Overall, the question of the link between a plate model and its 3D counterpart is of such practical and theoretical importance that it has given rise to many publications. This introduction focuses more specifically on the references of interest in the literature that we tried to incorporate into our formulation.
The quality of a plate model is usually assessed by comparison with a corresponding 3D reference model. This has led to a first group of works focusing on the development of a posteriori error estimators, initially neglecting **edge** effects [28, 14, 47, 29, 30], then taking them into account [31]. Thus, by the end of the 1980’s, the performance of the classical Kirchhoff-Love and Reissner-Mindlin plate and shell theories in 3D elasticity was relatively well-understood. Another group of works concerned the technique of asymptotic developments with the thickness as the small parameter [19, 24, 12, 17, 16, 6, 11]. In addition, using the same approach as Reissner [40, 42] in statics and Mindlin in dynamics [41], a shear factor was often introduced [49] in order to correct what would have been deduced from a pure kinematic assumption by taking into account some a priori information on the three-dimensional stress state. This mixed nature of the link between plate theory and 3D theory is the reason why the construction of improved theories [35, 48, 34, 39, 8] and associated **finite** elements [9, 2] relies heavily on mixed formulations.

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In the present paper, a unique model, **based** on Discrete Ply Model, allows to simulate **edge** impact and compression after impact (CAEI) of two stacking sequences highly oriented, representative of a real aeronautical stiffener structure. A relatively good correlation is found between experiment and model regarding the important number of phenomena which is taken into account: permanent indentation after impact, force–displacement impact curve, crack length after impact, delamination after impact, stress- displacement CAEI curve, stress-out-of-plane displacement curve, CAEI final failure. Moreover only 14 parameters are needed to feed this model, and all are obtained using standard tests and have physical relevance. This concept is important because it allows considering running similar tests with other materials without performing complex and special identification tests but only using physical parameters of standard tests. The good results of this approach could allow to define a new design method to improve the **edge** impact damage tolerance.

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or elasticity. The degrees of freedom (dofs) associated with Whitney differential forms have a direct physical relevance. When considering polynomial inter- polation of higher degree, dofs can be chosen in different ways. In [17] and its extension [15] (see also [16]), higher moments are used. They are also considered in the general framework of the **finite** **element** exterior calculus [3]. In [20], the localization issue has been addressed, namely, the relationship between dofs and measurable quantities (such as circulations, fluxes, densities) for the field they are related with. In the framework of high order Whitney **finite** **element** spaces, integrals on suitable subsimplices of the mesh are a valid alternative as dofs to the classical moments. Their definition is **based** on the introduction of the small simplices, that are subsimplices resulting from homothetic contractions of the

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4.1. Three-point bending test
A single-**edge** notched beam under three-point bending is considered (see Figure 5). The material parameters used in the nonlocal damage model are given in Table I.
The beam is computed by means of an arc-length procedure using the crack mouth opening displacement (CMOD) as a control parameter. We have chosen an unstructured mesh to avoid potential simpli ﬁcations due to the symmetry of the problem and to keep a rather general solution processing. The ﬁnal damage ﬁeld is given in Figure 6. The load–deﬂection curve is shown in Figure 7. The crack is located on the subset de ﬁned by the elements whose damage is greater than 0.8: This subset of the mesh and the idealized crack are given in Figure 8. The idealized crack is almost straight, even if the tracking area is nonsymmetric with respect to the center of the beam. One can also notice that the crack bend outward near the top of the beam: This stems from the in ﬂuence of the loading platen that modi ﬁes the strain distribution in this area.

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or elasticity. The degrees of freedom (dofs) associated with Whitney differential forms have a direct physical relevance. When considering polynomial interpo- lation of higher degree, dofs can be chosen in different ways. In [9] and its extension [7] (see also [8]), higher moments are used. They are also considered in the general framework of the **finite** **element** exterior calculus [3]. In [11], the localization issue has been addressed, namely, the relationship between dofs and measurable quantities (such as circulations, fluxes, densities) for the field they are related with. In the framework of high order Whitney **finite** **element** spaces, integrals on suitable subsimplices of the mesh are a valid alternative as dofs to the classical moments. Their definition is **based** on the introduction of the small simplices, that are subsimplices resulting from homothetic contractions of the elements of the mesh 1 . New dofs are then the weights, integrals of a k-form on

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In the present paper, a unique model, **based** on Discrete Ply Model, allows to simulate **edge** impact and compression after impact (CAEI) of two stacking sequences highly oriented, representative of a real aeronautical stiffener structure. A relatively good correlation is found between experiment and model regarding the important number of phenomena which is taken into account: permanent indentation after impact, force–displacement impact curve, crack length after impact, delamination after impact, stress- displacement CAEI curve, stress-out-of-plane displacement curve, CAEI final failure. Moreover only 14 parameters are needed to feed this model, and all are obtained using standard tests and have physical relevance. This concept is important because it allows considering running similar tests with other materials without performing complex and special identification tests but only using physical parameters of standard tests. The good results of this approach could allow to define a new design method to improve the **edge** impact damage tolerance.

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The identification of modal parameters or, more generally, the system identification results of a structure in op- eration find an important application in the calibration of a numerical model of the investigated structure. **Finite** **element** (FE) models are used e.g. to verify design specifications, to assess stress fields in structures, to predict vibra- tion levels under prescribed harmonic excitations, to detect abnormal structural behavior in the context of structural health monitoring [11, 12], and so on. With model updating techniques, the parameters of the FE model are cal- ibrated such that some model properties are close to the truly observed structural properties. Vibration-**based** FE model updating techniques [13, 14] identify model parameters by minimizing a cost function involving the identi- fied and model-**based** modal parameters (or derived variables thereof). The involved experimental data are subject to uncertainties. In a broad sense, these uncertainties can be classified into two categories of aleatory (irreducible) and epistemic (reducible) uncertainties [15]. Aleatory uncertainty may result from geometric dimension variability due to manufacturing tolerances or inherent variability of materials such as concrete, while epistemic uncertainty is caused by lack of knowledge (e.g. due to **finite** number of data samples, undefined measurement noises, unknown excitations, and so on). These uncertainties can be considered in two ways in model updating. First, the uncertainty of the updated parameters can be evaluated **based** on the uncertainty of the experimental data. Second, the uncertainty of the experimental data can be taken into account in stochastic updating techniques as suggested in [12]. In this paper, we consider the first way.

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The two software are coupled through a co-simulation procedure giving the opportunity to perform various kinds of pantograph catenary operational cases since the technology developed includes specific pantograph adjustment capabilities and data exchange options to ease parametric studies achievement and post-processing. The global co-simulation process is described from the creation of communication entities to the time dynamic calculation. A focus is made on the iterative fixed time step procedure with a computer memory **based** data exchange. The method is assessed and demonstrates a high agreement with a pure OSCAR © simulation using an equivalent three lumped mass model.

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VI. C ONCLUSIONS AND FUTURE WORK
This paper presents a modeling methodology to obtain the kinematic relationships of soft manipulators. The kinematic equations are derived from a FEM model (or any equivalent physics **based** model) that can be obtained from the geometry and the material properties of a soft manipulator. After a pro- jection in a small constraint space, a set of coupled equations relate the position of the end-effector to the contribution of actuators and displacement of sensors. The validity of the method is demonstrated in two different manipulators with complex geometry. In the case of the CBHA, the results were compared to those obtained with two geometric models developed for the same robot. While the model of the material used does not take into account the properties of viscosity, this consideration is only due to the absence of knowledge of these specific properties for the material used. Indeed, the framework used allows for modeling viscoelasticity with Prony series [46]. In general, a viscoelastic model is characterized by a rate-independent term, which in this case is the shear modulus representing the elastic behavior, and a rate-dependent modu- lus. The rate-dependent modulus of the material is defined by the Prony series **based** on time; faster strain rates will induce higher modulus than static loads. The limitations on the use of Prony series come with the determination of the required coefficients, since it involves stress relaxation tests performed under controlled temperature and loading speed. Another way to model viscoelasticity behaviour is to introduce a rate- dependent damping effect using Rayleigh equation. Rayleigh damping is a viscous damping that is proportional to a linear combination of mass and stiffness. Using Rayleigh damping, The internal forces in the robot (equation 1) takes the form:

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Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des V[r]

3.5 Material data
In order to define material properties, a class for each material type is available. The procedure is the following : If a **finite** **element** lies in a material (e.g. Copper), then the **element** code (an integer number) for this **element** has to be associated to the material Copper. From a programming point of view Copper is a class, and the properties of copper can be accessed via the constructors of this class. Several con- structors of material classes are implemented in order to initialize data in several situations (constant data, temperature dependent data, ...). The used materials are to be defined in the mesh file. Note that a default material called GenericMaterial is defined with default values for coefficients.

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The aim of this paper is to derive a new displacement ﬁnite **element** for inﬂatable beams directly from the virtual work principle. The result will be a symmetrical stiffness matrix which holds for airtight tubes.
In the ﬁrst section of the paper, the ﬁnite **element** is derived for the static in-plane stretching and bending problem of an inﬂated cylindrical beam made of a membrane. First, the discretized nonlinear equations will be written by use of the virtual work principle with Timoshenko’s kinematics, ﬁnite rotations and small strains. The procedure will enable one to correctly account for the shear effect and the pressure in the governing equations. Then, the tangent stiffness matrix will be derived as the sum of a tangent matrix due to the internal strain work and another tangent matrix due to the internal pressure. Whereas it is essential to assume ﬁnite rotations in order to correctly exhibit all the terms due the internal pressure, it is sensible to use the inﬂatable beam in small deformations only. Thus, the nonlinear problem will be linearized around a well-deﬁned reference conﬁguration. This will lead to the inﬂatable beam ﬁnite **element** involving the Timoshenko shear coefﬁcient and the inﬂation pressure. The particular case of a 3-node beam ﬁnite **element** with quadratic shape functions will be considered.

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2 **Finite** **element** catenary modelling
The three dimensional catenary is modelled using OSCAR software. The model is mostly composed of pre-tensioned Euler-Bernoulli **Finite** Elements but also allows masses, bars, springs, etc. This allows every kind of catenary geometries: conventional or high speed designs, DC or AC lines with one or several contact wires, with or without stitch wires, etc. Several catenary sections can be considered including overlap sections where pantographs run under several contact wires. The pantograph considers lumped masses with possible bumpstops and friction elements in the pantograph. Typical non linearities are dropper compression and pantograph/contact wire contact. In the latter case, shape functions of the contact wire beams and a mass or flexible beam representation of the friction band are used to compute penetration at the moving interface and generate a contact force using unilateral linear contact stiffness. The coupling of these flexible structures was studied in the literature, especially about load moving on a FE mesh [14][15][16][17]. OSCAR has been deeply validated against inline measurements and the catenary model is considered as stable and efficient Erreur ! Source du renvoi introuvable..

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1 What is sparselizard
Sparselizard (Copyright (C) 2017- Alexandre Halbach and Christophe Geuzaine, University of Liege, Belgium) is an open source C++ **finite** **element** library provided under the terms of the GNU General Public License (GPL), version 2 or later. It is meant to be user friendly and concise while decently fast and parallelised. A working example solving an electrostatic problem on a 3D disk with volume charges and grounded at its boundary is shown below.

Context of the work Manufacturing Characterization Impact tests Fatigue Design of Experiments Modeling bi-axial Impact NDT Tomography RX... Aim of the work Manufacturing Ch[r]

2.4. Physical model of vocal folds
To validate the results from the mathematical model, a self-oscillating physical model of vocal folds was fabricated at ENSTA Paris. Detailed description of this model and results of the measurements can be found e.g. in [18, 19]. Let us just say here that it was designed as a vocal-fold-shaped **element** vibrating in the wall of a rectangular wind tunnel. The model is 4 : 1 scaled; to avoid difficulties with asymmetric vocal fold vibration, one of the vocal folds is static. Best possible effort was made to keep the important dimensionless parameters (Reynolds and Strouhal numbers) of the model close to the real situation. The shape of the vocal folds was specified in the same fashion as in the mathematical model.

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Robotic needle insertion in moving soft tissues using constraint-**based** inverse **Finite** **Element** simulation
Paul Baksic 1 , Hadrien Courtecuisse 1 , Christian Duriez 2 and Bernard Bayle 1 ,
Abstract— This paper introduces a method for robotic steer- ing of a flexible needle inside moving and deformable tissues. The method relies on a set of objective functions allowing to automatically steer the needle along a predefined path. In order to follow the desired trajectory, an inverse problem linking the motion of the robot end effector with the objective functions is solved using a **Finite** **Element** simulation. The main contribution of the article is the new constraint-**based** formu- lation of the objective functions allowing to: 1) significantly reduce the computation time; 2) increase the accuracy and stability of the simulation-guided needle insertion. The method is illustrated, and its performances are characterized in a realistic framework, using a direct simulation of the respiratory motion generated from in vivo data of a pig. Despite the highly non-linear behavior of the numerical simulation and the significant deformations occurring during the insertion, the obtained performances enable the possibility to follow the trajectory with the desired accuracy for medical purpose.

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ON DIFFUSION EQUATIONS
RACHELE ALLENA AND CHRISTOPHE CLUZEL
Heterogeneous materials such as bone or woven composites show mesostruc- tures whose constitutive elements are all oriented locally in the same direction and channel the stress flow throughout the mechanical structure. The interfaces between such constitutive elements and the matrix are regions of potential degra- dations. Then, when building a numerical model, one has to take into account the local systems of orthotropic coordinates in order to properly describe the damage behavior of such materials. This can be a difficult task if the orthotropic directions constantly change across the complex three-dimensional geometry as is the case for bone structures or woven composites. In the present paper, we propose a **finite** **element** technique to estimate the continuum field of orthotropic directions **based** on the main hypothesis that they are mainly triggered by the external surface of the structure itself and the boundary conditions. We employ two diffusion equations, with specific boundary conditions, to build the radial and the initial longitudinal unit vectors. Then, to ensure the orthonormality of the basis, we compute the longitudinal, the circumferential, and the radial vectors via a series of vector products. To validate the numerical results, a comparison with the average directions of the experimentally observed Haversian canals is used. Our method is applied here to a human femur.

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(a) (b)
Figure 3. Biot’s equations case. (a) The computational domain is only the subsurface Ω = Ω s , with exter- nal boundaries Γ j and internal boundaries Γ jk . The outer normals νj and ν jk are also indicated. (b) Scheme
for the twelve degrees of freedom associated with each **element**, eight to the solid displacements and four to the fluid displacements.

domains where the diffusion tensor S = Id. The cell-centered FV method with an
upwind discretization of the convective terms provides the stability and is extremely robust. However in this case, the mesh is assumed to be admissible [7, Definition 9.1]. In particular, this implies that the orthogonality condition has to be satisfied. As mentioned in [5], a difficulty in the implementation is to construct such admissi- ble meshes. Structured rectangular meshes are admissible, but they cannot be used for complex geometries arising in physical contexts. Furthermore, the **finite** **element** (FE) method allows for an easy discretization of diffusive terms with full tensors without imposing any restrictions on the meshes. However, some numerical insta- bilities may arise in the convection-dominated case.

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