Kagemoto and Yue Interaction Theory

This is an interaction theory which provides the exact solution (i.e. it is not based on a Wide Spacing Approximation). The theory uses the Cylindrical Eigenfunction Expansion and Graf's Addition Theorem to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated Diffraction Transfer Matrix.

The basic idea is as follows: The scattered potential of each body is represented in the Cylindrical Eigenfunction Expansion associated with the local coordinates centred at the mean centre position of the body. Using Graf's Addition Theorem, the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the Cylindrical Eigenfunction Expansion associated with its local coordinates. Using the Diffraction Transfer Matrix, which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in Kagemoto and Yue 1986 and in Peter and Meylan 2004.

We extend the finite depth interaction theory of kagemoto86 to water of infinite depth and bodies of arbitrary geometry. The sum over the discrete roots of the dispersion equation in the finite depth theory becomes an integral in the infinite depth theory. This means that the infinite dimensional diffraction transfer matrix in the finite depth theory must be replaced by an integral operator. In the numerical solution of the equations, this integral operator is approximated by a sum and a linear system of equations is obtained. We also show how the calculations of the diffraction transfer matrix for bodies of arbitrary geometry developed by goo90 can be extended to infinite depth, and how the diffraction transfer matrix for rotated bodies can be easily calculated. This interaction theory is applied to the wave forcing of multiple ice floes and a method to solve the full diffraction problem in this case is presented. Convergence studies comparing the interaction method with the full diffraction calculations and the finite and infinite depth interaction methods are carried out.

Finite Depth Interaction Theory

We will compare the performance of the infinite depth interaction theory with the equivalent theory for finite depth. As we have stated previously, the finite depth theory was developed by kagemoto86 and extended to bodies of arbitrary geometry by goo90. We will briefly present this theory in our notation and the comparisons will be made in a later section.

In water of constant finite depth [math]\displaystyle{ d }[/math], the scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_out_d) \phi_j^\mathrm{S} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd} \sum_{\nu = - \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\ &\quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\nu = - \infty}^{\infty} A_{m \nu}^j K_\nu (k_m r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ A_{m \nu}^j }[/math]. The positive wavenumber [math]\displaystyle{ k }[/math] is related to [math]\displaystyle{ \alpha }[/math] by the dispersion relation

[math]\displaystyle{ (eq_k) \alpha = k \tanh k d, }[/math]

and the values of [math]\displaystyle{ k_m }[/math], [math]\displaystyle{ m\gt 0 }[/math], are given as positive real roots of the dispersion relation

[math]\displaystyle{ (eq_k_m) \alpha + k_m \tan k_m d = 0. }[/math]

The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be also be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_in_d) \phi_j^\mathrm{I} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd} \sum_{\mu = - \infty}^{\infty} D_{0 \mu}^j J_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}\\ & \quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\mu = - \infty}^{\infty} D_{m\mu}^j I_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ D_{m\mu}^j }[/math]. A system of equations for the coefficients of the scattered wavefields for the bodies are derived in an analogous way to the infinite depth case. The derivation is simpler because all the coefficients are discrete and the diffraction transfer operator can be represented by an infinite dimensional matrix. Truncating the infinite dimensional matrix as well as the coefficient vectors appropriately, the resulting system of equations is given by

[math]\displaystyle{ {\bf a}_l = {\bf B}_l \Big( {\bf d}_l^\mathrm{In} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \, {\bf a}_j \Big), \quad l=1, \ldots, N, }[/math]

where [math]\displaystyle{ {\bf a}_l }[/math] is the coefficient vector of the scattered wave, [math]\displaystyle{ {\bf d}_l^\mathrm{In} }[/math] is the coefficient vector of the ambient incident wave, [math]\displaystyle{ {\bf B}_l }[/math] is the diffraction transfer matrix of [math]\displaystyle{ \Delta_l }[/math] and [math]\displaystyle{ {\bf T}_{jl} }[/math] is the coordinate transformation matrix analogous to (T_elem_deep).

The calculation of the diffraction transfer matrices is also similar to the infinite depth case. The finite depth Green's function

[math]\displaystyle{ (green_d) &G(r,\theta,z;s,\varphi,c)\\ &= \frac{\i}{2} \, \frac{\alpha^2-k^2}{d(\alpha^2-k^2)-\alpha}\, \cosh k(z+d) \cosh k(c+d) \sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(k r) J_\nu(k s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}\\ & \quad + \frac{1}{\pi} \sum_{m=1}^{\infty} \frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}, }[/math]

given by black75 and fenton78, needs to be used instead of the infinite depth Green's function (green_inf). The elements of [math]\displaystyle{ {\bf B}_j }[/math] are therefore given by

[math]\displaystyle{ (B_elem_d) \lt center\gt \lt math\gt \begin{matrix} ({\bf B}_j)_{pq} &= \frac{\i}{2} \, \frac{(\alpha^2-k^2)\cosh^2 kd}{d(\alpha^2-k^2)-\alpha} \int\limits_{\Gamma_j} \cosh k(c+d) J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}\\ =and= ({\bf B}_j)_{pq} &= \frac{1}{\pi} \frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha} \int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} \end{matrix} }[/math]

</math>

for the propagating and the decaying modes respectively, where [math]\displaystyle{ \varsigma_q^j(\mathbf{\zeta}) }[/math] is the source strength distribution due to an incident potential of mode [math]\displaystyle{ q }[/math] of the form

[math]\displaystyle{ (test_modes_d) \lt center\gt \lt math\gt \begin{matrix} \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cosh k_m(c+d)}{\cosh kd} H_q^{(1)} (k s) \mathrm{e}^{\mathrm{i}q \varphi}\\ =for the propagating modes, and= \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cos k_m(c+d)}{\cos kd} K_q (k_m s) \mathrm{e}^{\mathrm{i}q \varphi} \end{matrix} }[/math]

</math>

for the decaying modes.

Numerical Results

In this section we will present some calculations using the interaction theory in finite and infinite depth and the full diffraction method in finite and infinite depth. These will be based on calculations for ice floes. We begin with some convergence tests which aim to compare the various methods. It needs to be noted that this comparison is only of numerical nature since the interactions methods as well as the full diffraction calculations are exact in an analytical sense. However, numerical calculations require truncations which affect the different methods in different ways. Especially the dependence on these truncations will be investigated.

Convergence Test

We will present some convergence tests that aim to compare the performance of the interaction theory with the full diffraction calculations and to compare the performance of the finite and infinite depth interaction methods in deep water. The comparisons will be conducted for the case of two square ice floes in three different arrangements. In the full diffraction calculation the ice floes are discretised in [math]\displaystyle{ 24 \times 24 = 576 }[/math] elements. For the full diffraction calculation the resulting linear system of equations to be solved is therefore 1152. As will be seen, once the diffraction transfer matrix has been calculated (and saved), the dimension of the linear system of equations to be solved in the interaction method is considerably smaller. It is given by twice the dimension of the diffraction transfer matrix. The most challenging situation for the interaction theory is when the bodies are close together. For this reason we choose the distance such that the escribed circles of the two ice floes just overlap. It must be recalled that the interaction theory is valid as long as the escribed cylinder of a body does not intersect with any other body.

Both ice floes have non-dimensionalised stiffness [math]\displaystyle{ \beta = 0.02 }[/math], mass [math]\displaystyle{ \gamma = 0.02 }[/math] and Poisson's ratio is chosen as [math]\displaystyle{ \nu=0.3333 }[/math]. The wavelength of the ambient incident wave is [math]\displaystyle{ \lambda = 2 }[/math]. Each ice floe has side length 2. The ambient wavefield is of unit amplitude and propagates in the [math]\displaystyle{ x }[/math]-direction. Three different arrangements are chosen to compare the results of the finite depth interaction method in deep water and the infinite depth interaction method with the corresponding full diffraction calculations. In the first arrangement the second ice floe is located behind the first, in the second arrangement it is located beside, and the third arrangement it is both beside and behind. The exact positions of the ice floes are given in table (tab:pos).

\begin{table} \begin{center} \begin{tabular}{@{}ccc@{}}

arrangement & [math]\displaystyle{ O_1 }[/math] & [math]\displaystyle{ O_2 }[/math]\

[math]\displaystyle{ 3pt] 1 & \lt math\gt (-1.4,0) }[/math] & [math]\displaystyle{ (1.4,0) }[/math]\\

2 & [math]\displaystyle{ (0,-1.4) }[/math] & [math]\displaystyle{ (0,1.4) }[/math]\\ 3 & [math]\displaystyle{ (-1.4,-0.6) }[/math] & [math]\displaystyle{ (1.4,0.6) }[/math] \end{tabular} \caption{Positions of the ice floes in the different arrangements.} (tab:pos) \end{center} \end{table}

Figure (fig:tsf) shows the solutions corresponding to the three arrangements in the case of water of infinite depth. To illustrate the effect on the water in the vicinity of the ice floes, the water displacement is also shown. It is interesting to note that the ice floe in front is barely influenced by the floe behind while the motion of the floe behind is quite different from its motion in the absence of the floe in front.

\begin{figure} \begin{center}

\includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi2}\[math]\displaystyle{ 0.4cm] \includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi0}\\lt center\gt \lt math\gt 0.4cm] \includegraphics[width=8.2cm]{int_n11_x12_y28_v5_b02_chi2} \end{center} \caption{Surface displacement of the ice floes and the water in their vicinity,\newline arrangements 1, 2 and 3.} (fig:tsf) \end{figure} To compare the results, a measure of the error from the full diffraction calculation is used. We calculate the full diffraction solution with a sufficient number of points so that we may use it to approximate the exact solution. \lt center\gt \lt math\gt E_2 = \left( \, \int\limits_{\Delta} \big| w_{i}(\mathbf{x})-w_{f}(\mathbf{x}) \big|^2 \mathrm{d}\mathbf{x} \, \right)^{1/2}, }[/math]

where [math]\displaystyle{ w_{i} }[/math] and [math]\displaystyle{ w_{f} }[/math] are the solutions of the interaction method and the corresponding full diffraction calculation respectively. It would also be possible to compare other errors, the maximum difference of the solutions for example, but the results are very similar.

It is worth noting that the finite depth interaction method only converges up to a certain depth if used with the eigenfunction expansion of the finite depth Green's function (green_d). This is because of the factor [math]\displaystyle{ \alpha^2-k^2 }[/math] in the term of propagating modes of the Green's function. The Green's function can be rewritten by making use of the dispersion relation (eq_k) \cite[as suggested by][p. 26, for example]{linton01} and the depth restriction of the finite depth interaction method for bodies of arbitrary geometry can be circumvented.

The truncation parameters for the interaction methods will now be considered for both finite and infinite depth. The number of propagating modes and angular decaying components are free parameters in both methods. In finite depth, the number of decaying roots of the dispersion relation needs to be chosen while in infinite depth the discretisation of a continuous variable must be selected. In the infinite depth case we are free to choose the number of points as well as the points themselves. In water of finite depth, the depth can also be considered a free parameter as long as it is chosen large enough to account for deep water.

Truncating the infinite sums in the eigenfunction expansion of the outgoing water velocity potential for infinite depth with truncation parameters [math]\displaystyle{ T_H }[/math] and [math]\displaystyle{ T_K }[/math] and discretising the integration by defining a set of nodes, </math>0\leq\eta_1 < \ldots < \eta_m < \ldots <

\eta_{_{T_R}}[math]\displaystyle{ , with weights }[/math]h_m[math]\displaystyle{ , the potential for infinite depth can be approximated by \lt center\gt \lt math\gt \phi (r,\theta,z) &= \mathrm{e}^{\alpha z} \sum_{\nu = - T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (\alpha r) \mathrm{e}^{\mathrm{i}\nu \theta}\\ &\quad + \sum_{m=1}^{T_R} h_m \, \psi (z,\eta_m) \sum_{\nu = - T_K}^{T_K} A_{\nu} (\eta_m) K_\nu (\eta_m r) \mathrm{e}^{\mathrm{i}\nu \theta}. }[/math]

In the following, the integration weights are chosen to be </math>h_m = 1/2\,(\eta_{m+1}-\eta_{m-1})[math]\displaystyle{ , }[/math]m=2, \ldots, T_R-1[math]\displaystyle{ and }[/math]h_1 = \eta_2-\eta_1[math]\displaystyle{ as well as }[/math]h_{_{T_R}} =

\eta_{_{T_R}}-\eta_{_{T_R-1}}[math]\displaystyle{ , which corresponds to the mid-point quadrature rule. Different quadrature rules such as Gaussian quadrature could be considered. Although in general this would lead to better results, the mid-point rule allows a clever choice of the discretisation points so that the convergence with Gaussian quadrature is no better. In finite depth, the analogous truncation leads to \lt center\gt \lt math\gt \phi (r,\theta,z) &= \frac{\cosh k (z+d)}{\cosh kd} \sum_{\nu = - T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (k r) \mathrm{e}^{\mathrm{i}\nu \theta}\\ & \quad + \sum_{m = 1}^{T_R} \frac{\cos k_m(z+d)}{\cos k_m d} \sum_{\nu = - T_K}^{T_K} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i}\nu \theta}. }[/math]

In both cases, the dimension of the diffraction transfer matrix, [math]\displaystyle{ \mathbf{B} }[/math], is given by [math]\displaystyle{ 2 \, T_H+1+T_R \, (2 \, T_K+1) }[/math].

Since the choice of the number of propagating modes and angular decaying components affects the finite and infinite depth methods in similar ways, the dependence on these parameters will not be further presented. Thorough convergence tests have shown that in the settings investigated here, it is sufficient to choose [math]\displaystyle{ T_H }[/math] to be 11 and [math]\displaystyle{ T_K }[/math] to be 5. Further increasing these parameter values does not result in smaller errors (as compared to the full diffraction calculation with 576 elements per floe). We will now compare the convergence of the infinite depth and the finite depth methods if [math]\displaystyle{ T_H }[/math] and [math]\displaystyle{ T_K }[/math] are fixed (with the previously mentioned values) and [math]\displaystyle{ T_R }[/math] is varied. To be able to compare the results, the discretisation of the continuous variable will always be the same for fixed [math]\displaystyle{ T_R }[/math] and these are shown in table (tab:discr). It should be noted that if only one node is used the integration weight is chosen to be 1.

\begin{table} \begin{center} \begin{tabular}{@{}cl@{}}

[math]\displaystyle{ T_R }[/math] & discretisation of [math]\displaystyle{ \eta }[/math]\

[math]\displaystyle{ 3pt] 1 & \{ 2.1 \}\\ 2 & \{ 1.2, 2.7 \}\\ 3 & \{ 0.8, 1.8, 3.0 \}\\ 4 & \{ 0.4, 1.4, 2.2, 3.2 \}\\ 5 & \{ 0.2, 1.0, 1.8, 3.0, 4.6 \} \end{tabular} \caption{The different discretisations used in the convergence tests.} (tab:discr) \end{center} \end{table} Figures (fig:behind), (fig:beside) and (fig:shifted) show the convergence for arrangement 1, arrangement 2 and arrangement 3, respectively, for the infinite depth method and the finite depth method with depth 2 (plot (a)) and depth 4 (plot (b)). Since the ice floes are located beside each other in arrangement 2 the average errors are the same for both floes. As can be seen from figures (fig:behind), (fig:beside) and (fig:shifted) the convergence of the infinite depth method is similar to that of the finite depth method. Used with depth 2, the convergence of the finite depth method is generally better than that of the infinite depth method while used with depth 4, the infinite depth method achieves the better results. Tests with other depths show that the performance of the finite depth method decreases with increasing water depth as expected. In general, since the wavelength is 2, a depth of \lt math\gt d=2 }[/math] should approximate infinite depth and hence there is no

advantage to using the infinite depth theory. However, as mentioned previously, for certain situations such as ice floes it is not necessarily true that [math]\displaystyle{ d=2 }[/math] will approximate infinite depth.

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{behind_d2}&& \includegraphics[height=0.38\columnwidth]{behind_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 1.} (fig:behind) \end{center} \end{figure}

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{beside_d2}&& \includegraphics[height=0.38\columnwidth]{beside_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 2.} (fig:beside) \end{center} \end{figure}

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{shifted_d2}&& \includegraphics[height=0.38\columnwidth]{shifted_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 3.} (fig:shifted) \end{center} \end{figure}

Multiple ice floe results

We will now present results for multiple ice floes of different geometries and in different arrangements on water of infinite depth. We choose the floe arrangements arbitrarily, since there are no known special ice floe arrangements, such as those that give rise to resonances in the infinite limit. In all plots, the wavelength [math]\displaystyle{ \lambda }[/math] has been chosen to be [math]\displaystyle{ 2 }[/math], the stiffness [math]\displaystyle{ \beta }[/math] and the mass [math]\displaystyle{ \gamma }[/math] of the ice floes to be 0.02 and Poisson's ratio [math]\displaystyle{ \nu }[/math] is [math]\displaystyle{ 0.3333 }[/math]. The ambient wavefield of amplitude 1 propagates in the positive direction of the [math]\displaystyle{ x }[/math]-axis, thus it travels from left to right in the plots.

Figure (fig:int_arb) shows the displacements of multiple interacting ice floes of different shapes and in different arrangements. Since square elements have been used to represent the floes, non-rectangular geometries are approximated. All ice floes have an area of 4 and the escribing circles do not intersect with any of the other ice floes. The plots show the displacement of the ice floes at time [math]\displaystyle{ t=0 }[/math].

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.03\columnwidth}p{0.46\columnwidth}} \includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_in} &&

\includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_out}\[math]\displaystyle{ 0.2cm] \includegraphics[width=0.45\columnwidth]{mult_n15_rh_rh_rh} && \includegraphics[width=0.45\columnwidth]{mult_n15_sq_sq_sq_sq_sq}\\ \end{tabular} \end{center} \caption{Surface displacement of interacting ice floes of different geometries.} (fig:int_arb) \end{figure} ==Summary== The finite depth interaction theory developed by [[kagemoto86]] has been extended to water of infinite depth. Furthermore, using the eigenfunction expansion of the infinite depth free surface Green's function we have been able to calculate the diffraction transfer matrices for bodies of arbitrary geometry. We also showed how the diffraction transfer matrices can be calculated efficiently for different orientations of the body. The convergence of the infinite depth interaction method is similar to that of the finite depth method. Generally, it can be said that the greater the water depth in the finite depth method the poorer its performance. Since bodies in the water can change the water depth which is required to allow the water to be approximated as infinitely deep (ice floes for example) it is recommendable to use the infinite depth method if the water depth may be considered infinite. Furthermore, the infinite depth method requires the infinite depth single diffraction solutions which are easier to compute than the finite depth solutions. It is also possible that the convergence of the infinite depth method may be further improved by a novel to optimisation of the discretisation of the continuous variable. }[/math]