overhauser boundary elements solution for periodic water waves in the physical plane

Engineering Analysis with Boundary Elements 11 (1993) 47-54

Overhauser boundary elements solution for periodic water waves in the physical plane

Juan C. Ortiz & Scott L. Douglass University of South Alabama, Mobile, Alabama 36688, USA

(Received 27 November 1991; revised version received and accepted 15 April 1992)

An Overhauser boundary element method (BEM) for the modeling of nonlinear periodic waves in the physical plane is described. The Overhauser element, used to eliminate discontinuities of the slope on the free surface of the wave, is described in detail. Examples for non-linear steady and breaking waves are shown. The Overhauser element is compared to Lagrangian linear and cubic elements. It is noted that the Overhauser element system is very stable. Neither 'smoothing' nor any other manipulation of the BEM results are necessary within the time-stepping algorithm to achieve a stable solution.

Key words: boundary element methods, nonlinear water waves, computational fluid dynamics, Overhauser splines, coastal engineering.

1 INTRODUCTION

Breaking waves are the primary forcing function in surf zone dynamics. Sediment transport, erosion, wave- driven nearshore currents, forces on coastal structures, etc., are all affected by overturning waves in the surf zone. However, very little is known about the hydrodynamics of breaking waves.

Numerical solutions to the equations governing inviscid water waves using boundary integral techniques have provided efficient ways to study this phenomenon. The free surface of the wave has traditionally been modeled using linear or higher order Lagrangian polynomials which do not guarantee C ~ continuity along the curve. To compensate for solution instabilities, manipulation of the intermediate solutions as the wave propagates through time is needed. Smoothing of the free surface, 1 redistribution of the nodes, 2 and 'after-the-fact' 4th order curve fitting through the nodes, 3 are all methods that have been employed to eliminate the instabilities encountered. In this work, an Overhauser spline element 4-7 is used to eliminate the discontinuities of the slope on the free surface. It is seen that no other manipulation of the intermediate results must be employed to achieve a non-divergent solution to the breaking wave problem.

Engineering Analysis with Boundary Elements 0955-7997/93/$05.00 © 1993 Elsevier Science Publishers Ltd.

2 WAVE THEORY

47

The fluid particle velocities in the cartesian x- and y-directions of an inviscid, incompressible fluid in irrotational flow may be described in terms of a velocity potential q~ satisfying Laplace's equation

v2~=o (1)

where the particle velocities u and v (in the x- and y-directions, respectively) are related to ~b by:

06 u = 0---x (2)

06 v Oy (3)

With the appropriate boundary conditions, eqns (1), (2) and (3) may be used to describe the motion of an overturning water wave. This work concentrates on the description of periodic water waves. The notation used for a typical periodic wave throughout this work is shown in Fig. 1. The motion of the wave is completely defined by the free surface Ft. Thus, our primary goal is the determination of r f and its deformation in time.

The boundary condition at the bottom surface (rb) is one of no flow normal to the surface,

0~n~ rb = 0 (4)

where n is the outward pointing normal to the surface r which bounds the domain ~(t). The boundary

48 J.C. Ortiz, S.L. Douglass

, f f / f . o f f

5 E . .

Fig. 1. Periodic water wave computational domain f~(t) for BEM analysis. Fb is the bottom surface, I" I and r r a r e the left

and right surfaces, and Fr is the free surface.

conditions at the sides (1"1 and Fr) are those of spacial periodicity:

q~[r, =~ ' r , and O~nr=-O~n~rr (5)

The boundary conditions on the free surface I'f are twofold: a kinematic and a dynamic condition. Assum- ing that a particle at the free surface will remain at the surface, 8 the material (or total) derivative with respect to time, D/Dt, of the particle locations, x and y, yields the kinematic free surface boundary conditions 1

Dx = u (6)

Dt Dy - - = (7) Dt

The dynamic condition is the satisfaction of Bernoulli's equation, by which the total derivative of the velocity potential of a particle on the surface is given by l

Dq~ = _1 (u 2 + v2 ) - gy (8) Dt 2

where g is the acceleration of gravity. The pressure on the free surface is taken as zero.

3 BACKGROUND

Most of the numerical solutions of breaking waves have used a boundary integral technique for velocity potential. Other approaches that depend on solving the interior flow problem, (e.g. Harlow & Welch, 9 Nichols et alJ °) have been less successful at producing realistic looking solutions. Longuet-Higgins & Cokelet 1 pio- neered the use of the mixed Lagrangian-Eulerian boundary integral techniques on overturning waves.

Longuet-Higgins & Cokelet used a conformal mapping transformation with the complex form of the velocity potential. Vinje & Brevig 11'u and Douglass !3 used a boundary integral method based on the Cauchy integral for the complex velocity potential without mapping. They used linear elements to describe the surface location and velocity potential for integral calculations. Gravert 14 seems to have applied a boundary

element technique with higher order elements to the periodic wave problem in the physical plane, with results comparable to those of Vinje & Brevig. Mclver & Peregrine 15 found that results from Vinje & Brevig's technique were in agreement with results from Longuet- Higgins & Cokelet's technique. Dold & Peregrine, 16 New et al., 17 and Seo & Dalrymple is all used some form of conformal mapping with complex variables. Dommer- muth et al. m used a similar method to that of Vinje & Brevig with regridding to successfully compare potential theory and experimental results. A series of papers by Grilli et al. 3'2°-22 have presented applications of a boundary element method (BEM) to the non-periodic breaking wave problem in the physical plane in terms of the velocity potential and its gradient using '3quasi- spline' elements. Grilli, Skourup and Svendsen also present a categorized literature review of existing solutions to the non-linear overturning water wave problem.

4 BEM SOLUTION

4.1 Solution algorithm

The general algorithm used in this work for the description of the wave surface particles as they move thro/agh time follows Longuet-Higgins & Cokelet. 1 The solution of the governing equations is addressed in the physical plane as was done by Vinje & Brevig. u The surface particles correspond to the nodes used in the discretization of the surface into boundary elements.

Given an initial geometry and a velocity potential ~b for each node on the free surface F r, BEM is used to solve Laplace's equation (eqn (1)) for the normal derivatives of q~ at each node. The normal and tangential derivatives of q~, along with the kinematic boundary conditions (eqns (6) and (7)), are then used to find Dx/Dt and Dy/Dt.

Once Dx/Dt and Dy/Dt are found, an Adams- Bashford-Moulton method is used to calculate the location of the wave surface nodes at the next time step. Using the dynamic boundary condition (Eqn (8)), the value of 4~ at each wave surface node is also found for the next time step. This process is repeated until the wave breaks, the front of the plunging jet makes contact with the trough, and the wave equations are no longer valid.

4.2 BEM solution of Laplace's equation

The boundary element method utilizes Green's third for- mula to obtain a solution to Laplace's equation over the domain f~(t) as a function of boundary integrals 23

C4~(rt) = Ir4J(r)~-~dF- Ir Odv(r) W (9)

Overhauser boundary elements solution for periodic water waves in the physical plane 49

where C is a constant, rt is the location of the source point, r is a dummy point over which the integrations are performed, and W is the Green's function corresponding to Laplace's equation in two dimensions:

= ~-~ log Ir - rt[ W

By discretizing the boundary and evaluating eqn (9) at every node, a system of simultaneous equations of the form

I.l o,:Iol{ } /,0, is obtained, where [H] and [G] are the coefficient matrices, dependant 0nly on the boundary geometry and the vectors {~b} and {O~b/On} represent the values of the potential and its normal derivative at the nodes.

With the boundary conditions defined in eqns (4) and (5), an initial free surface potential, and eqn (10), is rearranged into a standard system of linear equations which is then solved for the unknown {~b} and {Oga/On}.

The only necessary values for the time-stepping scheme are the normal derivatives of the potential on the free surface of the wave which are used in eqns (6), (7) and (8) to match the surface in time.

4.3 Time-stepping scheme

The system developed uses an Adams-Bashford- Moulton (ABM) multi-step predictor-corrector time- stepping scheme, given by

At . r,+ l = r, + ~ (9r,+l + 19i'n - 5f,-I + i,-2)

where r i is the coordinate vector of the free surface at time step i, i'i represents the time derivative of r i at time step i, and At is the size of time step. The velocity vector ri = (ui, vi) is obtained for each ri, as explained pre- viously, using the BEM method.

The predictor velocity vector, in+l, is obtained from an approximation of rn+ 1 given by:

At r,+l = r, + - ~ (55i', - 59f,_1 + 37i',_2 - 9i'n_3)

The ABM scheme, which utilizes the previous three values of the node velocities, is initialized with a Runge-Kutta (R-K) scheme. The R - K method is also used to restart the ABM method whenever the time step is modified.

In order to keep the system from becoming unstable as the nodes near the wave tip get closer and closer together, the time steps are automatically subdivided using a Courant number-based algorithm. The Courant number (~0) used in this system, similar to that used by Grilli, 22 is defined by

At' ~¢o - Ax'

where At' is equal to the time step divided by the wave period and Ax' is the distance between the two closest nodes divided by the wave length. Throughout the process, if c~ 0 ever becomes greater than 0.5, the time step is halved until ~0 becomes less than 0-5.

4.4 Boundary discretization

The success of the solution scheme depends on an accurate representation of the normal derivatives of the potential on the wave surface. Previous solutions to the wave breaking problem have used discretization of the boundary with elements which do not guarantee continuity of the derivative of the velocity potential. These discontinuities affect the accuracy of the normal derivatives at the wave surface and lead to instabilities in the time-stepping scheme. Without additional manipulation of the results, such as 'smoothing' the wave surface or curve-fitting through the nodes to find normal derivatives at the element junctures, the solution algorithm often becomes unstable and begins to break down before the wave jet reaches the wave trough.

A more accurate method of representing the boundary and the normal derivatives at the free surface can be obtained using spline elements which guarantee derivative continuity throughout the boundary. In the BEM system used in this work, the wave surface is discretized with Overhauser spline elements. 4'5'7'24

The Overhauser spline is an interpolating curve created by a linear blend of two quadratics:

c(s) = (1 - s)p(~) + sq(() (11)

where (, (, and s are parameters, p(~) and q(() are parametric quadratic curves, and c(s) is the parametric cubic spline resulting from the linear blending (see Fig. 2).

By assigning to ~, ~ and s the parametric values shown in Fig. 2, we may rewrite the definition of the curve c(s) in the common finite element/boundary element format used for isoparametric Lagrangian elements

4

c(s) = E Ni(s)Fi (12) i=1

where the F,.s are the functional values corresponding to the four nodes defining the spline, and the Nis are the

~---0.5 ~0

S : S ) s :SS s t .. -.. /

~=o ~=l s=! ~--o.s

Fig. 2. Overhauser spline definition.

2 4

/ e l eml ~'- " " - . ~ . - " " ~" elem2 \ k

I 3 5 "3

Fig. 3. Overlapping of Overhauser elements.

parametric shape functions defined by:

Nl(S) = -½s 3 + S 2 - - I s

N2(s ) = 3s3-5s2 + 1

N 3(s) = - ~ s 3 + 2 s 2 +½s

N41sl = ½s 3 - k S: (13)

The Overhauser elements enforce C 1 continuity by overlapping the elements, as shown in Fig. 3. The first element is defined by nodes l, 2, 3 and 4; the second element by nodes 2, 3, 4, and 5. The parametric derivative at node 3 is the same for elements 1 and 2 because both are defined using the quadratic which interpolates nodes 2, 3, and 4.

In addition, the Overhauser element is a very stable element, practically eliminating the problems associated with using higher-order elementsY It is well known that highly irregular curves can be obtained when higher- order (non-linear) polynomials are used to fit irregu- larly or unevenly spaced data points. 26

As a water wave moves, the distances between fluid particles change, coming together at the crest and moving farther apart at the trough. In numerical schemes, this means that the nodal point spacing also changes, becoming closer at the crest and farther apart at the trough. This uneven node spacing can lead to numerical instabilities when using higher-order elements. 22 As described below, such instabilities did not occur in the present work when using Overhauser elements, even when the spacing between the nodes became highly uneven.

To illustrate the difference between Overhauser elements and Lagrangian cubic elements a typical node dis- tribution is shown in Fig. 4 (in fact, it is the same set of data points corresponding to the last wave profile seen in Fig. 9.) Overhauser and Lagrangian cubic elements have been used to fit the same set of nodal points. As can be seen, the two curves appear to be very similar except at the inside part of the wave. Here, there are two obvious points of discontinuity at the end points of one of the Lagrangian cubic elements. At the end points of this same element there are also two 'bends' in the curve which appear to be inconsistent with reality. The Overhauser fit appears to be more consis- tent with the shape of an actual water wave.

Despite the advantages just mentioned, the Over-


Fig. 4. Comparison of Overhauser and Lagrangian cubic elements. The solid line is the Overhauser element fit. The

dashed line is the Lagrangian cubic element fit.

hauser element is very simple to implement. The element shape functions are similar to those used for Lagrangian cubic elements. Thus, the needed integrations are performed in the same way as for higher order elements. ° The only noticeable difference between the Overhauser and Lagrangian higher order elements is that the spline elements must be overlapped.

This simplicity is an advantage over other spline elements which may be used. For example, B-Splines (a non-interpolating spline which has the advantage of being able to represent common geometric shapes exactly) have also been used in BEM analysis with success, 27 but they require control points in addition to the boundary nodes. For the type of wave problem in this work, these control points, which help define the elements, would have to be updated and recalculated at each time step even though they are not part of the free surface itself. Although B-Splines may be very effective in other applications, the additional computational time and storage required are apparently not necessary for water wave problems.

4.5 Integration methods

All integrations over the elements are performed using standard four-point Gaussian quadrature, except for the singular integrals which are evaluated using a loga- rithmically weighted Gaussian quadrature method with seven points. The accuracy of the integrations seems to be adequate for the current type of problems being solved. This is probably due to the lack of near-singular integrals arising in the problems. One would expect that such integrals would arise at the top of the jet formed when the wave breaks because nodes are coming closer together as the surface closes in on itself. It appears that this problem is compensated by the fact that the discretization becomes more refined on the wave jet as the wave propagates. Thus, the ratio of the distance between an element and a source node not on that element, and the size of the element does not decrease significantly.

5 RESULTS 0.07

5.1 Waves of permanent form

Theoretically, for a given water wave depth, height, and length, there exists only one periodic wave form which will propagate without changing shape. Dean, 25 using non-linear wave theory, developed a table of 40 different waves of permanent form for engineering design. The table provides the appropriate coefficients used in a truncated Taylor's series approximation of the actual wave shape and velocity potential. As a test for the model, several of Dean's waves were used as initial conditions.

Figure 5 shows the results for a relatively small wave (wave form 5-B as specified in Dean's table). The horizontal distance and the free surface level relative to the still water level (SWL) are both normalized with respect to the wave length (L). The wave amplitude is half the maximum amplitude for a wave of permanent form in that same depth of water and with that same wave length. The depth (d) is equal to 0.088L and the amplitude is equal to 0.39d. In Fig. 6 are shown the results for a relatively larger wave (wave form 7-C as specified in Dean's table.) Again, the horizontal distance and the free surface level with respect to the SWL are both normalized with respect to the wave length. The wave amplitude is three fourths the maximum amplitude for a wave of permanent form in the same depth of water and with the same wave length. The depth is equal to 0.20L and the amplitude is equal to 0.47d. The free surface of both waves was modeled using 60 nodes and Overhauser elements (with linear elements at the end). The sides and bottom were modeled using 25 and 60 linear elements, respectively. Double noding was used on all corners.

It is apparent, from Figs 5 and 6, that the BEM

0.030 0.025

1 0 0A A 0.015 l 1 11 0.010 / t

S W L . . . . . . . . . . ! ....

I-0.005 I!-~ ! Z-0.010 [ ~

-0 0150~ 0 0'5

i i 110 115 210 215 3~0 Normalized Horizon~ Distance

U 315 4.'o

Fig. 5. BEM solution of Dean's steady wave # 5-B. Wave profiles are given at every T/2 seconds, beginning at t = 0 and

ending at t = 3 T.

-O.Ot

0.06 • ~ 0.05 ,~ 0.04

0.03 0.02 ou 0.01

-0.01 -0.02 -0.03

315 410 015 lio t.5 210 215 310 Normalized Horizontal Distance


Fig. 6. BEM solution of Dean's steady wave # 7-C. Wave profiles are given at every T/2 seconds, beginning at t = 0 and

ending at t = 3T.

system accurately models a wave of permanent form. The figures show that the change in form or amplitude of each wave as it travels through three wave periods is negligible. To confirm this, three parameters were measured as the wave propagated through time: the relative amplitude change (Ea), the change in the mean water level with respect to the amplitude (ey), and the relative change in the area under the surface (CA). For both waves, after propagating for three wave periods, (i.e. t = 3T), E a was less than 2.2%, ey was less than 0"30%, and eA was less than 0"12%. These results indicate that the system is accurate and stable.

5.2 Breaking waves in shallow water

Waves break when the depth is too shallow for a given wave amplitude. The following example shows a breaking wave in relatively shallow water. The initial shape of the wave is a simple sine wave with the depth (d) equal to 0" 13L and the amplitude equal to half the depth. The initial potential values along the wave surface are obtained from linear wave theory and are given by

a T . t~[t= 0 = ~ sin (x/L)

where a is the amplitude. The wave period T is given by

27r T - - - eza-fi-fi-3

This wave is similar to that shown in the work of Vinje & Brevig.l 1

The wave is modeled using 60 nodal points on the surface, initially equally spaced in the x-direction. The sides and bottom are modeled using 25 and 60 equally spaced nodes, respectively. The initial time step is T/IO0. To compare the effectiveness of the Overhauser elements with the effectiveness of standard Lagrangian elements, different runs were made with the surface of the wave


I I 1 I I I I I -3 -2 -1 0 I 2 3 4

Horizontal Distance

Fig. 7. BEM solution for a shallow wave using linear elements. Wave profiles are given at every T/20 seconds, beginning at t = 0 and ending at t = 0-35T.

0

-1 I I I I I I I I

-3 -2 -1 0 1 2 3 4 Horizontal Distance

Fig. 8. BEM solution for a shallow wave using cubic elements. Wave profiles are given at every T/20 seconds, beginning at t = 0 and ending at t = 0.30T.

,.-a

. 1 I I I I I I I I

-3 -2 -1 0 1 2 3 4 Horizontal Distance

Fig. 9. BEM solution for a shallow wave using Overhauser elements. Wave profiles are given at every T/20 seconds, beginning at t = 0 and ending at t = 0.45T. For clarity, the nodes are shown only on the last profile.

modeled using linear, cubic, and Overhauser elements. The sides and bot tom were modeled using linear elements for all cases.

The results of the wave modeled using linear elements are shown in Fig. 7. The wave profiles are given at every T/20, starting at t = 0. The I,~st profile is at t = 0.35T. The calculations break dow . . . . hortly after t--: 0.38T, when instabilities developed. The instabilities observed were similar to the ' sawtooth ' instabilities first mentioned by Longuet-Higgins & Cokelet)

Figure 8 shows the results obtained using Lagrangian cubic elements. The wave profiles are also given at every T/20, starting at t = 0. Instabilities cause the calculations to break down at t = 0.327". The last profile shown in the figure is at t = 0"30T.

The results of the wave modeled using Overhauser elements are shown in Fig. 9. Here again, the wave profiles are given at every 20th of a wave period, starting at t = 0 and ending at t = 0.45T. The calculations break down when the tip of the wave actually makes contact with the wave trough, shortly after t - -0 .45T . No instabilities develop to cause a breakdown of calculations before this time.

Figures 7, 8 and 9 indicate that the Overhauser

elements are more effective at modeling the water wave surface than the Lagrangian linear and cubic elements. It was possible to obtain results using Lagrangian and cubic elements similar to the results obtained using Overhauser elements by 'smoothing' the results periodi- cally as the algorithm stepped through time. This is an indication that at least part of the cause of instabilities in numerical approaches to the water wave problem arise from an inadequate modeling of the wave surface geometry.

The results of this example compare very favorably with those of Vinje & Brevig. The point at which the front of the wave becomes vertical, often the definition of breaking, is at approximately t = 0-3T. This is the same breaking point found by Vinje & Brevig. Also, the point at which the wave jet comes into contact with the trough, approximately t = 0.45 T, is comparable to that found by Vinje & Brevig (t = 0.46T).

6 C O N C L U S I O N S

The BEM system created to simulate water waves has


been shown to provide an accurate model for steady waves and plunging breakers. The system has proven to be very robust, showing no signs of numerical instabilities as seen in previous work. No manipulation, (i.e. smoothing, regridding, or curve fitting) of the geometry and boundary conditions was needed as the wave propagates in time, showing that the Over- hauser element is ideal for the solution of the non-linear overturning water wave problem. An added advantage of the Overhauser element is that it may be easily incor- porated into existing codes since it is very similar to the Lagrangian elements now being used in other BEM systems.

ACKNOWLEDGEMENTS

This work was supported by Cray Research Inc. and a computer time allocation grant from the Alabama Supercomputer Network Authority.

REFERENCES

1. Longuet-Higgins, M.S. & Cokelet, E.D. The deformation of steep surface waves on water, I. A numerical method of computation, Proceedings of the Royal Society of London, 1976, 350A, 1-26.

2. Dommermuth, D.G. & Yue, D.K.P. Numerical simu- lations of nonlinear axisymmetric flows with a free surface. Journal of Fluid Mechanics, 1987, 178, 195-219.

3. Grilli, S.T., Skourup, J. & Svendsen, I.A. An efficient boundary element method for nonlinear water waves, Engineering Analysis with Boundary Elements, 1989, 6(2), 97-107.

4. Ortiz, J.C. An improved boundary element system for the solution of Poisson's equation, Master's thesis, Louisiana State University, USA, 1986.

5. Waiters, H.G. Techniques for boundary element analysis in elastostatics influenced by geometric modelling, Master's thesis, Louisiana State University, USA, 1986.

6. Ortiz, J.C., Waiters, H.G., Gipson, G.S. & Brewer, J.A. Development of Overhauser splines as boundary elements, Boundary Elements IX, Vol. 1, Proceedings of the 9th International Conference on Boundary Element Methods, ed. C.A. Brebbia, W.L. Wendland & G. Kuhn, Springer- Vedag, Berlin, 1987, pp. 401-407.

7. Hall, W.S. & Hibbs, T.S. A continuous boundary element for 3-D elastostatics, Advanced Boundary Element Methods, ed. T.A. Cruse, Springer-Verlag, New York, 1987, pp. 135-144.

8. Kinsman, B. Wind Waves: Their Generation and Propaga- tion on the Ocean Surface, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.

9. Harlow, F.H. & Welch, J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, The Physics of Fluids, 1965, 8(12).

10. Nichols, B.D., Hotchkiss, C.W. & Hurt, R.S. SOLA-VOF: A solution algorithm for transient fluid flow with multiple free boundaries, Technical Report LA-8355, Los Alamos Scientific Laboratory, 1980.

11. Vinje, T. & Brevig, P. Numerical simulation of breaking waves, Proceedings of the Third International Conference on Finite Elements in Water Resources, University of Mississippi, 1980, pp. 5.196-5.210.

12. Vinje, T. & Brevig, P. Numerical calculations of forces from breaking waves, International Symposium on Hydro- dynamics in Ocean Engineering, Norwegian Institute of Technology, 1981, pp. 547-565.

13. Douglass, S.L. The influence of wind on nearshore breaking waves, PhD thesis, Drexel University, Philadel- phia, PA, USA, 1989.

14. Gravert, P. Numerische simulation extremer schwerewel- len im zeitbereich mit direkter randelement-methode und zeitschrittverfahren, PhD thesis, Universit/it der Bundes- wehr Hamburg, Hamburg, Germany, 1987.

15. McIver, P. & Peregrine, D.H. Comparison of numerical and analytical results for waves that are starting to break, International Symposium on Hydrodynamics in Ocean Engineering, Norwegian Institute of Technology, 1981, pp. 203-215.

16. Dold, J.W. & Peregrine, D.H. Steep unsteady water waves: An efficient computational scheme, Proceedings of the 19th Coastal Engineering Conference, American Society, ed. B. Edge, of Civil Engineers, New York, 1984, pp. 955-967.

17. New, A.L., McIver, P. & Peregrine, D.H. Computation of overturning waves, Journal of Fluid Mechanics, 1985, 150, 233-251.

18. Seo, S.N. & Dalrymple, R.A. An efficient model for periodic overturning waves, Engineering Analysis with Boundary Elements, 1990, 7(4), 1966-204.

19. Dommermuth, D.G., Yue, D.K.P., Lin, W.M., Rapp, R.J., Chan, E.S. & Melville, W.K. Deep-water plunging breakers: A comparison between potential theory and experiments, Journal of Fluid Mechanics, 1988, 189, 423- 442.

20. Grilli, S. & Svendsen, I.A. The modelling of nonlinear water wave interaction with maritime structures, Advances in Boundary Elements, Vol. 2 Proceedings of the l lth International Conference on Boundary Element Methods, ed. C.A. Brebbia & J.J. Connor, Computational Mechan- ics Publications, Boston, 1989, pp. 253-268.

21. Grilli, S. & Svendsen, I.A. The modelling of highly nonlinear waves: Some improvements to the numerical wave tank, Advances in Boundary Elements, Vol. 2, Proceedings of the l lth International Conference on Boundary Element Methods, ed. C.A. Brebbia & J.J. Connor, Computational Mechanics Publications, Boston, 1989, pp. 269-281.

22. Grilli, S.T. & Svendsen, I.A. Corner problems and global accuracy in the boundary element solution of nonlinear wave flows, Engineering Analysis with Boundary Elements, 1990, 7(4), 178-195.

23. Gipson, G.S. Boundary element fundamentals - - basic concepts and recent developments in the Poisson equation, Topics in Engineering, Vol. 2, ed. C.A. Brebbia & J.J. Connor, Computational Mechanics Publications, Southampton, UK, 1987.

24. Waiters, H.G., Ortiz, J.C., Gipson, G.S. & Brewer, J.A. Overhauser splines as improved boundary element types, Advanced Boundary Element Methods, ed. T.A. Cruse, Springer-Verlag, New York, 1987, pp. 461-464.

25. Brewer, J.A. & Anderson, D.C. Visual interaction with Overhauser curves and surfaces, Computer Graphics II, 1977, 2, 132-137.

26. Chapra, S.C. & Canale, R.P. Numerical Methods for Engineers, McGraw Hill, U.S.A., 1988.

27. Cabral, J.J.S.P., Wrobel, L.C. & Brebbia, C.A. A BEM


formulation using B-splines: I-uniform blending equations, Engineering Analysis with Boundary Elements, 1990, 7(3), 136-144.

28. Dean, R.G. Evaluation and development of water wave

theories for engineering application - - SR 1, Technical Report, U.S. Army, Corps of Engineers Coastal Engineer- ing Research Center, Fort Belvoir, VA, USA, November 1974.

overhauser boundary elements solution for periodic water waves in the physical plane

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