A stabilised nodal spectral element method for fully nonlinear water waves

Engsig-Karup, Allan Peter; Eskilsson, Claes; Bigoni, Daniele

doi:10.1016/j.jcp.2016.04.060

Cited by 44 publications

(78 citation statements)

References 90 publications

(178 reference statements)

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“…Several approaches can be considered to solve the Zakharov equations, such as the Hamiltonian Coupled-Mode Theory (or HCMT) method (Athanassoulis and Papoutsellis, 2017;Papoutsellis et al, 2018a), the extension of the high-order spectral (or HOS) method to variable bottom cases (Gouin et al, 2016), or the direct use of finite difference schemes (Bingham and Zhang, 2007). Other commonly applied approaches for solving the fully nonlinear potential flow problems are, for example, the Boundary Element Method (Longuet-Higgins and Cokelet, 1976;Dold and Peregrine, 1986;Grilli et al, 1989;Dold, 1992), the Finite Element Method (Wu and Eatock Taylor, 1994), the Quasi-Arbitrary Lagrangian Eulerian Finite Element Method (Ma and Yan, 2006), the Spectral Element Method (Engsig-Karup et al, 2016) and the Spectral Boundary Integral Method (Fructus et al, 2005;Wang and Ma, 2015).…”

Section: Introductionmentioning

confidence: 99%

Comparing methods of modeling depth-induced breaking of irregular waves with a fully nonlinear potential flow approach

Simon

Papoutsellis

Benoît

et al. 2019

J. Ocean Eng. Mar. Energy

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The modeling of wave breaking dissipation in coastal areas is investigated with a fully nonlinear and dispersive wave model. The wave propagation model is based on potential flow theory, which initially assumes nonoverturning waves. Including the impacts of wave breaking dissipation is however possible by implementing a wave breaking initiation criterion and dissipation mechanism. Three criteria from the literature, including a geometric, kinematic, and dynamic-type criterion, are tested to determine the optimal criterion predicting the onset of wave breaking. Three wave breaking energy dissipation methods are also tested: the first two are based on the analogy of a breaking wave with a hydraulic jump, and the third one applies an eddy viscosity dissipative term. Numerical simulations are performed using combinations of the three breaking onset criteria and three dissipation methods. The simulation results are compared to observations from four laboratory experiments of regular and irregular waves breaking over a submerged bar, irregular waves breaking on a beach, and irregular waves breaking over a submerged slope. The different breaking approaches provide similar results after proper calibration. The wave transformation observed in the experiments is reproduced well, with better results for the case of regular waves than irregular waves. Moreover, the wave statistics and wave spectra are predicted well in general, and in particular for regular waves. Some differences are observed for irregular wave cases, in particular in the low-frequency range. This is attributed to incomplete absorption of the long waves in the numerical model. Otherwise, the wave spectra in the range [0.5fp, 5fp] are reproduced well, before, inside, and after the breaking zone for the three irregular wave experiments.

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Section: Introductionmentioning

confidence: 99%

Comparing methods of modeling depth-induced breaking of irregular waves with a fully nonlinear potential flow approach

Simon

Papoutsellis

Benoît

et al. 2019

J. Ocean Eng. Mar. Energy

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“…In the last two decades or so, the NLPF formulation has been treated numerically by means of various direct numerical methods (DNM). Important examples are the finite difference method (FDM) [18] and the finite element method (FEM) [19], [20], [21], [22] [23], that require the construction of computational grids covering the entire fluid domain. Another popular strategy is the use of boundary element method (BEM), that transfers the computations on the physical boundaries of the fluid domain [24], [25], [26], [27], [28].…”

Section: Introductionmentioning

confidence: 99%

Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

Papoutsellis

Charalampopoulos

Athanassoulis

2018

European Journal of Mechanics - B/Fluids

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1. Introduction 2. Formulation of the problem 2.1 Classical formulation of the problem 2.2 The Hamiltonian Coupled-Mode Theory 2.2.1.. Exact vertical series expansion of the wave potential 2.2.2 The Hamiltonian coupled-mode system 2 6. Interaction of solitary waves with bathymetry and vertical walls 6.1 Reflection of solitary waves on a vertical wall 6.2 Shoaling of solitary waves over a plane beach 6.3 Reflection of shoaling solitary waves on a vertical wall at the end of a sloping beach 6.4 Propagation of a solitary wave over a sinusoidal patch 6.5 Transformation of a solitary wave over a 3D bathymetry with banks and trenches 7. Discussion and conclusions Acknowledgments Appendix A: An outline of the derivation of the Hamiltonian Coupled Mode System Appendix B: Calculation of the basic vertical integrals (a) Vertical integrals of the form ( ; ) m J s Z and ( ; ) m J s W (b) Vertical integrals of the form ( ; ) n m J s Z Z for 0 ,1, 2 s = (c) Vertical integrals of the form ( ; ) n m J s W Z for 0 ,1 s = Appendix C. Fast and accurate calculation of local wavenumbers ( , ) n k t x Appendix D: Finite-difference linear system for the substrate problem Eqs. (17a,b) Appendix E. Description of the 3D bathymetry References List of abbreviationsBEM boundary element method CMS coupled-mode system DNM direct numerical methods DtN Dirichlet-to-Neumann (operator) FD finite difference FDM finite difference method FEM finite element method HCMS Hamiltonian coupled-mode system HCMT Hamiltonian Coupled-Mode Theory NLPF nonlinear potential flow NR Newton-Raphson (method) RK Runge-Kutta (method) SGN Serre-Green-Nagdhi (equations) AbstractThis paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over arbitrary bathymetry Papoutsellis and Athanassoulis (2017) (arxiv.org/abs/1704.03276). Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two-and three-dimensional environments. The essential novelty of HCMT is a new representation of the Dirichlet-to-Neumann operator, which is needed to close the Hamiltonian evolution equations. This new representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the instantaneous, irregular, fluid domain, established recently by Athanassoulis & Papoutsellis (2017) (https://doi.org/10.1098/rspa.2017.0017). The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat an arbitrary bathymetry as simply as the flat-bottom case, without domain transformation. A key point for the efficient implementation of the method is the fast and accurate evaluation of the space-time varying coefficients appearing in some of its equations. In this paper, all varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computationa...

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“…After solving for the velocity potential,w n+1 is obtained by a C 0 gradient recovery as w n+1 is needed to advance the free surface conditions another time step. Explicit time-stepping schemes are effective for FNPF as the time-step restriction is not dependent on the horizontal mesh size, but only on the vertical resolution and water depth, see [98]. A σ-transformed FNPF solver is well-suited for large-scale The use of σ-transformed domains excludes any truncated bodies in the domain.…”

Section: Fully Nonlinear Potential Flowmentioning

confidence: 99%

“…In order to handle arbitrarily shaped bodies in the domain the mixed-Eulerian-Lagrangian (MEL) approach should be used and there are ongoing efforts to implement a SEM based on the MEL approach. In [102] a work-around on the mesh asymmetry problem associated with the MEL was presented. By using hybrid meshes consisting of a single layer of vertically aligned quads (in 2D) at the free surface, the eigenvalues were shown to be purely imaginary for any triangulation of the inner field.…”

Section: Fully Nonlinear Potential Flowmentioning

confidence: 99%

Spectral/hp element methods: Recent developments, applications, and perspectives

Cantwell

Monteserin

et al. 2018

J Hydrodyn

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Abstract:The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a 0 -C continuous expansion. Computationally and theoretically, by increasing the polynomial order p , high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

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A stabilised nodal spectral element method for fully nonlinear water waves

Cited by 44 publications

References 90 publications

Comparing methods of modeling depth-induced breaking of irregular waves with a fully nonlinear potential flow approach

Comparing methods of modeling depth-induced breaking of irregular waves with a fully nonlinear potential flow approach

Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

Spectral/hp element methods: Recent developments, applications, and perspectives

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