A General Formulation of Equilibrium Macro-Elements With Control of Spurious Kinematic Modes: The Exorcism of an Old Curse

Maunder, E. A. W.; Almeida, J. P. Moitinho de; Ramsay, A. C. A.

doi:10.1002/(sici)1097-0207(19960930)39:18<3175::aid-nme978>3.0.co;2-3

Cited by 45 publications

(53 citation statements)

References 11 publications

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“…This is a well‐known situation for hybrid formulations, in the case of stress‐based approaches the multiple solutions are known as spurious kinematic modes (SKMs), whereas for displacement‐based formulations, they are normally referred to as spurious static modes (SSMs). We note that the concept of hybrid compatible elements, in which interelement displacement continuity is imposed a posteriori, has been presented for two‐dimensional (2D) and three‐dimensional (3D) problems as a dual to the hybrid equilibrium (HE) elements.…”

Section: A Semihybrid Formulation For Kirchhoff Platesmentioning

confidence: 99%

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

Almeida

Maunder

Tiago

2019

Numerical Meth Engineering

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Summary We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C1 continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C1 continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second‐order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.

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Section: A Semihybrid Formulation For Kirchhoff Platesmentioning

confidence: 99%

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

Almeida

Maunder

Tiago

2019

Numerical Meth Engineering

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“…The Lagrange multipliers corresponding to these constraints are then identified as the displacements at the interface where codiffusivity is being imposed. Only a minimal contextualisation of this approach is provided here; further details can be found for example in the works of Moitinho de Almeida et al,() which is also covered in detail in our work…”

Section: Finite Element Modelsmentioning

confidence: 99%

Improved eigenfrequencies of mechanical systems by combining complementary models

Almeida

Maunder

2018

Numerical Meth Engineering

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Summary In this paper, we propose a novel approach to combine two complementary numerical models of the dynamical behaviour of mechanical systems: primal (compatible) and dual (equilibrated). This combined model provides an improved estimate of the eigenfrequencies of the system by feeding each model with information from the other. Numerical solutions obtained from both the fundamental models and from the proposed approach are presented and studied. This approach will be applicable to any eigenvalue problem associated with a PDE that can be expressed by complementary numerical models.

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“…Ces modes correspondent à des déplacements relatifs non nuls des frontières des éléments de la structure non chargée. On peut montrer (Maunder et al, 1997) que pour un élément triangulaire primitif de degré p :…”

Section: Equations De Compatibilitéunclassified

Modèles équilibre pour l'analyse duale

Kempeneers

Beckers

Almeida³

et al. 2003

Revue Européenne des Éléments Finis

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L'analyse duale pure a été l'une des premières méthodes utilisées pour estimer l'erreur de discrétisation globale commise lors d'un calcul par éléments finis. Elle repose sur la comparaison de deux solutions éléments finis, la première obtenue à partir d'un modèle cinématiquement admissible classique et la seconde issue d'un modèle statiquement admissible (approche de type équilibre). Nous présentons dans ce travail deux méthodes permettant de créer des éléments équilibre. La première qui peut être qualifiée d'hybride, permet d'étudier des problèmes bidimensionnels avec des éléments équilibre de degré élevé. La seconde, pour laquelle nous présentons dans ce travail l'extension récente aux problèmes 3D à maillages tétraédriques, est de type équilibre pur. Ces deux approches nous ont permis de présenter quelques résultats d'estimation de l'erreur globale par analyse duale.ABSTRACT. The pure dual analysis is one of the first methods developed to perform the estimation of the global discretization error of finite elements analysis. It is based on the comparison of two finite elements solutions, one of which being of the displacement type (kinematically admissible), the second one being of the equilibrium type (statically admissible). This work presents two methods allowing to create equilibrium elements. The first one which can be seen as an hybrid method, allows to compute high-order equilibrium solutions of 2D problems. The second method is of the pure equilibrium type. Its recent extension to 3D problems with tetrahedral mesh is presented here. Those approaches enabled us to present some results of global error estimation by dual analysis. MOTS-CLÉS : estimation d'erreur globale, analyse duale, modèles équilibre.

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A General Formulation of Equilibrium Macro-Elements With Control of Spurious Kinematic Modes: The Exorcism of an Old Curse

Cited by 45 publications

References 11 publications

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

Improved eigenfrequencies of mechanical systems by combining complementary models

Modèles équilibre pour l'analyse duale

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