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New Eulerian methods for problems with interfaces, with emphasis in fluid mechanics


Two fundamental issues in the simulation of mechanical problems are the representation of the domains geometry, and the construction of a numerical approximation of the governing equations that respects the boundary conditions. These issues are shared by solid and fluid mechanics, electromagnetism, etc. In this project we focus on Eulerian formulations in which the geometry is represented as the zero-level set of a function. The boundary is thus easily defined from the values of the level set (LS) function. The drawback is that a partial differential equation needs to be solved whenever the boundary moves (as in a free surface with a moving wave, or when the domains shape evolves along an optimization process). The first objective of this project is to develop especially-tailored techniques to compute the evolution of the LS function, so as to maximize accuracy and robustness. Emphasis will be given to semi-Lagrangian formulations, which adapt well to the mathematical nature of the problem. One of the innovative ideas is to combine techniques issued from computer graphics (clouds of points, point-set representation) with the LS method, which would blend the geometrical accuracy of Lagrangian approaches to the well-known flexibility of the (eulerian) LS method. Turning now to the second issue mentioned in the first paragraph, if the mesh does not conform to the domains geometry, the method is said to be ans;immersed boundary methods; (IB). In IB methods the main difficulty is the imposition of the boundary conditions, especially the Dirichlet ones. They can be enforced by means of fictitious forces or by direct modification to the linear system. This last option leads to more stable methods, but a locking phenomenon (boundary locking) restricts its applicability. As second objective of the project we thus propose to pursue a novel idea to get general IB methods, which do not rely on fictitious forces and are locking-free. The idea is quite simple: To adopt an approximation space that consists of functions that are discontinuous in a neighborhood of the boundary. This type of spaces does not produce locking, so that the difficulty is transferred to the handling of discontinuities within the numerical method. However, significant advances have been achieved recently in the area of Discontinuous Galerkin (DG) methods, and several of these methods with excellent convergence properties are already available. Combining locally discontinuous spaces with DG formulations, it will be possible (in principle) to push forward the state of the art of IB methods, obtaining high accuracy (optimal convergence), simplicity and stability (the idea is to impose Dirichlet conditions strongly). Though the goals of this project are of fundamental nature, their field of application is very vast (practically the whole of computational mechanics could take advantage of it). A significant socio-economic impact is thus foreseen. (AU)

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