To find a spatial discretization,
\(C^0\)-Galerkin finite-element expansions of the variables are, given an appropriate mesh tessellation of the fixed fluid and beam domains, substituted directly into the VP. The basis functions are
\(\tilde{\varphi }_i(x,y,z)\) in the fluid domain with the limiting basis function
\(\tilde{\varphi }_\alpha (x,y) = \tilde{\varphi }_\alpha (x,y,z=H_0)\) at the free surface
\(z=H_0\), and
\(\tilde{X}_k(x,y,z)\) in the structural domain. Both the fixed fluid and beam domains have coordinates
\(\mathbf {x} = (x,y,z)=(x_1,x_2,x_3)\). At the common interface
\(x=L_s\) (see Fig.
3), we assume that the respective meshes join up with common nodes. However, since there are two meshes, these nodes are denoted by indices
m and
n on the fluid mesh and by
\(\widetilde{m}\) and
\(\tilde{n}\) on the solid mesh. There is a mapping between these two node sets, namely
\(m=m(\widetilde{m})\). Here,
i and
j denote nodes in the fluid domain,
\(\alpha \) and
\(\beta \) nodes at its surface,
m and
n or
\(\widetilde{m}\) and
\(\tilde{n}\) nodes at the common fluid–structure boundary, and
k and
l nodes in the structure domain. Primed indices refer to the nodes below the water surface, and
\(\alpha _n\) denotes the surface nodes at the common boundary. Indices
\(a,b=1,2,3\) are the coordinate indices used for
\(\mathbf X \) and
\(\mathbf x \). The Einstein summation convention is assumed for all indices. Finally, with the subscript
h denoting the numerical approximations, the expansions are
$$\begin{aligned} \begin{aligned}&\phi _h(\mathbf {x},t) = \phi _i(t) \tilde{\varphi }_i(\mathbf {x}), \qquad \phi _{fh}(x,y,t) = \phi _\alpha (t) \tilde{\varphi }_\alpha (x,y), \qquad \eta _h(x,y,t) = \eta _\alpha (t) \tilde{\varphi }_\alpha (x,y), \\&X_h^a(\mathbf {x},t) = X_k^a(t) \tilde{X}_k(\mathbf {x}), \qquad U_h^a(\mathbf {x},t) = U_k^a(t) \tilde{X}_k(\mathbf {x}). \end{aligned} \end{aligned}$$
(32)
Substitution of (
32) into (
31) yields the spatially discrete Lagrangian function
$$\begin{aligned} {L} = \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta + \dot{X}^a_k N_{kl} U^a_l + \dot{X}^1_{\widetilde{m}} W_{\widetilde{m} n} \phi _n - H(\eta , \phi , X, U) , \end{aligned}$$
(33)
with Hamiltonian
$$\begin{aligned} H(\eta , \phi , X, U) = \frac{1}{2} \eta _\alpha M_{\alpha \beta } \eta _\beta + \frac{1}{2} \phi _i A_{ij} \phi _j + \frac{1}{2} U^a_k N_{kl} U^a_l- \frac{1}{2} X^a_k E^{ab}_{kl} X^b_l , \end{aligned}$$
(34)
wherein a superscript dot indicates a time derivative, and in which the matrices are given by
$$\begin{aligned} \begin{aligned}&M_{\alpha \beta } = \int _x \int _y \tilde{\varphi }_\alpha \tilde{\varphi }_\beta \,{\mathrm d}y\,{\mathrm d}x,\quad A_{ij} = \int _\varOmega \nabla \tilde{\varphi }_i \cdot \nabla \tilde{\varphi }_j\,{\mathrm d}V,\\&W_{\widetilde{m} n} = \int _y \int _{0}^{H_0} \tilde{X}_{\widetilde{m}} \tilde{\varphi }_n \,{\mathrm d}z{\mathrm d}y,\quad N_{kl} = \rho _0 \int _{\varOmega _0} \tilde{X}_k \tilde{X}_l\,{\mathrm d}V,\\&B_{kl}^{ab} = \int _{\varOmega _0} \frac{\partial \tilde{X}_k }{ \partial x_a } \frac{\partial \tilde{X}_l }{ \partial x_b }\,{\mathrm d}V, \quad E^{ab}_{kl} = \lambda B^{ab}_{kl} + \mu \left( B^{cc}_{kl} \delta _{ab} + B^{ba}_{kl} \right) . \end{aligned} \end{aligned}$$
(35)
Given that in both fluid and beam domains the basis functions come from the same function space, we can identify
\(\tilde{X}_{\widetilde{m}} \equiv \tilde{\phi }_{ m (\widetilde{m})}\). In other words, if the numbering is taken into account, at the fluid–beam inferface, basis functions are the same in both the fluid and the beam. The matrices in (
35) are symmetric; in particular, we highlight that
$$\begin{aligned} \begin{aligned} B^{ab}_{kl} = B^{ba}_{lk} \qquad \text {and}\qquad E^{ab}_{kl} = E^{ba}_{lk} . \end{aligned} \end{aligned}$$
(36)
Unlike in the continuous case,
cf. remarks after (
4), the Dirichlet boundary condition can be incorporated directly into the Lagrangian, i.e., by imposing
\(X_{k_b}^a=0\) and
\(U_{k_b}^a=0\), with
\((\cdot )_{k_b}\) denoting the structure–base nodes. Then (
33) becomes
$$\begin{aligned} \begin{aligned}&{L} = \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta + \dot{X}^a_{k'} N_{k'l'} U^a_{l'} + \dot{X}^1_{\widetilde{m}'} W_{\widetilde{m}' n} \phi _n - H(\eta , \phi , X, U) ,\\&H(\eta , \phi , X, U) = \frac{1}{2} \eta _\alpha M_{\alpha \beta } \eta _\beta + \frac{1}{2} \phi _i A_{ij} \phi _j + \frac{1}{2} U^a_{k'} N_{k'l'} U^a_{l'}- \frac{1}{2} X^a_{k'} E^{ab}_{k'l'} X^b_{l'} , \end{aligned} \end{aligned}$$
(37)
with primed structural indices denoting nodes excluding those at the beam bottom. The next step is to compute the momentum conjugate to
\(X^a_{k'}\) ,
$$\begin{aligned} R^a_{k'} = \frac{\partial {L}}{\partial {\dot{X}^a_{k'}}} = N_{k'l'} U^a_{l'} + \delta _{a1} \delta _{k'\widetilde{m}'} W_{\widetilde{m}' n} \phi _n \, , \end{aligned}$$
(38)
in which
\(\delta \) is the Kronecker delta symbol. Rearrangement of (
38) yields
$$\begin{aligned} U^a_{k'} = N_{k'l'}^{-1} R^a_{l'} - \delta _{a1} N_{k'l'}^{-1} \delta _{l'\widetilde{m}'} W_{\widetilde{m}' n} \phi _n , \end{aligned}$$
(39)
in which it is to be noted that
\(N_{k'l'}^{-1}\) is the inverse not of the full matrix
\(N_{kl}\), but of the system excluding the nodes at the beam bottom. Therefore, after using
\(R^a_{k'}\) instead of
\(U^a_{k'}\), the Lagrangian takes the form
$$\begin{aligned} {L} = \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta + \dot{X}^a_{k'} R^a_{k'} -H(\phi _\alpha ,\eta _{\alpha },X^a_{k'},R^a_{k'}), \end{aligned}$$
(40)
in which the Hamiltonian (computed using the Lagrangian
L in (
33) and (
39)) is given by
$$\begin{aligned} \begin{aligned} H(\phi _\alpha ,\eta _{\alpha },X^a_{k'},R^a_{k'})&= \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta + \dot{X}^a_{k'} R^a_{k'} - L =\frac{1}{2} \eta _\alpha M_{\alpha \beta } \eta _\beta +\frac{1}{2} \phi _i A_{ij} \phi _j + \frac{1}{2} \phi _{m} \widetilde{M}_{mn} \phi _{n}\\&\quad - R^1_{k'} N^{-1}_{k' l'} \delta _{l'\widetilde{m}'} W_{\widetilde{m}' n} \phi _n +\frac{1}{2} R^a_{k'} N^{-1}_{k'l'} R^a_{l'} +\frac{1}{2} X^a_{k'} E^{ab}_{k'l'} X^b_{l'} , \end{aligned} \end{aligned}$$
(41)
in which
$$\begin{aligned} \widetilde{M}_{mn} = (N^{-1})_{\widetilde{m}'\tilde{n}'} W_{\widetilde{m}'m} W_{\tilde{n}'n} . \end{aligned}$$
(42)
To facilitate the computations, we introduce the vector
P defined by
$$\begin{aligned} R^a_{k'} = N_{k'l'} P^a_{l'} , \end{aligned}$$
(43)
which obviates the need to compute the inverse of the full matrix
N, instead requiring only the part in the definition of
\(\widetilde{M}_{mn}\). The inverse
\((N^{-1})_{\widetilde{m}' \tilde{n}'}\) in (
42) is the submatrix of the inverse of
\(N_{k'l'}\) including interface but excluding beam-bottom nodes. Therefore, the substitution of (
43) into (
40) using (
41) yields
$$\begin{aligned} L = \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta + \dot{X}^a_{k'} N_{k'l'} P^a_{l'} - H(\phi _\alpha ,\eta _{\alpha },X^a_{k'},P^a_{k'}) , \end{aligned}$$
(44)
with the Hamiltonian
$$\begin{aligned} \begin{aligned} H(\phi _\alpha ,\eta _{\alpha },X^a_{k'},P^a_{k'})&= \frac{1}{2} \eta _\alpha M_{\alpha \beta } \eta _\beta + \frac{1}{2} \phi _i A_{ij} \phi _j + \frac{1}{2} \phi _m \widetilde{M}_{{m}n} \phi _n\\&\quad - P^1_{\widetilde{m}'} W_{\widetilde{m}' n} \phi _n + \frac{1}{2} P^a_{k'} N_{k'l'} P^a_{l'} + \frac{1}{2} X^a_{k'} E^{ab}_{k'l'} X^b_{l'} . \end{aligned} \end{aligned}$$
(45)
The fact that not all terms in (
45) are positive definite will be discussed in more detail later. Note that the Hamiltonian depends explicitly on only the surface degrees of freedom
\(\phi _\alpha \). Therefore, we are able to eliminate the interior degrees of freedom
\(\phi _{i'}\), with the primed index
\(i'\) denoting the nodes in the interior of the fluid excluding those on the free surface, in order to reduce the system to the general Hamiltonian form. Therefore, we derive the equations of motion by applying the VP to the Lagrangian (
44); after rearranging and using arbitrariness of respective variations as well as suitable end-point conditions, we obtain
$$\begin{aligned} 0 =&\int _0^{t_1} L\,{\mathrm d}t\nonumber \\ =&\int _0^{t_1} \bigg \lbrace \dot{\eta }_\alpha M_{\alpha \beta } \delta \phi _\beta - M_{\alpha \beta } \dot{\phi }_\beta \delta \eta _\alpha - \eta _\alpha M_{\alpha \beta } \delta \eta _\beta - \phi _i A_{ij} \delta \phi _j -\phi _{m} \widetilde{M}_{mn} \delta \phi _{n}\nonumber \\&\qquad \quad + \left( W_{\widetilde{m}' n} \, \phi _{n} \, \delta P^1_{\widetilde{m}'} + P^1_{\widetilde{m}'} \, W_{\widetilde{m}' n} \, \delta \phi _n \right) \nonumber \\&\qquad \quad + \left( \dot{X}^a_{k'} \, N_{k'l'} \, \delta P^a_{l'} - N_{k'l'} \, \dot{P}^a_{l'} \, \delta X^a_{k'} - P^a_{k'} \, N_{k'l'} \, \delta P^a_{l'} \right) - X^a_{k'} E^{ab}_{k'l'} \delta X^b_{l'} \bigg \rbrace {\mathrm d}t . \end{aligned}$$
(46)
Hence, by renaming certain indices, the following equations are obtained
$$\begin{aligned}&\delta \eta _\beta {:} \quad \dot{\phi }_\alpha = -\eta _\alpha , \end{aligned}$$
(47a)
$$\begin{aligned}&\delta \phi _\alpha {:} \quad M_{\alpha \beta } \dot{\eta }_\beta = \phi _i A_{i \alpha } \underline{\,+\,(\phi _m\widetilde{M}_{m n} - P^1_{\widetilde{m}'} \,W_{\widetilde{m}' n} )\delta _{\alpha n} }, \end{aligned}$$
(47b)
$$\begin{aligned}&\delta \phi _{j'}{:} \quad \phi _i A_{i j'} = \underline{(-\phi _m\widetilde{M}_{mn} + P^1_{\widetilde{m}'} W_{\widetilde{m}' n} ) \delta _{n j'}} , \end{aligned}$$
(47c)
$$\begin{aligned}&\delta P^a_{k'} : \quad N_{k'l'} \dot{X}^a_{l'} = N_{k'l'} P^a_{l'} \underline{- \delta _{a1} \delta _{k' \widetilde{m}'} W_{\widetilde{m}' n} \phi _n }, \end{aligned}$$
(47d)
$$\begin{aligned}&\delta X^a_{k'} : \quad N_{k'l'} \dot{P}^a_{l'} = -E^{ab}_{k'l'} X^b_{l'}, \end{aligned}$$
(47e)
in which the new coupling terms introduced by the present formulation are underlined. If we define the matrix
$$\begin{aligned} C_{i'j'} = A_{i'j'} +{\delta _{i'm}\widetilde{M}_{mn}\delta _{n j'}}, \end{aligned}$$
(48)
(
47c) can be split into internal and surface degrees of freedom and inverted to express internal ones in terms of surface ones and
P at the interface
$$\begin{aligned} \phi _{i'} = C_{i'j'}^{-1} \left( -\phi _{\alpha } A_{\alpha j'} {+ P^1_{\widetilde{m}'} W_{\widetilde{m}' n} \delta _{nj'} -\phi _{\alpha }\delta _{\alpha m} \widetilde{M}_{m n}\delta _{n j'}} \right) . \end{aligned}$$
(49)
The interior degrees of freedom are removed from the Lagrangian by substituting (
49) into (
40) to obtain
$$\begin{aligned} \begin{aligned} L = \dot{\eta }_\alpha M_{\alpha \beta } \phi _\beta - \frac{1}{2} \eta _\alpha M_{\alpha \beta } \eta _\beta - \frac{1}{2} \phi _\alpha D_{\alpha \beta } \phi _\beta + P^a_{k'} G^a_{k' \alpha } \phi _\alpha + P^a_{k'} N_{k'l'} \dot{X}^a_{l'} - \frac{1}{2} P^a_{k'} F^{ab}_{k'l'} P^b_{l'} - \frac{1}{2} X^a_{k'} E^{ab}_{k'l'} X^b_{l'} , \end{aligned} \end{aligned}$$
(50)
where Schur decomposition matrices
B,
F and
G have been introduced; their explicit forms are omitted. The structure of (
50) is as follows: the first line describes the fluid, the second the coupling, and the third the beam. In a more visual matrix notation, (
50) has the structure
$$\begin{aligned} \begin{aligned} L = (\dot{\eta } , \dot{\mathbf {X}}) \begin{pmatrix} M \, \phi \\ N \, \mathbf {P} \end{pmatrix} - \frac{1}{2} (\eta , \mathbf {X}) \begin{pmatrix} M &{} 0 \\ 0 &{} E \end{pmatrix} \begin{pmatrix} \eta \\ \mathbf {X} \end{pmatrix} - \frac{1}{2} (\phi , \mathbf {P}) \begin{pmatrix} D &{} -G^T\\ -G &{} F \end{pmatrix} \begin{pmatrix} \phi \\ \mathbf {P} \end{pmatrix} . \end{aligned} \end{aligned}$$
(51)
The classical Hamilton’s equations of an abstract system emerge when we introduce a generalized coordinate vector and its conjugate vector, i.e.
$$\begin{aligned} \begin{aligned} \mathsf{Q}&= \big ( \eta _1, \dots , \eta _{N_f}, X^1_1, \dots , X^1_{N_b}, X^2_1, \dots , X^2_{N_b}, X^3_1, \dots , X^3_{N_b} \big ),\\ \mathsf{P}&= \big (M_{1 \alpha } \phi _\alpha , \dots , M_{N_f \alpha } \phi _\alpha , N_{1k'} P^1_{k'}, \dots , N_{N_b k'} P^1_{k'}, N_{1k'} P^2_{k'}, \dots , N_{N_b k'} P^2_{k'}, N_{1k'} P^3_{k'}, \dots , N_{N_b k'} P^3_{k'} \big ), \end{aligned} \end{aligned}$$
(52)
with
\(N_f\) degrees of freedom at the free surface and
\(N_b\) degrees of freedom in the beam (recall, fixed-bottom nodes are excluded), using which the Lagrangian can be written in the form:
$$\begin{aligned} \mathsf{L} = \frac{{\mathrm d}\mathsf{Q}}{{\mathrm d}t} \cdot \mathsf{P} - \mathsf H(\mathsf{Q},\mathsf{P}) \end{aligned}$$
(53)
with Hamiltonian
\(\mathsf H(\mathsf{P},\mathsf{Q})\). After introducing the following (symmetric) matrices
$$\begin{aligned} \begin{aligned} \mathsf{M}_\mathsf{Q}&= \begin{pmatrix} M &{} 0 \\ 0 &{} E \end{pmatrix},\\ \mathsf{M}_\mathsf{P}&= \begin{pmatrix} M^{-1}D M^{-1} &{} -M^{-1}G^T N^{-1}\\ - N^{-1} G M^{-1} &{} N^{-1} F N^{-1} \end{pmatrix}, \end{aligned} \end{aligned}$$
(54)
we can write the Hamiltonian in (
53) as
$$\begin{aligned} \mathsf H(\mathsf{Q},\mathsf{P}) = \frac{1}{2}\mathsf{Q}^T M_\mathsf{Q} \mathsf{Q} + \frac{1}{2} \mathsf{P}^T M_\mathsf{P}\mathsf{P} . \end{aligned}$$
(55)