and the integral formulation is reduced. The accuracy and convergence rate are demonstrated on a three-dimensional antenna model. Research limitations/implications This approach introduces additional degrees of freedom compared to the classical coupling approach. The benefit is a noticeable reduction in the number of iterations when the arising linear equation systems are solved separately. Practical implications This paper focuses on multiple heterogeneous objects surrounded by a homogeneous medium. Hence, the method is suited for a wide range of applications. Originality/value The novelty of the paper is the proposed formulation for the coupling of both methods.
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Zoalkfl, Danial, Anton Chepurnenko, Batyr Yazyev, Aleksandr Ishchenko, and Stepan Litvinov. "Determination of temperature fields and stresses during the construction of a massive monolithic foundation slab of a wind turbine tower." E3S Web of Conferences 402 (2023): 12002. http://dx.doi.org/10.1051/e3sconf/202340212002.

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The article proposes a method for determining temperature fields and stresses during the construction of massive monolithic structures in a two-dimensional axisymmetric formulation. The solution is performed using the finite element method. The calculation takes into account the shrinkage of concrete, as well as the change in its physical and mechanical characteristics over time. The problem of calculating a massive monolithic foundation of a wind turbine is presented. Recommendations are given to reduce the risk of early cracking.
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Liu, Chun Jie, Xi Wang, and De’an Wan. "Study on Angular Stiffness of Monolithic Flexible Joint." Advanced Materials Research 189-193 (February 2011): 1816–21. http://dx.doi.org/10.4028/www.scientific.net/amr.189-193.1816.

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The monolithic flexible joint is introduced as a novel component of inertial guidance instrument. The angular stiffness of the joint is investigated by employing theory of flexure hinges considering the key part of the joint is a variation of the common circular flexure hinge. Closed-form equation is formulated to unify the typical angular stiffness equations developed by other authors in terms of the circular hinge, and the main variables of the formulation are also discussed. Finite element models of the monolithic joint are built to confirm the analytical model predictions. A measuring system controlled by computer is also developed to evaluate the angular stiffness of the monolithic flexible joint. Checked against finite element analysis and experimental measurement data, the analytical model predictions are within 7% error margins. The study results indicate that the angular stiffness is more sensitive to the minimal thickness of hinge and less sensitive to the notch angle and the oblique angle of hinge.
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Dissertations / Theses on the topic "Monolithic finite element formulation"

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Chiang, Chen-Yu. "Transport in biological systems. Monolithic method for fluid-structure interaction." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS477.

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Le travail de cette thèse a pour objectif de développer un solveur dédié aux problèmes d'interaction fluide-structure (IFS), en particulier ceux rencontré en biologie, tels que la dynamique d'un écoulement sanguin à travers des tronçons veineux munis de valves. La circulation du sang est étudiée à l'aide de modèles pertinents sur les plans anatomique et physique. Le premier aspect des problèmes d'IFS concerne la gestion de la stabilité. Une formulation monolithique eulérienne basée sur la méthode des caractéristiques assure la stabilité inconditionnelle et introduit une approximation du premier ordre en temps avec deux modèles distincts de matériaux hyper élastiques. Le second aspect est relatif au contact entre deux parties du domaine solide, tel celui apparaissant entre deux valvules au cours de la fermeture de la valve et à l'état fermé sur un surface valvulaire relativement importante. Un algorithme de contact est proposé et validé à l'aide de tests de référence. L'étude computationnelle de l'écoulement sanguin à travers des tronçons veineux munis de valves est mené, une fois le solveur IFS vérifié et validé. Le domaine computationnel bidimensionnel est soit constitué d'une simple unité de base, soit du modèle de circuit veineux en forme d'échelle avec une veine superficielle et une profonde, communicant par une série de veines perforantes. Un maillage tridimensionnel de l'unité de base a été construit. Les simulations dans ce domaine tridimensionnel nécessite le recours au calcul haute performance. La dynamique de l'écoulement sanguin est fortement couplée à la mécanique de la paroi vasculaire. La paroi déformable des veines et artères de gros calibre est composée de trois couches principales (l'intima, la media, et l'adventitia) constituées de matériaux composites ayant une composition spécifique dans chaque couche. Dans ce travail, la rhéologie de la paroi est supposée être représentée par un matériau du type Mooney-Rivlin
The present work aims at developing a numerical solver for fluid-structure interaction (FSI) problems, especially those encountered in biology such as blood circulation in valved veins. Blood flow is investigated using anatomically and physically relevant models. The first aspect of FSI problems is related to management of algorithm stability. An Eulerian monolithic formulation based on the characteristic method unconditionally achieves stability and introduce a first order in time approximation with two distinct hyperelastic material models. The second aspect deals with between-solid domain contact such as that between valve leaflets during closure and in the closed state over a finite surface, which avoid vcusp tilting and back flow. A contact algorithm is proposed and validated using benchmarks. Computational study of blood flow in valved veins is investigated, once the solver was verified and validated. The 2D computational domain comprises a single basic unit or the ladder-like model of a deep and superficial veins communicating by a set of perforating veins. A 3D mesh of the basic unit was also built. Three-dimensional computation relies on high performance computing. Blood flow dynamics is strongly coupled to vessel wall mechanics. Deformable vascular walls of large veins and arteries are composed of three main layers (intima, media, and adventitia) that consist of composite material with a composition specific to each layer. In the present work, the wall rheology is assumed to be a Mooney-Rivlin material
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El, Feghali Stéphanie. "Nouvelle formulation monolithique en élément finis stabilisés pour l'interaction fluide-structure." Phd thesis, Ecole Nationale Supérieure des Mines de Paris, 2012. http://pastel.archives-ouvertes.fr/pastel-00743488.

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L'Interaction Fluide-Structure (IFS) décrit une classe très générale de problème physique, ce qui explique la nécessité de développer une méthode numérique capable de simuler le problème FSI. Pour cette raison, un solveur IFS est développé qui peut traiter un écoulement de fluide incompressible en interaction avec des structures différente: élastique ou rigide. Dans cet aspect, le solveur peut couvrir une large gamme d'applications.La méthode proposée est développée dans le cadre d'une formulation monolithique dans un contexte Eulérien. Cette méthode consiste à considérer un seul maillage et résoudre un seul système d'équations avec des propriétés matérielles différentes. La fonction distance permet de définir la position et l'interface de tous les objets à l'intérieur du domaine et de fournir les propriétés physiques pour chaque sous-domaine. L'adaptation de maillage anisotrope basé sur la variation de la fonction distance est ensuite appliquée pour assurer une capture précise des discontinuités à l'interface fluide-solide.La formulation monolithique est assurée par l'ajout d'un tenseur supplémentaire dans les équations de Navier-Stokes. Ce tenseur provient de la présence de la structure dans le fluide. Le système est résolu en utilisant une méthode élément fini et stabilisé suivant la formulation variationnelle multiéchelle. Cette formulation consiste à décomposer les champs de vitesse et pression en grande et petite échelles. La particularité de l'approche proposée réside dans l'enrichissement du tenseur de l'extra contraint.La première application est la simulation IFS avec un corps rigide. Le corps rigide est décrit en imposant une valeur nul du tenseur des déformations, et le mouvement est obtenu par la résolution du mouvement de corps rigide. Nous évaluons le comportement et la précision de la formulation proposée dans la simulation des exemples 2D et 3D. Les résultats sont comparés avec la littérature et montrent que la méthode développée est stable et précise.La seconde application est la simulation IFS avec un corps élastique. Dans ce cas, une équation supplémentaire est ajoutée au système précédent qui permet de résoudre le champ de déplacement. Et la contrainte de rigidité est remplacée par la loi de comportement du corps élastique. La déformation et le mouvement du corps élastique sont réalisés en résolvant l'équation de convection de la Level-Set. Nous illustrons la flexibilité de la formulation proposée par des exemples 2D.
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Sivess, Andrew Gregory. "Chebyshev polynomial based finite element stiffness matrix formulation." Diss., Connect to online resource, 2005. http://wwwlib.umi.com/cr/colorado/fullcit?p1427762.

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Liu, Guanhui, and 刘冠辉. "Formulation of multifield finite element models for Helmholtzproblems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44204875.

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PASTOR, JORGE AURELIO SANTA CRUZ. "RESERVOIR MODELING THROUGH A COUPLED FINITE ELEMENT FORMULATION." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2001. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=2082@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
A produção de hidrocarbonetos resulta na redução da pressão do reservatório( depletação ). À medida que a pressão do reservatório diminui, as tensôes efetivas aplicadas na matriz rochosa aumentam, provocando reduções na porosidade e na permeabilidade da rocha assim como redução de volume, esta última conhecida como compactação.A compactação do reservatório pode provocar subsidência da superfície com conseqüentes impactos ambientais e problemas em equipamentos localizados no poço, tais como revestimentos, e outros problemas associados, tais como produção de sólidos. No entanto, compactação não é sempre prejudicial porque ajuda a manter a pressão do reservatório e, conseqüentemente, a produtividade. O acoplamento fluxo-deformação é fundamental na análise deste problema. Este trabalho tem com objetivo discutir a teoria e as equações que modelam este processo acoplado, suas limitações e sua capacidade de representar corretamente os fenômenos físicos envolvidos.Além disto, foi desenvolvido um simulador numérico baseado no método dos elementos finitos, para a modelagem transiente de um fluxo monofásico através de um meio poroso, considerando-se o acoplamento fluxo do fluido,deformações e temperatura.O material rochoso é modelado segundo um modelo poroelástico. O simulador foi testado comparando resultados com resultados obtidos através de soluções analíticas.Além disto, uma simulção de fluxo em reservatório foi efetuada para avaliar a capacidade do simulador, tendo-se comparado os resultados com resultados encontrados na literatura. A análise foi feita considerando o overburden e o sideburden.Foi verificada uma ótima concordância entre os resultados.O simulador mostrou-se capaz de representar as variações de pressão não apenas decorrentes da difusão do fluido, mas também aquelas provocadas por variações de tensões totais. Em alguns casos, a variação de tensões totais no topo do reservatório é significativa,demonstrando que simuladores convencionais podem induzir erros significativos em termos de variações das pressões no fluido.
Prodution ofhydrocarbon often to a reduction in reservoir pressure. Depending upon the rock compressibility, this reduction in reservoir pressure causes substancial strains and eventual shear collapse. While reservoir pressure decreases the effective stress increases, induting porosity and permeability reduction changes and an overall volume decrease known as compaction. Compaction of reservoir may eventually be transmitted to the surface and cause vertical movements, known as subsidence. Compaction may have serious consequences upon well casing,and other associated problems, such as solid production. However, compaction is not always detrimental because it helps maitaining reservoir pressure and consequently, reservoir productivity. Hydromechaninical coupling is essential to analyze this problem.The aim of this work is to discuss the theory and develop the equations that governthis coupled process. The limitations and possibilities in representing the associated phenomena are highlighted. A numerical, finite element based, simulator was developed to model the single-phase flow through porous media taking into accout the hydrothermo-mechanical coupling. The rock material is assumed to behave as a poroelastic material.The results obtained by the computer simulator were compared with theorical solutions for the classical problem of uniaxial deformation test and for the stress concentration aroun inclined welbores in porous media. The results showed excellent agreement. A idealized reservoir simulation was carrierd out using the computer model and the results of pore pressure, total stresses and displacement changes were compared with results published in the literature, obtained by similar approaches. The comparisons showed very good agreement. In the simulations the presences of overburden, sideburden and underburden were recognized. The simulator represented well the changes in fluid pressure associated with both the diffusion process and the changes in total stresses. In some cases, the changes in total stresses at the top of the reservoir are significant which demonstrates the partial flaw of the conventional flow simulators that are not able to take this effect into account.
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Liu, Guanhui. "Formulation of multifield finite element models for Helmholtz problems." Click to view the E-thesis via HKUTO, 2010. http://sunzi.lib.hku.hk/hkuto/record/B44204875.

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FREY, SERGIO LUIZ. "A FINITE ELEMENT FORMULATION FOR THE NAVIER-STOKES PROBLEM." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1991. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=19805@1.

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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Métodos estabilizados de elementos finitos são propostos e analisados para problemas de fluidos, com particular ênfase nas equações de Navier-Stokes incomprenssível. Após a apresentação da mecânica dos escoamentos dos fluidos, introduzimos no Capítulo 3, no contexto de problema de Stokes, as dificuldadas numéricas associadas ao método de Galerkin em problemas de fluidos e simulamos em sucesso alguns escoamentos lentos através de formulações finitos para estabilizadas. No capítulo 4, propomos uma nova formulação de elementos finitos para a equação da energia, mais precisamente para o modelo da advecção-difusão do calor. Graças a um novo desenho do parâmetro de estabilidade T, o qual permite adicionar difusão às regiões advectivas e difusivas-dominadas do escoamento de maneira diferemciada, obtivemos um bom desempenho novo método mesmo em situações de altíssimo número de Péclet (10(2) menor que Pe menor que 10 e (6)), conforme ilustram os testes numéricos realizados. Coletando as experiências adquiridas com modelos lineares de Stokes e da advencção-difusão, nos foi possível propor, analisar o erro e testar dois novos métodos estabilizados para o problema de Navier-Stokes transiente. Construídos de maneira a herdar as boas características de estabilidade dos métodos propostos apresentam bom desempenho em escoamentos fortemente advectivos, bem como não necessitam atender a priori à condição de Baduska-Brezzi. Através de um algoritmo preditor/ multi-corretor de integração do termo inercial da equação de movimento, estes ,métodos foram capazes de de simular de maneira precisa escoamentos de interesse em Mecânica(400 menor que Re< menor que 500), captando escoamentos secundários, tais como recirculações de fluido.
Stabilized methods for fluid problems are proposed and analysed with particular emphasis to the incompressible Navier-Stokes equations. We Begin in Chapter 2 introducing the balance equations of fluid Mechanics. Next. In Chapter 3, we discuss the numerical difficulties of the Galerkin method in fluids(in the contexto f the Stokes problem) and performance some succeful simulations of creeping flows, employing stabilized formulations. In Chapter 4, we propose a new finite element formulation for the energy equation, or more preciselly for the advective-diffusive model. Taking advantage of new design of the stability parameter T, which permits to add diffusion to advective and diffusive regions of the flow in a different way, we success to obtain a good performance of the new method in flows with very high Péclet numbers (10(2) lass than Pe lessa than 10(6)), as illustred at numerical testes performed. By collecting the Stokes and advective-diffusive experiences,it was possible to propose, analyse and test two new stabilized methods for the transient Navier-Stokes problem. These methods were built in a way to heritage the good characteristics showed by the stabilized methods introduced for the Stokes and adventive-diffusive models. The new methods propoposed have a good performance in high advective flows, besides there is no need to satisfy the Babuska-Brezzi condition. Employing a predictor/multi-corretor algorithm, we were able to simulate accruratly some useful flows(400 less than Re less than 500), such as fluid recirculations.
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Pratap, Rudra 1964. "A NEW RESIDUAL FINITE-ELEMENT FORMULATION FOR ELASTODYNAMIC PROBLEMS." Thesis, The University of Arizona, 1987. http://hdl.handle.net/10150/276552.

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In the research undertaken a finite element formulation has been developed for an elastodynamic problem using a least squares approach. The special requirements of the problem demanded a study of suitability of various elements. The emergence of the final element is a result of both theoretical and numerical study of three different elements. The approximation function is assumed on the basis of the order of the governing differential equations. Then the square of the error resulting from the approximate solution is minimized over the entire domain as well as the boundaries in the same functional. The element equation emerging from the formulation does not yield a singular stiffness matrix, since the boundary conditions are already taken into account in the element equation. The formulation presented in this thesis is only for the normal propagation of phi-wave. A finite element code has been developed based on the new formulation.
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Wang, Haitao, and 王海濤. "Formulation of finite element methods for determining singular stress fields." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2002. http://hub.hku.hk/bib/B31243708.

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PORTO, ANA CRISTINA DOS SANTOS DA SILVA. "CAVITY BACKED SLOT ANTENNAS: A FINITE ELEMENT-BOUNDARY INTEGRAL FORMULATION." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2004. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=5270@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Devido às propriedades de banda larga e polarização circular, as antenas espirais tornaram-se bastante atrativas para diversos serviços de telecomunicações modernos e móveis. Neste trabalho uma antena espiral tipo fenda apoiada em uma cavidade cilíndrica metálica foi analisada. Por causa de sua geometria complexa, optou-se por um método numérico híbrido, realizado através de uma implementação de elementos mistos do método dos elementos finitos - integral de fronteira. Baseado nesta formulação, um programa computacional foi desenvolvido. Um grande esforço foi realizado a fim de escrever o programa de maneira que o armazenamento e as necessidades computacionais fossem mínimos, boa parte conseguida pela aplicação das condições de contorno na superfície metálica. Com o objetivo de melhor explorar a geometria do problema, definiu-se uma malha de elementos finitos onde a parte que cabia a abertura foi dividida em quadriláteros enquanto que no restante da superfície, elementos triangulares foram utilizados. Repetiu-se então esta malha ao longo da altura da cavidade, de maneira que hexaedros e prismas formaram a malha final. Para cada espécie de elemento, funções de base vetoriais específicas foram aplicadas. A fim de verificar a precisão do algoritmo, o mesmo programa computacional foi utilizado na análise de uma antena tipo fenda retangular, também apoiada em uma cavidade cilíndrica metálica.
Known for the properties of broadband and circular polarization, spiral antennas have become sufficiently attractive for services of modern and mobile telecommunications. In this work a cavity-backed slot spiral antenna was analyzed. Because of its complex geometry, it was chosen a hybrid numerical method, carried through an implementation of mixing elements of the finite elements - boundary integral method. Based in this formulation, a computer program was developed. A great effort was carried out in order to write the program thus the computational storage and necessities were minimum, good part obtained by application of the boundary conditions on the metallic surface. With the goal of better exploring the geometry of the problem, a mesh of finite elements was defined where the part that fit the aperture was divided into quadrilaterals, while that in the rest of the surface, triangular elements were used. By repeating this mesh along the height of the cavity, hexahedrals and prisms formed the final mesh. For each kind of element, specific vector base functions were applied. In order to verify the accuracy of the algorithm, the same computational program was used in the analysis of a cavity-backed slot rectangular antenna.
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Books on the topic "Monolithic finite element formulation"

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Carrera, Erasmo. Finite element analysis of structures through unified formulation. Chichester, West Sussex: John Wiley & Sons, Inc., 2014.

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Carrera, Erasmo, Maria Cinefra, Enrico Zappino, and Marco Petrolo. Finite Element Analysis of Structures Through Unified Formulation. Chichester, UK: John Wiley & Sons, Ltd, 2014. http://dx.doi.org/10.1002/9781118536643.

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Szabo, B. A. Introduction to finite element analysis: Formulation, verification, and validation. Hoboken, N.J: Wiley, 2011.

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Pramote, Dechaumphai, Wieting A. R, and Langley Research Center, eds. Thermal-structural finite element analysis using linear flux formulation. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1990.

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Pramote, Dechaumphai, Wieting A. R, and Langley Research Center, eds. Thermal-structural finite element analysis using linear flux formulation. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1990.

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Collins, J. D. A combined finite element-boundary element formulation for solution of axially symmetric bodies. Ann Arbor, Mich: University of Michigan, Radiation Laboratory, Dept. of Electrical Engineering and Computer Science, 1991.

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United States. National Aeronautics and Space Administration., ed. A variational justification of the assumed natural strain formulation of finite elements. [Washington, DC]: National Aeronautics and Space Administration, 1991.

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Waldman, W. A penalty element formulation for calculating bulk stress. Melbourne, Australia: Aeronautical Research Laboratory, 1989.

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Golla, David Frank. Dynamics of viscoelastic structures: a time-domain finite element formulation. [Downsview, Ont.]: [Institute for Aerospace Studies], 1985.

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Golla, David Frank. Dynamics of viscoelastic structures: A time-domain finite element formulation. [Downsview, Ont.]: Institute for Aerospace Studies, 1986.

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Book chapters on the topic "Monolithic finite element formulation"

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Ida, Nathan. "Finite element formulation." In Numerical Modeling for Electromagnetic Non-Destructive Evaluation, 276–352. Boston, MA: Springer US, 1995. http://dx.doi.org/10.1007/978-1-4757-0560-7_8.

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Służalec, Andrzej. "Finite-Element Formulation." In Introduction to Nonlinear Thermomechanics, 135–47. London: Springer London, 1992. http://dx.doi.org/10.1007/978-1-4471-1906-7_14.

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Marques, Severino P. C., and Guillermo J. Creus. "Viscoelastic Finite Element Formulation." In Computational Viscoelasticity, 77–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-25311-9_9.

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Pariseau, William G. "Finite element seepage formulation." In Notes on Numerical Modeling in Geomechanics, 63–66. London: CRC Press, 2022. http://dx.doi.org/10.1201/9781003166283-12.

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Tzou, H. S. "Finite Element Formulation and Analyses." In Piezoelectric Shells, 405–56. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-1783-8_10.

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Nielsen, C. V., W. Zhang, L. M. Alves, N. Bay, and P. A. F. Martins. "Coupled Finite Element Flow Formulation." In Modeling of Thermo-Electro-Mechanical Manufacturing Processes, 11–36. London: Springer London, 2012. http://dx.doi.org/10.1007/978-1-4471-4643-8_3.

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Tzou, Hornsen. "Finite Element Formulation and Analyses." In Piezoelectric Shells, 409–45. Dordrecht: Springer Netherlands, 2018. http://dx.doi.org/10.1007/978-94-024-1258-1_13.

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Gan, Buntara S. "Finite Element Formulation of Beam Elements." In An Isogeometric Approach to Beam Structures, 61–126. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-56493-7_3.

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Gopalakrishnan, Srinivasan. "Introduction to Spectral Finite Element Formulation." In Elastic Wave Propagation in Structures and Materials, 357–94. Boca Raton: CRC Press, 2022. http://dx.doi.org/10.1201/9781003120568-12.

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Chattopadhyay, Adhir Baran, Shazia Hasan, and Snehaunshu Chowdhury. "Finite Element Formulation of Field Problems." In Advances in Intelligent Systems and Computing, 113–26. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-19-7558-5_6.

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Conference papers on the topic "Monolithic finite element formulation"

1

Ebna Hai, Bhuiyan Shameem Mahmood, Markus Bause, and Paul Kuberry. "Finite Element Approximation of the Extended Fluid-Structure Interaction (eXFSI) Problem." In ASME 2016 Fluids Engineering Division Summer Meeting collocated with the ASME 2016 Heat Transfer Summer Conference and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/fedsm2016-7506.

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This contribution is the second part of three papers on Adaptive Multigrid Methods for the eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. To the best of our knowledge, such a model is new in the literature. This model is used to design an on-line structural health monitoring (SHM) system in order to determine the coupled acoustic and elastic wave propagation in moving domains and optimum locations for SHM sensors. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research focuses on the newly developed mathematical model of a new FSI problem, which is referred to as extended Fluid-Structure Interaction (eXFSI) problem in the ALE framework. The eXFSI is a strongly coupled problem of typical FSI with a coupled wave propagation problem on the fluid-solid interface (WpFSI). The WpFSI is a strongly coupled problem of acoustic and elastic wave equations, where wave propagation problems automatically adopts the boundary conditions from the FSI problem at each time step. The ALE approach provides a simple but powerful procedure to couple solid deformations with fluid flows by a monolithic solution algorithm. In such a setting, the fluid problems are transformed to a fixed reference configuration by the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-solid mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as a one step-θ scheme, from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with a Newton-like method where the discretization is done with a Galerkin finite element scheme. The implementation is accomplished via the software library package DOpElib based on the deal.II finite element library for the computation of different eXFSI configurations.
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Rochus, Ve´ronique, Gaetan Kerschen, and Jean-Claude Golinval. "Dynamic Analysis of the Nonlinear Behavior of Capacitive MEMS Using the Finite Element Formulation." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84994.

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The dynamic response of a mass-spring model of electrostatically actuated MEMS is analyzed when a voltage step is applied. The dynamics of a clamped-clamped beam representing a micro-bridge system are also considered. To model and simulate the strong coupling between the electric and mechanical fields, a general finite element program has been developed based on a new monolithic formulation. The methodology proposed in this program does not require any iterative numerical scheme based on separate electrostatic and mechanical solvers, respectively.
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Ebna Hai, Bhuiyan Shameem Mahmood, and Markus Bause. "Adaptive Finite Elements Simulation Methods and Applications for Monolithic Fluid-Structure Interaction (FSI) Problem." In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting collocated with the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/fedsm2014-21379.

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Will an aircraft wing have the structural integrity to withstand the forces or fail when it’s racing at a full speed? Fluid-structure interaction (FSI) analysis can help you to answer this question without the need to create costly prototypes. However, combining fluid dynamics with structural analysis traditionally poses a formidable challenge for even the most advanced numerical techniques due to the disconnected, domain-specific nature of analysis tools. In this paper, we present the state-of-the-art in computational FSI methods and techniques that go beyond the fundamentals of computational fluid and solid mechanics. In fact, the fundamental rule require transferring results from the computational fluid dynamics (CFD) analysis as input into the structural analysis and thus can be time-consuming, tedious and error-prone. This work consists of the investigation of different time stepping scheme formulations for a nonlinear fluid-structure interaction problem coupling the incompressible Navier-Stokes equations with a hyperelastic solid based on the well established Arbitrary Lagrangian Eulerian (ALE) framework. Temporal discretization is based on finite differences and a formulation as one step-θ scheme, from which we can extract the implicit euler, crank-nicolson, shifted crank-nicolson and the fractional-step-θ schemes. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of FSI problem and the analysis of various fluid-mesh motion techniques, a comparison of different second-order time-stepping schemes. The time discretization is based on finite difference schemes whereas the spatial discretization is done with a Galerkin finite element scheme. The nonlinear problem is solved with Newton’s method. To control computational costs, we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaption during the computation. The implementation using the software library package DOpElib and deal.II serves for the computation of different fluid-structure configurations.
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Ebna Hai, Bhuiyan Shameem Mahmood, and Markus Bause. "Adaptive Multigrid Methods for Extended Fluid-Structure Interaction (eXFSI) Problem: Part I — Mathematical Modelling." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-53265.

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This contribution is the first part of three papers on Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art of recent developments in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research will focus on the newly developed mathematical model of a new FSI problem which is called eXtended Fluid-Structure Interaction (eXFSI) problem in ALE framework. This model is used to design an on-live Structural Health Monitoring (SHM) system in order to determine the wave propagation in moving domains and optimum locations for SHM sensors. eXFSI is strongly coupled problem of typical FSI with a wave propagation problem on the fluid-structure interface, where wave propagation problems automatically adopted the boundary conditions from of the typical FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as an one step-θ scheme; from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with Newton’s method whereas the spatial discretization is done with a Galerkin finite element scheme. To control computational costs we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaptation during the computation. The implementation is accomplished via the software library package DOpElib and deal.II for the computation of different eXFSI configurations.
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Miller, Samuel J., and Hakan Ozaltun. "Evaluation of U10Mo Fuel Plate Irradiation Behavior via Numerical and Experimental Benchmarking." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-89588.

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This article analyzes dimensional changes due to irradiation of monolithic plate-type nuclear fuel and compares results with finite element analysis of the plates during fabrication and irradiation. Monolithic fuel plates tested in the Advanced Test Reactor (ATR) at Idaho National Lab (INL) are being used to benchmark the performance of proposed fuel for several high power research reactors. Post-irradiation metallographic images of plates sectioned at the mid-plane were analyzed to determine dimensional changes of the fuel and the cladding response. A constitutive model of the fabrication process and irradiation behavior of the tested plates was developed using the general purpose commercial finite element analysis package, ABAQUS. Using calculated burn-up profiles of irradiated plates to model the power distribution and including irradiation behaviors such as swelling and irradiation enhanced creep, model simulations allow analysis of plate parameters that are either impossible or infeasible in an experimental setting. The development and progression of fabrication induced stress concentrations at the plate edges was of primary interest, as these locations have a unique stress profile during irradiation. Additionally, comparison between 2D and 3D models was performed to optimize analysis methodology. In particular, the ability of 2D and 3D models to account for out of plane stresses which result in 3-dimensional creep behavior that is a product of these components. Results show that assumptions made in 2D models for the out-of-plane stresses and strains cannot capture the 3-dimensional physics accurately and thus 2D approximations are not representative. Stress-strain fields are dependent on plate geometry and irradiation conditions, thus, if stress based criteria is used to predict plate behavior (as opposed to material impurities, fine micro-structural defects, or sharp power gradients), unique 3D finite element formulation for each plate is required.
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Jadaan, Osama M., Lynn M. Powers, and John P. Gyekenyesi. "Multiaxial Creep Life Prediction of Ceramic Structures Using Continuum Damage Mechanics and the Finite Element Method." In ASME 1998 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/98-gt-489.

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High temperature and long duration applications of monolithic ceramics can place their failure mode in the creep rupture regime. A previous model advanced by the authors described a methodology by which the creep rupture life of a loaded component can be predicted. That model was based on the life fraction damage accumulation rule in association with the modified Monkman-Grant creep rupture criterion. However, that model did not take into account the deteriorating state of the material due to creep damage (e.g., cavitation) as time elapsed. In addition, the material creep parameters used in that life prediction methodology, were based on uniaxial creep curves displaying primary and secondary creep behavior, with no tertiary regime. The objective of this paper is to present a creep life prediction methodology based on a modified form of the Kachanov-Rabotnov continuum damage mechanics (CDM) theory. In this theory, the uniaxial creep rate is described in terms of stress, temperature, time, and the current state of material damage. This scalar damage state parameter is basically an abstract measure of the current state of material damage due to creep deformation. The damage rate is assumed to vary with stress, temperature, time, and the current state of damage itself. Multiaxial creep and creep rupture formulations of the CDM approach are presented in this paper. Parameter estimation methodologies based on nonlinear regression analysis are also described for both, isothermal constant stress states and anisothermal variable stress conditions This creep life prediction methodology was preliminarily added to the integrated design code. CARES/Creep (Ceramics Analysis and Reliability Evaluation of Structures/Creep), which is a postprocessor program to commercially available finite element analysis (FEA) packages. Two examples, showing comparisons between experimental and predicted creep lives of ceramic specimens, are used to demonstrate the viability of this methodology and the CARES/Creep program.
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Fatima, Arooj, Stefan Turek, Abderrahim Ouazzi, and Muhammad Aaqib Afaq. "An adaptive discrete Newton method for regularization-free Bingham model." In VI ECCOMAS Young Investigators Conference. València: Editorial Universitat Politècnica de València, 2021. http://dx.doi.org/10.4995/yic2021.2021.12389.

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Developing a numerical and algorithmic tool which correctly identifies unyielded region in the yield stress fluid flow is a challenging task. Two approaches are commonly used to handle the singular behaviour at the yield surface, i.e. Augmented Lagrangian approach and the regularization approach, respectively. Generally in the regularization approach, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress [1]. The three field formulation of yield stress fluid corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is solved efficiently and accurately by a monolithic finite element method. The velocity and pressure are discretized by higher order stable FEM pair $Q_2/P^{\text{disc}}_1$ and the auxiliary stress is discretized by $Q_2$ element.Furthermore, this problem is highly nonlinear and presents a big challenge to any nonlinear solver. We developed a new adaptive discrete Newton's method, which evaluates the Jacobian with the directional divided difference approach [2]. The step length in this process is an important key: We relate this length to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton's method. The resulting linear sub problems are solved using the geometrical multigrid solver. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for two different prototypical configurations, i.e. "Viscoplastic fluid flow in a channel" and "Lid Driven Cavity", respectively [2].REFERENCES[1] Aposporidis, A., Haber, E., Olshanskii, M. A. and Veneziani, A. A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution. Comput. Methods Appl. Mech. Engrg, Vol. 200, pp. 2434–2446, (2011).[2] Fatima, A., Turek, S., Ouazzi, A. and Afaq, M. A. An adaptive discrete Newton method for regularization-free Bingham model. Ergebnisberichte des Instituts fuer Angewandte Mathematik Nummer 635, Fakultaet fuer Mathematik, TU Dortmund University, 635, 2021.
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Kim, Cheol, and Dong-Yeub Lee. "Design Optimization of a Piezoelectric Fiber Actuator With a Natural Curvature." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-33989.

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Piezoelectric Fiber Composite with Interdigitated Electrodes (PFCIDE) was previously introduced as an alternative to monolithic wafers with conventional electrodes for applications of structural actuation. This paper is an investigation into the performance improvement of piezoelectric fiber composite actuators by optimizing the stacking sequence and changing the matrix material. This paper presents the numerical optimization of a piezoelectric fiber/piezoelectric matrix composite actuator with IDE (PFPMIDE). Various concepts from different backgrounds, including three-dimensional linear elastic and dielectric theories, have been incorporated into the present linear piezoelectric model. To see the structural responses of the actuator integrated with the PFPMIDE, three-dimensional finite element formulations were derived. Numerical analysis shows larger center displacement of the curved actuator with the PFPMIDE due to optimization of the piezoelectric fiber angles. This paper presents the concept of a curved actuator that occurs naturally via thermal residual stress during the curing process, as well as the optimization of the maximum curved actuator displacement, which is accomplished using the Davidon-Fletcher-Powell (DFP) method.
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Kim, Cheol, and Kun-Hyung Koo. "Development of a PZT Fiber/Piezo-Polymer Composite Actuator With Interdigitated Electrodes." In ASME 2001 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/imece2001/ad-23749.

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Abstract Piezoelectric Fiber Composites with Interdigitated Electrodes (PFCIDE) were previously introduced as an alternative to monolithic wafers with conventional electrodes for applications of structural actuation. This paper is an investigation into the performance improvement of piezoelectric fiber composite actuators by changing the matrix material. This paper presents a modified micro-electromechanical model and numerical analyses of piezoelectric fiber/piezopolymer matrix composite actuator with interdigitated electrodes (PFPMIDE). Various concepts from different backgrounds including three-dimensional linear elastic and dielectric theories have been incorporated into the present linear piezoelectric model. The rule of mixture and the modified method to calculate effective properties of fiber composites were extended to apply to the PFPMIDE model. The new model was validated comparing with available experimental data and other analytical results. To see the structural responses of a composite plate integrated with the PFPMIDE, three-dimensional finite element formulations were derived. Numerical analyses show that the shape of the graphite/epoxy composite plate with the PFPMIDE may be controlled by judicious choice of voltages, piezoelectric fiber angles, and elastic tailoring of the composite plate.
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Landis, Chad M. "New finite element formulation for electromechanics." In SPIE's 9th Annual International Symposium on Smart Structures and Materials, edited by Christopher S. Lynch. SPIE, 2002. http://dx.doi.org/10.1117/12.475012.

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Reports on the topic "Monolithic finite element formulation"

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Curfman, L. V. A new finite element formulation for incompressible flow. Office of Scientific and Technical Information (OSTI), February 1995. http://dx.doi.org/10.2172/26516.

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Hong, Jung-Wuk. Coupling of Peridynamics and Finite Element Formulation for Multiscale Simulations. Fort Belvoir, VA: Defense Technical Information Center, October 2012. http://dx.doi.org/10.21236/ada582696.

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Bochev, Pavel Blagoveston. Control volume finite element method with multidimensional edge element Scharfetter-Gummel upwinding. Part 1, formulation. Office of Scientific and Technical Information (OSTI), June 2011. http://dx.doi.org/10.2172/1020517.

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Landers, Joseph A., and Robert L. Taylor. An Augmented Alagrangian Formulation for the Finite Element Solution of Contact Problems. Fort Belvoir, VA: Defense Technical Information Center, March 1986. http://dx.doi.org/10.21236/ada166649.

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Romkes, Albert, Serge Prudhomme, and J. T. Oden. Convergence Analysis of a Discontinuous Finite Element Formulation Based on Second Order Derivatives. Fort Belvoir, VA: Defense Technical Information Center, November 2004. http://dx.doi.org/10.21236/ada439718.

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Glowinski, R., W. Kinton, and M. F. Wheeler. A Mixed Finite Element Formulation for the Boundary Controllability of the Wave Equation. Fort Belvoir, VA: Defense Technical Information Center, October 1990. http://dx.doi.org/10.21236/ada226066.

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Alexander, A., J. T. Tzeng, W. H. Drysdale, and B. P. Burns. Effective Three-Dimensional (3-D) Finite Element Material Stiffness Formulation for Modeling Laminated Composites. Fort Belvoir, VA: Defense Technical Information Center, April 1996. http://dx.doi.org/10.21236/ada306454.

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Masud, Arif, and Mohammad Panahandeh. A Finite Element Formulation of Multi-Layered Shells for the Analysis of Laminated Composites. Fort Belvoir, VA: Defense Technical Information Center, January 1995. http://dx.doi.org/10.21236/ada348947.

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Yang, T. Y., and Alex T. Chen. Static and Dynamic Formulation of a Symmetrically Laminated Beam Finite Element for a Microcomputer. Fort Belvoir, VA: Defense Technical Information Center, February 1985. http://dx.doi.org/10.21236/ada170885.

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Juanes, Ruben, and Tadeusz W. Patzek. Stabilized numerical solutions of three-phase porous media flow using a multiscale finite element formulation. Office of Scientific and Technical Information (OSTI), January 2003. http://dx.doi.org/10.2172/834625.

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