Unfitted Finite Element Methods for Floating Structure Hydrodynamics

Abstract

There is a growing need for new floating structures for renewable energy and offshore infrastructure, driven by abundant offshore resources and limited land space. Current design processes rely heavily on physical experiments and conventional grid-based numerical tools, which require mesh generation for each design iteration. This meshing step is identified as a major bottleneck, as seen in aerospace and automotive industries, due to its cost and complexity. The same challenge is presumed to apply to offshore floating-structure design. To circumvent this, one can use unfitted, embedded, or immersed boundary methods that reduce human intervention in the meshing step. These unfitted methods have seen limited application in modeling offshore structures. In this thesis specifically, we investigate unfitted finite element methods as an alternative simulation approach for floating offshore structures.

We introduce and justify the fundamental assumptions of the physical modeling for both the fluid and the structure. Specifically, we derive the linearized potential flow formulation in reduced form for the fluid and model the structure as a rigid body. We demonstrate how we couple the fluid and the structure in a single system of equations in the time domain. In this work, the structure is only incorporated as a boundary condition, due to the rigid body assumption. As we work with linearized potential flow, we also derive the system of equations in the frequency domain representation. We transform the systems of equations to their corresponding weak formulations such that we can deal with a challenging integral on the dynamic boundary condition for the structure more easily by assuming a constant wetted surface of the structure.

The traditional finite element method is introduced, and the time and frequency domain weak formulations are transformed to their corresponding discrete weak formulations suitable for the finite element method. A concise historical overview of unfitted, embedded and immersed boundary methods is presented, which highlights the motivation behind their development and challenges encountered. We then compare the newest unfitted methods and select the cut finite element method (CutFEM), aggregated unfitted finite element method (AgFEM), shifted boundary method (SBM), and weighted shifted boundary method (WSBM), because of their suitable properties for moving domains and complex geometries. After which, each of the four aforementioned methods is explained in detail. We describe the discrete spaces and discrete weak forms for each method. And we derive the shifted boundary condition for the kinematic and dynamic boundary conditions at the structural boundary.

To assess and compare the four unfitted methods, we perform a benchmarking study in 2D and 3D, using both implicit and explicit geometry representations. Convergence rates, condition numbers, computational performance, and implementation complexity are compared. CutFEM and AgFEM exhibit the expected convergence rates, whereas SBM and WSBM converge one order lower than CutFEM and AgFEM because no gradient recovery is used. Note that this is only because the problem considered here is Neumann type boundaries. For Dirichlet type boundaries, all four methods should have optimal convergence rates. Regarding numerical conditioning, SBM has the smallest condition numbers overall; for \(p_e = 1\), AgFEM is comparable to SBM, and for \(p_e = 2\), AgFEM’s condition numbers are similar to CutFEM’s. WSBM has the largest condition numbers. Performance tests in 2D show consistent trends in runtime and memory allocations: SBM is fastest and most memory-efficient, followed by CutFEM, then WSBM, and finally AgFEM. AgFEM’s main bottleneck is the costly initialization of the finite element space due to its aggregation algorithm. In terms of implementation complexity, AgFEM is the most complex. CutFEM and WSBM both pose challenges, with CutFEM further complicated by its reliance on a tessellation algorithm for integration. SBM is easiest to implement, although its discrete weak formulation is relatively more involved.

We demonstrate the time domain capabilities of the numerical framework for both 2D and 3D problems, including multi-body simulations, using simple and realistic floating structures. All four unfitted finite element methods are shown to agree with a body-fitted method and to reproduce experimental heave free-decay results, provided sufficient mesh resolution and polynomial order. An adaptive mesh refinement strategy is proposed, because Cartesian grids cannot efficiently represent fine geometric details in 3D. This strategy is completely implemented for CutFEM and AgFEM, but for SBM and WSBM, the Hessian operator on the reference element for adaptively refined grids is still missing, so a term in the shifting operator is currently neglected. With appropriate mesh refinement and polynomial degree, all unfitted methods give consistent results; CutFEM and AgFEM are less sensitive to coarse meshes, while SBM and WSBM may require gradient recovery to reach high accuracy with fewer elements. WSBM in particular, can show spurious oscillations or divergence on non-adaptive meshes, highlighting the critical role of mesh adaptivity and the need to complete the Hessian operator definition for a full comparison of SBM and WSBM.

The numerical framework’s performance for frequency domain problems is demonstrated by applying CutFEM, AgFEM, and SBM to 2D cases to compute added mass and added damping coefficients, and extending AgFEM to 3D realistic floating structures using unstructured refined meshes from GMSH. The methods successfully reproduce these coefficients for simple geometries and, with AgFEM, for realistic ones. The accuracy of added mass solutions appears more sensitive to mesh refinement near the structure than that of added damping. A remaining challenge is optimally balancing domain size and local mesh refinement for each wavelength, rather than relying on a single background mesh.


For future research, the dissertation’s completeness can be improved by implementing gradient recovery for SBM and WSBM and by defining the Hessian on the reference element. For frequency domain problems, an optimal, wavelength-dependent mesh strategy should be developed using the adaptive approach from the time domain studies, balancing domain size, refinement level, and automated background grid generation. An extension of the solvers to parallel computing is underway to enable large-scale multi-structure simulations, the challenge of which is parallal adaptive AgFEM. The numerical framework should have validated wave generation and damping modules, enabling longer simulations within a limited domain. For large-scale simulations, the first priority is validating the multi-body frequency domain formulation. Comparable large-scale validation studies must also be identified and performed for time domain simulations. For the time domain studies, mooring systems should be incorporated, either via linear stiffness matrices or more advanced models. The current framework can be expanded upon by incorporating higher fidelity models. This also introduces the challenges of moving free surfaces and moving structures that require remeshing. One step avoiding free surfaces with higher fidelity models, is modeling of fluid structure interaction of submerged risers or mooring lines which have marine growth. Lastly, we have proposed the generalized SBM (GSBM) to address challenges encountered during this project. This method should be subjected to the benchmark initiative and it can be a promising tool for the time domain simulations.