The modelling of error localization uses the existing error of constitutive law. In par- ticular, the applicability of such a **method** to large industrial test cases such as JASON2 is adressed. In order to improve the qual- ity of modal **updating** of the low frequency behaviour, a comparison between FRF and modal approaches of the error of constitu- tive law is made. Error location is exten- sively studied in the case of the dynamical behaviour of a clamped structure, with realis- tic limitations in order to find a better linear equivalent **model**. Design variables are typi- cally thickness of panels and section or num- ber of beams. Particular attention is paid to automatic selection of relevant parameters during **updating** in order to keep the sensitiv- ity matrix well-conditioned and to obtain an accurate solution for parameters fitting.

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Imbault and Arminjon (1993) propose a semi-analytical **method** to handle the texture evolution effect in their analytical expression of the yield locus. The principle is simple: they assume that a linear operator identified by a polycrystalline **model** can express the evolution of the ODF coefficients with strain.
Van Houtte (1994a) uses the dual plastic potentials to derive convenient formulas of yield loci of rate insensitive anisotropic materials. Van Bael et al. (1996) and Munhoven et al. (1997) have checked that 6th order polynomial series yield locus are required to reproduce the accuracy of polycrystalline approaches. They propose to define this sixth order series yield locus by a least square fit on a large number of points (typically 70 300) in the deviatoric stress space. These points are calculated by the Taylor **model** based on an assumed constant texture (Van Bael, 1994; Munhoven et al., 1996). This identification is performed once, outside the FE code (Winters, 1996). It provides 210 coefficients to describe the whole yield locus. This **model** called 'Ani3vh' is compared to the stress-strain interpolation **model** in Section 5. Unfortunately, taking into account the texture evolution effect with this yield locus means the computation of the 210 coefficients of the sixth order series for each integration point, each time a texture **updating** is necessary. Recently, Hoferlin et al. (1999) have extended this approach in the strain rate space and coupled it with the microstructural hardening **model** from Teodosiu and Hu (1995). Li et al. (2003) apply this constitutive **model** to investigate the effects of texture and strain-path changes on the plastic anisotropy in sheet metal forming.

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The modelling of error localization uses the existing error of constitutive law. In par- ticular, the applicability of such a **method** to large industrial test cases such as JASON2 is adressed. In order to improve the qual- ity of modal **updating** of the low frequency behaviour, a comparison between FRF and modal approaches of the error of constitu- tive law is made. Error location is exten- sively studied in the case of the dynamical behaviour of a clamped structure, with realis- tic limitations in order to find a better linear equivalent **model**. Design variables are typi- cally thickness of panels and section or num- ber of beams. Particular attention is paid to automatic selection of relevant parameters during **updating** in order to keep the sensitiv- ity matrix well-conditioned and to obtain an accurate solution for parameters fitting.

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of structures in operation, which is due to the fact that they have excellent theoretical and computational properties, for instance numerical efficiency and robustness, see e.g. [6, 7].
For any system identification **method**, the estimated parameters are afflicted with variance errors due to **finite** data length, unknown excitation and measurement noise. The variance of the modal parameter estimates is a most relevant information for assessing their accuracy. It depends on the chosen system identification algorithm. A practical approach for the variance estimation of modal parameters was developed in [8], where an estimated covariance on the measurements is propagated to the desired parameters by considering a sensitivity analysis. The required sensitivities are derived analytically through first-order perturbation theory, from the data to the identified parameters, and are then computed using the system identification estimates. In [9], the covariance computation scheme for the covariance- driven subspace **method** (SSI-cov) has been developed, and in [10] a fast and memory efficient implementation of the covariance computation for SSI-cov has been proposed.

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4.2. PARAMETER IDENTIFICATION FROM FULL FIELD MEASUREMENTS
with u ∗ a kinematically admissible displacement field. The principle of virtual fields **method** is simple. It aims at verifying the equilibrium equation in a weak sense for a prescribed strain field. To do so, the stress tensor on the left term is evaluated from the constitutive law (explicitly for a linear elastic problem, implicitly for an incremental plastic problem) as function of the material parameters and of the experimental strain: σ ∼ = f ( θ , ε ∼ ex p ). Hence, it is possible to build a system of equations where the unknowns are the n parameters to be identified. In order to be solved, the system must contain at least n linearly independent equations, which are obtained by carefully choosing the virtual displacement fields. This latter requirement represents the main difficulty with this **method**. Several works within the literature concern the choice (possibly automatic) of the virtual fields [52, 55, 129]. This Virtual Fields **Method** is a promising approach, since it allows the identification of dynamic properties by adding an acceleration term to the principle of virtual work. Furthermore, since there is no need to build an underling **Finite** **Element** **model**, the VFM does not require to know the exact boundary conditions. However, a particular attention to the experimental uncertainty is necessary, since the required quantities (strain and acceleration) are derivatives of the measured displacement: the derivation operation might amplify the uncertainty. Furthermore, as for the CEG methods, the majority of the applications within the literature concern situations for which the plane stress hypothesis is valid.

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Introduction
The **finite** **element** (FE) subproblem **method** (SPM) provides advantages in repetitive analyses and helps improving the solution accuracy [1-6]. It allows to benefit from previous computations in- stead of starting a new complete FE solution for any geometrical, physical or **model** variation. It al- so allows different problem-adapted meshes and computational efficiency due to the reduced size of each SP. A general framework allowing a wide variety of refinements is herein presented. It is a FE SPM based on canonical magnetostatic and magnetodynamic problems solved in a sequence, with at each step volume sources (VSs) and surface sources (SSs) originated from previous solu- tions. VSs express changes of material properties. SSs express changes of boundary conditions (BCs) or interface conditions (ICs). Common and useful changes from source to reaction fields, ideal to real flux tubes (with leakage flux), 1-D to 2-D to 3-D models, perfect to real materials, and statics to dynamics, can all be defined via combinations of VSs and SSs. The developments are per- formed for the magnetic vector potential FE formulation, paying special attention to the proper dis- cretization of the constraints involved in each SP. The **method** is illustrated and validated on vari- ous problems.

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[2] H. Barbosa and T. Hughes. **Finite** **element** **method** with lagrange multipliers on the boundary. circumventing the babuska-brezzi condition. Comp. Meth. in Applied Mech. and Engrg., 85(1):109–128, 1991.
[3] H. Ji and J.E. Dolbow. On strategies for enforcing interfacial constraints and evaluat- ing jump conditions with the extended **finite** **element** **method**. International Journal for Numerical Methods in Engineering, 61:2508–2535, 2004.

4 Perspectives and applications
The enrichment of the contact geometry by an arbitrary function permits : 1. to take into account a complicated geometry within one contact **element** ; 2. to account for a change of the local geometry due to loading conditions. As mentioned, if the enrichment is chosen to be localized within NTS contact elements, the choice of the enrichment function is limited : its value must be zero at the edges of the master segments. It implies a strong connection between the **finite** **element** mesh and the enrichment. A possible application of this approach is the modeling of periodic structures using a regular mesh, Fig. 5 . Enrichment of thin-walled or beam structures by a constant enriching function seems to be meaningfull, since the predominant deformation of such structures does not affect the geometry of the surface (Fig. 5 ,a-b). A possible applications is a modeling of contact with grid structures, micro contact with fiber, etc. The proposed approach is also valid for the case, when the deformation of the surface geometry is small in comparison to the deformation of the bulk material (Fig. 5 ,c-d), for instance, hard coating on a soft substrate. An anisotropic friction can be simulated implicitly by a special enrichment of the master surface (Fig. 6 ).

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

High order stabilized finite element method for MHD plasma modeling.. J Tarcisio-Costa, Boniface Nkonga.[r]

Figure 4-2: Four quadralateral elements used in construction of the local patch. The
resulting patch **element** is an approximation of a larger quadralateral **element**.
, It is important to note that VVkl is treated as data in eq. 4.8. By looking at the
linear elements contributing to the patch, one can easily see the quadralateral form that arises from P (fig. 4-2). Considering the case of a patch constructed from four bilinear rectangular elements illustrated in figure 4-2, one can easily see that a biquadratic **element** results, described by a biquadratic basis function. Although these four-**element** patches are the primary choice for implementation in the viscoelastic solver, it is instructive to

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which were in agreement with regions of the largest error in the current density. Unfortunately, for the case of moving front with low regularity, we observed large oscillations for both the uniform and adaptive space-time **method**. These numerical oscillations may be due to inherent instabil- ity associated with the continuous Galerkin formulation for low regularity solutions of the p-curl problem. One possible resolution for the instability is to instead use a SUPG-type formulation for the p-curl problem. While we were unable to demonstrate efficiency gains of the adaptive space- time **method** versus the uniform space-time **method**, our results show that the uniform space-time **method**, and more specifically implicit methods, merits a closer look by the engineering community due to its inherent stability in time. Moreover, we believe adaptive FE methods can still yield effi- ciency gains provided the convergence rate are of the same order for both space and time. This can be accomplished for example by an adaptive space-time **method** using first-order edge elements which refines in space by a factor of 4 in space and a factor of 2 in time. Though, for practical purposes, we believe an adaptive space FE **method** using first-order edge elements with first-order time-stepping schemes can readily lead to the improvement gains in efficiency using the proposed a posteriori error estimators for semi-discretization.

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1 Introduction
We consider a bone growth **model** based on [1]. It describes the evolution of the concentrations of the following quantities: the mesenchymal stem cells s, the os- teoblasts b, the bone matrix m and the osteogenic growth factor g. Bone healing begins by the migration of the stem cells to the site of the injury. Then along the bone, these cells differentiate into osteoblasts which start to synthetize the bone ma- trix. This cell differentiation is only possible in presence of the growth factor. The proposed **model** takes into account several phenomena: the diffusion of the stem cells and the growth factor, the migration of the stem cells towards the bone matrix, the proliferation and the differentiation of stem cells. The osteoblasts are considered without movement since they are fixed at the bone matrix. Moreover, the **model** in- cludes the case of heterogeneous domains, with possibly anisotropic diffusions. It is

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Abstract
Numerical simulations of fluid-structure interaction (FSI) are of first interest in numer- ous industrial problems: aeronautics, heat treatments, aerodynamic, bioengineering.... Because of the high complexity of such problems, analytical study is in general not suf- ficient to understand and solve them. FSI simulations are then nowadays the focus of numerous investigations, and various approaches are proposed to treat them. We pro- pose in this thesis a novel monolithic approach to deal with the interaction between an incompressible fluid flow and rigid/ elastic material. This **method** consists in consider- ing a single grid and solving one set of equations with different material properties. A distance function enables to define the position and the interface of any objects with complex shapes inside the volume and to provide heterogeneous physical properties for each subdomain. Different anisotropic mesh adaptation algorithms based on the varia- tions of the distance function or on using error estimators are used to ensure an accurate capture of the discontinuities at the fluid-solid interface. The monolithic formulation is insured by adding an extra-stress tensor in the Navier-Stokes equations coming from the presence of the structure in the fluid. The system is then solved using a **finite** **element** Variational MultiScale (VMS) **method**, which consists of a decomposition, for both the velocity and the pressure fields, into coarse/resolved scales and fine/unresolved scales. The distinctive feature of the proposed approach resides in the efficient enrichment of the extra constraint. In the first part of the thesis, we use the proposed approach to assess its accuracy and ability to deal with fluid-rigid interaction. The rigid body is prescribed under the constraint of imposing the nullity of the strain tensor, and its movement is achieved by solving the rigid body motion. Several test case, in 2D and 3D with simple and complex geometries are presented. Results are compared with existing ones in the literature showing good stability and accuracy on unstructured and adapted meshes. In the second part, we present different routes and an extension of the approach to deal with elastic body. In this case, an additional equation is added to the previous system to solve the displacement field. And the rigidity constraint is replaced with the corresponding behaviour law of the material. The elastic deformation and motion are captured using a convected level-set **method**. We present several 2D numerical tests, which are considered as classical benchmarks in the literature, and discuss their results.

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Let us define a **model** problem. Consider a bounded and regular domain Ω of R 2 with boundary Γ and let γ denote a regular closed curve that separates Ω into two bounded domains Ω + and Ω − , i.e. Ω = Ω + ∪ γ ∪ Ω − . We define the boundary value problem:
− ∇ · (a∇u) = f in Ω,

respect, explicit numerical time schemes such as the well-known central differ- ence scheme have been widely used as they do not require numerical iterations at each time step, and also for their good properties in term of accuracy and robustness with possible nonlinearities. The main drawback of such approaches concerns their conditional stability, and in particular the Courant-Friedrich- Lewy (CFL) condition (see Courant et al. [14]), which involves rather small time steps in practice. However, for transient dynamic simulations (possibly non-linear), the value of the critical time step is of the order of the physical phenomena that are involved. In other words, the CFL condition is not so restrictive in practice as it corresponds to the pertinent time scale of the tran- sient phenomenon. For transient dynamic simulations, it is also often needed to introduce in the numerical **model** specific interfaces, or singularities which can evolve with time (dynamic crack growth simulations, contact/impact sur- faces, etc...).

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We applied our **method** to sequences of cardiac ultrasound 2D images, in order to track the endocardium of the left ven- tricle. To this end, we propagated initial manual segmenta- tions by successively applying the displacement fields com- puted from each pair of consecutive frames. We used a 4-level pyramid starting from 5 × 5 nodes for the coarsest level to 33×33 nodes for the finest, the image being of size 608×428 pixels. The local bases are chosen quadratic.

This section introduces the principles of the **method** inherited from [ DGM ∗ 09 ]. The needle is described as a set of serially linked beams composed of two nodes, each node having 6 DoFs (posi- tion and rotation). The **model** of the needle is based on the Timo- shenko’s formulation, which relies on the beam’s theory. The vo- lumic **model** is meshed with linear tetrahedral elements and fol- lows the Hook’s law. In addition, to handle large displacements both models are combined with the co-rotational approach. After the discretization, the internal forces can be written as a nonlinear function F (q) where q are the nodal positions of the FE meshes. Note that the proposed solution is generic and could be applied to other nonlinear models.

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