Oriol Colomés* ([email protected]), Jan Modderman ([email protected]), Guglielmo scovazzi ([email protected])
Unfitted/Immersed or Embedded Finite Element methods have gained significant attention in the past decade for their effectiveness in simulating problems in complex geometries. However, many existing methods in the literature necessitate ad-hoc modifications of standard Finite Element data structures. This often involves tessellation of elements intersected by the embedded boundary, the construction of special quadrature rules, or the definition of geometry-dependent surrogate boundaries. These ad-hoc modifications often prevent the use of these approaches for problems where the geometry is a varying parameter of the PDE at hand. In this work, we propose a novel approach: a generalization of the Shifted Boundary method [1,2], which eliminates the need for special data structures or quadrature rules, and enables the solution of unfitted problems with geometry-independent Finite Element spaces. This newly introduced framework holds particular significance for problems involving evolving geometries and potential topological changes.
Applications extend to scenarios such as topology optimization or fluid-structure interaction problems where the ability to handle dynamic geometries is crucial. In addition, the proposed methodology can also be relevant for machine-learning based surrogate models where the geometry is a parameter. In this talk, we will present the formulation of the Generalized Shifted Boundary method and showcase its application to a variety of problems with evolving domains.
REFERENCES: [1] Main, A. and Scovazzi, G. (2018). The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems. Journal of Computational Physics, 372, 972-995. [2] O. Colomés, A. Main, L. Nouveau and G. Scovazzi (2021). A weighted shifted boundary method for free surface flow problems. Journal of Computational Physics, 424, 109837.
[3] Xu, D., Colomés, O., Main, A., Li, K., Atallah, N. M., Abboud, N., & Scovazzi, G. (2024). A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow. Journal of Computational Physics, 510, 113095.
[4] Scovazzi, G., Xu, D., Colomés, O., Main, A., Li, K., Atallah, N. M., & Abboud, N. A Weighted Shifted Boundary Method for the Navier-Stokes Equations with Immersed Moving Boundaries. Available at SSRN 5251653. (preprint)