computing a solution thanks to successive forward and backward substitutions. **In** **solid** **mechanics**, as **in** other applications, these methods have proved to be efficient **in** the two-dimensional case, but a significant fill-**in** phenomenon usually occurs when factor- izing large-scale three-dimensional problems [11, 99]. Hence, the memory requirement generally compromises their use, even more when each left-hand side **in** (1) needs to be factorized (i.e. when the matrices change all along the sequence). On the other hand, iterative methods provide a sequence of (improving) approximations of the solution of the linear system. Contrary to the direct **solvers**, these methods only require the knowl- edge of the action of the matrix on a vector. Furthermore, they are particularly well adapted to parallel computing (see, e.g., Chapter 11 **in** [90]). Among all these methods available **in** the literature, it is known that Krylov subspace methods are the method of choice for large-scale problems, especially when solving sequences as (1) **in** the case where left-hand sides are changing [90]. However, Krylov subspace methods are gener- ally efficient when they are are combined with preconditioning [11]. This concept, whose design remains an active domain of research, consists conceptually **in** multiplying the matrix of the original system by another matrix called preconditioner, while maintaining the same solution. The aim is to obtain a new operator with better properties (detailed later); the preconditioner generally corresponds either to an approximation of the inverse of the original matrix or to the inverse of an approximation of the original matrix. It is worth mentioning that the computation of a preconditioner can be rather expensive, and **in** the context of the solution of (1) with slowly varying matrices, a preconditioner is often reused for several successive linear systems.

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Keywords Rotating shaft · breathing crack · **nonlinear** dynamics · stability · Floquet theory · period doubling · bifurcation diagram · Poincar´e section
1 Introduction
The development and propagation of a crack represents the most common and trivial beginning of integrity losses **in** engineering structures. For rotating shafts, a propagat- ing fatigue crack can have detrimental effects on the reliability of a process or utility plant where theses vital parts are subjected to very arduous working conditions **in** harsh environment. It is one of the most serious causes of accidents and, an early warning is essential to extend the durability and increase the reliability of these machines. Ac- cording to Bently and Muszynska [1986], **in** the 70s and till the beginning of the 80s, at least 28 shaft failures due to cracks were registered **in** the USA energy industry. It is

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CEA, DEN, DEC, SESC, F-13108 Saint-Paul Lez Durance, France
b
LMA, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille, 31, Chemin Joseph Aiguier, F-13402 Marseille Cedex 20, France
**In** this paper an adaptive multilevel mesh reﬁnement method, coupled with the Zienkiewicz and Zhu a posteriori error estimator, is applied to **solid** **mechanics** with the objective of conduct reliable **nonlinear** studies **in** acceptable computational times and memory space. Our automatic approach is ﬁrst veriﬁed on linear behaviour, on 2D and 3D simulations. Then a **nonlinear** material behaviour is studied. Advantages and limitations of the local defect correction method **in** **solid** **mechanics** problems **in** terms of reﬁnement ratio, error level, CPU time and memory space are discussed. This kind of resolution is also compared to a global h-adaptive resolution.

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knowledge of p y without an additional closure relation.
The computational results obtained from this approximation are shown on Figs. 6.2 and 6.3 . These ﬁgures are obtained with the set of parameter ﬁtted to muscle data (see Tab. 7.1 on p. 139 ). We see that the model based on mean ﬁeld approximation fails to reproduce neither the stationary state (measured after 5ms, compare with ) nor the transients. **In** the hard device, this is particularly visible for the mean trajectories (see Fig. 6.2 (A)): the equivalent model (**solid** lines) never ﬁts the Langevin simulations (dashed lines) except during the application of the step because it does not allow the change **in** the distribution of the cross-bridges (purely elastic response, see the T 1 curve on Fig. 6.2 (B)). Also, one can see that the shape of the T 2 curve corresponds to the local minimum of the energy deﬁned by the initial fraction of cross-bridges **in** post-power-stroke n 0 1 (thick dot-dashed line). However, the upper and lower boundaries of this local minimum branch correspond to the upper and lower limits for conﬁgurations (1, 0, 0) and (0, 0, 1) respec- tively, rather that the upper and lower limits for the conﬁgurations n 0 1 , 0, 0 and 0, 0, n 0

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6. CONCLUSION
**In** this paper we proposed several improvements to the current standard algorithm for solving variational fracture models. Over-relaxation is extremely cheap and simple to implement, but can greatly reduce the number of iterations required for convergence. Composing over-relaxed alternate minimization with Newton-type methods yields a further decrease **in** runtime, although at a more significant development cost. Together, these improvements to alternate minimization reduce the time to solution by a factor of 5–6 × for the surfing and thermal shock test cases. Lastly, we proposed and tested preconditioners for the linear subproblems **in** alternate minimization and the coupled Jacobian arising **in** the Newton iterations when solving the whole problem with a monolithic active set method. These efforts are complementary to other approaches recently proposed **in** the literature, such as adaptive remeshing [ 44 ], adaptive time-stepping, continuation algorithms [ 39 ], or refined line-search techniques [ 40 ] that were not considered **in** this work. Our tests focus only on the simplest settings for variational fracture **mechanics** assuming small deformations and a simple rate-independent material behaviour. However, the developed techniques can be readily adapted to more complex contexts, including hyperelasticity, viscoelasticity, and inertial effects.

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configuration is considered. This task is here achieved using the Monte Carlo algorithm of Section 2 , implemented within the EDStar development environment [10 , 13] , using the Mcm3D library [25] . This implementation deals with three- dimension geometries using advanced computer-graphics tools. The input fields are the output of AVBP. Unlike **in** the benchmark simulations of Section 3 where the input fields were analytic, the input fields are now provided using the LES grid of AVBP (4.74 million tetrahedrons) together with an interpolation procedure provided by the combustion specia- lists to reflect the spatial integration schemes involved **in** the fluid **mechanics** and chemistry **solvers**. As radiative transfer specialists, we therefore make no choice: we strictly accept what would be, ideally, the input fields that PRISSMA should reflect, **in** its coupling with AVBP, if no computation constraint was taken into account. Ideally, along the same line, our Monte Carlo simulations should use the best gaseous line- absorption properties available, i.e. the detailed line profiles provided by spectroscopic databases such as HITEMP [26] and CDSD [27] . However, at the present stage, only few attempts were reported **in** which such line-by-line Monte Carlo strate- gies were tested and none of them are compatible with our requirements **in** terms of three-dimension geometry and heterogeneity. As described **in** Section 2.3 , our “reference” simulation makes therefore only use of a narrow band k- distribution strategy. The corresponding spectral data were produced using the SNB-ck approach of [28–31] , separating CO 2 and H 2 O thanks a decorrelation assumption described at the end of Section 2.3 . Three hundred sixty-seven spectral narrowbands are used, each of width Δν ¼ 25 cm −1 , and the

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This disadvantage is overcome to some extent by so-called ‘ho- mogenization’ models, **in** which the detailed spatial distribution of the grains is ignored and the aggregate is treated as a finite number of grains with different orientations and material properties. **In** this mean-field approach compatibility of stress and strain equilibrium is not enforced between spatially contiguous grains, but rather be- tween each grain and a ‘homogeneous effective medium’ defined by the average of all the other grains. For viscoplastic behaviours as considered here, a well-known member of this class makes use of the so-called ‘tangent’ anisotropic scheme of Molinari et al. ( 1987 ) and Lebensohn & Tome ( 1993 ). **In** this model the local stress and strain rate tensors vary among the grains. **In** the geophys- ical literature this approach is generally known as the viscoplastic self-consistent (VPSC) model, and we use this name for this first- order approximation. The VPSC model has been widely used **in** **solid**-earth geophysics including studies of CPO development **in** the upper mantle (e.g. Wenk et al. 1991 ; Tommasi et al. 1999 , 2000 ,

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MODEL-FREE DATA-DRIVEN METHODS **IN** **MECHANICS**:
MATERIAL DATA IDENTIFICATION AND **SOLVERS**
LAURENT STAINIER, ADRIEN LEYGUE, AND MICHAEL ORTIZ
Abstract. This paper presents an integrated model-free data-driven approach to **solid** **mechanics**, allowing to perform numerical simulations on structures on the basis of measures of displacement fields on representative samples, without postulating a specific constitutive model. A material data identification procedure, allowing to infer strain-stress pairs from displacement fields and boundary conditions, is used to build a material database from a set of mutiaxial tests on a non- conventional sample. This database is **in** turn used by a data-driven solver, based on an algorithm minimizing the distance between manifolds of compatible and balanced mechanical states and the given database, to predict the response of structures of the same material, with arbitrary geometry and boundary conditions. Examples illustrate this modelling cycle and demonstrate how the data- driven identification method allows importance sampling of the material state space, yielding faster convergence of simulation results with increasing database size, when compared to synthetic material databases with regular sampling patterns.

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4.3 Conclusion
The Discontinuous Galerkin Material Point Method has been applied to hyperbolic problems of **solid** **mechanics**. It has first been shown on a Riemann problem **in** a linear elastic bar under small strains **in** section 4.1.1 , that the method is able to capture the exact solution that consists of two elastic discontinuities propagating **in** the medium. Indeed, the stability properties of the one-dimensional schemes derived **in** section 3.3.1 enable, for particular space discretizations, the use of a CFL number set to unity. Nevertheless, once the optimal stability condition is lost, that is CF L < 1, the method is slightly more diffusive than the MPM. Next, the solution of problems **in** history-dependent solids (sections 4.1.2 ) have shown that efficient tools can be embedded into the method **in** order to deal with (visco)plastic flows. **In** particular, approximate elastic-plastic Riemann **solvers** can be employed, provided that the characteristic structure of the problem is known. **In** addition, the results of section 4.2.1 highlight that the total Lagrangian formulation of the DGMPM allows circumventing the eventual grid-crossing occurring **in** updated Lagrangian MPM for problems involving waves **in** finite deforming solids. As a consequence, the numerical scheme also provides solutions that are close to exact ones for non-linear problems. Moreover, the arbitrariness of the grid can be fully exploited by employing adaptive mesh techniques on the reference configuration so as to track accurately waves **in** the current configuration for problems involving complex geometries. At last, the two-dimensional simulations performed **in** sections 4.1.3 , 4.1.4 and 4.2.2 show that DGMPM results are **in** good agreement with FEM while eliminating spurious oscillations.

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8
4 Conclusion and Future Work
We present a multi decoupled modeling method to simulate 3D muscles with complex architecture **in** real-time (15 FPS), that is a compromise between highly detailed 3D FEM and 1D Hill-type existing approaches. This leads to higher flex- ibility on the modeling side (separate thus optimized discretizations of active, isotropic and anisotropic parts). As expected, uniaxial simulations fit validated 1D Hill wire-segments behavior. **Fast** computation time allows to run the sim- ulation **in** real-time using activation input based on real EMG measurements. We plan to validate this method with an extended set of subject-specific mus- cles reconstructed from MRI, and combine it with a skeletal model, to achieve comprehensive musculoskeletal simulations. We currently work on validating de- formations using MRI data showing the knee **in** various postures combined with EMG measurements.

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(2) the family of elastoplastic material models contains both simple (linear elasticity) and more complex, **nonlinear**, C 0 -continuous descriptions (elastoplasticity with **nonlinear** hardening). Our contribution focuses on results of uniaxial tensile tests **in** order to be as straightforward as possible. **In** addition to introducing the general idea of identification approaches based on BI as gently as possible (to our abilities), we also discuss some more complex extensions, such as not only incorporating the error **in** the stress, but also incorporating 65

Figure 6: Saint-Venant-Kirchhoff hyperelastic bar undergoing large deformations (positions - Piola stress) plot at different times. Material parameters are the same as **in** the linear elastic case.
The DGMPM is no longer exactly on the analytical solution of this **nonlinear** problem and the reason might be the use of an approximate Riemann solver instead of the **nonlinear** one (which is more expensive). Once again classical MPM fails to represent the discontinuity and introduces oscillations **in** the results.

The aim of this work is to compare two existing multilevel compu- tational approaches coming from two different families of multi- scale methods **in** a **nonlinear** **solid** **mechanics** framework. A locally adaptive multigrid method and a numerical homogenization technique are considered. Both classes of methods aim to enrich a global model representing the structure’s behavior with more sophisticated local models depicting ﬁne localized phenomena. It is clearly shown that even being developed with different vocations, such approaches reveal several common features. The main con- ceptual difference relying on the scale separation condition has ﬁnally a limited inﬂuence on the algorithmic aspects. Hence, this comparison enables to highlight a uniﬁed framework for multiscale coupling methods.

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gourinat@ensica.fr
ABSTRACT
This paper presents developments performed on photoelastodynamic bench of ENSICA's Department of Mechanical Engineering. Classical wave, vibrating, shock and rotating parts theories, were compared with colour pictures of isochromatic lines obtained with rapid camera and urethane resin specimens. For non-linear shock and large deflection, explicit code LS- DYNA has been used. Then, the facility has been used to analyse dynamic work of gears for power transmission, **in** comparison with numerical computations. These developments have lead to demonstrations now included **in** engineering general courseware, about stress analysis : theory of elasticity and dynamics of structures. Gear visualisation have been included **in** integrated France-Canada developments concerning dynamics of transmission, as a complement with theoretical models and experimental acoustic analysis of functioning gears.

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2.6.3 MESHFREE METHODS NOT REQUIRING A BACKGROUND MESH (TRUE MESHFREE)
The second type of meshfree methods does not require any background mesh and encompasses collocation type methods (solution of the strong form of the PDEs). Some examples are the finite point method (FPM, by Onate et al. [123]) as well as the meshless local Petrov-Galerkin method (MLPG, Atluri and Zhu [124]). The smoothed particle hydrodynamics (SPH) method is one of the older and more mature meshless methods. The method was first introduced by Gingold and Monaghan [125] and by Lucy [126] **in** 1977. SPH was introduced for the simulation of strength of materials problems by Libersky and Petschek [127] **in** 1991. SPH has enjoyed a great deal of development over the past decades. It has been successfully used for many short-to-medium duration transient **solid** **mechanics** problems. However, the standard SPH approach is not well suited to long-duration strength of material problems. The main difficulty lies **in** that the standard SPH formulation uses an explicit (forward difference) time integration scheme that is conditionally stable. For **solid** **mechanics** problems, the stability criterion leads to very small time steps (inversely proportional to the speed of stress wave propagation **in** the **solid**). This leads to unreasonably long computation times for longer duration simulations.

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interacting case. It is an interesting open problem to extend our result to more physical interactions and to k > 2.
Open problems and the link with QFT. **Nonlinear** Gibbs measures have also played an important role **in** constructive quantum field theory (QFT) [24, 47, 64]. By an argument similar to the Feynman-Kac formula, one can write the (formal) grand-canonical partition function of a quantum field **in** space dimension d, by means of a (classical) **nonlinear** Gibbs measure **in** dimension d + 1, where the additional variable plays the role of time [48, 49]. The rigorous construction of quantum fields then sometimes boils down to the proper definition of the corresponding **nonlinear** measure. This so-called Euclidean approach to QFT was very successful for some particular models and the literature on the subject is very vast (see, e.g., [48, 1, 55, 28] for a few famous examples and [61] for a recent review). Like here, the problem becomes more and more difficult when the dimension grows. The main difficulties are to define the measures **in** the whole space and to renormalize the (divergent) physical interactions. We have not yet tried to renormalize physical interactions **in** our context or to take the thermodynamic limit at the same time as T → ∞. These questions are however important and some tools from constructive QFT could then be useful.

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A. RBF interpolation error
**In** this section, studies on the accuracy of the RPIM shape functions used for curve fitting are conducted . The fitting of functions is based on the nodal function value sets that are generated at regularly as well as at irregularly distributed nodes. The procedure carried out for curve fitting is : first we create a set of field nodes **in** the domain where the function is to be fitted, then for a given test node x (usually different from the set of field nodes) where the function is to be fitted, we choose n nodes **in** the influence domain of x, now we can construct RPIM shape functions, and finaly, using these shape functions we can calculate the function value at x and compare it with the real value.

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