S.T. Grilli, R. Subramanya Ocean Engineering Department ...

ABSTRACT
An existing model based on fully nonlinear potential flow equations is used to study wave propagation. The solution approach combines a higher-order Boundary Element Method (BEM) for solving Laplace's equation at a given time, and Lagrangian Taylor expansions for the time updating of the free surface position and potential^ <\ Shoaling and breaking of solitary waves are calculated in very shallow water, which requires solving the problem in computational domains with sharp geometry and large aspect ratios. Accuracy of the solution is improved by using a new interpolation technique (first introduced in**), and accurate quasi-singular integration techniques based on modified Telles and Lutz methods. Appli- cations are presented that demonstrate the accuracy and efficiency of the new approaches.
INTRODUCTION
Efficient two-dimensional models have been developed over the last decade for the solution of fully nonlinear potential flows with a free surface (see*' ™* *• *). Most of the existing approaches have been based on solving Laplace's equation at a given time, using a Boundary Integral Equation or a Boundary Element Method (BEM), and on updating both the free surface geometry and potential, using a time stepping method based on an Eulerian-Lagrangian description of the free surface.
Among the important phenomena affecting ocean waves in coastal areas, shoaling up to breaking over a sloping bottom is of prime importance. The shape and kinematics of waves close to breaking, directly control sediment transport by coastal currents, shaping of beaches, and design of coastal structures used for
Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
194 Boundary Elements
beach protection. In the present paper, the fully nonlinear wave model by Grilli, el al?" *' * is used to calculate shoaling and breaking of small incident solitary waves over very gentle slopes (1:35 and smaller). This model has successfully been used in earlier applications^- *\ for calculating solitary wave runup and breaking over steep slopes, and over gentle slopes (down to 1:35). It has also been used in a variety of situations with wave-structure interaction^*. Solitary waves have been selected in these applications, both for simplicity, and because they approximate well extreme design waves.
Wave breaking occurs over gentle slopes, roughly, when wave height be- comes equal to the local depth. Hence, small amplitude waves shoal-up and propagate all the way up the slope before they break. In the model, this requires solving the problem in computational domains with very sharp geometry and large aspect ratio (see Figs.2, and 6). For such domains, unless a (prohibitive) refined discretization is used, the narrowing of the geometry over the slope leads to quasi-singular integrals and loss of accuracy in the BEM.
To improve the accuracy of the BEM model in such situations, a "sliding" cubic interpolation technique (first introduced by Otta et al. ) has been used on the free surface, in replacement of the quasi-spline elements used in earlier applications^, and accurate and efficient integration techniques have been developed for quasi-singular integrals, based on modified Telles and Lutz methods, in replacement of the adaptive integration used earlier .
Equations and numerical procedures are presented in the next two sections, and applications of the model are then presented that demonstrate the accuracy and efficiency of the new approaches.
THE MATHEMATICAL AND NUMERICAL MODEL
Governing equations and boundary conditions
Equations for the two-dimensional potential model by Grilli et al^ * are briefly outlined in the following. The velocity potential < (z,Z) is used to describe inviscid irrotational 2D flows in the vertical plane (z,z), where the velocity is defined by, u = V<£ = (u,w) (Fig. 1). Continuity equation in the fluid domain
with boundary T(t) is a Laplace equation for the potential,
0 infi(t) (1)
Using the free space Green's function, G(x, zj) = — ~ log | x — x/ |, equation (1) transforms into the following Boundary Integral Equation (BIE),
= jf' [(x)G(x, x,) - fla)"] dT(x) (2) T(z)
where x = (x, z), and xi = (x;, zj) are position vectors for points on the boundary, n is the unit outward normal vector, and a(z/) is a geometric coefficient.
Boundary Elements 195
-1.5 x/h 0 10 20 30 40 50
Figure 1: Sketch of typical computational domain for solitary wave shoaling over a gentle 1:35 slope.
On the free surface F/(t), boundary conditions,
D<f>
(3)
(4)on Tf(t)
respectively, with r the position vector of a free surface fluid particle, g the acceleration due to gravity, z the vertical coordinate (positive upwards, and z = 0 at the undisturbed free surface), p* the atmospheric pressure, and p the fluid density. The material derivative is defined as, = ~ + u - V.
Waves are generated by simulating a piston wavemaker motion on the "open sea" boundary IVi(t) of the computational domain. Motion and normal velocity are specified over the paddle as,
X = X,, (5)
where the overlines denote specified values, and [xp, Up] are prescribed wavemaker motion and velocity, respectively (see for detail).
Along the stationary bottom I\, and other fixed boundaries I\-2, a no-flow condition is prescribed as,
"55" = 0 on and (6)
Numerical Model
Free surface boundary conditions (3) and (4) are integrated at time t, to establish both the new position and the boundary conditions on the free surface at sub- sequent time (t -f- At) (with At being a small time step). Second-order Taylor expansions are used in this process, to express both the new position r(t + At) and the potential <j)(r(t + At)) on the free surface, in an Eulerian-Lagrangian
formulation based on the material derivative. Taylor series coefficients can then be expressed as functions of the potential, its time derivative, and the normal and tangential derivatives of both of these, at the free surface (see for detail).
These quantities are obtained from the solution of two BIE's of the form (2), for (f> and |£, over F(t). Tangential derivatives are computed using 4th-order "sliding" polynomials along the free surface. At corners, special "compatibility conditions" are used, to improve the accuracy of the solution (see ).
The BIE's are solved by a BEM, using a set of collocation nodes on the boundary, and higher-order elements to interpolate between the collocation nodes. In the earlier version of the model by*\ quasi-spline elements were used on the free surface (cubic spline interpolation of the geometry, and linear shape functions), and isoparametric elements elsewhere (up to 4th-order). The reason for using spline elements is, in highly curved regions of the free surface like in the crest of a breaking wave, continuity of the tangent must be ensured between elements, or numerical errors and instability occur (seef). In the present paper, the cubic mid-interval interpolation method, first introduced by Otta, et a/.**, has been implemented and used on the free surface. In this method, the mid-section of a 4-node isoparametric element is used for the interpolation between each pair of nodes on the free surface. This 4-node element is constructed on two nodes delimiting each interval on the free surface, plus two more nodes : either one node on each side of the interval, or, for both extremities on the free surface, no node on the corner side, and two nodes on the other side. After each interpolation, the 4-node element is "slided" forward on the free surface, which ensures local continuity of the slope.
Each integral in (2) is transformed into a sum of integrals over each boundary element. In the model by\ non-singular integrals are calculated by standard Gauss quadrature rules, and a kernel transformation is applied to weakly singular integrals, which are then integrated by a numerical quadrature exact for the logarithmic singularity. An adaptive integration technique^ is used for improving the accuracy of regular integrations near corners and other locations, like the overturning jet in breakers, where elements on different parts of the boundary are close to each other. This adaptive integration, however, requires calculation of intersect angles, and distance from all collocation nodes to each element on the boundary. These calculations are computationally expensive and prevent the technique from being systematically used over the whole boundary. Hence, adaptive integration is only used for pre-specified elements.
In the present study, to be able to use quasi-singular integration methods with sufficient efficiency over the whole boundary, a conditional algorithm has been implemented based on the methods by Telles and Lutz . In the latter methods, quasi-singular integration is performed when the minimum relative distance, d' = y, from the collocation node to the considered element is small (in which / denotes the element length). In the method by Telles, a transformation of coordinates is performed on the reference element, thus clustering more Gauss points around the singularity. In the the method by Lutz, a more accurate
0.05 0.1 0.15 0.2 0.25 0.3 0
Figure 2: Relative error of a quasi-singular integral as a function of d' = d/l, the distance of the collocation node to the element. Symbols represent different integration methods (10 Gauss points per interval): regular Gauss (4-); modified Telles (MT) (•); Telles (D); Adaptive integration * (AI) with 2% segments (o); idem 2* (x); idem 2* (o); idem 2* (A).
transformation is performed for very small distance, and Lutz also suggests a way to account for highly curved elements. In the present study, Telles' method has been modified by estimating d' based on a curve, fitted to pre-calculated distances at each Gauss point along the element, instead of using Newton's iterations. This modification results in a speed-up of the integration (see below). The modified Telles method (MT) is used in the model for, 0.05 < d' < 1. The method by Lutz has been extended to curved elements, and is used in the model for d' < 0.05.
A comparison of various integration methods has been made by calculating the relative error, | e |, of the integral of a logarithmic Green's function over a (straight) 4-node element, as a function of d'. An adaptive integration method with 10 Gauss points per segment, and 2™ segments (see ^ for detail) is taken as the "exact" reference result for this integral, to which other integration methods are compared. Results in Fig. 2 show, integration methods referred to are increasingly more accurate. The AI method outperforms all the other methods, even with only a few segments, and, for d' > 0.25, both Telles and MT methods converge to the regular Gauss method with 10 points on one segment. The important factor, however, in selecting the optimum method for a similar accuracy, is the corresponding CPU time. For cases in Fig. 2, the CPU time is 1,2, 4, 5, 22, 102, and 474 times the CPU time corresponding to the regular Gauss method, respectively. Obviously, the MT method offers a good accuracy/speed ratio, as well as the AI method with 2* segments.
8 12 16 20 24 28
Figure 3: Computational domain for the propagation of a solitary wave over 5 time units for = 0.3 — 0.6. Results are given in Fig. 4 and 5.
Optimum accuracy of computations
Global accuracy of computations with the model is checked by computing the change in volume and total energy of the propagating wave at each time step. Errors are function of both the size and degree of the elements of the BEM discretization, and of the size of the time step.
Grilli & Svendsen^ used their model in various spatio-temporal discretizations, with quasi-spline elements on the free surface and quadratic elements elsewhere. Results of calculations with a constant time step, At*, showed optimum accuracy for a mesh Courant number, Co = x/ TT ~ 0.5 (in which Aio denotes the initial distance between nodes on the free surface). For highly transient waves (e.g., close to breaking), however, the distance between nodes significantly changes at every time step, and an adaptive time step procedure must be used. In*, the time step was calculated based on the optimum mesh Courant number, and on the minimum distance between nodes on the free surface for the given time. In the present case, however, the mid-interval interpolation is used on the free surface, and the optimum Co must be re-calculated. This is done in the first application in the next section.
APPLICATIONS
Global accuracy for solitary wave propagation over constant depth
The model is used, like in *, to propagate fully nonlinear solitary waves of height, ~ = 0.3 — 0.6, over constant depth A, using various spatio-temporal discretiza- "tions (Fig. 3). Waves are generated into the model using the "numerically exact" method by Tanaka ** (see *), and absolute errors with respect to Tanaka's results are calculated.
Computational data are identical to those for the application in *. The domain aspect ratio is £ = 28 (with h = 1). Three spatial discretizations are used in the computations, with initial free surface grid steps Ax^ = : 0.15, 0.20, 0.25. In each discretization, Az' = 0.25 on the lateral boundaries, and, on the bottom,
Boundary Elements
-logl Ael (b)
0.2 0.4 0.6 0.8 1.2 1.4
Figure 4: Absolute errors (A's) with respect to theoretical values by Tanaka, for wave volume m (a); energy e (b); and height H (c), as a function of Co, for a wave with initial height = 0.5. Cases are with Mil ( ), or QSE ( ) on the free surface, and with Az' = 0.25 (O), 0.20 (o), or 0.15 (o).
H/h 0.2 0.3 0.4 0.5
Figure 5: Absolute errors as in Fig. 4, as a function of initial wave height , for Ax'= 0.15, and Co = 0.35.
Ax' = 0.25 in the first set of results, and 0.40 in the next two ones. Mid-interval interpolation (Mil) is used on the free surface, and three-node quadratic elements are used on the other three boundaries. Ten Gauss points are used per element, and adaptive integration is used in the corner elements.
Propagation of a steep solitary wave with height, ~ = 0.5, is first tested in the BEM model. The initial wave is introduced in the model with its crest at x' = 14, and is propagated over 5 time units (Fig. 3). Maximum absolute changes (A's) in wave volume m, total energy e, and height H, are calculated. Tanaka's solution for this height gives : m — 1.7914787, and e = 0.6157121. Four different constant time steps, AtJ, = At«,«/J, are used with each discretization : 0.025, 0.05, 0.10, 0.20. This corresponds to propagating the solitary wave over 201, 101, 51 and 26 time steps respectively. Hence 12 different cases are calculated. Errors on wave volume, energy, and height are plotted in Fig. 4, as a function of Co = ^- and Ax|, (solid lines). Results obtained in using quasi-spline elements (Q E) on the free surface are also plotted in Fig. 4 for comparison (dashed lines). One sees, with the ME, errors are minimum for, Co c± 0.30-0.35, whereas the minimum is 0.5 for QSE. In overall, errors are lowered by at least two order of magnitude when using Mil instead of QSE.
The influence of wave height on model accuracy has also been tested. Waves with height, = 0.3-0.6, are propagated in the same domain as above (Fig. 3), with AxJ, = 0.15, Co = 0.35, and using the Mil method on the free surface. Results are plotted in Fig. 5, and show that the model is accurate over the whole range of tested wave heights.
Wave shoaling and breaking over a gentle slope
Grilli et al. and Otta et ai ^ used their earlier model to calculate shoaling of solitary waves over a slope, up to initiation of breaking. A similar case is presented in the following, for incident solitary waves of initial height, ~ =
-1
0 5 -
r r • r • • j....l:a5 _
35 36 37
36.5
Figure 6: Shoaling of a solitary wave with initial height H'^ = 0.20, over a 1:35 slope. PLots correspond to : (a) six profiles at time t' = 17.46, 37.21, 39.93,40.40,43.27, and 43.92 (left to right); (b) blow-up in full scale of last four profiles in (a), ( ) shoaling curve; (c) blow-up of last profile in (a), (o) BEM discretization nodes.
0.10,0.15,0.20, shoaling over a 1:35 slope (Fig. 1). New numerical techniques described in the previous section have been implemented in the present model, both for quasi-singular integrations, and for the interpolation on the free surface (Mil or QSE). Since computing time is not a problem, however, the (more accurate) adaptive integration method (AI) has been used in all cases when necessary, and only the Mil method has been tested and compared to the QSE method, as far as global and local accuracy are concerned.
The computational domain is as sketched in Fig. 1. After some trials, it was found that all tested waves break before they reach the top of the slope. Hence, to avoid unnecessary refinement of the discretization and of the integration methods in the upper part of the slope, where the domain geometry is very sharp, a small shelf has been added to the right of the domain, for depth h = 0.1 A*, i.e., x' > 41.5. The free surface discretization has 180 two-node intervals, with AzJ, = 0.25, and there are 100 quadratic elements on the bottom and lateral boundaries. The total number of nodes is 384. The distance between nodes on the bottom is 0.5 in the constant depth region, and is progressively reduced over the slope, to 0.40, 0.25, 0.20, 0.15 and 0.10, in order to get increased resolution where depth decreases. On the shelf bottom, the distance between nodes is 0.15. Adaptive integration with up to 2™ subdivisions (as function of the geometry) is used on the free surface and on the bottom, for the elements located between x' = 36 and 45. AI is also used in all corner elements. The mesh Courant number is selected as C<, = 0.50 and, hence, AtJ, = 0.125. With these data, the CPU time is 10.2sec (IBM9000) per time step.
Incident solitary waves are generated on the leftward lateral boundary of the domain, using the numerical piston wavemaker. Fig. 6 shows stages of wave shoaling and breaking calculated with the Mil on the free surface, for ~ = 0.20. In Fig. 6a, free surface profiles are shown at six different times, up to the instant of wave instability by spilling breaking (last profile). Fig. 6a and 6b show blow- ups of the region over the slope where breaking occurs. The time step reduces during propagation, down to At' = 0.020 at the time of breaking (f = 43.92). The total number of time steps is 680 and the average time step is 0.065. Fig. 6b shows, breaking occurs at z£ = 35.8, with a wave height H\> = 0.364, and a local ratio wave height over depth ("breaking index") = 1.38. This index is much larger than the usual design value for gentle slopes (~ 0.80), and agrees to within 5% with measurements by Grilli & Svendsen (1992, personal communication). Such an agreement can only be obtained when full nonlinearity is used in the equations.
Detailed results of calculations for the three wave heights, and for both QSE and Mil methods show, maximum relative errors on volume and total energy are less than 0.01% for x' < 28., i.e., more than half the way up the slope (Fig. 7). In the last stages of shoaling over the slope, however, discretization nodes gather in highly curved regions of the boundary where hydrodynamic jets are forming (e.g., crest of an impending breaking wave in Fig. 6c), and scatter at some other areas (e.g., wave troughs), leading to a less accurate description of
lA ml/m
x/h
x/h
Figure 7: Relative errors on wave volume m (a), and total energy e (b), for the calculations in Fig.l, 6, with solitary waves of initial height H' . Free surface is interpolated using QSE ( ), or Mil method ( ). Arrows in Fig. 7a indicate locations at which errors on total energy in Fig. 7b reach 1.5%.
1.6 'H = ' k.zo 'oil's' b
32 34 36 38 40
Figure 8: Shoaling curves for solitary waves over a 1:35 slope. Lines and symbols are as in Fig. 7.
the flow, and to larger errors. Fig. 6c shows, due to the rather long computational domain, only a few nodes end up approximating the breaking wave jet on the free surface. This, hence, limits the jet resolution or, more exactly, the period of time over which calculations are able to accurately follow the jet further than the overturning point. To improve on this, it would be necessary to either use regridding, or a finer initial discretization.
In the present application, computations have been interrupted when relative errors become greater than 1.5%. Fig. 7b shows, the energy error first reaches this threshold. One also sees, this occurs earlier with the (less accurate) QSE method. Arrows in Fig. 7a show locations of the points of maximum energy error. For the Mil method, errors on volume are about one order of magnitude smaller than corresponding errors on total energy (Fig. 7a versus 7b). The last wave profile shown all Figs. 6 corresponds to the time of maximum energy error, in the Mil case.
Finally, Fig. 8 gives a detailed comparison of the relative wave height , as a function of x', for the three tested wave heights. Comparison is made between results with the QSE method (dashed lines), and results with the Mil method (solid lines). Arrows again indicate locations at which the energy errors reach 1.5%. One sees, the shoaling curves are smoother with the Mil method than with the QSE method, and calculations can be carried out for a longer time within the .required accuracy. These results agree to within 5% with measurements by Grilli & Svendsen (1992, personal communication), up to the breaking point.
CONCLUSIONS
Improved interpolations and integration methods have been implemented in an existing fully nonlinear wave propagation BEM model. Results show improve-
ments in both the accuracy and the efficiency of the calculations, particularly in cases with very shallow water for which the domain geometry is very sharp and leads to quasi-singular integrals. Results for solitary wave shoaling over other slopes, smaller than 1:35, and also for periodic waves, will be presented at the conference.
ACKNOWLEDGMENTS
This publication is the result of research sponsored by the U.S. National Science Foundation, Engineering/ Earthquake, Hazards and Mitigation Program, under Grant BCS-9111827, and by the University of Rhode Island Sea Grant College Program, U.S. Department of Commerce, under Grant NA-89AA/D/SG/082. The U.S. Government is authorized to produce and distribute reprints for govern- mental purposes notwithstanding any copyright notation that may appear hereon.
References
[1] Dold, J.W. & Peregrine, D.H. 'An Efficient Boundary Integral Method for Steep Unsteady water Waves' In Numerical methods for Fluid Dynamics II (ed. K.W. Morton &M.J. Baines), pp. 671-679, Clarendon Press, Oxford, 1986.
[2] Grilli, S. Modeling of Nonlinear Wave Motion in Shallow Water. Chapter 3 in Com- putational Methods for Free and Moving Boundary Problems in Heat and Fluid Flow (eds. L.C. Wrobel & C.A. Brebbia), pps. 37-65, Computational Mechanics Publication, Elsevier Applied Sciences, London, UK, 1993.
[3] Grilli, S. Losada, M.A. & Martin, F. The Breaking of a Solitary Wave over a Step : Modeling and experiments. In Proc. 4th Intl. Conf. on Hydraulic Engi- neering Software (HYDROSOFT92, Valencia, Spain, July 92) (eds. W.R. Blain and E. Cabrera), Fluid Flow Modelling, pp. 575-586. Computational Mechanics Publications, Elsevier Applied Science (invited paper), 1992.
[4] Grilli, S. Losada, M.A. & Martin, F. Wave Impact Forces on Mixed Breakwaters. In Proc. 23rd Intl. Conf. on Coastal Engineering (ICCE23, Venice, Italy, October 92) 14 pps., ASCE edition (in press), 1993.
[5] Grilli, S., Skourup, J. & Svendsen, LA. An Efficient Boundary Element Method for Nonlinear Water Waves. Engineering Analysis with Boundary Elements 6 (2) 97-107, 1989.
[6] Grilli, S. & Svendsen, LA. Corner Problems and Global Accuracy in the Boundary Element Solution of Nonlinear Wave Flows. Engineering Analysis with Boundary Elements, 7 (4), 178-195, 1990.
[7] Grilli, S. & Svendsen, LA. Long Wave Interaction with Steeply Sloping Structures. In Proc. 22nd Intl. Conf. on Coastal Engineering (ICCE22, Delft, The Netherland, July 90) Vol. 2, pps. 1200-1213. ASCE edition, 1991.
[8] Longuet-Higgins, M.S. & Cokelet, E.D. The Deformation of Steep Surface Waves on Water - I. A Numerical Method of Computation. Proc. R. Soc. Lond. A350, 1-26, 1976.
[9] Lutz, E. Exact Gaussian Quadrature Methods for Near-Singular Integrals in the Boundary Element Method. Engineering Analysis with Boundary Elements 9, 233-245, 1992.
[10] New, A.L., Mclver, P. & Peregrine, D.H. Computation of Overturning Waves. J. FluidMech. 150, 233-251, 1985.
[11] Otta, A.K., Svendsen, LA. & Grilli, S.T. Unsteady Free Surface Waves in Region of Arbitrary Shape. CACR, University of Delaware, Research Report 92-10, 153pp, 1992.
[12] Otta, A.K., Svendsen, LA. & Grilli, S.T. The Breaking and Runup of Solitary Waves on Beaches. In Proc. 23rd Intl. Conf. on Coastal Engineering (ICCE23, Venice, Italy, October 92) 14 pps., ASCE edition (in press), 1993.
[13] Svendsen, LA. & Grilli, S. Nonlinear Waves on Steep Slopes, J. Coastal Research SI 7 185-202,1990.
[14] Tanaka, M. The Stability of Solitary Waves, Phys. Fluids 29 (3), 650-655, 1986.
[15] Telles, J.C.F. A Self-Adaptive Coordinate Transformation for Efficient Numer- ical Evaluation of General Boundary Element Integrals. Intl. J. Num. Meth. in Engineering 24, 959-973, 1987.
[16] Vmje, T. & Brevig, P. Numerical Simulation of Breaking Waves. Adv. Water Res. 4, 77-82, 1981.

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