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Development of Hyperelastic Reduced-Order Models For Biomimetic Material Applications Using Proper Generalised Decomposition
This thesis forms part of Work Package 4 of the ANR-RHU EndoVx project. The aim of this work is to simulate 3D-printed models of abdominal aortic aneurysms (AAA) in order to develop surgical simulators for training and practising procedures. We propose to use model reduction (ROM), specifically the Proper Generalised Decomposition (PGD) technique. PGD is an a priori ROM technique based on the separation of variables, designed to solve complex problems more efficiently by eliminating the need for redundant calculations over time or parameter ranges. This technique has been used in various applications, with a range of different separations. Here, we use the position of the load, as endovascular repair (EVAR) of AAA involves the introduction of guide wires into the model, resulting in complex loads at various locations.We present preliminary results in 1D and 2D for a linear elastic material. However, the main objective of this study is to extend the method to a hyperelastic model, a choice consistent with the models typically used for biological tissues as well as most of the 3D printing materials under consideration. Hyperelasticity introduces terms of different orders of non-linearity into the weak form of the PGD equation, which requires an incremental formulation and a linearisation strategy. Our main contribution lies in the application of a strategy previously used to deal with other forms of non-linearity, reducing higher-order terms to first order by introducing a known approximation. The question of what is used in this approximation is central. We therefore present several possible strategies. Results in 1D and 2D for a rectangular domain as well as for a more complex Y-shaped domain using these different strategies are presented, and their advantages and disadvantages are analysed. Numerical considerations regarding the implementation of the method, and a brief analysis of a PGD decomposition in which the amplitude of the load is an additional parameter—either replacing or supplementing the position of the load—are presented.
Composition du jury :
- Pedro DÍEZ, Professor, Universitat Politècnica de Catalunya, Rapporteur & Examinateur,
- Udo NACKENHORST, Professor, Leibniz Universitat Hannover, Rapporteur & Examinateur,
- Aline BEL-BRUNON, Directrice de Recherche, Université Gustave Eiffel, Examinatrice
- Alexandre DABY-SEESARAM, Maître de conférences, ENSTA, Examinateur