A high-order adaptive harmonic polynomial cell method with immersed boundaries for weakly/fully nonlinear wave-structure interaction simulations

Chao Tong

Research output: Book/ReportPh.D. thesis

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With the objective of pushing forward the state of the art of nonlinear potential flow (PF) models to accurately simulate wave-structure interaction in marine hydrodynamics, which is essential for marine structure design and environmental risk assessment, a high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is developed, verified and validated in two-dimensions (2D), and subsequently extended to three-dimensions (3D) with success.

In this work, the present method combines the immersed boundary method (IBM) and adaptive cell refinement (ACR) technique with the harmonic polynomial cell (HPC) method for different purposes. The use of the IBM makes it easier to deal with arbitrarily shaped boundaries or moving structures in regular grids, while the ACR technique applied to boundaries or regions of interest enables significant reduction of the total number of unknowns in the global equation system meanwhile maintaining high accuracy.

Matrix-based stability analyses are carried out for linear wave propagation in 2D and 3D rectangular numerical wave tanks, which provide mathematical guidance for evaluating the stability of immersed-boundary (IB) based strategies in time-domain simulations. It is found that the cell geometry has a strong influence on the robustness of numerical models based on the HPC method. Square cells lead to the best stability and accuracy in 2D models. Differently in the 3D HPC method, compressed cells in the vertical direction lead to better stability than stretched cells, while aspect ratio close to one on the horizontal plane is superior. Nevertheless, a cubic cell is still preferred in most situations. An accuracy and convergence study demonstrates that the proposed IB-AHPC method maintains a level of accuracy higher than third order, which is consistent with the standard HPC method proposed by Shao and Faltinsen [1, 2].

Given the limitations of second-order PF models in addressing the nonlinearities in wave-structure-interaction problems beyond third order, and the prohibitive computational cost associated with Navier-Stokes equation solvers for large-scale problems, it is crucial to bridge the gap between these two approaches. Under this circum- stance, both weakly nonlinear and fully nonlinear PF models have been developed and investigated in the present work. In contrast to the traditional perturbation- based weakly nonlinear PF models, a weak-scatterer approximation is adopted and studied in this work. Moreover, a generalized weak-scatterer hypothesis, in which the approximated scattered free surface boundary conditions are linearized based on the Taylor-expansion along an arbitrary direction from the incident wave profile, has been proposed and validated in our work. Accordingly, an Arbitrary Lagrangian-Eulerian (ALE) approach is adopted to track the free-surface markers, the time evolution of which is updated by the explicit fourth-order Runge-Kutta scheme.
The present 2D and 3D IB-AHPC models are systematically verified and validated. It shows that the IB-AHPC solvers effectively model highly nonlinear propagating waves and accurately predict up to third-order hydrodynamic loads on structures. The comparisons between the results of our present models and published theoretical, numerical, and experimental solutions demonstrate good agreement and consistency, suggesting that the proposed IB-AHPC method is a promising potential-flow method in marine hydrodynamics.
Original languageEnglish
Place of PublicationKongens Lyngby
PublisherTechnical University of Denmark
Number of pages240
Publication statusPublished - 2023


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