Quantum and Classical Fidelity for Singular Perturbations of the Inverted and Harmonic Oscillator

Let us consider the quantum/versus classical dynamics for Hamiltonians of the form \beq \label{0.1} H\_{g}^{\epsilon} := \frac{P^2}{2}+ \epsilon \frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $\epsilon = \pm 1$, $g$ is a real constant. We shall in particular study the Quantum Fidelity between $H\_{g}^{\epsilon}$ and $H\_{0}^{\epsilon}$ defined as \beq \label{0.2} F\_{Q}^{\epsilon}(t,g):= < \exp(-it H\_{0}^{\epsilon})\psi, exp(-itH\_{g}^ {\epsilon})\psi > \edq for some reference state $\psi$ in the domain of the relevant operators. We shall also propose a definition of the Classical Fidelity, already present in the literature (\cite{becave1}, \cite{becave2}, \cite{ec}, \cite{prozni}, \cite{vepro}) and compare it with the behaviour of the Quantum Fidelity, as time evolves, and as the coupling constant $g$ is varied.Comment: To be published in Journal of Mathematical Analysis and Application

Improved formulation for the stabilized enhanced strain solid-shell element (SHB8PS): geometric linear and nonlinear applications

In this study, the formulation of the SHB8PS solid-shell element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. The resulting physically stabilized and locking-free finite element consists in a continuum mechanics shell element based on a purely three-dimensional formulation. In fact, this is a hexahedral element with eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modelling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

A new locking-free formulation for the SHB8PS solid–shell element: non-linear benchmark problems

In this work, a new physically stabilized and locking-free formulation of the SHB8PS element is presented. This is a solid-shell element based on a purely 3D formulation. It has eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

Validation d’une nouvelle version de l’élément solide/coque “SHB8PS” sur des cas tests non linéaires

L’intérêt de disposer d’éléments finis volumiques capables de modéliser des structures minces est motivé par de nombreux problèmes industriels. Ainsi, ces dernières années, plusieurs travaux ont été réalisés dans ce domaine. Ces éléments coques épaisses ont de nombreux avantages : ils sont capables de représenter le comportement de structures minces avec une bonne prise en compte des phénomènes à travers l’épaisseur et avec un gain de temps de calcul significatif, ils permettent de mailler des géométries complexes où coques et solides doivent cohabiter sans les problèmes connus de raccordement de maillages. L’élément SHB8PS a été développé dans ce sens à partir d’une formulation purement tridimensionnelle. Récemment, une nouvelle version, libre de verrouillage (en membrane et cisaillement), a été formulée et validée en linéaire. Dans la présente étude, cette version revisitée est validée à travers de nombreux cas tests non linéaires.Contrat EDF R&

New prismatic solid-shell element : Assumed strain formulation and hourglass mode analysis

The formulation of a six-node solid-shell called SHB6, which is a linear, isoparametric element, is discussed. An eigenvalue analysis of the element stiffness matrix is first carried out. Several modifications are introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko and Bindeman. SHB6's coordinates and displacements are related to the nodal coordinates and displacements through the linear shape functions. Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, the formula is obtained. The newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges well and performs much better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested

An improved assumed strain solid-shell element formulation with physical stabilization for geometric non-linear applications and elastic-plastic stability analysis

In this paper, the earlier formulation of the SHB8PS finite element is revised in order to eliminate some persistent membrane and shear locking phenomena. This new formulation consists of a solid-shell element based on a purely three-dimensional approach. More specifically, the element has eight nodes, with displacements as the only degrees of freedom, as well as an arbitrary number of integration points, with a minimum number of two, distributed along the 'thickness' direction. The resulting derivation, which is computationally efficient, can then be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. A reduced integration scheme is used to prevent some locking phenomena and to achieve an attractive, low-cost formulation. The spurious zero-energy modes due to this in-plane one-point quadrature are efficiently controlled using a physical stabilization procedure, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the assumed strain method. In addition to the extended and detailed formulation presented in this paper, particular attention has been focused on providing full justification regarding the identification of hourglass modes in relation to rank deficiencies. Moreover, an attempt has been made to provide a sound foundation to the derivation of the co-rotational coordinate frame, on which the calculations of the stabilization stiffness matrix and internal load vector are based. Finally to assess the effectiveness and performance of this new formulation, a set of popular benchmark problems is investigated, involving geometric non-linear analyses as well as elastic-plastic stability issues

Improved formulation for the stabilized enhanced strain solid-shell element (SHB8PS): geometric linear and nonlinear applications

In this study, the formulation of the SHB8PS solid-shell element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. The resulting physically stabilized and locking-free finite element consists in a continuum mechanics shell element based on a purely three-dimensional formulation. In fact, this is a hexahedral element with eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modelling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

A new locking-free formulation for the SHB8PS solid–shell element: non-linear benchmark problems

In this work, a new physically stabilized and locking-free formulation of the SHB8PS element is presented. This is a solid-shell element based on a purely 3D formulation. It has eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

Locking-free formulation for the stabilized enhanced strain solid-shell element (SHB8PS): geometrically non-linear applications

In this work, a new locking-free and physically stabilized formulation of the SHB8PS solid-shell element is presented. The resulting finite element consists of a continuum mechanics shell element based on a purely three-dimensional approach. This eight-node hexahedron is integrated with a set of five Gauss points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase its computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

A physically stabilized and locking-free formulation of the (SHB8PS) solid-shell element

In this work, the formulation of the SHB8PS finite element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. This is a solid-shell element based on a purely three-dimensional formulation. In fact, the element has eight nodes as well as five integration points, all distributed along the "thickness" direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Dans cette étude, la formulation de l’élément SHB8PS est revisitée dans le but d’éliminer certains blocages persistants en membrane et cisaillement transverse. Rappelons que cet élément est de type coque épaisse obtenue à partir d’une formulation purement tridimensionnelle. Il possède donc huit noeuds et cinq points d’intégration répartis selon la direction de l’épaisseur. Ainsi, il peut être utilisé pour modéliser des structures minces tout en prenant correctement en compte les différents phénomènes à travers l’épaisseur. Afin d’améliorer ses performances de calcul et d’éviter certains blocages, l’intégration réduite a été employée. Les modes de hourglass générés par la sous-intégration sont efficacement stabilisés et les modes de blocages persistants sont éliminés par une technique de projection pouvant se mettre sous le formalisme de la « méthode de déformation postulée »