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Unstructured nodal DG-FEM solution of high-order Boussinesq-type equations

Goal: The main objective of the present study has been to develop a numerical model and inves- tigate solution techniques for solving the recently derived high-order Boussinesq equations of Madsen et al. (2002) in irregular domains in one and two horizontal dimensions. The Boussinesq-type methods are simple alternatives to solving full three-dimensional wave problems by e.g. Navier-Stokes equations, which can capture all the important wave phenomena such as diffraction, refraction, nonlinear wave-wave interactions and interaction with structures. The OceanWave3D project started after this project.

Methods: Time Domain Simulation, Unstructured Nodal Discontinuous Spectral Element Method

Date: 15 August 2003 - 15 August 2006

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Allan Peter Engsig-Karup
added a project goal
The main objective of the present study has been to develop a numerical model and inves- tigate solution techniques for solving the recently derived high-order Boussinesq equations of Madsen et al. (2002) in irregular domains in one and two horizontal dimensions. The Boussinesq-type methods are simple alternatives to solving full three-dimensional wave problems by e.g. Navier-Stokes equations, which can capture all the important wave phenomena such as diffraction, refraction, nonlinear wave-wave interactions and interaction with structures. The OceanWave3D project started after this project.
 
Allan Peter Engsig-Karup
added 4 research items
This abstract describes recent work on applying the Discontinuous Galerkin Finite El- ement Method (DG-FEM) to solve a set of highly accurate Boussinesq-type equations for the interaction of nonlinear waves with bottom mounted structures. We present here some validation results in 2-D which establish the accuracy and convergence properties of the scheme, along with some preliminary results in 3-D for the diraction of waves around bottom mounted structures. The mathematical model being applied is derived in (7, 6), and we consider both the Pad e (4,4) and the Pad e (2,2) versions of this method which can accurately propagate nonlinear waves in relative water depths up to kh 25 & 10 respectively, (k the wavenumber, h the water depth) up to something near the stable breaking limit. The associated wave kinematics (vertical distribution of pressure and velocity) are accurate up to kh 12 & 4 respectively. A nite dierence based solution to these equations has been used to obtain solutions for highly nonlinear wave run-up on piecewise rectangular structures as described in (4, 5) and references therein. This solution, as implemented, is limited to a rectangular, uniformly spaced grid and hence piecewise rectangular structures. The main goal of the present work is to remove this limitation and allow the treatment of more general geometries. The choice of a DG-FEM methodology provides the geometric exibilit y of an unstructured grid while also providing exibilit y in the local order of accuracy of the numerical scheme. This combination could lead to improved computational eciency , although this has yet to be demonstrated. To briey outline the mathematical model, we consider water waves in 3-D using a coordinate system with origin in the still-water plane and the z-axis vertically upwards. The uid domain is bounded by the sea bed at z = h(x), with x = (x; y), and the free surface at z = (x; t), where t is time. The free surface boundary conditions are written in terms of velocity components at the free surface ~ u = (~ u; ~ v) = u(x; ; t) and ~ w = w(x; ; t):
A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.
We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated and absorbed in the interior of the computational domain using a flexible relaxation technique applied on the free surface variables.