Yan-Lin Shao

A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics

We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods, e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.

A numerical study of the second-order wave excitation of ship springing in infinite water depth:

Assuming the wave steepness of the incident waves and the ship motions are small, the second-order weakly-nonlinear hydrodynamic problem of a ship moving with constant forward speed is studied numerically in a consistent way. The boundary value problem is formulated in a body-fixed coordinate system and the perturbation scheme is used. This formulation does not include any derivatives of the velocity potential on the right-hand side of the body-boundary conditions, and thus avoid the difficulties associated with the terms similar to the so-called mj-terms and their derivatives. The second-order sum-frequency wave excitation of ship springing is studied in both monochromatic and bichromatic head-sea waves. Different Froude numbers are considered. A time-domain Higher-Order Boundary Element Method based on cubic shape function is used as a numerical tool. An upstream finite difference scheme is used for longitudinal derivative terms in the free-surface conditions. For a modified Wigley hull in head-sea waves, it is found that the second-order velocity potential gives dominant contribution to second-order wave excitation of ship springing in the wave frequency region where sum-frequency springing occurs. Quadratic velocity terms in the Bernoulli equation have a relatively small contribution. The numerical results also demonstrate strong dependency of the second-order wave excitation of ship springing on the Froude numbers for small wave lengths. The effect of beam and draft is investigated.

Towards Efficient Fully-Nonlinear Potential-Flow Solvers in Marine Hydrodynamics

Solving potential-flow problems using the Boundary Element Method (BEM) is a strong tradition in marine hydrodynamics. An early example of the application of BEM is by Bai & Yeung [1]. The bottleneck of the conventional BEM in terms of CPU time and computer memory arises as the number of unknowns increases. Wu & Eatock Taylor [2] suggested that the Finite Element Method (FEM) field solver is much faster than the BEM based on their comparisons in a wave making problem. In this paper, we aim to find a highly efficient method to solve fully-nonlinear wave-body interaction problems based on potential-flow theory. We compare the efficiency and the accuracy of five different methods for the potential flows in two dimensions (2D), two of which are BEM-based while the other three are field solvers. The comparisons indicate that it is beneficial to use either an accelerated matrix-free BEM, e.g. Fast Multipole Method accelerated BEM (FMM-BEM), or any field solvers whose resulting matrix are sparse.Another highlight of this paper is that an efficient numerical potential-flow method named the harmonic polynomial cell (HPC) method is developed. The flow in each cell is described by a set of harmonic polynomials. The presented procedure has approximately 4th order accuracy, while its resulting matrix is sparse similarly as the other field solvers, e.g. Finite Element Method (FEM), Finite Difference Method (FDM) and Finite Volume Method (FVM). The method is verified by a linear wave making problem for which the steady-state analytical solution is available, and the forced oscillation of a semi-submerged circular cylinder for which the frequency-domain added mass and damping coefficients are compared. The fully-nonlinear wave making problem and nonlinear propagating waves over a submerged bar are also studied for validation purposes. Only 2D cases are studied in this paper.Copyright

Fully-Nonlinear Wave-Current-Body Interaction Analysis by a Harmonic Polynomial Cell Method

A new numerical 2D cell method has been proposed by Shao & Faltinsen [1] based on representing the velocity potential in each cell by harmonic polynomials. The method was named the Harmonic Polynomial cell (HPC) method. The method was later extended to 3D to study potential-flow problems in marine hydrodynamics [2]. With the considered number of unknowns that are typical in marine hydrodynamics, the comparisons with some existing boundary element based methods including the Fast Multipole Accelerated Boundary Element Methods showed that the HPC method is very competitive in terms of both accuracy and efficiency. The HPC method has also been applied to study fully-nonlinear wave-body interactions [1, 2], for example, sloshing in tanks, nonlinear waves over different sea-bottom topographies and nonlinear wave diffraction by a bottom-mounted vertical circular cylinder. However, no current effects were considered. In this paper, we study the fully-nonlinear time-domain wave-body interaction considering the current effects. In order to validate and verify the method, a bottom-mounted vertical circular cylinder which has been studied extensively in the literature will first be examined. Comparisons are made with published numerical results and experimental results. As a further application, the HPC method will be used to study multiple bottom-mounted cylinders. An example of the wave diffraction of two bottom-mounted cylinders is also presented.

Second-Order Diffraction and Radiation of a Floating Body With Small Forward Speed

DTU Orbit (09/08/2019) Second-Order Diffraction and Radiation of a Floating Body With Small Forward Speed The formulation of the second-order wave-current-body problem in the inertial coordinate system involves higher-order derivatives in the body boundary condition. A new method taking advantage of the body-fixed coordinate system in the near field is presented to avoid the calculation of higher-order derivatives in the body boundary condition. The new method has an advantage over the traditional method when the body surface has a sharp corner or high curvature. The nonlinear wave diffraction and forced oscillation of floating bodies are studied up to second order in wave slope. A small forward speed is taken into account. The results of the new method are compared with that of the traditional method based on a formulation in the inertial coordinate system. When the traditional method applies, good agreement has been obtained.

Challenges in Wave Force Modelling for Mooring Design in High Seas

(18/11/2019) Challenges in wave force modelling for mooring design in high seas Line breakage events have been experienced on moored structures during recent years. These are often occurring in heavy weather and overload is one of the reasons pointed out. The present paper identifies posible physical phenomena that may lead to wave forces higher than predicted by state-of-the-art hydrodynamic tools and procedures, and thereby higher mooring lineloads, in high and steep waves. In particular, a need to re-explore wave-group induced slowly varying, low-frequency (LF)drift forces has been identified. Both mobile offshore units (MODU’s) and permanently moored floaters are considered, semisubmersibles and FPSOs. Empirical corrections are sometimes being applied in design of mooring lines, while not ingeneral, and there is no established common industry practice on such corrections. More advanced tools and knowledge do exist in research communities, while they still need further development for robust engineering use. A brief overview is given of state-of-the-art methods and tools in modelling of the hydrodynamic forces on large-volume floaters, with particular focus on slowly varying wave forces. Full scale experiences from real sea events and from a variety of earlier case studies including model tests are reviewed. It is found that several items may be critical in the proper prediction of LF wave forces in high seas and combined current and should be investigated further, in particular: Wave-current interaction Viscous wave drift forces Large and nonlinear wave -frequency vessel motions. Based upon these preliminary investigations, the paper gives recommendations for actions and further developments for improved predictions in industry practice. Copyright 2015, Offshore Technology Conference

Fully-Nonlinear Wave-Current-Body Interaction Analysis by a Harmonic Polynomial Cell (HPC) Method

In the Ronald W. Yeung Honoring Symposium on Offshore and Ship Hydrodynamics in OMAE2012 hold in Rio de Janeiro, Shao & Faltinsen [1] have proposed a new numerical 2D cell method based on representing the velocity potential in each cell by harmonic polynomials. The method was named the Harmonic Polynomial cell (HPC) method. The method was later extended to 3D to study potential-flow problems in marine hydrodynamics [2]. With the considered number of unknowns that are typical in marine hydrodynamics, the comparisons with some existing boundary element based methods including the Fast Multipole Accelerated Boundary Element Methods showed that the HPC method is very competitive in terms of both accuracy and efficiency. The HPC method has also been applied to study fully-nonlinear wave-body interactions [1, 2], for example, sloshing in tanks, nonlinear waves over different sea-bottom topographies and nonlinear wave diffraction by a bottom-mounted vertical circular cylinder. However, no current effects were considered.In this paper, we study the fully-nonlinear time-domain wave-body interaction considering the current effects. In order to validate and verify the method, a bottom-mounted vertical circular cylinder which has been studied extensively in the literature will first be examined. Comparisons are made with published numerical results and experimental results. As a further application, the HPC method will be used to study multiple bottom-mounted cylinders. An example of the wave diffraction of two bottom-mounted cylinders is also presented.Copyright

Numerical Analysis of Second-Order Wave Loads on Large-Volume Marine Structures in a Current

A time-domain Higher-Order Boundary Element Method (HOBEM) based on cubic shape functions for second-order wave-current-body interaction developed by Shao & Faltinsen [1] is further refined by investigating the feasibility of adopting the unstructured meshes on the free surface and body surfaces from an open source mesh generator [2]. When the steady local flow effect is considered in the time-domain boundary-value-problem formulation, the advection terms in the free surface are part of the sources of numerical instability. In this paper, the advection terms are taken care of in an implicit way in a 4th order Runge-Kutta scheme with much better stability. Some numerical examples extensively studied in the literature are studied in order to validate the present numerical model.Copyright

Stochastic Linearization and its Application in Motion Analysis of Cylindrical Floating Structure with Bilge Boxes

Structure with Bilge Boxes DTU Orbit (03/11/2019) Stochastic Linearization and its Application in Motion Analysis of Cylindrical Floating Structure with Bilge Boxes To account for the viscous effects of damping devices, for instance, bilge keels or bilge boxes, on the motions of ships and offshore structures, Morison’s equation is often adopted as an empirical but practical approach in the design process. In order to combine the standard engineering panel method with the drag term in Morison’s equation, and remain in the frequency domain, the drag term has to be linearized based on, for instance, stochastic linearization. In this paper, the stochastic linearization scheme is implemented in an in-house code and verified through the comparison with the DNV GL software WADAM. The model test results of a large cylindrical FPSO with bilge box are used to calibrate the drag coefficients in the Morison’s equation. When the linearized drag forces are included, heave motion RAOs correspond better to the model test results. However, the predicted natural periods of heave motions are seen to be smaller than those obtained from model tests. It is suspected that the viscous flow separation around the bilge box increases the added mass of the unit beyond what is predicted by potential flow alone. Discussions are made on the effect of viscous added mass on the heave natural period. It is quite common to only include the damping effects in the motion analysis for large offshore structures and ignore the contribution of the viscous effects on the excitation force. For the considered cylindrical FPSO, this paper demonstrates that the viscous excitation force can be important in survival conditions.

The harmonic polynomial cell method for moving bodies immersed in a Cartesian background grid

In numerical simulations with moving bodies, and often with complex geometries, generation of high-quality body-fitted grids is a cumbersome and time-consuming task. An alternative is to use a fixed (Cartesian) background grid, and allow the body to move freely over this. The challenge in such methods is to transfer the body-boundary conditions of the moving body to fixed grid nodes in a rational manner. In this paper, an Immersed Boundary Method (IBM) is proposed to simulate potential flow about a moving body on a Cartesian background grid. The recently developed Harmonic Polynomial Method, proven both accurate and computationally efficient, is used to represent the velocity potential in the fluid. The body-boundary conditions are interpolated by using ghost nodes inside the body with mirror interpolation points in the fluid.The method is first tested for a fixed cylinder in oscillatory flow to determine the accuracy of the proposed IBM, before considering the equivalent case of an oscillating cylinder in still fluid. Finally, a steadily-advancing cylinder is studied, which is considered as the most challenging case with respect to spurious pressure oscillations. These are known to be a challenge in many IBMs, and special attention is therefore devoted to this aspect.Copyright