An adaptive harmonic polynomial cell method with immersed boundaries: Accuracy, stability, and applications

Tong, Chong Wen; Shao, Yanlin; Bingham, Harry B.; Hanssen, Finn Christian W.

doi:10.1002/nme.6648

Cited by 16 publications

(11 citation statements)

References 64 publications

(96 reference statements)

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“…When the data points are equally spaced, analytical solutions to the least-square equations can be found, in the form of convolution coefficients that can be applied to all sub-sets (also called stencils later in this paper) of the data, to obtain an estimation of the smoothed signal and its derivatives at the central point of each subset. The SG filters have been successfully applied, among others, by Ducrozet et al (2014), Engsig-Karup et al (2009), andTong et al (2021), in modeling fully nonlinear water waves. Despite becoming more and more popular in time-domain simulations related to ocean waves as well as wave-structure interactions, it is worth mentioning that the SG filters cannot eliminate the saw-tooth instabilities.…”

Section: Low-pass Filter Based On Optimized Wlsmentioning

confidence: 99%

“…To model the nonlinear wave loads on and the hydrodynamic responses of the offshore structures, advanced fully-nonlinear models exist, such as the time-domain fullynonlinear potential flow models or the Navier-Stokes solvers with proper turbulence modeling. Among others, the fullynonlinear potential flow models include those based on Boundary Element Method (Grilli, 1997;Ferrant et al, 2003;Bai and Eatock Taylor, 2006Zhou et al, 2016;Zhang and Teng, 2021), Finite Element Method (Wu and Eatock Taylor, 2003;Wu, 2006 2007;Yan and Ma, 2007;Sun et al, 2015;Huang and Wang, 2020), Finite Difference Method (Bingham and Zhang, 2007;Engsig-Karup et al, 2009;Ducroze et al, 2014;Hicks et al, 2021), and Harmonic Polynomial Cell method Faltinsen, 2012 2014a;Hanssen et al, 2018;Liang et al 2020;Tong et al, 2021). Navier-Stokes solvers have also become widely used in various hydrodynamic analyses for offshore structures.…”

Section: Introductionmentioning

confidence: 99%

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A consistent second‐order hydrodynamic model in the time domain for floating structures with large horizontal motions

Shao

Zheng

Liang

et al. 2021

Computer aided Civil Eng

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Floating offshore structures often exhibit low-frequency oscillatory motions in the horizontal plane, with amplitudes in the same order as their characteristic dimensions and larger than the corresponding wave-frequency responses, making the traditional formulations in an inertial coordinate system inconsistent and less applicable. To address this issue, we explore an alternative formulation completely based on a non-inertial body-fixed coordinate system. Unlike the traditional seakeeping models, this formulation consistently allows for large-amplitude horizontal motions. A numerical model based on a higher-order boundary element is applied to solve the resulting boundary-value problems in the time domain. A new set of explicit time-integration methods, which do not necessitate the use of upwind schemes for spatial derivatives, are designed to deal with the convectivetype free-surface conditions. To suppress the weak saw-tooth instabilities on the free surface in time marching, we also present novel low-pass filters based on optimized weighted-least-squares, which are in principle applicable for both structured and unstructured meshes. For ship seakeeping and added resistance analyses, we show that the present computational model does not need to use softsprings for surge and sway, in contrast to the traditional models. For a spar floating offshore wind turbine (FOWT), the importance of consistently taking into account the effects of large horizontal motions is demonstrated considering the bichromatic incident waves. The present model is also referred to as a complete 2nd order wave-load model, as all the 2nd order wave loads, including the sumfrequency and difference-frequency components, are solved simultaneously.

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Section: Low-pass Filter Based On Optimized Wlsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A consistent second‐order hydrodynamic model in the time domain for floating structures with large horizontal motions

Shao

Zheng

Liang

et al. 2021

Computer aided Civil Eng

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“…This problem was later solved by Hanssen et al [24] by using two overlapping grids, with a local grid attached to the fixed or movable structure and the other Cartesian background grid fixed on the Earth. Tong et al [25] developed an immersed-boundary adaptive HPC (IB-AHPC) method in 2D to deal with complex geometries, and eliminated the spurious pressure oscillation by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. More investigations can be found in [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…Tong et al [25] developed an immersed-boundary adaptive HPC (IB-AHPC) method in 2D to deal with complex geometries, and eliminated the spurious pressure oscillation by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. More investigations can be found in [24][25][26][27]. Focusing on the accuracy properties of the 2D HPC method, the work of [28] reveals that the HPC method provides an accuracy higher than fourth order when the cells are square-shaped, and they also reported the loss of accuracy when stretched/compressed meshes are applied.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions of Two-Dimensional Navier–Stokes Equations Based on a Generalized Harmonic Polynomial Cell Method With Non-Uniform Grid

Shao

Fuhrman

2022

Journal of Offshore Mechanics and Arctic Engineering

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It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier-Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed boundary method is used to deal with general geometries involving the fluid-structure-interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier-Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier-Stokes equations solver uses second-order FDM for the discretization of the diffusion and advection terms, which may be replaced by other higher-order schemes to further improve the accuracy.

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“…Shao and Faltinsen (2012) and Shao and Faltinsen (2014) proposed high-order Harmonic Polynomial Cell (HPC) methods in both 2D and 3D respectively to study water waves and their interaction with structures. Some recent extensions were made to utilize immersed boundary strategies and overset meshes to achieve better accuracy and stability (e.g., see Hanssen et al, 2018;Tong et al, 2019Tong et al, , 2021Law et al, 2020;Liang et al, 2020). Compared to the BEMs, field solvers deal with spare matrices, and the computational costs are roughly linearly dependent on the number of unknowns.…”

Section: Introductionmentioning

confidence: 99%

Accurate and efficient hydrodynamic analysis of structures with sharp edges by the Extended Finite Element Method (XFEM): 2D studies

Wang¹,

Shao²,

Chen³

et al. 2021

Preprint

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Achieving accurate numerical results of hydrodynamic loads based on the potential-flow theory is very challenging for structures with sharp edges, due to the singular behavior of the local-flow velocities. In this paper, we introduce, perhaps the first time in the literature on marine hydrodynamics, the Extended Finite Element Method (XFEM) to solve fluidstructure interaction problems involving sharp edges on structures. Compared with the conventional FEMs, the singular basis functions are introduced in XFEM through the local construction of shape functions of the finite elements. Four different FEM solvers, including conventional linear and quadratic FEMs as well as their corresponding XFEM versions with local enrichment by singular basis functions at sharp edges, are implemented and compared. To demonstrate the accuracy and efficiency of the XFEMs, a thin flat plate in an infinite fluid domain and a forced heaving rectangle at the free surface, both in two dimensions, will be studied. For the flat plate, the mesh convergence studies are carried out for both the velocity potential in the fluid domain and the added mass, and the XFEMs show apparent advantages thanks to their local enhancement at the sharp edges. Three different enrichment strategies are also compared, and suggestions will be made for the practical implementation of the XFEM. For the forced heaving rectangle, the linear and 2nd order mean wave loads are studied. Our results confirm the previous conclusion in the literature that it is not difficult for a conventional numerical model to obtain convergent results for added mass and damping coefficients. However, when the 2nd order mean wave loads requiring the computation of velocity components are calculated via direct pressure integration, the influence of singularity is significant, and it takes a tremendously large number of elements for the conventional FEMs to get convergent results. On the contrary, the numerical results of XFEMs converge rapidly even with very coarse meshes, especially for the quadratic XFEM. Unlike other methods based on domain decomposition when dealing with singularities, the FEM framework is more flexible to include the singular functions in local approximations. Only 2D steady and frequency-domain solutions are discussed in this study, while an extensions to the time domain and 3D problems are straightforward.

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An adaptive harmonic polynomial cell method with immersed boundaries: Accuracy, stability, and applications

Cited by 16 publications

References 64 publications

A consistent second‐order hydrodynamic model in the time domain for floating structures with large horizontal motions

A consistent second‐order hydrodynamic model in the time domain for floating structures with large horizontal motions

Numerical Solutions of Two-Dimensional Navier–Stokes Equations Based on a Generalized Harmonic Polynomial Cell Method With Non-Uniform Grid

Accurate and efficient hydrodynamic analysis of structures with sharp edges by the Extended Finite Element Method (XFEM): 2D studies

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