Patrice Cartraud

A computational approach to handle complex microstructure geometries

In multiscale analysis of components, there is usually a need to solve microstructures with complex geometries. In this paper, we use the extended finite element method (X-FEM) to solve scales involving complex geometries. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing a new enrichment strategy. Although the mesh does not need to conform to the physical surfaces, it needs to be fine enough to capture the geometry of these surfaces. A simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement. Numerical experiments on the periodic homogenization of two-phase complex cells demonstrate the accuracy and simplicity of the X-FEM

Homogenization of corrugated core sandwich panels

The present work is devoted to the computation of the effective properties of corrugated core sandwich panels. Due to their periodic structure, the homogenization theory is used, based on the asymptotic expansion method. At the leading order, an equivalent Kirchhoff-Love homogeneous plate is derived, with an overall behavior obtained from basic cell problems posed on the three-dimensional period of the panel. The finite element computation of these effective properties is presented in this paper. The accuracy of the homogenization method is proved, since the real panel and equivalent plate responses are very close for membrane and pure bending loadings. However, a discrepancy appears for simple bending loading, underlining that transverse shear effects cannot be neglected. Therefore, a specific study is developed in order to derive the transverse shear stiffness, thus enabling to determine an equivalent Reissner-Mindlin homogeneous plate.

Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method

This paper is concerned with the elastostatic behavior of heterogeneous beams with a cross-section and elastic moduli varying periodically along the beam axis. By using the two-scale asymptotic expansion method, the interior solution is formally derived up to an arbitrary desired order. In particular, this method is shown to constitute a systematic way of improving Bernoullis theory by including higher-order terms, without any assumption, in contrast to Timoshenkos theory or other refined beam models. Moreover, the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized, and the necessity of a specific treatment of edge effects is thus underlined.

Higher-order effective modeling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution

Abstract This paper constitutes the second part of a study devoted to the 1D modeling of beam-like structures, of arbitrary cross-section with periodic properties along the axis. In the first part, the asymptotic expansion method has been described, and the equations involving the successive terms of the outer solution have been given. In particular, it has been shown that neither the leading term nor the full outer solution are able to satisfy arbitrary prescribed edge data. A specific study is thus necessary and it is presented in this paper. The method used is the decay analysis initially proposed for homogeneous plates by Gregory and Wan (J. Elast. 14 (1984) 27). This approach consists in deriving the appropriate set of boundary conditions for the outer expansion solution, for any arbitrary edge data (stress, displacement or mixed data). These boundary conditions are obtained from the solution of canonical beam problems, without computing any inner solution. The method is illustrated through numerical examples on a lattice structure. The results obtained prove the importance of a correct treatment of the edge effects, in order that the inclusion of higher-order terms in the beam model serves actually to increase the accuracy of the outer expansion solution.

n-dimensional Harmonic Balance Method extended to non-explicit nonlinearities

The harmonic balance method is widely used for the analysis of strongly nonlinear problems under periodic excitation. The concept of hypertime allows for the generalization of the usual formulation to multi-tone excitations. In this article, the method is applied to a system involving a nonlinearity which cannot be explicitly expressed in the multi-frequency domain in terms of harmonic coef_cients. The direct and inverse Discrete Fast Fourier Transforms are then necessary to alternate between time and frequency domains in order to take into account this nonlinearity. The results show the efficiency and the precision of the method.

A micro-Macro strategy for ship structural analysis with FETI-DP method

In the analysis of ship structures at small scale, with structural details heterogeneities and because there is only one prototype produced, which is the final product, the designers rely on finite element simulations. The finite element discretization of such structure, leads to a huge global numerical model, that suffers for computational cost and memory resource that may be unaffordable. In such a case, a multi-scale analysis should be performed. The classical local-global analysis that is used by engineers has several limitations such as: structure details are not periodic, therefore classical homogenization methods are not easily applicable; edge effects are not taken into account; zooming techniques are not easy to use: the gluing they require with the global scale often introduces artificial edge effects.

Prediction of transient engine loads and damage due to hollow fan blade-off

The loss of a fan blade causes serious damages on an engine and can endanger the aircraft integrity and the safety of passengers. Commercial aircraft engines must then meet the FAA (Federal Aviation Administration) and JAA (Joint Aviation Authorities) certification requirements concerning the fan blade containment. The certification is validated through a Fan Blade-Off (FBO) test on a whole engine. The success in this test requires destructive and expensive development tests performed at the different stages of the design process. To reduce the number of these experiments and thus, the costs and the time of development, the engine behaviour under FBO can be understood and even predicted thanks to finite element (FE) analysis. This paper shows a comparison between a FBO simulation of hollow blades, computed with an explicit integration FE code, and experimental data obtained during an intermediate FBO test carried out by Snecma Moteurs. The results of the load levels and the similarity on the sequence of events show good agreement.

Study of component mode synthesis methods in a rotor-stator interaction case

The study of rotor-stator interactions between blade-tips and outer casings through direct contact in modern turbomachines is very time-consuming if the classical finite element method is used. In order to improve the knowledge over these interaction phenomena, faster methods have to be applied. The construction of reduced-order models using component mode synthesis methods generally allows for dramatic increase in computational efficiency. Two of these methods, namely a fixed interface method and a free interface methods are considered in an original manner to reduce the size of a realistic two-dimensional model. They are then compared in a very specific contact case-study. The equations of motion are solved using an explicit time integration scheme with the Lagrange multiplier method where friction is accounted for. The primary goal of the present study is to investigate the general behavior of such approaches in the presence of contact nonlinearities. It will be shown that in our contact case, a good accuracy can be obtained from a reduced models with very limited number of modes.

Thermal properties of composite materials : effective conductivity tensor and edge effects

The homogenization theory is a powerful approach to determine the effective thermal conductivity tensor of heterogeneous materials such as composites, including thermoset matrix and fibres. Once the effective properties are calculated, they can be used to solve a heat conduction problem on the composite structure at the macroscopic scale. This approach leads to good approximations of both the heat flux and temperature in the interior zone of the structure, however edge effects occur in the vicinity of the domain boundaries. In this paper, following the approach proposed in [10] for elasticity, it is shown how these edge effects can be corrected. Thus an additional asymptotic expansion is introduced, which plays the role of a edge effect term. This expansion tends to zero far from the boundary, and is assumed to decrease exponentially. Moreover, the length of the edge effect region can be determined from the solution of an eigenvalue problem. Numerical examples are considered for a standard multilayered material. The homogenized solutions computed with a finite element software, and corrected with the edge effect terms, are compared to a heterogeneous finite element solution at the microscopic scale. The influences of the thermal contrast and scale factor are illustrated for different kind of boundary conditions.

Development of Beam-To-Beam Contact Detection Algorithms for Rotor-Stator Rubbing Applications

In nuclear power plant turbosets, the design-basis accident consists of a blade-off on the low pressure turbine last stage. During the accidental shutdown, a severe rotor-casing interaction may occur at critical speeds due to large shaft line displacements originated by a high unbalance excitation. The contact between the shaft and the stator, also called the diaphragm in this study, induces an important angular deceleration rate and greatly modifies the turbogenerator dynamics including the amplitude of the loads in the bearings. Therefore the main objective is to verify that the designed turbine is capable of going through critical speeds without catastrophic consequences for the shaft line. To this end, a model of a turbogenerator has been developed to compute rotor speed transients by considering the rotating speed of the rotor as an unknown, which allows for the angular deceleration due to rubbing to be calculated in a more realistic fashion. Lagrange multipliers method is applied to compute contact forces. The diaphragm, which is a non-rotating bladed disks assembly, is modeled by curved and straight beams, and different assumptions for the contact detection are studied to find a compromise between CPU time and accuracy. Results of the numerical tool show that the contact forces are sensitive to the retained assumptions only when heavy rub occurs. Nevertheless, the rotating speed and bearing loads are computed with a satisfactory accuracy, even with an approximation on contact detection that saves CPU time.