A boundary integral equation method for water wave-structure interaction problems S. Liapis Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA ABSTRACT It is well known that at certain frequencies the boundary integral equation describing water wave-structure interactions does not have a unique solution. Large numerical errors occur in a frequency bandwidth close to each "irregular" frequency and the fact that these frequencies are not a priori known for a given body adds to the severity of the problem. The present work describes a new method to suppress the influence of the irregular frequencies. The boundary integral equation is supplemented with a set of moment equations-the null-field equations. The resulting overdetermined system is then solved by a least squares procedure. It is shown that this procedure will produce a unique solution. Numerical results for a sphere, a circular cylinder and a box barge demonstrate the effectiveness of the method. 1. INTRODUCTION Consider a rigid body floating on the free surface of deep water and undergoing small oscillations about a mean position. When calculating the hydrodynamic coefficients of the body the boundary integral equation formulation is most commonly used. This approach is based on Green's theorem with an appropriate Green function and has the major advantage of reducing the problem to solving an integral equation on the body boundary and therefore the number of unknowns is reduced by an order of magnitude. The integral equation is solved numerically using a panel method. Following Hess and Smith [1] who were the first to develop the method to a practical stage, the body surface is approximated by a number of plane quadrilateral elements of constant singularity strength and the integral equation is solved by collocation resulting in a linear system of equations for the singularity strengths.
556 Boundary Elements A major complication of the integral equation in the presence of a free surface is that it does not have a unique solution at certain characteristic frequencies also known as 'irregular' frequencies. These frequencies correspond to eigenvalues where the interior Dirichlet problem has a nontrivial solution. They may be pictured as eigenfrequencies for a 'fictitious' sloshing motion inside the body. This phenomenon was first pointed out by F.John [2] and verified numerically by Frank [3] for oscillating cylinders in two dimensions. It must be emphasized that the irregular frequencies are not caused by any physical phenomenon but are related to the solution procedure. Large numerical errors occur in a frequency bandwidth close to each irregular frequency and this phenomenon has a severe effect especially in the high frequency range where the irregular frequencies are closer to each other. A major complication is that the irregular frequencies cannot be determined analytically for a given body. In several cases the irregular frequency effects may be difficult to detect. An example is a catamaran where the strong interaction between the hulls can be mixed up with irregular frequency effects. The existence of the irregular frequencies represents the most serious drawback of the boundary integral method. There exist a few methods for removing the irregular frequencies some of them motivated from acoustics where a similar non-uniqueness occurs. Generaly speaking, one can either modify the integral operator or extend the computational domain. The method of modifying the integral operator was adopted by Ursell [4], Sayer[5], Ogilvie and Shin[6] and Wu and Price [7]. The Green function is modified by adding a source at the origin which absorbs all the energy of the interior eigenmodes. This approach introduces a very small additional cost but its applications have been limited to two-dimensional problems. A similar argument was used by Ursell [8] and Jones [9] to overcome the irregular frequency problem in acoustics. An alternative method was proposed by Burton and Miller [10] in acoustics and adopted by Kleinman [11] and Lee and Sclavounos [12] for water-wave problems. The boundary integral equation is combined with the integral equation obtained by normal differentiation with respect to the field point along the boundary. A linear combination of these two equations moves the irregular frequencies away from the real axis therefore producing a unique solution. The term containing the second derivatives of the Green function is best handled by using a hypersingular boundary integral formulation (see for example Liu and Rizzo [13]). There exist a few methods of extending the computational domain to suppress the irregular frequencies. It is possible to place a lid on the interior free surface thus suppressing the interior sloshing modes (e.g. Ohmatsu [14]). While this method has been successful for two-dimensional problems, its
Boundary Elements 557 application for three-dimensional problems has been limited by the additional computational effort required to discretize the interior free surface. A different method was proposed by Schenck [15] in acoustics, Rezayat et al [16] in elastodynamics and Lau and Hearn [17] in water wave problems. The solution to the boundary integral equation is supplemented by the requirement that the interior potential be zero at certain interior points. The choise of these interior points is arbitrary and special care must be taken so that they do not coincide with the nodes of the interior eigenmodes. The present work describes a new method for eliminating the detrimental effects of the irregular frequencies. It is based on supplementing the original boundary integral equation with a set of moment-like equations known as the null-field equations. Since both approaches are compatible this procedure produces accurate results for all frequencies. It is proved that this procedure guarantees a unique solution in a range depending on the number of extra equations. Furthermore for bodies that have two planes of symmetry only one extra equation is sufficient to guarantee uniqueness for all frequencies. Numerical results obtained for a sphere, a right circular cylinder and a box barge in heave demonstrate the effectiveness of the method. In all cases considered, the irregular frequency effects have been suppressed by using only one extra equation. 2. MATHEMATICAL FORMULATION a) The Boundary Value Problem Consider a rigid body performing forced oscillations on the free surface of deep water at a frequency <o. An xyz coordinate system is defined with z positive upwards and the xy plane coincident with the calm free surface. Under the usual assumptions of an inviscid, irrotational flow the problem may be defined by a velocity potential @(x,y,z,t). It is convenient to decompose * into the sum of the six radiation potentials as : - Re [( 4>.(x,yd S)e -'"' ] (1)
558 Boundary Elements where L is the complex amplitude of the body motion in each of the six degrees of freedom of the body. In the fluid domain each of the potentials must satisfy the Laplace equation: - 0 (2) The boundary condition on the free surface is in its linear form: -v<fry + (4>y), - 0 on z - 0 (3) On the body surface S each of the potentials must satisfy: i - -icon. - K(P) ; -1,2...6 (4).rJM J J where n% are the components of the generalized normal directed out of the fluid domain defined by the relations : - /f, (n^^g) - f x /T, with r - (x,yj) in addition to the above, appropriate radiation conditions at infinity are required to make the solution unique. b) Derivation of the Integral Equation The Green function for this problem which is the potential of a submerged source is given by Wehausen and Laitone [18,13.17] as Re[G(P,Q)e-^<] with
Boundary Elements 559 G(P,Q) - r + - + 2 v J,(kR)dk + 2*ive^VjJvK) r, ^o jt - v (5) is the field point - (n,0 & ** source point -S)^ + (y-r )\ v - is the wavenumber g JQ is the Bessel Junction of the first kind 4 denotes principal value integral Application of Green's theorem provides a Fredholm equation of the second kind for the values of the potential on the body surface S: [J */0)Y'0)<*(, - fj Vj(Q)G(P,Q)dSQ (6) From the values of the potential on the body surface, the added mass and damping coefficients may be obtained. Following the standard definition given in Newman [19] the added mass and damping coefficients are: A.j - Re [( -p [J 4>/Q)»,dS<P/iw] (7) B.. - Re [p // */0)n.(0)dy ij -U...6 The integral equation (6) is solved numerically using a panel method. The body surface is approximated by an ensemble offlatquadrilateral panels.the value of the potential is assumed constant over each panel reducing the problem of finding a continuous potential distribution to determining a finite number of unknown potential strengths. These potential strengths are determined by collocation where the integral equation (6) is satisfied at one point for each panel. The integral equation (6) possesses a unique solution unless the corresponding homogeneous integral equation has a non-trivial solution. It is known that the homogeneous integral equation does have nontrivial solutions whenever G> is an eigenvalue of the interior Dirichlet problem. These values of w called irregular frequencies, form a discrete infinite set and for a body of arbitrary shape are not known a priori. Large numerical errors occur in a frequency bandwidth close to each irregular frequency and this phenomenon has a severe effect especially in the high frequency range where the irregular frequencies are closer to each other.
560 Boundary Elements c) Null-Field Equations A different approach to solve the boundary value problem for <p was proposed by Martin [20] (also see Martin and Ursell [21]). A point in the interior of the body is selected and designated as the origin. Then the Green function G may be expanded as: m-0 n-0 o-l where the water wave multipoles $*mn(0) ^ * complete set of harmonic potentials satisfying the free-surface condition (3) and a*mn(p) &rc regular coefficients. The multipole potentials * mn(q) ^e defined in Martin [20]. This approach leads to an infinite set of moment-like equations called the null-field equations for water-waves: g$* (Q) /nnwv r,/^\^o /^\i jg _ g J - 1,2 (9) m,/i -0,1,2,.. It is possible to show that the null-field equations possess a unique solution at all frequencies (Martin [20]). However their numerical implementation is not always successful. They exhibit convergence problems and their use has been limited to bodies of simple geometrical shape such as two-dimensional circular and elliptical cylinders. 3. SUPPRESSION OF THE IRREGULAR FREQUENCIES The existence of the irregular frequencies represents the most serious drawback of the boundary integral method. A major complication is that these frequencies are not known a priori for a given problem. A review of existing methods for removing the irregular frequencies was given at the introduction. The effectiveness of each method may be judged using two criteria: a) The method must be robust and general enough to handle threedimensional problems involving bodies of arbitrary shape in all degrees of freedom b) The extra computational cost must be small when compared to that of the original boundary integral method.
Present Method Boundary Elements 561 In order to eliminate the detrimental effects of the irregular frequencies the system of M algebraic equations is supplemented by thefirstn null-field equations (9). The result is an overdetermined system of M + N equations for the unknown potential strengths which is solved by a least squares procedure. Let v%+i be the N+ 1^ irregular wavenumber. Then we shall prove: Theorem 1: The combination of equations (6) and (9) has a unique solution Proof: Denote by (p; the solution to the interior Dirichlet problem. We represent <p; by a distribution of dipoles: may - IJ Using the expansion (8) for G we may write : 3*'««2) -EEE«'^)fi^ m-0 n-0 a-l (ID On the body surface the boundary condition <pj = 0 gives: It is seen that equation (12) is identical to the homogeneous form of equation (6).Therefore if equation (12) for the dipole moment y has a nontrivial solution so will the homogeneous form of equation (6). Let (p^bea solution of equations (6) and (9). Then any other solution of equation (6) is of the form: 4>2-4>i + Y (13)
562 Boundary Elements Substitution into equation (9) gives: a@* - 0 m + n * N (14) Consistency of equations (11) and (13) is possible only if: B*^ " 0 (m + n z N) It follows that <pj and its first N-l partial derivatives vanish at the origin From the shape of the eigenfuctions it may be concluded that the solution is unique in the range Therefore given an estimate of the N+ 1^ irregular wavenumber (see for example Wu and Price [22] where the location of the irregular frequencies is found approximately using an equivalent box approximation ), the present method will give accurate results for all v^v^^. In practice the method can perform much better. If for example thefirstnull-field equation in heave is used, this will enforce (p- = 0 at the origin. Therefore, in addition to the first irregular frequency in heave all the irregular frequencies for which <p- * 0 at the origin are suppressed. A similar argument can be used for more additional equations where the partial derivatives of (p$ are forced to vanish at the origin. In fact it will be shown that if the body has two planes of symmetry, then using only thefirstnull-field equation suppresses the effects of all the irregular frequencies producing a unique solution for all frequencies. Theorem 2 If afloatingbody has two symmetry planes then using the first null-field equation will suppress the effects of all the irregular frequencies. Proof: Only the case of heave will be considered. The proof for the other degrees of freedom is similar. If L is the body length and B the breadth, then the interior potential may be expanded locally around the origin in a Fourier series: *i - EE #) cos(zh) cos() (15) ;-0 Jb-0 ^ &
Boundary Elements 563 since <pj is harmonic Wz) are given as: c e" + c e-" with Y* - #)' + (*)* As a result of the free-surface condition (3) we must have ^(0)^0 and in view of equation (15) we have <p.*0 at the origin for all interior eigenmodes. Therefore using the first null-field equation which enforces (p- = 0 at the origin will suppress the effects of all the irregular frequencies resulting to a unique solution at all frequencies. The present method is related to the combined boundary integral equation method (CBIEM) proposed in Lau and Hearn [17] where the solution of the integral equation is supplemented by the requirement that the interior potential be zero at certain interior points. The selection of these interior points is arbitrary and special care must be exercised so that they do not coincide with the nodes of the interior eigenmode because then the method fails. The effectiveness of the present method stems from the fact that it removes the arbitrariness in selecting the number and location of these interior points. Instead, given an estimate of the location of the irregular frequencies, it provides a definite rule for the number of extra equations that must be used to guarantee uniqueness. Especially in the high frequency range, where the number of nodes of the interior eigenmode increase, the present method is believed to have a clear advantage. Regarding the extra computational effort required, the computational penalty consists of evaluating NM extra influence coefficients and solving a system of N + M equations with M unknowns as opposed to an M by M system. Although a precise estimate of the extra CPU time has not been obtained, the extra CPU for the modified method should not exceed (M + N) /M as a percentage of the original method. In the special case when the body has two symmetry planes, which is quite common especially in the case of offshore platforms, using only the first equation will suppress the irregular frequency effects. As a conclusion, it is believed that the present method is effective in the spirit of the two criteria previously set, that is it is general enough to handle problems involving bodies of arbitrary shape and at the same time the extra computational cost is small. Furthermore it may be easily incorporated in existing boundary integral codes.
564 Boundary Elements 4. NUMERICAL RESULTS To illustrate the performance of the numerical method, the added mass and damping coefficients for a sphere,a vertical circular cylinder and a box barge have been computed. Because of the simple geometry there exist published results using simplified approaches. Figure 1 shows the nondimensional added mass and damping coefficients for a heaving sphere versus the nondimensional parameter vr where R is the radius of the sphere. 65 plane quadrilaterals were used to discretize one-quarter of the underwater surface. The asterisks are results obtained by Baracat [23] using the method of multipoles. For a sphere in heave the first irregular wavenumbers are v^r = 2.56 and v%r = 5.52. The dashed line results were obtained Lrom the numerical solution of equation (b). In the vicinity of the irregular frequencies substantial numerical errors occur. If however, the system of equations is supplemented by the first symmetric null-field equation, in the resulting solution which is the solid line in figure 1, the irregular frequency effects are suppressed. For this particular example only the first symmetric null-field equation which enforces <pj = 0 at the origin is sufficient to guarantee uniqueness given that all interior eigenmodes corresponding to the irregular frequencies have (pj * 0 at the origin. The additional computational time introduced is very small since only M additional coefficients must be evaluated and the linear system of equations is augmented by only one equation. Figure 2 shows the nondimensional added mass and damping coefficients for a vertical circular cylinder versus the nondimensional parameter vr where R is the radius of the cylinder. The radius to draft ratio is 2. 61 plane quadrilateral panels were used to discretize one-quarter of the underwater surface. The asterisks are results obtained by Breit et al [24] using ring sources and a one-dimensional integral equation. For a cylinder in heave the first irregular wavenumbers are v^r = 2.88 and v%r = 5.56. The dashed line results were obtained from the numerical solution of the system of equations (6). As in the case of a sphere,in the vicinity of the irregular frequencies substantial numerical errors occur. If however, the system of equations is supplemented by the first symmetric null-field equation, the resulting solution, which is the solid line on figure 1, is free of the irregular frequency effects. Figure 3 presents the hydrodynamic coefficients of a rectangular box barge oscillating in heave. The box has dimensions 10 by 10 by 2 and 60 panels were used to discretize one-quarter of its underwater surface. As in the previous cases, close to the irregular frequencies the numerical solution of equations (6) suffers from large numerical errors. By supplementing the integral equation by the first null-field equation the irregular frequency
effects are suppressed. 5. CONCLUSIONS Boundary Elements 565 A new method for suppressing the irregular frequency effects has been proposed. This method is very general and introduces a small computational cost compared to the original boundary integral method. It also eliminates the need of evaluating the term containing the second derivatives of the Green function or adding panels at the free surface as required by other methods. Numerical results for a sphere, a right circular cylinder and a box barge oscillating in heave demonstrate the effectiveness of the method. The proposed method can be easily implemented in existing boundary integral codes resulting in accurate numerical predictions for all frequencies. 6. REFERENCES 1. Hess, J.L. and Smith, A.M.'Calculation of nonliving potential flow about arbitrary three-dimensional bodies' J. Ship Res. 8, No 4, 22-44,1964. 2. John, F.'On the motion of floating bodies. II. Simple harmonic motions' Com. Pure and Appl. Math. 3, 45-101,1950. 3. Frank, W.'Oscillation of cylinders in or below the free surface of deep fluids' Rep. No 2375, David Taylor Research Center, Bethesda,1967. 4. Ursell, F.'lrregular frequencies and the motion of floating bodies' J. Fluid Mech. 105, 143-156, 1981. 5. Sayer, P. 'An integral-equation method for determining the Quid motion due to a cylinder heaving on water of finite depth' Proc. Roy. Soc. London A 372, 93-110, 1980. 6. Ogilvie, T.F. and Shin, Y.S.'Integral-equation solutions for time-dependent free-surface problems' J. Soc. Nav. Arch. Japan 143, 86-96, 1978. 7. Wu, X-J and Price, W.G. 'A multiple Green's function expression for the hydrodynamic analysis of multi-hull structures' Appl. Ocean Res. 9,58-66,1987. 8. Ursell, F. 'On the exterior problems of acoustics' Proc. Camb. Phil. Soc. 74,117-125, 1973. 9. Jones, D.S.'Integral equations for the exterior acoustic problem' Q.J. Mech. Appl. Math. 37, 119-133,1973. 10. Burton, A.J. and Miller, G.F.'Tlie application of integral equation methods to the numerical solution of some exterior boundary-value problems' Proc. R. Soc. Lond. A 323, 201-220, 1971. 11. Klefnman, R. E.'On the mathematical theory of the motion of floating bodies-an update' David Taylor Rep. 82/074,1982. 12. Lee, C.H. and Sclavounos, P.D.'Removing the irregular frequencies from integral equations in wave-body interactions'j.fiuici Mech. 207,393-418,1989. JJ. Liu, Y. and Rizzo, F.'A weakly singular form of the hypersinguiar
566 Boundary Elements boundary integral equation applied to 3-D acoustic wave problems' Comp. Meth. Appl. Mech. Engng. 96, 271-287,1992. 14. Ohmatsu, S.'On the irregular frequencies in the theory offloatingbodies' Papers Ship Res. Inst.,No 48, 1975. 15. Schenk, H.A. Improved integral formulation for acoustics radiation problems' J. Acoust. Soc. Amer. 44, 41-58, 1968. 16. Rezayat, M.; Shippy, D.J. and Rizzo, F.J. On time-harmonic elastic-wave analysis by the boundary element method for moderate to high frequencies' Comp. Meth. Appl. Mech. Engin. 55, 349-367, 1986. 17. Lau, S.M. and Hearn, G.E.'Suppression of irregular frequency effects in fluid-structure interaction using a combined boundary integral equation method' Int. J. Num. Fluids 9, 763-782,1989. 18. Wehausen, J.V. and Laitone, E.V. Surface Waves. Handbuch der Physik 9,446-778, Springer, 1960. 19. Newman, J.N. Marine Hydrodynamics. 3d Ed., MIT Press,Cambridge Mass., 1980. 20. Martin, P.A. 'On the null-field equations for water-wave radiation problems' J. Fluid Mech. 113, 315-332,1981. 21. Martin, P.A. and Ursell, F. 'On the null-field equations for water-wave radiation problems' Proc. 3d Intl. Conf. Num. Ship Hydrodyn., Paris, 1980. 22. Wu, X-J and Price, W.G.'An equivalent box approximation to predict irregular frequencies in arbitrarily shaped three-dimensional marine structures' Appl. Ocean Res. 8, 223-232, 1986. 23. Baracat,R. 'Vertical motion of a floating sphere in a sine-wave sea' J.Fluid Mech., 13,540-556, 1962. Also corrections in an unpublished report entitled 'Forced periodic heaving of a semi-immersed sphere'. 24. Breit, S.R.; Newman, J.N. and Sclavounos, P.D. 'A new generation of panel programs for radiation-diffraction problems' Conference on the Behaviour of Offshore Structures (BOSS) Delft,1985.