Table 1 Definition of indices for the formulation of compatible interface wave–structure interaction model

Fluid particle:

i

\(\qquad ( i \in \Lambda _{\textrm{particle}} = \{ 1, \cdots , N_{\textrm{particle}} \})\)

Neighboring particle within effective radius:

j

Polygon wall:

s

\(\qquad (s \in \Lambda _{\textrm{pol}} = \{ 1, \cdots , N_{\textrm{pol}} \})\)

Polygon wall on the fluid–structure (FS) interface:

\(\tilde{s}\)

\(\qquad (\tilde{s} \in \tilde{\Lambda }_{\textrm{pol}} \subset \Lambda _{\textrm{pol}})\)

Global node of the polygon wall:

L

\(\qquad (L \in \Lambda _{\textrm{pnode}} = \{ 1, \cdots , N_{\textrm{pnode}} \})\)

Global node of the polygon wall on the FS interface:

\(\tilde{L}\)

\(\qquad (\tilde{L} \in \tilde{\Lambda }_{\textrm{pnode}} \subset \Lambda _{\textrm{pnode}})\)

Finite element (FE):

e

\(\qquad (e \in \Lambda _{\textrm{elem}} = \{ 1, \cdots , N_{\textrm{elem}} \})\)

FE surface:

b

\(\qquad (b \in \Lambda _{\textrm{surface}} = \{ 1, \cdots , N_{\textrm{surface}} \})\)

FE surface on the FS interface:

\(\tilde{b}\)

\(\qquad (\tilde{b} \in \tilde{\Lambda }_{\textrm{surface}} \subset \Lambda _{\textrm{surface}})\)

Global node of FE:

K

\(\qquad (K \in \Lambda _{\textrm{fnode}} = \{ 1, \cdots , N_{\textrm{fnode}} \})\)

Global node of the FE on the FS interface:

\(\tilde{K}\)

\(\qquad (\tilde{K} \in \tilde{\Lambda }_{\textrm{fnode}} \subset \Lambda _{\textrm{fnode}})\)

Timestep:

n

\(\qquad (1 \le n \le N_{\textrm{timestep}})\)