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Theses

Éléments finis isogéométriques massifs coque sans verrouillage pour des simulations en mécanique non linéaire des solides

Abstract : With the introduction of IsoGeometric Analysis (IGA), the calculation of shell has become possible using the exact geometry for coarse meshes. In order to that, Lagrange polynomials are replaced by NURBS functions, the most commonly used technology in Computer-Aided Design, to perform the analysis. In addition, NURBS functions have a higher order of continuity, which leads to higher per-degree-of-freedom accuracy of the shell solution than with classical Finite Elements Analysis (FEA). IGA has now been widely applied in shell formulations. Nevertheless, it has still rarely been studied in the context of solid-shell models. This second shell approach is, however, very useful for engineers, since it enables to calculate thin structures using 3D solid elements, i.e. involving only displacements as degrees of freedom. The difficulty in shell analysis is to deal with locking which highly deteriorates the convergence of the solution. The NURBS framework does not enable to solve the problem directly. Then, to really benefit from NURBS in shells, specific strategies need to be implemented to answer the locking issue. This is the goal of the thesis in the context of solid-shell elements. The first work has consisted, on a curved beam problem, in extending the locking-free methods usually encountered in FEA to the NURBS context. The study resulted in the development of two new strategies for NURBS: the first one is based on a selective reduced integration technique and the second one makes use of a B-bar projection. The global formalism offered by the B-bar method appearing more suitable for NURBS, it has then been investigated for solid-shell elements. More precisely, a mixed formulation has first been elaborated from which, it has been possible to derive the equivalent B-bar projection. From a theoretical point of view, this strategy constitutes the most important result of this work: a systematic method to construct a consistent B-bar projection is to write a mixed formulation. With regards to the implementation, the main idea to treat locking of the solid-shell elements has been to modify the average of the strain and stress components across the thickness of the shell. Hourglass control has also been added to stabilize the element in particular situations. The resulting element is of good quality for low order approximations and coarse meshes: the quadratic version seems to be more accurate than basic NURBS elements of order 4. The proposed method leads to a global stiffness matrix of small size but full. This problem is inherent to NURBS functions. It has been limited here by using a local least squares procedure to approach the B-bar projection. Finally, the mixed element has been successfully extended to geometric non-linearity which reflects the ability of the methodology to be used in complex simulations.
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  • HAL Id : tel-01149917, version 1

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Robin Bouclier. Éléments finis isogéométriques massifs coque sans verrouillage pour des simulations en mécanique non linéaire des solides. Mécanique des structures [physics.class-ph]. INSA de Lyon, 2014. Français. ⟨NNT : 2014ISAL0090⟩. ⟨tel-01149917⟩

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