For several of years, our research has focused on finite element methods that fall under the class of unfitted (also known as immersed boundary) methods. Such methods are particularly suited to the development of digital twins, as they facilitate the automatic generation of patient-specific simulations on complex geometries, due to the fact that they do not require a discretisation that strictly conforms to the domain boundary.
Our current results have focused on the development, numerical analysis and mathematical foundations of a novel method called ɸ-FEM. The main benefit of our method is that it uses the classical finite element tools on unfitted meshes. Furthermore, we also demonstrated that the method significantly improves convergence when compared to a similar fitted discretisation of the domain.
The benefit and applicability of the ɸ-FEM method have already been demonstrated in the context of diverse problems, including heat transfer, crack propagation, interface between two materials, and fluid-structure interaction. The method is currently being extended to three-dimensional non-linear elasticity with both Neumann and Dirichlet boundary conditions.