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Historical development

BEM models

Boundary element methods can be traced back to the work of Longuet-Higgins and Cokelet (1976) who considered the evolution of two-dimensional space-periodic waves in deep water. They work in complex variables and use conformal mapping to transform a semi-infinite periodic domain into internals of a closed contour representing the mapped free surface. The material time derivatives of surface coordinates and of surface potential are then expressed via the normal derivative of the potential. To find the normal derivative, the Dirichlet problem of finding the normal gradient of a harmonic function from its values on a closed contour is formulated and solved by using Green’s theorem. This leads to an integral equation relating the normal gradient of the potential with its values at the mapped free surface. The integral equation is solved numerically and the normal derivative of the potential is used to calculate the time derivatives. Then material coordinates of the free surface and the surface potential at the incremented time are calculated by using a fourth-order finite difference technique. The numerical scheme demonstrated a weak saw-toothed instability which was suppressed by applying polynomial smoothing. The method was applied to simulate the evolution of high periodic waves and development of overturning profiles at initial stages of wave breaking. Since the method uses the Lagrangian approach to track the evolution of the free surface, it is referred to as a mixed Eulerian-Lagrangian method (MEL). This becomes a common feature of boundary-element methods. Later, the method was extended to the case of constant finite depth (New et ah, 1985). Vinje and Brevig (1981) suggested an alternative method where Cauchy’s integral theorem is applied to a complex flow potential in physical space. Working in physical variables allowed to construct solutions with vertical solid boundaries (fixed or moving) and to simulate waves in a wave flume with a piston wavemaker (Dommermuth et ah, 1988).

Using complex variables restricts the application of the boundary integral formulation to 2D problems. A more flexible formulation was therefore developed with Green’s theorem applied in physical space. Apart from being applicable to both 2D and 3D problems, such formulation also allows flexible treatment of fixed and moving solid boundaries of arbitrary shape and is suitable for development of efficient numerical wave tanks. Examples of BEM-based 2D numerical wave tanks, which differ by details of numerical realisation and methods of wave generation and absorption can be found in Grilli et ah (1989); Ohyama and Nadaoka (1991); Wang et ah (1995). Though the first works on application of BEM to 3D waves appeared relatively early (e.g., Isaacson, 1982), it took long time to develop robust and flexible models suitable for wide range of applications. Apart from the more difficult formulation for 3D geometry, this was caused by the drastic increase of computational cost for such models. Simplified formulations were often suggested to deal with these problems, which restricted models applicability. For example, Isaacson (1982) used a Green function that assumes the symmetry of the solution about the flat sea bed (method of images). This restricts the application of the method to constant depth. Xue et ah (2001) considered deep water waves periodic in both horizontal directions. This simplification allowed to perform high-resolution simulations and to obtain valuable results on dynamics and evolution of 3D breaking waves. Considerable efforts have finally resulted in the development of 3D numerical wave tanks capable of simulating general highly nonlinear waves on arbitrary bathymetry Grilli et ah (2001).

The solution method used by the models mentioned above and in fact by most contemporary BEM models can be briefly summarised as follows. By applying Green’s theorem with an appropriate selection of the Green function, the Laplace equation can be reduced to an integral equation defined on a boundary of the computational domain. The equation relates the values of the potential and its normal derivative at the boundary. The free surface boundary conditions are written in the mixed Eulerian-Lagrangian form and connect the material time derivatives of the surface potential and of the surface position with the gradient of the potential (velocity) at the free surface. If both the potential and the normal derivative are known, the time derivatives can be calculated. The boundary conditions on solid surfaces specify the normal derivatives of the potential at these boundaries (Neumann boundary condition). For the free surface, the surface potential is known either from the initial conditions or from a previous calculation step (Cauchy boundary condition). Then, the boundary integral equation can be used to calculate the normal derivatives at the free surface and the potential at the solid boundary. After discretisation of the integral equation, a system of linear algebraic equations is solved to find the unknown values of the potential and the normal derivative at the surface nodes. The free surface boundary conditions can then be used to update the free surface position and the surface potential by applying an appropriate time stepping technique.

Major improvements had been made in terms of the method accuracy and stability (e.g., Grilli and Svendsen, 1990). More recently, considerable efforts are concentrated on improving computational efficiency of BEM models (Fochesato and Dias, 2006; Yan and Liu, 2011; .liang et al., 2012) and to the adaptation of the method for parallel computing (Nimmala et al., 2013). Over the years BEM models have been applied to a wide range of water wave problems including problems of propagation of extreme waves directly relevant to this chapter (e.g., Fochesato et al., 2007; Ning et al., 2009). For further reference, the up-to-date formulation for a 3D BEM numerical wave tank with a review of earlier work and examples of applications can be found in Grilli et al. (2010).

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