# A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

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## Abstract

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global \(L^2\) projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

## Keywords

Nonlinear and dispersive free surface waves Wave–structure interaction Fully nonlinear potential flow Spectral element method Mixed Eulerian–Lagrangian Marine hydrodynamics Numerical wave tank## 1 Introduction

Over the last decades, numerical simulations of free surface flows have become an indispensable tool in the analysis and design of marine structures. Ships, floating production systems, offshore wind turbines and wave energy converters all rely on numerical estimates of loads and motions. For reliable estimates of environmental loads, the numerical tools need to accurately and efficiently account for nonlinear wave–wave and wave–structure interactions in large and representative marine regions.

The development of numerical models for time-domain simulations of fully nonlinear and dispersive water wave propagation has been a research topic since the 1960s and with real engineering applications since the 1970s [2]. Models based on fully nonlinear potential flow (FNPF) have been considered relatively mature for about two decades, see the review papers [55, 66] and the references therein. More recent researches focus on improved modelling by incorporating more physics and finally enabling large-scale simulation of nonlinear waves [6, 40]. In many applications, computational speed is far more important than the cost of hardware and, therefore, algorithms that provide speed and scalability of work effort is of key interest.

### 1.1 On Numerical FNPF Models

In broad terms, FNPF models can be divided into two categories: (i) models for wave propagation and (ii) models for wave–body interaction. The models focused on wave propagation include models based on finite difference methods (FDM) [15, 18, 23, 52], the high-order spectral (HOS) method [13, 28, 61] and other pseudo-spectral methods [7, 9]. The models can utilize \(\sigma \)-transformed methods to map the physical domain into a fixed computational domain [4, 56] to increase the efficiency of the model. While being very different models in terms of formulation and implementation, they all share the treat of high-order accuracy in space. It is well known that high-order discretisation methods can give significant reduction in computational effort compared to the use of conventional low-order methods [35]. This is especially true for long-time, large-scale simulations where the wave propagation is greatly influenced by the accuracy in the dispersion properties of the model. Indeed, it has been shown that the combination of efficient high-order algorithms [17] with different means of acceleration (such as multi-domain approaches implemented on modern many-core hardware [21]) can render FNPF models practically feasible for analysis of wave propagation on standard work stations—even with a real-time perspective within reach for applications with appreciable domain sizes [23].

The models focused on wave–body interaction naturally favour geometrical flexibility with respect to handling the bodies over high-order convergence rates. The boundary element method (BEM) is widely used for flow around bodies due to the ease in handling complex geometries [24, 33]. Today quadratic BEM are most often used [30, 44, 65]. However, scalability has been shown to favour finite element methods (FEM) [52, 64]. The use of FEM for fully nonlinear potential flow has also received significant attention, starting with the seminal work of Wu and Taylor [63]. Studies of FNPF solvers based on second-order FEM are plentiful, see, e.g. [26, 29, 41, 42, 58, 64, 70]. FNPF solvers can be derived from Luke’s variational principle [39], leading to robust formulations [26, 57]. These formulations come with a computational cost due to the need for solving iteratively a nonlinear algebraic system of equations as a part of the time-stepping. Nevertheless, most of the wave–body models adopt the Mixed Eulerian–Lagrangian method (MEL) due to Longuet–Higgins and Cokelet [38]. In the MEL approach, the free surface boundary conditions are solved in the Lagrangian frame while the Laplace problem is solved in Eulerian form. Please note that the widely used MEL formulation is a subclass of the general Arbitrary-Lagrangian–Eulerian (ALE) framework [14], with the notable difference that in MEL the velocities of changing mesh features in the fluid volume do not need to be accounted for during the time integration. However, the MEL approach does require re-meshing of the elements, and development of techniques that improves the efficiency, e.g. QALE-FEM [43], has been presented.

### 1.2 Spectral Element Models for FNPF

High-order finite element methods, such as the Spectral Element Method (SEM) due to Patera [48], have historically received the least attention for FNPF equations. SEM is thought to be highly attractive as it combines the high-order accuracy of spectral methods for problems with sufficiently smooth solutions with the geometric flexibility via adaptive meshing capability of finite element methods. Additionally, efficient iterative solvers are available, e.g. see [47, 51]. In other words, with SEM both wave propagation and wave–body interaction problems can be handled efficiently within a single model.

An early study employing SEM for the FNPF model is described by Robertson and Sherwin [50], where an ALE technique was used to track the free-surface motion. Importantly, this study identified that numerical instabilities were caused by asymmetry in the mesh topology of the fluid domain (as also discussed in Ref. [62]). To stabilise the model, the authors added a diffusive term proportional to the mesh skewness to the kinematic free surface condition. More recently, an FNPF-SEM model based on \(\sigma \)-transformation was presented in Ref. [19]. It was illustrated that the quartic nonlinear terms present in the Zakharov form [67] of the free surface conditions generally caused severe aliasing problems. This problem was mitigated by over-integration of the free surface equations and application of a gentle elemental spectral filter. This approach was later extended to 3D in Ref. [20].

Finally, in the context of SEM for FNPF models we also mention the high-order spectral element (HOSE) method [68]. Remark that it is only the boundary domains that are discretised with HOSE, not the interior domain.

### 1.3 Paper Contribution

The key scientific challenge addressed in this work is to extend the 2D (vertical plane) FNPF-SEM model presented in Ref. [19] to also include wave–body interaction. To this end, we discard the \(\sigma \)-transformation in favour of the MEL approach. We propose a remedy for the mesh asymmetry instability of the MEL approach based on using hybrid meshes. It is numerically shown that a layer of quadrilateral elements below the free surface and unstructured triangles in the remaining domain provides a stable solution. Due to the high-order nature of the proposed scheme, it is necessary to introduce a re-meshing strategy based on combining local and global techniques. This extends the re-meshing strategies used in established FEM solvers, e.g. QALE-FEM [43], and the re-meshing strategy is found to improve both temporal robustness and accuracy of the solver. The combination of SEM, MEL, re-meshing, over-integration and mild filtering is shown to yield a high-order, accurate and robust model capable of handling non-trivial wave–structure interaction problems, as evidenced by several strenuous numerical benchmarks provided in Sects. 4 and 5.

## 2 Mixed Eulerian–Lagrangian (MEL) Formulation

### 2.1 Governing Equations

*g*is the acceleration of gravity. Please note that the Zakharov form contains up to quartic nonlinear terms. The presence of these nonlinear terms needs proper treatment to deal with aliasing effects that may arise in numerical schemes.

*X*define a point on the reference domain \(\varGamma ^\mathrm {FS}_0\) at time \(t=t_0\). There is a mapping \({\mathscr {F}}\) that relates the material points \(X\in \varGamma ^\mathrm {FS}_0\) to the spatial points \(x=x(X,t)\) on the current domain \(\varGamma ^\mathrm {FS}\). Hence, a function \(f:\varGamma ^{\text {FS}} \times T \rightarrow {\mathbb {R}}\) in the Eulerian frame is related to the corresponding function in the Lagrangian frame \({\hat{f}}:\varGamma ^\mathrm {FS}_0 \times T \rightarrow {\mathbb {R}}\) by \({\hat{f}}(X,t)=f({\mathscr {F}}(X),t)\). The time derivative in the Lagrangian frame can be expressed in the spatial coordinates as

*f*in the Lagrangian frame as

### 2.2 Forces

## 3 Numerical Discretisation

Following [19], we present the discretisation of the governing equations based on the method of lines, where first a semi-discrete system of ordinary differential equations is formed by the spatial discretisation in two space dimensions using a nodal SEM. The resulting semi-discrete system is integrated in time using an explicit fourth-order Runge–Kutta method.

### 3.1 Unsteady Free Surface Boundary Equations

#### 3.1.1 Domain Partitioning and SEM Approximation

*k*denoting the

*k*th element. We introduce for any tessellation \({\mathscr {T}}_h\) the spectral element approximation space of continuous, piece-wise polynomial functions of degree at most

*P*

#### 3.1.2 Galerkin Formulation

### 3.2 Laplace Problem

#### 3.2.1 Domain Partitioning and SEM Approximation

*P*,

#### 3.2.2 Curvilinear Iso-parametric Elements

*k*th element \({\mathscr {H}}_{h}^k\subset {\mathscr {H}}_h\). On this element, we form a local polynomial expansion expressed as

Using transfinite interpolation with linear blending [27], the affine transformations from the element reference domains to physical domain are done via local-to-the element mappings of the form \({\varvec{\varPsi }}_k(r,s) = g(r,s,\varGamma _1,...,\varGamma _m)\), with \(g(\cdot )\) defining the mapping (cf. [5]), \(m=3\) (triangles) or \(m=4\) (quadrilaterals).

### Remark

The use of curvilinear elements implies that the Jacobian of the mapping is no longer constant as for straight-sided triangular elements. Thus, to avoid quadrature errors higher order quadrature are employed in the discrete Galerkin projections and this increases the cost of the scheme proportionally for the elements in question.

#### 3.2.3 Galerkin Formulation

### 3.3 Re-meshing Strategy

Wave propagation naturally produces deformations in all node positions in the free surface elements and subsequently also affects the element sizes. Thus, to maintain temporal stability when using explicit time-stepping methods for the nonlinear Lagrangian free surface formulations, it is necessary to employ re-meshing [13]. The re-meshing for the FNPF-SEM solver consists of two operations: (i) a local re-meshing technique (applied to the quadrature points) to retain well-conditioned global operators [3], and (ii) a global re-meshing technique (applied to the elemental vertices) to maintain balanced resolution, avoid large element aspect ratios and obtain a good mesh quality.

Initially, the physical nodes of a free surface element are positioned at the Legendre–Gauss–Lobatto (LGL) quadrature points. However, the deformation of the free surface will cause the physical nodes to deviate from the LGL points, negatively affecting the conditioning of Vandermonde matrices as well as the accuracy of the numerically evaluated integrals. To avoid this, a local re-mesh operation is required. Keeping the positions of the vertex nodes, the \(P-1\) internal nodes are simply interpolated from the particle nodes to the LGL points of the element. This operation does not change the initial mesh connectivity and is performed at every Runge–Kutta sub-step.

The locally re-meshed free-surface elements are then used to form the 2D hybrid mesh. In general, we do not re-generate the mesh at every step but keep the mesh connectivity and move the mesh points (similar to QALE-FEM [43]). The elements in the quadrilateral layer move with the free surface elements, while the vertices of the triangles in the fluid domain are subjected to a Laplacian smoothing technique [31] to obtain a good mesh quality. To keep good mesh quality, a global re-meshing operation is invoked whenever a free surface element goes below 75% or above 125% of its original size as a result of nodes following the Lagrangian motions. The global re-meshing operation re-positions the free surface physical nodes to their initial horizontal position. Global re-meshing is often used in FNPF-FEM models, typically at every time step, e.g. [63], whereas the local re-meshing is needed in high-order FNPF-SEM modelling.

## 4 Numerical Properties

We start out by considering the numerical properties of the model related to the temporal stability and convergence of the numerical MEL scheme. Results of comparison with the stabilised Eulerian formulation [19] are included since the Eulerian and Lagrangian formulations are complementary.

### 4.1 Temporal Linear Stability Analysis of Semi-discrete System

*b*’ refer to the free surface nodes and the ‘

*i*’ refers to all interior nodes. The eigenspectrum of \(\lambda ({\mathscr {J}})\) determines the temporal stability of the linear system given by Eq. (22).

The starting point for our eigenanalysis is a confirmation of the results presented in Robertson and Sherwin [50]. Using a triangulated asymmetric mesh, we also find eigenvalues with non-negligible real parts leading to temporal instability. To fix this problem, we need to avoid asymmetric meshes near the free surface layer. Following [50], we can also consider a triangulated symmetric mesh in Fig. 4 which turns out to be stable for all polynomial orders except orders 3 and 4 used in the analysis. This is partly in line with the results and conclusions presented by Robertson and Sherwin, but reveals that numerical issues may arise even for such triangulated meshes in specific configurations.

Furthermore, since our objective is to introduce arbitrarily shaped bodies inside the fluid domain, in the next experiment we use a hybrid mesh. This mesh consists of a layer of quadrilateral below the free surface of the fluid similar to the meshes used in Ref. [19] and this layer is combined with an additional triangulated layer used to represent complex geometries such as a submerged cylinder as illustrated in Fig. 5. The eigenanalysis shows a set of purely imaginary eigenvalues (to machine precision) from which we conclude that temporal stability can be reached for arbitrary polynomial expansion orders. These results are in line with numerous other experiments and other mesh configurations we have carried out that are not presented here. Thus, the results of numerical analysis are confirming that by introducing a quadrilateral layer at the free surface level, we can fix the temporal stability problem. Using instead a similar mesh but with slightly skewed quadrilaterals, we again find that the mesh asymmetry leads to temporal instability.

### Remark

The temporal instability is associated with the accuracy of the vertical gradient approximation that is used to compute \({\tilde{w}}\) at the free surface since this determines the dispersive properties of the model. Poor accuracy in \({\tilde{w}}\) destroys the general applicability of the model since the wave propagation cannot be resolved accurately. A quadrilateral layer with vertical alignment of nodes close to the free surface provides accurate recovery of the vertical free surface velocities that determines the dispersion properties of the scheme and fixes the temporal instability problem described in Ref. [50]. We stress that the eigenvalue analysis is specific to the chosen mesh and discretisation, and as such is not a general proof of temporal stability.

### 4.2 Temporal Stability and Stabilisation Techniques

In the following, we consider the strenuous benchmark of long-time propagation of a steep nonlinear stream function wave in a periodic domain [49]. The domain is one wavelength wide and discretised into a mesh with \(8 \times 1\) elements of polynomial order \(P_h=P_{\text {v}}=6\). The time step (\(\varDelta t\)) used corresponds to \(T/\varDelta t=80\), where *T* is the wave period.

In Fig. 6, we show numerical results for a stream function wave with dispersion parameter \(kh=1\) with a steepness of \((H/L)_{{\text {max}}}=70\%\) of maximum steepness. The wave is integrated in time for 50 wave periods. Following [19], exact integration of the free surface equations as well as a 1% top mode spectral filter is used. In addition, the global re-meshing strategy is enforced. The numerical result in Fig. 6a is seen to completely overlap with the analytical solution even after 50*T*, illustrating the excellent nonlinear and dispersive properties of the numerical scheme. The conservation of energy and mass is presented in Fig. 6b. The rapid initial loss of energy and mass is due to difficulties in generating high-accuracy initial conditions for very steep stream function waves. After the initial loss mass end energy is largely conserved.

The global re-meshing discussed in Sect. 3.3 was found to be essential in maintaining temporal stability for longer integration times for steep nonlinear waves when using the Lagrangian free surface formulation. Figure 7 shows the error in free surface elevation for stream functions waves at \(kh=1\) with respect to steepness and integration time. For short integration times (up to one wave period) and for linear waves (\(<20\%\) of maximum steepness), there is no major effect of using the global re-meshing. But for steep waves and long-time integration, the global re-meshing strategy is vital for improving the stability.

### 4.3 Convergence Tests

## 5 Numerical Experiments

We examine different test cases considering solitary waves interacting with submerged and surface piercing objects. The computational results are compared to experimental data and numerical results from literature. In all numerical experiments, we generate the solitary waves using the expression of Dutykh and Clamond [16]. The conservation of mass and energy will be examined compared to the integral equations in Ref. [16].

### 5.1 Reflection of High-Amplitude Solitary Waves from a Vertical Wall

*h*from the wall. The water level at the wall position grows leading to the formation of a thin jet shooting up along the wall surface. When the maximum of free surface elevation is reached at the wall position it is said that the wave is attached to the wall, at this moment \(t=t_a\) and the height of the crest is \(\eta _a\). Thereafter, the jet forms and reaches its maximum run up \(\eta _0\) at time \(t_0\). After this event, the jet collapses slightly faster than it developed. The detachment time \(t_d\) corresponds to the wave crest leaving the wall. The height of the wave at that instant is \(\eta _d\), always smaller than the attachment height (i.e. \(\eta _d< \eta _a < \eta _0\)). Then, a reflected wave propagates in the opposite direction with the characteristics of a solitary wave of reduced amplitude. This adjustment in the height produces a dispersive trail behind the wave and is characteristic for nonlinear solitary waves. The depression becomes more abrupt for increasing wave steepness. See Fig. 10.

The numerical experiments are carried out using a domain size \(x\in [-22.5, 22.5]\) m, with the initial position of the centre of the solitary wave at \(x_0=0\) m. We employ a structured mesh consisting of quadrilaterals with \(N_{{\text {el}}}^{2D}\) elements in a single layer, where \(N_{{\text {el}}}^{2D}\in [40,120]\) is varied proportional to the wave height to resolve the waves in the range \(a/h\in [0.2,0.6]\). Each element is based on the polynomial expansion orders \((P_h,P_{\text {v}})=(6,7)\). The time step size is chosen in the interval \(\varDelta t\in [0.01, 0.02]\) s. A mild spectral filtering using a \(1\%\) top mode spectral filter every time step using the MEL scheme.

### 5.2 Solitary Wave Propagation Over a Submerged Semi-circular Cylinder

We consider the numerical experiment described in Wang et al. [59] on the interaction between a solitary wave and a submerged semi-circular bump. We investigate the case of a mildly nonlinear (\(a/h=0.2\)) solitary wave and a cylinder with a small radius \(R/h = 0.3\). According to Cooker et al. [11], this setting is in the wave–train region; the interaction is weak and the solitary wave is only slightly perturbed by the obstacle. There is a small train of reflected waves, but not a reflected solitary wave.

In Fig. 13a, it is seen how the solitary wave propagates undisturbed from the initial condition until it starts the interaction with the semi-circular cylinder. There is some minor shoaling of the solitary wave, and after the bump a reflective depression wave is created. After the interaction, the solitary wave is restored to its original form. This is clearly seen in the waterfall plot of the time evolution of the free surface (Fig. 13b). The results are in good agreement with results of Wang et al. [59] using a BIM. The solitary waves mass and energy, relative to the reference values [16], are well preserved during the interaction as illustrated in Fig. 13c. The reason for the oscillations in the beginning is due to size of the domain, as the initial solitary wave becomes a bit truncated. Nevertheless, the errors in mass and energy are very small. The computed horizontal force is presented in Fig. 13d. The force is symmetric, and similar to the force reported in Ref. [59]. As the solution is inviscid, both the numerical FNPF-SEM solution and the BIM solution differ from the horizontal force estimated by the well-known Morison equation that includes a parametrization of drag and inertia forces [34, 59].

### 5.3 Solitary Wave Propagation Over a Fixed Submerged Cylinder

Numerous studies, both experimental and numerical, can be found regarding the interactions of solitary waves with a submerged horizontal cylinders, see e.g. [8, 10, 53]. We will consider the recent study by Aristodemo et al. [1]. The experiments, both experimentally and numerically (using a smoothed particle method), investigated the horizontal and vertical forces for setups where the inertia forces should be dominating over the lift and drag forces (the experiments have Reynolds numbers in the order of \(10^4\) and Keulegan–Carpenter number between 3.3 and 9.4).

Here, however, we will investigate a case where viscous effects are important. Consider a wave with amplitude \(a/h=0.177\) and a cylinder of diameter \(D/h=0.3175\) with a submergence depth of \(z_0/h=0.5\) (see Fig. 1 for definition of \(z_0\)). The computational domain is \(x\in [-10,\,10]\) m and \(z\in [0,-0.4]\) m. Figure 14 shows the initial mesh for the experiments. The mesh follows the outline described in the previous section: the mesh consists of a thin layer of quadrilateral elements near the free surface. Away from the cylinder there are zones of quadrilateral elements only. These zones flank the central part consisting of unstructured triangles. The mesh has a total of 315 elements of orders \(P_h=P_{\text {v}}=6\) (117 triangles and 198 quadrilaterals). The time step used is \(\varDelta t = 0.02\) s and the filter drains off 5% of the energy in the highest mode.

### 5.4 Solitary Wave Interaction with a Truncated Vertical Cylinder

Lin [37] proposed a numerical method for the Navier–Stokes equations based on the transformation of the fluid domain using a multiple-layer \(\sigma \)-coordinate model. Wave–structure interactions of a weakly nonlinear solitary waves (\(a/h=0.1\)) with a rectangular obstacle fixed at different positions (seated, mid-submergence, truncated surface piercing) were investigated in a two-dimensional wave flume of depth \(h=1\) m. The free surface elevation time series at three different wave gauge locations were found to be in excellent agreement with computed Navier–Stokes solutions based on the volume of fluid (VOF) approach. We consider here the experiment corresponding to the obstacle piercing the free surface (width: 5 m, height: 0.6 m, draught: 0.4 m) for validation of the present methodology dealing with truncated surface piercing bodies.

## 6 Conclusions

We have presented a new stabilised nodal spectral element model for simulation of fully nonlinear water wave propagation and wave–structure interaction based on an MEL formulation. In the MEL approach, we avoid the use of a \(\sigma \)-transformed vertical coordinate, and bodies of arbitrarily shaped geometry are handled by high-order curvilinear elements.

A main result is that the stability issues associated with mesh asymmetry as reported in Ref. [50] can be mitigated using a hybrid mesh; a quadrilateral layer with interfaces aligned with the vertical direction is used to resolve the free surface layer and an unstructured triangular/quadrilateral mesh can be used elsewhere. The aliasing effects that typically arise from the nonlinear free surface boundary conditions are addressed using the hybrid mesh strategy combined with the ideas described in Ref. [19]: (i) exact quadrature rules and (ii) mild spectral filtering to add some artificial viscosity to secure robustness for marginally resolved water wave motion. In addition, (iii) a local re-meshing strategy is used to counter element deformations that may lead to numerical ill-conditioning and in the worst cases breakdown if not used. The local re-meshing strategy is combined with a global re-meshing strategy that seeks to maintain element sizes close to the original sizes and the relative aspect ratios between neighbouring elements.

Our numerical analysis confirms this strategy to work well for steep nonlinear water waves when using the proposed stabilised nonlinear MEL formulation. We validate the model by revisiting known benchmarks for nonlinear wave models. The numerical results obtained are excellent compared with other published results and demonstrate the high accuracy that can be achieved with the proposed high-order spectral element method. It was noted that the FNPF-SEM model yielded high-order solutions for both wave propagation and wave–body interaction problems, illustrating that we can address both these problem within a single numerical model. The present paper focused on high-order simulations. However, it is well known that non-smooth solutions can cause high-order methods to experience reduced convergence rates or even numerical instabilities. For such cases, it can be advantageous to use a low-order model. Fortunately, inside the SEM model there is also a low-order model available.

In ongoing work, we are extending the proposed stabilised FNPF-SEM solver towards more realistic nonlinear hydrodynamics applications in three space dimensions (cf. the model based on an Eulerian formulation in three space dimensions [20]). Additionally, we are incorporating moving and floating bodies. These extensions will include also the design of efficient multi-grid algorithms that can deliver scalable work effort in line with the preliminary work described in Ref. [36].

## Notes

### Acknowledgements

This work contributed to the activities in the research project Multi-fidelity Decision making tools for Wave Energy Systems (MIDWEST) that is supported by the OCEAN-ERANET program. The DTU Computing Center (DCC) supported the work with access to computing resources. Claes Eskilsson was partially supported by the Swedish Energy Agency through Grant no. 41125-1.

## References

- 1.Aristodemo, F., Tripepi, G., Davide Meringolo, D., Veltri, P.: Solitary wave-induced forces on horizontal circular cylinders: laboratory experiments and SPH simulations. Coast. Eng.
**129**, 17–35 (2017)CrossRefGoogle Scholar - 2.Beck, R., Reed, A.: Modern computational methods for ships in a seaway. Trans. Soc. Nav. Archit. Mar. Eng.
**109**, 1–52 (2001)Google Scholar - 3.Bjøntegaard, T., Rønquist, E.M.: Accurate interface-tracking for arbitrary Lagrangian–Eulerian schemes. J. Comput. Phys.
**228**(12), 4379–4399 (2009)MathSciNetzbMATHCrossRefGoogle Scholar - 4.Cai, X., Langtangen, H.P., Nielsen, B.F., Tveito, A.: A finite element method for fully nonlinear water waves. J. Comput. Phys.
**143**, 544–568 (1998)MathSciNetzbMATHCrossRefGoogle Scholar - 5.Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods—fundamentals in single domains. Springer, New York (2006)zbMATHCrossRefGoogle Scholar
- 6.Cavaleri, L., Alves, J.-H.G.M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T.H.C., Hwang, P., Janssen, P.A.E.M., Janssen, T., Lavrenov, I.V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W.E., Sheremet, A., McKee Smith, J., Tolman, H.L., van Vledder, G., Wolf, J., Young, I.: Wave modelling—the state of the art. Prog. Oceanogr.
**75**(4), 603–674 (2007)CrossRefGoogle Scholar - 7.Chern, M .J., Borthwick, A .G .L., Eatock Taylor, R.: A pseudospectral \(\sigma \)-transformation model of 2-D nonlinear waves. J. Fluids Struct.
**13**(5), 607–630 (1999)CrossRefGoogle Scholar - 8.Chian, C., Ertekin, R.C.: Diffraction of solitary waves by submerged horizontal cylinders. Wave Motion
**15**(2), 121–142 (1992)MathSciNetzbMATHCrossRefGoogle Scholar - 9.Christiansen, T., Engsig-Karup, A.P., Bingham, H.B.: Efficient pseudo-spectral model for nonlinear water waves. In Proceedings of the 27th International Workshop on Water Waves and Floating Bodies (IWWWFB) (2012)Google Scholar
- 10.Clement, A., Mas, S.: Hydrodynamic forces induced by a solitary wave on a submerged circular cylinder. Proc. Int. Offshore Polar Eng. Conf.
**1**, 339–347 (1995)Google Scholar - 11.Cooker, M.J., Peregrine, D.H., Vidal, C., Dold, J.W.: The interaction between a solitary wave and a submerged cylinder. J. Fluid Mech.
**215**(1), 1–22 (1990)MathSciNetzbMATHCrossRefGoogle Scholar - 12.Cooker, M.J., Weidman, P.D., Bale, D.S.: Reflection of a high-amplitude solitary wave at a vertical wall. Oceanogr. Lit. Rev.
**7**, 1279 (1998)zbMATHGoogle Scholar - 13.Dommermuth, D.G., Yue, D.K.P.: A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech.
**184**, 267–288 (1987)zbMATHCrossRefGoogle Scholar - 14.Donea, J., Huerta, A., Ponthot, J.P., Rodriguez-Ferran, A.: Arbitrary Lagrangian–Eulerian methods, encyclopedia of computational mechanics, Chapter 14, vol. 1. Wiley, Hoboken (2004)Google Scholar
- 15.Ducrozet, G., Bingham, H.B., Engsig-Karup, A.P., Ferrant, P.: High-order finite difference solution for 3D nonlinear wave–structure interaction. J. Hydrodyn. Ser. B
**22**(5), 225–230 (2010)CrossRefGoogle Scholar - 16.Dutykh, D., Clamond, D.: Efficient computation of steady solitary gravity waves. Wave Motion
**51**(1), 86–99 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 17.Engsig-Karup, A.P.: Analysis of efficient preconditioned defect correction methods for nonlinear water waves. Int. J. Numer. Method. Fluids
**74**(10), 749–773 (2014)MathSciNetCrossRefGoogle Scholar - 18.Engsig-Karup, A.P., Bingham, H.B., Lindberg, O.: An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys.
**228**, 2100–2118 (2009)MathSciNetzbMATHCrossRefGoogle Scholar - 19.Engsig-Karup, A.P., Eskilsson, C., Bigoni, D.: A stabilised nodal spectral element method for fully nonlinear water waves. J. Comput. Phys.
**318**(6), 1–21 (2016)MathSciNetzbMATHCrossRefGoogle Scholar - 20.Engsig-Karup, A.P., Eskilsson, C., Bigoni, D.: Unstructured spectral element model for dispersive and nonlinear wave propagation. In Proceedings 26th International Ocean and Polar Engineering Conference (ISOPE), Greece (2016)Google Scholar
- 21.Engsig-Karup, A.P., Glimberg, S.L., Nielsen, A.S., Lindberg, O.: Fast hydrodynamics on heterogenous many-core hardware. In: Raphäel, C. (ed.) Designing Scientific Applications on GPUs, Lecture Notes in Computational Science and Engineering. CRC Press, Cambridge (2013)Google Scholar
- 22.Engsig-Karup, A.P., Hesthaven, J.S., Bingham, H.B., Madsen, P.A.: Nodal DG-FEM solutions of high-order Boussinesq-type equations. J. Eng. Math.
**56**, 351–370 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 23.Engsig-Karup, A.P., Madsen, M.G., Glimberg, S.L.: A massively parallel GPU-accelerated model for analysis of fully nonlinear free surface waves. Int. J. Numer. Method. Fluids
**70**(1), 20–36 (2011) MathSciNetzbMATHCrossRefGoogle Scholar - 24.Ferrant, P.: Simulation of strongly nonlinear wave generation and wave-body interactions using a 3-D MEL model. In: Proceedings of the 21st ONR Symposium on Naval Hydrodynamics, Trondheim, Norway, pp. 93–109 (1996)Google Scholar
- 25.Fischer, P., Mullen, J.: Filter-based stabilization of spectral element methods. Comptes Rendus De L Academie Des Sciences Serie I-mathematique
**332**, 265–270,02 (2001)MathSciNetzbMATHGoogle Scholar - 26.Gagarina, E., Ambati, V.R., van der Vegt, J.J.W., Bokhove, O.: Variational space-time (dis)continuous Galerkin method for nonlinear free surface water waves. J. Comput. Phys.
**275**, 459–483 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 27.Gordon, W.J., Hall, C.A.: Construction of curvilinear coordinate systems and their applications to mesh generation. Int. J. Numer. Method. Eng.
**7**, 461–477 (1973)zbMATHCrossRefGoogle Scholar - 28.Gouin, M., Ducrozet, G., Ferrant, P.: Propagation of 3D nonlinear waves over an elliptical mound with a High-Order Spectral method. Eur. J. Mech. B Fluids
**63**, 9–24 (2017)MathSciNetzbMATHCrossRefGoogle Scholar - 29.Greaves, D .M., Wu, G .X., Borthwick, A .G .L., Eatock Taylor, R.: A moving boundary finite element method for fully nonlinear wave simulations. J. Ship Res.
**41**(3), 181–194 (1997)Google Scholar - 30.Harris, J.C., Dombre, E., Benoit, M., Grilli, S.T.: A comparison of methods in fully nonlinear boundary element numerical wave tank development. In: 14émes Journées de l’Hydrodynamique (2014)Google Scholar
- 31.Herrmann, L.R.: Laplacian-isoparametric grid generation scheme. J. Mar. Sci. Appl.
**102**(5), 749–756 (1976)Google Scholar - 32.Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)zbMATHCrossRefGoogle Scholar
- 33.Kim, C.H., Clément, A.H., Tanizawa, K.: Recent research and development of numerical wave tanks—a review. Int. J. Offshore Polar Eng.
**9**(4), 241–256 (1999)Google Scholar - 34.Klettner, C., Eames, I.: Momentum and energy of a solitary wave interacting with a submerged semi-circular cylinder. J. Fluid Mech.
**708**, 576–595 (2012)MathSciNetzbMATHCrossRefGoogle Scholar - 35.Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus
**24**, 199–215 (1972)MathSciNetCrossRefGoogle Scholar - 36.Laskowski, W., Bingham, H.B., Engsig-Karup, A.P.: Modelling of wave-structure interaction for cylindrical structures using a spectral element multigrid method. In Proceedings of the 34th International Workshop on Water Waves and Floating Bodies (2019)Google Scholar
- 37.Lin, P.: A multiple-layer \(\sigma \)-coordinate model for simulation of wave-structure interaction. Comput. Fluids
**35**(2), 147–167 (2006)zbMATHCrossRefGoogle Scholar - 38.Longuet-Higgins, M .S., Cokelet, E .D.: The deformation of steep surface waves on water. I. A numerical method of computation. Proc. R. Soc. Lond. Ser. A
**350**(1660), 1–26 (1976)MathSciNetzbMATHCrossRefGoogle Scholar - 39.Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech.
**27**(2), 395–397 (1967)MathSciNetzbMATHCrossRefGoogle Scholar - 40.Ma, Q.W. (ed.): Advances in numerical simulation of nonlinear water waves, vol. 11. World Scientific Publishing Co. Pte. Ltd., Singapore (2010)zbMATHGoogle Scholar
- 41.Ma, Q .W., Wu, G .X., Eatock Taylor, R.: Finite element simulation of fully non-linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. Int. J. Numer. Method. Fluids
**36**(3), 265–285 (2001)zbMATHCrossRefGoogle Scholar - 42.Ma, Q .W., Wu, G .X., Eatock Taylor, R.: Finite element simulations of fully non-linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation. Int. J. Numer. Method Fluids
**36**(3), 287–308 (2001)CrossRefGoogle Scholar - 43.Ma, Q.W., Yan, S.: Quasi ALE finite element method for nonlinear water waves. J. Comput. Phys.
**212**(1), 52–72 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 44.Ma, Q.W., Yan, S.: QALE-FEM for numerical modelling of non-linear interaction between 3d moored floating bodies and steep waves. Int. J. Numer. Method Eng.
**78**(6), 713–756 (2009)zbMATHCrossRefGoogle Scholar - 45.Madsen, P.A., Bingham, H.B., Liu, H.: A new boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech.
**462**, 1–30 (2002)MathSciNetzbMATHCrossRefGoogle Scholar - 46.Maxworthy, T.: Experiments on the collision between solitary waves. J. Fluid Mech.
**76**(1), 177–185 (1976)CrossRefGoogle Scholar - 47.Olson, L.: Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM J. Sci. Comput.
**29**, 2189–2209, 01 (2007)MathSciNetzbMATHCrossRefGoogle Scholar - 48.Patera, A.T.: A Spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys.
**54**, 468–488 (1984)zbMATHCrossRefGoogle Scholar - 49.Rienecker, M.M., Fenton, J.D.: A Fourier approximation method for steady water waves. J. Fluid Mech.
**104**, 119–137 (1981)zbMATHCrossRefGoogle Scholar - 50.Robertson, I., Sherwin, S.J.: Free-surface flow simulation using hp/spectral elements. J. Comput. Phys.
**155**, 26–53 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 51.Rønquist, E .M., Patera, A .T.: Spectral element multigrid. i. Formulation and numerical results. J. Sci. Comput.
**2**(4), 389–406 (1987)MathSciNetzbMATHCrossRefGoogle Scholar - 52.Shao, Y.-L., Faltinsen, O.M.: A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics. J. Comput. Phys.
**274**, 312–332 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 53.Sibley, P., Coates, L.E., Arumugam, K.: Solitary wave forces on horizontal cylinders. Appl. Ocean Res.
**4**(2), 113–117 (1982)CrossRefGoogle Scholar - 54.Tanizawa, K.: A nonlinear simulation method of 3-D body motions in waves (1st report). J. Soc. Nav. Archit. Jpn.
**178**, 179–191 (1995)CrossRefGoogle Scholar - 55.Tsai, W.-T., Yue, D.K.P.: Computation of nonlinear free-surface flows. In Annual review of fluid mechanics, Vol. 28, pp. 249–278. Annual Reviews, Palo Alto, CA (1996)MathSciNetCrossRefGoogle Scholar
- 56.Turnbull, M .S., Borthwick, A .G .L., Eatock Taylor, R.: Numerical wave tank based on a \(\sigma \)-transformed finite element inviscid flow solver. Int. J. Numer. Method Fluids
**42**(6), 641–663 (2003)zbMATHCrossRefGoogle Scholar - 57.van der Vegt, J.J.W., Tomar, S.K.: Discontinuous galerkin method for linear free-surface gravity waves. J. Sci. Comput.
**22**(1), 531–567 (2005)MathSciNetzbMATHCrossRefGoogle Scholar - 58.Wang, C.-Z., Wu, G.-X.: A brief summary of finite element method applications to nonlinear wave–structure interactions. J. Mar. Sci. Appl.
**10**, 127–138 (2011)CrossRefGoogle Scholar - 59.Wang, L., Tang, H., Wu, Y.: Study on interaction between a solitary wave and a submerged semi-circular cylinder using acceleration potential. In Proceedings of The Annual International Offshore and Polar Engineering Conference (ISOPE) (2013)Google Scholar
- 60.Warburton, T.: An explicit construction for interpolation nodes on the simplex. J. Eng. Math.
**56**(3), 247–262 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 61.West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L.: A new numerical method for surface hydrodynamics. J. Geophys. Res. Oceans
**92**(C11), 11803–11824 (1987)CrossRefGoogle Scholar - 62.Westhuis, J.-H.: The numerical simulation of nonlinear waves in a hydrodynamic model test basin. PhD thesis, Department of Mathematics, University of Twente, The Netherlands (2001)Google Scholar
- 63.Wu, G .X., Eatock Taylor, R.: Finite element analysis of two-dimensional non-linear transient water waves. Appl. Ocean Res.
**16**(6), 363–372 (1994)CrossRefGoogle Scholar - 64.Wu, G .X., Eatock Taylor, R.: Time stepping solutions of the two-dimensional nonlinear wave radiation problem. Ocean Eng.
**22**(8), 785–798 (1995)CrossRefGoogle Scholar - 65.Yan, H., Liu, Y.: An efficient high-order boundary element method for nonlinear wave–wave and wave–body interactions. J. Comput. Phys.
**230**(2), 402–424 (2011)MathSciNetzbMATHCrossRefGoogle Scholar - 66.Yeung, E.W.: Numerical methods in free-surface flows. In: Annual Review of Fluid Mechanics, Vol. 14, pp. 395–442. Annual Reviews, Palo Alto, Calif. (1982)MathSciNetCrossRefGoogle Scholar
- 67.Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys.
**9**, 190–194 (1968)CrossRefGoogle Scholar - 68.Zhu, Q., Liu, Y., Yue, D.K.P.: Three-dimensional instability of standing waves. J. Fluid Mech.
**496**, 213–242 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 69.Zienkewitch, O.C.: The finite element method in engineering science, 2nd edn. McGraw-Hill, New York (1971)Google Scholar
- 70.Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: Chapter 6—free surface and buoyancy driven flows. In: Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P. (eds.) The Finite Element Method for Fluid Dynamics, 7th edn, pp. 195–224. Butterworth-Heinemann, Oxford (2014)zbMATHCrossRefGoogle Scholar