Date of Award


Degree Type


Degree Name

Master of Science in Ocean Engineering


Ocean Engineering

First Advisor

Stephan Grilli


A numerical model for fully nonlinear water waves was developed and validated over the past five years, and used as a research tool for calculating various wave propagation and wave-structure interaction problems in coastal areas (Grilli 1989 etc.).

This model solves unsteady two-dimensional potential flow equations in domains of arbitrary geometry, and can be used to calculate wave shoaling and runup on slopes, and wave interaction with coastal structures. The Eulerian-Lagrangian description of the free surface enables the model to compute wave overturning over slopes and structures, up to the instant the tip of the breaking wave hits the free surface.

Waves are generated in the model, by imposing waves directly on the free surface, by simulating a piston or flap wavemaker motion, as in laboratory experiments, or by using a line of internal sources. Except for the assumption of potential flow, no further approximation is made in the model, unlike in most wave theories. The model, therefore, is not restricted to special types of waves (e.g., short, long, periodic, non-periodic,...), and can be used for arbitrary incident wave conditions. Submerged or emerged structures of arbitrary shape, like obstacles on the bottom or breakwaters, can be introduced in the model, as well as gentle or steep bottom slopes.

Thesis Objectives

The existing model was however restricted in its ability to accurately predict wave kinematics in very shallow water, i.e. in domains with large aspect ratios, owing to the accumulation of numerical errors attributable to quasi-singular integrations. Furthermore, the representation of the field variables on the free surface based on quasi-spline boundary elements (although very accurate in many cases) could reduce the accuracy of the computations close to breaking jets. Resolution, or the ability to “zoom into” regions of interest such as the breaking jets of shoaling waves and to compensate for “stokes drift” effects on the free surface (a side-effect of the Lagrangian representation of the free surface) was also lacking in the earlier model.

The objectives of the thesis are to address these problems and enhance the applicability and predictive abilities of the model both in terms of accuracy and efficiency. More specifically, the objectives are:

  • Improved quasi-singular integrals: To develop new methods for quasi-singular integration based on several state-of-the-art techniques providing improved overall accuracy and efficiency of calculation of the integral terms.

  • Improved free surface representation: To implement a special type of boundary element - namely the Mid, Cubic Interpolation element [MCI element] developed by Grilli and Subramanya (1994) in the model and thereby to improve the free-surface representation of field variables, which earlier were represented by linear shape-functions.

  • Nodal regridding: To develop and implement a technique for adaptivenodalregridding, so as to limit errors due to insufficient nodal resolution (stokes drift effects), or to quasi-singularities resulting from node concentration (nodes drifting very close to each other).

A brief discussion of the numerical procedure, and their application, follows in the next section.

Manuscript 11

The application of the BEM model developed by Grilli et al. to calculate wave shoaling and breaking in very shallow water requires using computational domains with very sharp geometry and large aspect ratio. For such domains, in addition to comer problems, the narrowing of the geometry over the slope, the shallower the water, creates quasi-singular situations for the nodes that, on one side of the boundary, are getting closer to the other side of the boundary (eg. bottom or free surface). The formation of narrow jets in breakers, in which BEM nodes, equivalent to Lagrangian particles, accumulate, is another cause for such quasi-singular situations. Since it is often prohibitive or impractical to refine the free surface discretization in jets or in narrow regions of the domain, it is usually necessary to address the problem of quasi-singular integrations. An adaptive integration technique was introduced by Grilli & Svendsen in earlier applications of the model, to improve the accuracy of integrations in comer elements and in other quasi-singular situations. This technique, however, proved computationally expensive when systematically used over the whole boundary. In this manuscript, new more accurate and efficient integration techniques for quasi-singular integrals are introduced, based on modified Telles and Lutz methods. The modified Telles method collocates the integration points closer to the approximate point of logarithmic singularity, using a double transformation technique. The quasi-singularity of the integrand is therefore better described and the resulting integrations more accurate. The generalized Lutz method has been extended to account for curved elements. It essentially computes new Gaussian weights and points from a preconstructed table based on the minimum distance from the element of the integration point. The new integration techniques are presented as well as applications that demonstrate the accuracy and efficiency of the new approaches.

Manuscript 22

The fully nonlinear wave model by Grilli, et al. will be used, in combination with the new integration techniques (Manuscript 1), and improved free surface representation (Manuscript 3), to calculate shoaling up to breaking of solitary waves over a mild 1:35 slope. This model was successfully used in earlier applications for calculating wave propagation over a broad range of both steep and gentle slopes. It was also used in a variety of situations with wave-structure interaction. Solitary waves are selected in the present case, as in earlier applications, both for simplicity, and because they approximate well extreme design waves.

Shoaling of solitary waves on both gentle (1:35) and steeper slopes (≤ 1:6.50) is analyzed up to breaking using both a fully nonlinear wave model and results from high accuracy laboratory experiments. The experimental results cited in the paper, were obtained in the experimental facilities of the University of Delaware, by J. Veeramony & lb. A. Svendsen. The experimental and numerical results are part of a joint paper submitted to ASCE (see footnote below). For the mildest slope, close agreement is obtained between both approaches up to breaking, where waves become very asymmetric and breaking indices reach almost twice the value of the largest stable symmetric wave. Shoaling characteristics of solitary waves are then discussed.

Manuscript 3

In this paper, several numerical aspects of the model for fully nonlinear waves are improved and validated to study wave breaking due to shoaling over a gentle plane slope and wave breaking induced by a moving lateral boundary.

An improved numerical treatment of the boundary conditions at the intersection between moving lateral boundaries and the free surface (comer) is implemented and tested in the model, and the free surface interpolation method is improved to better model highly curved regions of the free surface that occur in breaking waves. A node regridding technique is introduced to improve the resolution of the solution close to moving boundaries and in breaking jets.

Examples are presented for solitary wave propagation, shoaling, and breaking over a gentle slope and for wave breaking induced by a moving vertical boundary. In all cases, results are more accurate and computations can be carried out for a longer time than with the earlier model.


The underlying theory and mathematical details are explained in Appendix A. Appendix B discusses the improvements in the model and the results therefrom, Some future directions of study are suggested. A glossary of scientific notations explaining the conventions used in the manuscripts is included in Appendix C.



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