Many engineering and scientific problems require the solution of partial differential equations in complex geometries. Often, these problems involve parametrized geometries, e.g. design optimization, or moving domains, e.g. fluid-structure interaction problems. For such cases, traditional methods based on body-fitted grids require time-consuming mesh generation or re-meshing techniques. Unfitted finite element methods, e.g. CutFEM of AgFEM, are appealing techniques that address these challenges. However, they require ad-hoc integration methods and stabilization techniques to prevent instabilities for small cut cells. Recently, the Shifted Boundary Method (SBM), was introduced to prevent integration over cut cells and small cut-cell instabilities. An extension of the SBM was recently introduced, the Weighted Shifted Boundary Method (WSBM), where the variational form is weighted by the elemental active volume fraction, improving discrete mass/momentum conservation properties in simulations with moving domains. In this work we introduce the Generalized Shifted Boundary Method (GSBM), a geometry-agnostic generalization of the SBM and WSBM formulations that avoids the need of redefinition of integration domains and finite element spaces. The GSBM enables a unified formulation for problems with evolving geometries, supports gradient-based optimization of problems with varying geometries including topological changes, and unifies SBM, WSBM, and optimal-surrogate variants within a single framework. In this work we describe the formulation, and corresponding tests, for three model problems, namely: the Poisson problem, linear elasticity and transient Stokes flow.

The development of accurate and efficient methods for hydrodynamic analysis of floating structures is essential for advancing offshore renew-able energy technologies. In this work, we evaluate three unfitted Finite Element methods: the Shifted Boundary Method, the Cut Finite Element Method, and the Aggregated Unfitted Finite Element Method. These three methods are assessed for the estimation of added mass and damping coefficients of floating structures in two dimensions. These methods eliminate the need for traditional meshing, simplifying the analysis of complex geometries, particularly those with sharp edges, in the frequency domain using linear potential flow theory. We present a novel implementation of these techniques, highlight-ing their ability to handle multiple geometries with a single background mesh while maintaining high accuracy. Results are validated against experimental, numerical, and analytical benchmarks, demonstrating good agreement. This work not only highlights the potential of unfitted Finite Element methods for efficient and accurate hydrodynamic analysis but also identifies key challenges and knowledge gaps to guide future advancements in wave-structure interaction modeling.

This work presents a novel application of an Aggregated unfitted Finite Element Method (AgFEM) to solve the linear radiation potential flow problem in the frequency domain to estimate added mass and added damping for floating structures of arbitrary geometry. The flexibility of AgFEM in handling complex geometries makes it a compelling alternative to conventional techniques. The governing equations of the flow problem and the dynamics of the structure are fully coupled. Two case studies are conducted, estimating the loads on a spar and semisubmersible. The results demonstrate that AgFEM captures the general trends of the added mass and damping.