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Thesis
Home   /   Thesis   /   Semi-implicit numerical method for the hyperelastic model

Semi-implicit numerical method for the hyperelastic model

Abstract

The fluid-structure interaction motivates the development of numerical methods in many fields of application: simulating the wind resistance of a bridge, the flow of blood in arteries, or the calculation of the lift of an aircraft are only a few examples. The nature of the problems considered leads to the development of radically different methods. One way to simulate this is to consider a global model for continuum mechanics valid from the elasticity of solid materials to the dynamics of gases (monolithic approach). Such a strategy has been implemented in Hera. In fast dynamics, the main problem is that waves travel through solids much faster than gases, which strongly penalizes the time step of the simulation. Indeed, the numerical methods involved are explicit, and a CFL criterion that depends on the local sound speed guarantees stability.

In order to get rid of this constraint, one solution is to use an implicit time discretization for the system of Euler equations. In practice, the indiscriminate use of this solution can be counterproductive if it leads to the inversion of extensive linear systems. The price of the inversion can become more critical than the multiplication of explicit steps. One way to get around this difficulty is to use implicit integration only for the solid part.

In this thesis, we are interested in imagining, analyzing, and implementing a numerical method for this problem. The model considered is hyperelasticity. We will study the possibility of using locally (in the solid) and possibly partially (IMEX) implicit numerical methods. Abgrall and Torlo have recently proposed a semi-implicit method without a matrix of any order in 1D on a Cartesian mesh. In this work, they transform a system of nonlinear conservation laws into a linear system by a relaxation method. A deferred correction method ensures the integration in time. It allows reaching a CFL condition of a few units without having a matrix to invert. We will explore the opportunity to use this method for the hyperelasticity system and extend it in dimensions higher than 1 on unstructured meshes.

Laboratory

DSSI
DSSI
Paris Sud
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