A simplified analysis of plate structures using Trefftz‐functions

Hochard, Christian; Proslier, L.

doi:10.1002/nme.1620340111

Cited by 16 publications

(15 citation statements)

References 11 publications

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“…N , it has been shown numerically that the minimum of R N is reached for v D OM , where O is the homothetic center and M 2 @! N ; furthermore, for a circular and spherical geometric shapes, it has been shown in [33] (see Appendix D); it follows that ".v/ D I d . Let us note that in the 2D case of a cracked circle and of a double square || , as well as in the 3D case of a double unit parallelepiped || , the same minimum eigenfunction for R N has been obtained numerically.…”

Section: Remarkmentioning

confidence: 86%

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New bounding techniques for goal‐oriented error estimation applied to linear problems

Ladevèze

Pled

Chamoin

2012

Numerical Meth Engineering

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The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems, thanks to new bounding techniques based on Saint-Venant's principle. The main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains that enables to cleverly derive guaranteed and accurate bounding of contributions to the global error estimate over a local region of the domain. Performances of these techniques are illustrated through several numerical experiments.h can be recovered from the data and the FE stress field h alone. Starting from an admissible solution . O u h , O h / provided by one of the existing techniques, one can measure the global residual on constitutive relation (3), called the CRE and hereafter

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Section: Remarkmentioning

confidence: 86%

“…Eventually, differentiation of (34) with respect to variable completes the proof of relation (32). Similarly, the derivation of relation (33) can be proved in a straightforward manner.…”

Section: Proofmentioning

confidence: 98%

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New bounding techniques for goal‐oriented error estimation applied to linear problems

Ladevèze

Pled

Chamoin

2012

Numerical Meth Engineering

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“…A basis focusing on the overall response of the structure was constructed to approximate the loading conditions. This construction was based on the Trefftz‐like solutions developed in by approximating the displacement field with polynomial functions satisfying equilibrium equations. The basis of the loading conditions was obtained from the projections of these Trefftz‐like solutions onto the boundaries of the structure.…”

Section: Structural Monitoring: An Inverse Problemmentioning

confidence: 99%

Load identification for full‐field reconstruction: applications to plates under tension loads

Martini

Hochard

Charles

2012

Numerical Meth Engineering

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International audienceThe full-field reconstruction method presented here is suitable for real-time structural monitoring purposes, such as improving the performances of structures. The aim is to characterize mechanical fields and boundary conditions while the structure is in service, using just a few on-line measurements. This characterization is an ill-posed inverse problem, the solution of which requires making some prior assumptions. The structures studied are assumed to be in steady-state configurations and to show linear mechanical behavior, and the loading zones are assumed not to be overstressed; the mechanical fields are monitored only inside the structures. The full-field identification is regularized using the boundary conditions, which are identified from strain measurements, and the mechanical fields are then reconstructed from these identified boundary conditions. On the basis of Saint-Venant's principle, the boundary conditions are approximated with a few parameters, the basis functions of which are obtained from the projections of Trefftz-like solutions onto the boundaries of the structures. These approximate boundary conditions are linked to the global mechanical responses of the structures. Lastly, this full-field reconstruction method is applied to plate-like structures with holes subjected to in-plane loads. The results show that the load identification procedure efficiently regularizes the full-field reconstruction method, and that this method is suitable for structural monitoring purposes. The sensitivity of this method to errors is similar to that of the load identification procedure, and the maximum errors in the solutions are located at the boundaries

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“…In the Galerkin method, the Trefftz functions are also used as the weighting functions such that [17,18]:…”

Section: Indirect Trefftz Formulationmentioning

confidence: 99%

Free vibration analysis of thin circular plates by the indirect Trefftz method

Ghannadiasl

Noorzad

2009

Mesh Reduction Methods

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The vibration of plates is important in many applications pertaining to mechanical, civil and aerospace engineering. Therefore, the vibration of plates is an important research area that has been studied by many researchers. To date, there are abundant plate vibration solutions available in the open literature based on Kirchhoff assumptions. The purpose of the present paper is free vibration analysis of thin circular plates by the indirect Trefftz method. In thin plate vibration problems, we will deal with the governing equation with the homogeneous boundary conditions. The Trefftz method basically employs the complete set of solutions satisfying the governing equation as the first step. To derive the boundary integral equation, either the reciprocity law, which is similar to those used in conventional BEMs, or the weight residual method can be used. The proposed approach has some merits when compared with other regular boundary element formulations reported so far. A main benefit for the Trefftz method is that it does not involve singular integrals due to the properties of its solution basis functions (T functions); thus, it can be categorized into the regular boundary element method. Besides, this advocated approach yields a solution that offers simultaneously the advantages of the classical FEM and BEM solutions, without having their drawbacks. Finally, several numerical examples are demonstrated to show the validity of the current approach.

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A simplified analysis of plate structures using Trefftz‐functions

Cited by 16 publications

References 11 publications

New bounding techniques for goal‐oriented error estimation applied to linear problems

New bounding techniques for goal‐oriented error estimation applied to linear problems

Load identification for full‐field reconstruction: applications to plates under tension loads

Free vibration analysis of thin circular plates by the indirect Trefftz method

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