Dynamic model of open shell structures buried in poroelastic soils

Abstract

This paper is concerned with a three-dimensional time harmonic model of open shell structures buried in poroelastic soils. It combines the dual boundary element method (DBEM) for treating the soil and shell finite elements for modelling the structure, leading to a simple and efficient representation of buried open shell structures. A new fully regularised hypersingular boundary integral equation (HBIE) has been developed to this aim, which is then used to build the pair of dual BIEs necessary to formulate the DBEM for Biot poroelasticity. The new regularised HBIE is validated against a problem with analytical solution. The model is used in a wave diffraction problem in order to show its effectiveness. It offers excellent agreement for length to thickness ratios greater than 10, and relatively coarse meshes. The model is also applied to the calculation of impedances of bucket foundations. It is found that all impedances except the torsional one depend considerably on hydraulic conductivity within the typical frequency range of interest of offshore wind turbines.

Appendices

Appendix 1: Description of matrices $\mathbf {I}_\mathrm {S}^\mathrm {i}$, $\mathbf {I}_\mathrm {H}^\mathrm {i}$, $\mathbf {U}^*$, $\mathbf {T}^*$, $\mathbf {D}^*$ and $\mathbf {S}^*$

Let $\mathbf {x}$ and $\mathbf {n}$ be the position and unit normal vectors of the observation point, while $\mathbf {x}^\mathrm {i}$ and $\mathbf {n}^\mathrm {i}$ are those of the collocation point. The distance vector between both points is $\mathbf {r}=\mathbf {x}-\mathbf {x}^\mathrm {i}$, its norm is $r=|\mathbf {r}|$, and the distance derivative is denoted as $r_{,j}=\partial r / \partial x_j$. The partial derivatives of the distance with respect to the unit normal vectors are $\partial r/\partial n = r_{j}n_j$ and $\partial r/\partial n^\mathrm {i} = -r_{j}n_j^\mathrm {i}$. For the sake of brevity, the wavenumbers are rewritten as $k_j=k_{\mathrm {P}j}$ and $k_3=k_\mathrm {S}$, and the following frequency-dependent parameters are defined:

$$\begin{aligned}&J = \frac{1}{\hat{\rho }_{22}\omega ^2}, \; Z = \frac{\hat{\rho }_{12}}{\hat{\rho }_{22}}, \; \alpha _j = k_j^2-\frac{\mu }{\lambda +2\mu }k_3^2, \nonumber \\&\beta _j = \frac{\mu }{\lambda +2\mu }k_j^2-\frac{k_1^2 k_2^2}{k_3^2} \end{aligned}$$

(55)

The matrices $\mathbf {I}_\mathrm {S}^\mathrm {i}$ and $\mathbf {I}_\mathrm {H}^\mathrm {i}$ appearing respectively in SBIE and HBIE are:

$$\begin{aligned} \mathbf {I}_\mathrm {S}^\mathrm {i}= \left[ \begin{array}{cc} J &{} 0 \\ 0 &{} \delta _\textit{lk} \end{array}\right] ,\; \mathbf {I}_\mathrm {H}^\mathrm {i}= \left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} \delta _\textit{lk} \end{array} \right] \end{aligned}$$

(56)

The fundamental solution matrix $\mathbf {U}^*$ is:

$$\begin{aligned} \mathbf {U}^*= \left[ \begin{array}{cc} -\tau ^*_{\textit{00}} &{} u^*_{\textit{0k}} \\ -\tau ^*_{\textit{l0}} &{} u^*_\textit{lk} \end{array} \right] \end{aligned}$$

(57)

where

$$\begin{aligned} \tau ^{*}_\textit{00}&= \frac{1}{4\pi } \eta \end{aligned}$$

(58)

$$\begin{aligned} \eta&= \frac{1}{k_1^2-k_2^2} \left[ \frac{\alpha _1}{r}e^{-ik_1r}- \frac{\alpha _2}{r}e^{-ik_2r}\right] \end{aligned}$$

(59)

$$\begin{aligned} u^{*}_\textit{0k}&= -\frac{1}{4\pi } \varTheta r_{,k}\end{aligned}$$

(60)

$$\begin{aligned} \varTheta&= \frac{Q/R-Z}{\lambda +2\mu }\frac{1}{k_1^2-k_2^2}\nonumber \\&\quad \times \left[ \left( \frac{1}{r^2}+\frac{ik_1}{r}\right) e^{-ik_1r}- \left( \frac{1}{r^2}+\frac{ik_2}{r}\right) e^{-ik_2r}\right] \end{aligned}$$

(61)

$$\begin{aligned} \tau ^{*}_\textit{l0}&= \frac{1}{4\pi J} \varTheta r_{,l}\end{aligned}$$

(62)

$$\begin{aligned} u^{*}_\textit{lk}&= \frac{1}{4\pi \mu }\left( \delta _\textit{lk}\psi - \chi r_{,l} r_{,k} \right) \end{aligned}$$

(63)

$$\begin{aligned} \psi&= \frac{1}{r}e^{-ik_3r} + \frac{1}{r}\left( \frac{1}{ik_3r}+\frac{1}{(ik_3r)^2}\right) e^{-ik_3r} - \frac{1}{k_1^2-k_2^2} \cdot \nonumber \\&\left[ \frac{\beta _1}{r}\left( \frac{1}{ik_1r}+\frac{1}{(ik_1r)^2}\right) e^{-ik_1r}\right. \nonumber \\&\quad \left. -\frac{\beta _2}{r}\left( \frac{1}{ik_2r}+\frac{1}{(ik_2r)^2}\right) e^{-ik_2r}\right] \end{aligned}$$

(64)

$$\begin{aligned} \chi&= \frac{1}{r}\left( 1+\frac{3}{ik_3r}+\frac{3}{(ik_3r)^2}\right) e^{-ik_3r} - \frac{1}{k_1^2-k_2^2}\cdot \nonumber \\&\left[ \frac{\beta _1}{r}\left( 1+\frac{3}{ik_1r}+\frac{3}{(ik_1r)^2}\right) e^{-ik_1r}\right. \nonumber \\&\quad \left. -\frac{\beta _2}{r}\left( 1+\frac{3}{ik_2r}+\frac{3}{(ik_2r)^2}\right) e^{-ik_2r}\right] \end{aligned}$$

(65)

The fundamental solution matrix $\mathbf {T}^*$ is:

$$\begin{aligned} \mathbf {T}^*= \left[ \begin{array}{cc} -(\textit{U}^*_{n\textit{00}}+JX'^*_jn_j) &{} t^*_{\textit{0k}} \\ -\textit{U}^*_{n\textit{l0}} &{} t^*_\textit{lk} \end{array} \right] \end{aligned}$$

(66)

where

$$\begin{aligned} U^{*}_\textit{n00}+JX'^*_j n_j&= \frac{1}{4\pi } W_0 \frac{\partial r}{\partial n}\end{aligned}$$

(67)

$$\begin{aligned} W_0&= Z \varTheta - J \frac{\partial \eta }{\partial r}\end{aligned}$$

(68)

$$\begin{aligned} t^{*}_\textit{0k}&= \frac{1}{4\pi } \left[ \textit{T}_{01} r_{,k} \frac{\partial r}{\partial n} + \textit{T}_{02} n_k \right] \end{aligned}$$

(69)

$$\begin{aligned} \textit{T}_{01}&= -2\mu \left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) \end{aligned}$$

(70)

$$\begin{aligned} \textit{T}_{02}&= -\lambda \left( \frac{\partial \varTheta }{\partial r}+\frac{2}{r}\varTheta \right) - 2\mu \frac{1}{r}\varTheta + \frac{Q}{R}\eta \end{aligned}$$

(71)

$$\begin{aligned} U^{*}_\textit{nl0}&= \frac{1}{4\pi \mu } \left[ W_1 r_{,l} \frac{\partial r}{\partial n} + W_2 n_l \right] \end{aligned}$$

(72)

$$\begin{aligned} W_1&= Z\chi - \mu \left( \frac{\partial \varTheta }{\partial r} - \frac{1}{r}\varTheta \right) \end{aligned}$$

(73)

$$\begin{aligned} W_2&= -Z\psi - \mu \frac{1}{r}\varTheta \end{aligned}$$

(74)

$$\begin{aligned} t^{*}_\textit{lk}&= \frac{1}{4\pi } \left[ T_1 r_{,l} r_{,k} \frac{\partial r}{\partial n} \right. \nonumber \\&\quad \left. +\,T_2\left( \delta _\textit{lk}\frac{\partial r}{\partial n} + r_{,k} n_l \right) + T_3 r_{,l} n_k\right] \end{aligned}$$

(75)

$$\begin{aligned} T_1&= -2\left( \frac{\partial \chi }{\partial r} - \frac{2}{r}\chi \right) \end{aligned}$$

(76)

$$\begin{aligned} T_2&= \frac{\partial \psi }{\partial r} - \frac{1}{r}\chi \end{aligned}$$

(77)

$$\begin{aligned} T_3&= -\frac{2}{r}\chi +\frac{\lambda }{\mu }\left( \frac{\partial \psi }{\partial r} - \frac{\partial \chi }{\partial r} -\frac{2}{r}\chi \right) \nonumber \\&\quad + \frac{Q}{R}\frac{1}{J}\varTheta \end{aligned}$$

(78)

The fundamental solution matrix $\mathbf {D}^*$ is:

$$\begin{aligned} \mathbf {D}^* = \left[ \begin{array}{cc} -d^*_{\textit{00}} &{} d^*_{\textit{0k}} \\ -d^*_{\textit{l0}} &{} d^*_{lk} \end{array} \right] \end{aligned}$$

(79)

where

$$\begin{aligned} d^{*}_\textit{00}&= \frac{1}{4\pi J} W_0 \frac{\partial r}{\partial n^\mathrm {i}}\end{aligned}$$

(80)

$$\begin{aligned} d^{*}_\textit{0k}&= \frac{1}{4\pi \mu }\left( -W_1r_{,k}\frac{\partial r}{\partial n^\mathrm {i}}+W_2n_k^\mathrm {i}\right) \end{aligned}$$

(81)

$$\begin{aligned} d^{*}_\textit{l0}&= \frac{1}{4\pi J}\left( -T_{01} r_{,l} \frac{\partial r}{\partial n^\mathrm {i}} + T_{02} n_l^\mathrm {i} \right) \end{aligned}$$

(82)

$$\begin{aligned} d^{*}_\textit{lk}&= \frac{1}{4\pi } \left[ T_1 r_{,l} r_{,k} \frac{\partial r}{\partial n^\mathrm {i}} - T_2\left( -\delta _\textit{lk}\frac{\partial r}{\partial n^\mathrm {i}}+r_{,l} n_k^\mathrm {i}\right) - T_3 r_{,k} n_l^\mathrm {i}\right] \end{aligned}$$

(83)

The fundamental solution matrix $\mathbf {S}^*$ is:

$$\begin{aligned} \mathbf {S}^*= \left[ \begin{array}{cc} -s^*_\textit{00} &{} s^*_{\textit{0k}} \\ -s^*_\textit{l0} &{} s^*_\textit{lk} \end{array} \right] \end{aligned}$$

(84)

where

$$\begin{aligned} s^{*}_\textit{00}&= \frac{1}{4\pi }\left[ Q_1\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}}+Q_2\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) \right] \end{aligned}$$

(85)

$$\begin{aligned} Q_1&= \frac{Z^2}{\mu }\chi - 2Z\left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) + J\left( \frac{\partial ^2 \eta }{\partial r^2}-\frac{1}{r}\frac{\partial \eta }{\partial r}\right) \end{aligned}$$

(86)

$$\begin{aligned} Q_2&= \frac{Z^2}{\mu }\psi +2Z\frac{1}{r}\varTheta -J\frac{1}{r}\frac{\partial \eta }{\partial r} \end{aligned}$$

(87)

$$\begin{aligned} s^{*}_\textit{0k}&= \frac{1}{4\pi }\left\{ S_{01}r_{,k}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}}+S_{02}n_k\frac{\partial r}{\partial n^\mathrm {i}}\right. \nonumber \\&\left. \quad +\,S_{03}\left[ n_k^\mathrm {i}\frac{\partial r}{\partial n}+r_{,k}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) \right] \right\} \end{aligned}$$

(88)

$$\begin{aligned} S_{01}&= -2Z\left( \frac{\partial \chi }{\partial r}-\frac{2}{r}\chi \right) \nonumber \\&\quad -2\mu \left[ -\frac{\partial ^2 \varTheta }{\partial r^2}+\frac{3}{r}\left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) \right] \end{aligned}$$

(89)

$$\begin{aligned} S_{02}&= \frac{Q}{R}\left( \frac{Z}{J}\varTheta -\frac{\partial \eta }{\partial r}\right) \nonumber \\&\quad +\,Z\left[ \frac{\lambda }{\mu }\left( \frac{\partial \psi }{\partial r}-\frac{\partial \chi }{\partial r}-\frac{2}{r}\chi \right) -\frac{2}{r}\chi \right] \nonumber \\&\quad +\,\lambda \left[ \frac{\partial ^2 \varTheta }{\partial r^2}+\frac{2}{r}\left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) \right] \nonumber \\&\quad +\, 2\mu \frac{1}{r}\left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) \end{aligned}$$

(90)

$$\begin{aligned} S_{03}&= -Z\left( \frac{\partial \psi }{\partial r}-\frac{1}{r}\chi \right) -2\mu \frac{1}{r}\left( \frac{\partial \varTheta }{\partial r}-\frac{1}{r}\varTheta \right) \end{aligned}$$

(91)

$$\begin{aligned} s_\textit{l0}^*&= \frac{1}{4\pi }\Bigg \{ -S_{01} r_{,l}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}} +\, S_{02} n_l^\mathrm {i}\frac{\partial r}{\partial n}\nonumber \\&\quad - S_{03}\left[ -n_l\frac{\partial r}{\partial n^\mathrm {i}}+r_{,l}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) \right] \Bigg \} \end{aligned}$$

(92)

$$\begin{aligned} s^{*}_\textit{lk} =&\frac{\mu }{4\pi } \Bigg \{S_1\left[ r_{,l}n_k^\mathrm {i}\frac{\partial r}{\partial n} - r_{,k}n_l\frac{\partial r}{\partial n^\mathrm {i}} \right. \nonumber \\&\left. \quad -\,\delta _\textit{lk}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}} + r_{,k}r_{,l}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) \right] \nonumber \\&\quad +\,S_2\left( r_{,k}n_l^\mathrm {i}\frac{\partial r}{\partial n} - r_{,l}n_k\frac{\partial r}{\partial n^\mathrm {i}}\right) + S_3 r_{,l}r_{,k}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}} \nonumber \\&\quad +S_4\left[ \delta _\textit{lk}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) +n_ln_k^\mathrm {i}\right] + S_5 n_k n_l^\mathrm {i}\Bigg \} \end{aligned}$$

(93)

$$\begin{aligned} S_1&= -\frac{\partial ^2 \psi }{\partial r^2} + \frac{1}{r}\left( \frac{\partial \psi }{\partial r} + 3\frac{\partial \chi }{\partial r} - \frac{6}{r}\chi \right) \end{aligned}$$

(94)

$$\begin{aligned} S_2&= 2\frac{\lambda }{\mu }\left[ -\frac{\partial ^2 \psi }{\partial r^2} + \frac{\partial ^2 \chi }{\partial r^2}+ \frac{1}{r}\left( \frac{\partial \psi }{\partial r} + \frac{\partial \chi }{\partial r}- \frac{4}{r}\chi \right) \right] \nonumber \\&\quad +\frac{4}{r}\left( \frac{\partial \chi }{\partial r}-\frac{2}{r}\chi \right) - 2\frac{Q}{R}\frac{1}{J}\left( \frac{\partial \varTheta }{\partial r} - \frac{1}{r}\varTheta \right) \end{aligned}$$

(95)

$$\begin{aligned} S_3&= 4 \left[ -\frac{\partial ^2 \chi }{\partial r^2} + \frac{1}{r}\left( 5\frac{\partial \chi }{\partial r}-\frac{8}{r}\chi \right) \right] \end{aligned}$$

(96)

$$\begin{aligned} S_4&= \frac{2}{r}\left( -\frac{\partial \psi }{\partial r}+\frac{1}{r}\chi \right) \end{aligned}$$

(97)

$$\begin{aligned} S_5&= \frac{4}{r^2}\chi +\frac{\lambda }{\mu }\frac{4}{r}\left( -\frac{\partial \psi }{\partial r} + \frac{\partial \chi }{\partial r} + \frac{2}{r}\chi \right) \nonumber \\&\quad +\frac{\lambda ^2}{\mu ^2}\left[ -\frac{\partial ^2 \psi }{\partial r^2} + \frac{\partial ^2 \chi }{\partial r^2} + \frac{2}{r}\left( -\frac{\partial \psi }{\partial r} + 2\frac{\partial \chi }{\partial r}+\frac{1}{r}\chi \right) \right] \nonumber \\&\quad +\frac{Q}{R}\frac{1}{J}\left[ -2\frac{\lambda }{\mu }\left( \frac{\partial \varTheta }{\partial r}+\frac{2}{r}\varTheta \right) -\frac{4}{r}\varTheta +\frac{Q}{R \mu }\eta \right] \end{aligned}$$

(98)

Appendix 2: Transformation of some fundamental solution terms

The following relationships and the Stokes’ theorem let turn strongly singular and hypersingular surface integrals into weakly singular surface integrals and nearly singular line integrals. Most of them may also be seen in Domínguez et al. [17]. Note that Equation (B20) of [17] contained an erratum as it lacks two terms. The present Eq. (104) corrects this.

$$\begin{aligned} \frac{\mathbf {n}\cdot \mathbf {n}^\mathrm {i}}{r^3}= & {} -\frac{3}{r^3}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}} +\left( \varvec{\nabla }\times \frac{\mathbf {r}\times \mathbf {n}^\mathrm {i}}{r^3}\right) \cdot \mathbf {n} \end{aligned}$$

(99)

$$\begin{aligned} \frac{r_{,l}n_{,k}-r_{,k}n_{,l}}{r^2}= & {} \epsilon _\textit{lkj}\left( \varvec{\nabla }\times \frac{\mathbf {e}_j}{r}\right) \cdot \mathbf {n} \end{aligned}$$

(100)

$$\begin{aligned} \frac{r_{,k}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) }{r^2}= & {} -\frac{n_k}{r^2}\frac{\partial r}{\partial n^\mathrm {i}} +\left( \varvec{\nabla }\times \frac{\mathbf {e}_k\times \mathbf {n}^\mathrm {i}}{r}\right) \cdot \mathbf {n} \end{aligned}$$

(101)

$$\begin{aligned} \frac{r_{,l}r_{,k}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) }{r^3}= & {} -\frac{5 r_{,l} r_{,k}}{r^3}\frac{\partial r}{\partial n}\frac{\partial r}{\partial n^\mathrm {i}} -\frac{r_{,l}n_k^\mathrm {i}+r_{,k}n_l^\mathrm {i}}{r^3}\frac{\partial r}{\partial n} \nonumber \\&+\left[ \varvec{\nabla }\times \left( r_{,l}r_{,k}\frac{\mathbf {r}\times \mathbf {n}^\mathrm {i}}{r^3}\right) \right] \cdot \mathbf {n} \end{aligned}$$

(102)

$$\begin{aligned} \frac{n_k}{r^3}= & {} \frac{3 r_{,k}}{r^3}\frac{\partial r}{\partial n} +\left( \varvec{\nabla }\times \frac{\mathbf {r}\times \mathbf {e}_k}{r^3}\right) \cdot \mathbf {n} \end{aligned}$$

(103)

$$\begin{aligned} \frac{r_{,l}r_{,k}r_{,j}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) }{r^2}= & {} -\frac{1}{3}\frac{r_{,j} n_k n_l^\mathrm {i}}{r^2} -\frac{1}{3}\frac{r_{,l} n_k n_j^\mathrm {i}}{r^2}\nonumber \\&+\,\delta _\textit{lk}\frac{1}{3}\frac{r_{,j}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) }{r^2} \nonumber \\&+\,\delta _\textit{jk}\frac{1}{3}\frac{r_{,l}\left( \mathbf {n}\cdot \mathbf {n}^\mathrm {i}\right) }{r^2} -\frac{r_{,l}r_{,j}n_k}{r^2}\frac{\partial r}{\partial n^\mathrm {i}} \nonumber \\&+\,\frac{1}{3}\left[ \varvec{\nabla }\times \left( r_{,l}r_{,j}\frac{\mathbf {e}_k\times \mathbf {n}^\mathrm {i}}{r}\right) \right] \cdot \mathbf {n} \end{aligned}$$

(104)