In the proposed hybrid MPM–FEM, MPM is applied for deformation analyses of the solid skeleton using both Lagrangian and Eulerian frames, the latter of which is used together to apply FEM for flow simulations of the fluid phase. This section presents both of the discretized equations in some detail. For reference, Fig. 2 shows a schematic diagram illustrating the combination of these separate discretization schemes.
Spatial discretization
An arbitrary variable \(\Phi (\varvec{x})\) in the solid skeleton and the fluid phase is approximated as integrated over every FE (finite element) and every solid material point as, respectively,
$$\begin{aligned}&\int _{\varOmega ^\text {f}}\Phi (\varvec{x})d\varOmega \approx \sum _{e=1}^{N^{e}}\int _{\varOmega ^{e}}\Phi (\varvec{x})d\varOmega , \end{aligned}$$
(22)
$$\begin{aligned}&\int _{\varOmega ^\text {s}}\Phi (\varvec{x})d\varOmega \approx \sum _{p=1}^{N^{p}}\int _{\varOmega ^{p}}\Phi (\varvec{x})d\varOmega \approx \sum _{p=1}^{N^{p}}\Phi (\varvec{x}^{p})\varOmega ^{p}, \end{aligned}$$
(23)
where superscripts ‘e’ and ‘p’ represent a FE and material point, respectively. Also, \(\varOmega ^{e}\) is the domain of an element and \(\varOmega ^{p}\) is the volume of a solid material point. The field variables such as velocity, pressure and their test functions can be approximated using the shape functions \(N_{\alpha }(\varvec{x})\) associated with the Eulerian grid as
$$\begin{aligned} \Phi (\varvec{x})\approx \Phi ^h(\varvec{x})=\sum _{\alpha =1}^{N^n}N_{\alpha }(\varvec{x})\Phi _{\alpha }, \end{aligned}$$
(24)
where \(N^n\) is the total number of nodes and \(\Phi _{\alpha }\) is the nodal value at node \(\alpha \). Here and hereafter, variables with superscript ‘h’ indicates their approximations in this manner. It should be noted that the same shape function and the same Eulerian grid are utilized to discretize both the variables associated with the solid skeleton and fluid phase. In this study, we employ the 2nd-order B-spline basis functions as shape functions, the details of which are presented in Appendix A.
The stabilized FEM with the SUPG/PSPG stabilization schemes [7, 34] is applied to the governing equation of the fluid phase, Eq. (20), to obtain the following discretized equation:
$$\begin{aligned}&\int _{\varOmega ^\text {f}}\omega _i^h n\rho ^\text {f} \left( \frac{\partial v_{i}^{\text {f},\,h}}{\partial t}+\bar{v}_{j}^{\text {f}, \, h}\frac{\partial v_{i}^{\text {f}, \, h}}{\partial x_j}-b_i^\text {f} \right) d\varOmega +\int _{\varOmega ^\text {f}}\omega ^h_i\hat{P}_i^hd \varOmega \nonumber \\&\quad -\int _{\varOmega ^\text {f}}\frac{\left( \partial \omega _i^h n\right) }{\partial x_i}p^{\text {f}, \, h}d\varOmega +\int _{\varOmega ^\text {f}}q^h\frac{\partial }{\partial x_i}\left( (1-n)v_i^{\text {s}, \, h}+nv_i^{\text {f}, \, h}\right) d\varOmega \nonumber \\&\quad \quad +\sum _{e=1}^{N^e}\int _{\varOmega ^{e}}\tau _\text {m}^e\left( \bar{v}_j^{\text {f}, \, h}\frac{\partial \omega _i^h}{\partial x_j}+\frac{1}{\rho ^\text {f}}\frac{\partial q^h}{\partial x_i}\right) \nonumber \\&\quad \quad \quad \cdot \left[ n\rho ^\text {f} \left( \frac{\partial v_{i}^{\text {f}, \, h}}{\partial t}+\bar{v}_{j}^{\text {f}, \, h}\frac{\partial v_{i}^{\text {f}, \, h}}{\partial x_j}-b_i^\text {f} \right) +\hat{P}_i^h + n\frac{\partial p^{\text {f}, \, h}}{\partial x_i}\right] d\varOmega \nonumber \\&\quad \quad +\sum _{e=1}^{N^e}\int _{\varOmega ^{e}}\tau _\text {cont}^e\frac{\partial \omega _i^h}{\partial x_i}\rho ^\text {f} \frac{\partial }{\partial x_j}\left( (1-n)v_j^{\text {s}, \, h}+nv_j^{\text {f}, \, h}\right) d\varOmega \nonumber \\&\quad \quad =-\int _{\partial \varOmega ^\text {f}}\omega _i^h n\bar{p}^{\text {f}, \, h}d\Gamma . \nonumber \\ \end{aligned}$$
(25)
The first four integral terms on the left-hand side and the integral term on the right-hand side are Galerkin items. On the other hand, the fifth term involves the SUPG and PSPG terms, which are introduced to stabilize the advection-induced unstable behavior and to suppress pressure oscillation, respectively. Also, the sixth term is the shock-capturing term[2] introduced to avoid the numerical instability of free surfaces. The stabilization parameters, \(\tau _\text {m}^e\) and \(\tau _\text {cont}^e\), are defined as, respectively,
$$\begin{aligned}&\tau _\text {m}^e\equiv \left[ \left( \frac{2}{\varDelta t}\right) ^2+\left( \frac{2\Vert \bar{v}_i^{\text {f}, \, h}\Vert }{h_e}\right) ^2+\left( \frac{4\nu }{h_e^2}\right) ^2\right] ^{-\frac{1}{2}} \end{aligned}$$
(26)
$$\begin{aligned}&\tau _\text {cont}^e\equiv \frac{h_e}{2}\Vert \bar{v}_i^{\text {f}, \, h}\Vert \xi (Re_e) \end{aligned}$$
(27)
$$\begin{aligned}&Re_e\equiv \frac{\Vert \bar{v}_i^{\text {f}, \, h}\Vert h_e}{2\nu } \end{aligned}$$
(28)
$$\begin{aligned}&\xi (Re_e)\equiv {\left\{ \begin{array}{ll} \dfrac{Re_e}{3}, &{}Re_e\le 3\\ 1, &{}Re_e > 3 \end{array}\right. } \end{aligned}$$
(29)
where \(\varDelta t\), \(h_e\), \(\nu \), \(\bar{v}_i^{\text {f}, \, h}\) and \(Re_e\) are the time increment, the characteristic element length, the kinematic viscosity, the advection velocities, and the element Reynolds number, respectively.
Substituting Eqs. (22) and (24) into Eq. (25), we obtain the following FE equations for the fluid phase:
$$\begin{aligned}&(\varvec{M}+\varvec{M}_{\text {s}})\frac{\partial \varvec{v}_i^\text {f}}{\partial t}+(\varvec{A}+\varvec{A}_{\text {s}})\varvec{v}_i^\text {f}+(\varvec{G}_{\text {s},\,i}-\varvec{G}_i)\varvec{p}^\text {f}\end{aligned}$$
(30)
$$\begin{aligned}&\quad +(\varvec{Q}_{(i)}+\varvec{Q}_{\text {s}, \,{{(i)}}})(\varvec{v}_i^\text {f}-\varvec{v}_i^\text {s})+\varvec{S}_{\text {c},\, ij}^\text {s}\varvec{v}_{j}^\text {s}+\varvec{S}_{\text {c},\, ij}^\text {f}\varvec{v}_{j}^\text {f}=(\varvec{F}+\varvec{F}_\text {s})\varvec{b}_i^\text {f},\nonumber \\&\varvec{C}^\text {s}_{i}\varvec{v}_i^\text {s}+\varvec{C}^\text {f}_{i}\varvec{v}_i^\text {f}+\varvec{M}_{\text {p},\,i}\frac{\partial \varvec{v}_i^\text {f}}{\partial t}+\varvec{A}_{\text {p},\,i}\varvec{v}_i^\text {f}+\varvec{G}_{\text {p}}\varvec{p}^\text {f}+\varvec{Q}_{\text {p},\,i}(\varvec{v}_i^\text {f}-\varvec{v}_i^\text {s})\nonumber \\&\quad =\varvec{F}_{\text {p},\,i}\varvec{b}_i^\text {f}, \end{aligned}$$
(31)
which constitutes the set of the momentum balance and the continuity (mass conservation) equations, respectively. These discretized equations are to be solved for the nodal fluid velocity and pressure vectors at nodes, provided that the nodal solid velocity vectors are given as data. Here, \(\varvec{M}\), \(\varvec{A}\), \(\varvec{G}\), \(\varvec{Q}\), \(\varvec{S}_\text {c}\), \(\varvec{F}\) and \(\varvec{C}\) are the mass matrix, advection matrix, pressure matrix, interaction matrix, shock-capturing matrix, force vector, and continuity matrix, respectively. Also, subscript ‘i’ indicates the component of a nodal velocity vector in the \(x_i\)-direction, and subscripts ‘s’ and ‘p’ indicate the SUPG and PSPG matrices, respectively. The specific forms of these matrices are written as follows:
$$\begin{aligned}&[\varvec{M}]^{e}_{\alpha \beta }=\int _{\varOmega ^{e}}\rho ^\text {f} n N_{\alpha }^{e}N_{\beta }^{e}d\varOmega , \end{aligned}$$
(32)
$$\begin{aligned}&[\varvec{A}]^{e}_{\alpha \gamma }=\int _{\varOmega ^{e}}\rho ^\text {f} n N_{\alpha }^{e}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}\frac{\partial N_{\gamma }^{e}}{\partial x_{j}}d\varOmega , \end{aligned}$$
(33)
$$\begin{aligned}&[\varvec{G}_i]^{e}_{\alpha \beta }=\int _{\varOmega ^{e}}\left( n \frac{\partial N_{\alpha }^{e}}{\partial x_i}+N_{\alpha }^{e}\frac{\partial n}{\partial x_i}\right) N_{\beta }^{e}d\varOmega , \end{aligned}$$
(34)
$$\begin{aligned}&[\varvec{Q}_i]^{e}_{\alpha \beta }=\int _{\varOmega ^{e}} \vartheta n^2N_{\alpha }^{e}\left( \frac{\mu ^\text {f}}{k}+\frac{1.75\rho ^\text {f}}{\sqrt{150 k}}\frac{|v^\text {f}_{i\beta }-v^\text {s}_{i\beta }|}{n^{3/2}}\right) N_{\beta }^{e}d\varOmega , \end{aligned}$$
(35)
$$\begin{aligned}&[{\varvec{C}_i^\text {s}}]^{e}_{\alpha \beta }=\int _{\varOmega ^{e}}\left( (1-n)N_{\alpha }^{e}\frac{\partial N_{\beta }^{e}}{\partial x_i}-N_{\alpha }^{e}\frac{\partial n}{\partial x_i}N_{\beta }^{e}\right) d\varOmega , \end{aligned}$$
(36)
$$\begin{aligned}&[{\varvec{C}_i^\text {f}}]^{e}_{\alpha \beta }=\int _{\varOmega ^{e}}\left( n N_{\alpha }^{e}\frac{\partial N_{\beta }^{e}}{\partial x_i}+N_{\alpha }^{e}\frac{\partial n}{\partial x_i}N_{\beta }^{e}\right) d\varOmega , \end{aligned}$$
(37)
$$\begin{aligned}&[{\varvec{F}}]^{e}_{\alpha }=\int _{\varOmega ^{e}}\rho ^\text {f} n N_{\alpha }^{e}d\varOmega , \end{aligned}$$
(38)
$$\begin{aligned}&[\varvec{M}_{\text {s}}]_{\alpha \gamma }^{e}=\tau _\text {m}^e\int _{\varOmega ^{e}}\rho ^\text {f} n \frac{\partial N_{\alpha }^{e}}{\partial x_{j}}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}N_{\gamma }^{e}d\varOmega , \end{aligned}$$
(39)
$$\begin{aligned}&[\varvec{A}_\text {s}]^{e}_{\alpha \delta }=\tau _\text {m}^e\int _{\varOmega ^{e}}\rho ^\text {f} n \frac{\partial N_{\alpha }^{e}}{\partial x_{j}}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}N_{\gamma }^{e}\bar{v}_{l\gamma }^{\text {f}, \, e}\frac{\partial N_\delta ^{e}}{\partial x_l}d\varOmega , \end{aligned}$$
(40)
$$\begin{aligned}&[\varvec{G}_{\text {s},\,i}]^{e}_{\alpha \gamma }=\tau _\text {m}^e\int _{\varOmega ^{e}} n \frac{\partial N_{\alpha }^{e}}{\partial x_{j}}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}\frac{N_{\gamma }^{e}}{\partial x_i}d\varOmega , \end{aligned}$$
(41)
$$\begin{aligned}&[\varvec{Q}_{\text {s},\,i}]^{e}_{\alpha \gamma }=\tau _\text {m}^e\int _{\varOmega ^{e}} \vartheta n^2\frac{\partial N_{\alpha }^{e}}{\partial x_{j}}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}\nonumber \\&\qquad \qquad \ \ \qquad \left( \frac{\mu ^\text {f}}{k}+\frac{1.75\rho ^\text {f}}{\sqrt{150 k}}\frac{|v^\text {f}_{i\gamma }-v^\text {s}_{i\gamma }|}{ n^{3/2}}\right) N_{\gamma }^{e}d\varOmega ,\end{aligned}$$
(42)
$$\begin{aligned}&[{\varvec{F}_\text {s}}]^{e}_{\alpha \beta }=\tau _\text {m}^e\int _{\varOmega ^{e}}\rho ^\text {f} n \frac{\partial N_{\alpha }^{e}}{\partial x_{j}}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}d\varOmega , \end{aligned}$$
(43)
$$\begin{aligned}&[\varvec{M}_{\text {p},\,i}]_{\alpha \gamma }^{e}=\tau _\text {m}^e\int _{\varOmega ^{e}} n \frac{\partial N_{\alpha }^{e}}{\partial x_i}N_{\beta }^{e}d\varOmega , \end{aligned}$$
(44)
$$\begin{aligned}&[\varvec{A}_{\text {p},\,i}]^{e}_{\alpha \gamma }=\tau _\text {m}^e\int _{\varOmega ^{e}} n \frac{\partial N_{\alpha }^{e}}{\partial x_i}N_{\beta }^{e}\bar{v}_{j\beta }^{\text {f}, \, e}\frac{\partial N_{\gamma }^{e}}{\partial x_{j}}d\varOmega , \end{aligned}$$
(45)
$$\begin{aligned}&[\varvec{G}_{\text {p}}]^{e}_{\alpha \beta }=\tau _\text {m}^e\int _{\varOmega ^{e}}\frac{n}{\rho ^\text {f}}\frac{\partial N_{\alpha }^{e}}{\partial x_{j}}\frac{\partial N_{\beta }^{e}}{\partial x_{j}}d\varOmega , \end{aligned}$$
(46)
$$\begin{aligned}&[\varvec{Q}_{\text {p},\,i}]^{e}_{\alpha \beta }=\tau _\text {m}^e\int _{\varOmega ^{e}}\frac{\vartheta n^2}{\rho ^\text {f}}\frac{\partial N_{\alpha }^{e}}{\partial x_i}\nonumber \\&\left( \frac{\mu ^\text {f}}{k}+\frac{1.75\rho ^\text {f}}{\sqrt{150k}}\frac{|v^\text {f}_{i\beta }-v^\text {s}_{i\beta }|}{ n^{3/2}}\right) N_{\beta }^{e}d\varOmega ,\end{aligned}$$
(47)
$$\begin{aligned}&[{\varvec{F}_{\text {p},\,i}}]^{e}_{\alpha }=\int _{\varOmega ^{e}} n \frac{\partial N_{\alpha }^{e}}{\partial x_i}d\varOmega , \end{aligned}$$
(48)
$$\begin{aligned}&[{\varvec{S}_{{\text {c},\,ij}}^\text {s}}]^{e}_{\alpha \beta }=\tau _\text {cont}^e\int _{\varOmega ^{e}}\rho ^\text {f}\frac{\partial N_{\alpha }^{e}}{\partial x_i}\left( (1-n)\frac{\partial N_{\beta }^{e}}{\partial x_{j}}-\frac{\partial n}{\partial x_{j}}N_{\beta }^{e}\right) d\varOmega , \end{aligned}$$
(49)
$$\begin{aligned}&[{\varvec{S}_{{\text {c},\,ij}}^\text {f}}]^{e}_{\alpha \beta }=\tau _\text {cont}^e\int _{\varOmega ^{e}}\rho ^\text {f}\frac{\partial N_{\alpha }^{e}}{\partial x_i}\left( n \frac{\partial N_{\beta }^{e}}{\partial x_{j}}+\frac{\partial n}{\partial x_{j}}N_{\beta }^{e}\right) d\varOmega . \end{aligned}$$
(50)
Here, subscripts ‘j’ and ‘l’ indicate the components of the nodal velocity vectors, ‘\(\alpha \)’, ‘\(\beta \)’, ‘\(\gamma \)’, and ‘\(\delta \)’ denote the values of the grid nodes. Also, the volume fraction of the solid material points in an element, \(\vartheta \), has been introduced, because there is a possibility that the fluid phase and the solid skeleton do not overlap with each other. In these discretized equations, n, k, and \(\vartheta \) are functions of space to be interpolated by use of the shape functions as
$$\begin{aligned} n \approx \sum _{\alpha =1}^{N^{n}} n_{\alpha } N_{\alpha }^{e}, \quad k \approx \sum _{\alpha =1}^{N^{n}} k_{\alpha } N_{\alpha }^{e}, \quad \vartheta \approx \sum _{\alpha =1}^{N^{n}} \vartheta _{\alpha } N_{\alpha }^{e}, \end{aligned}$$
(51)
where \(n_{\alpha }\) and \(k_{\alpha }\) are the nodal values of the porosity and permeability, respectively.
On the other hand, substituting Eqs. (22)\(\sim \)(24) into Eq. (21), we obtain the semi-discrete momentum balance equation for the solid skeleton as
$$\begin{aligned} \varvec{M}^\text {s}\varvec{a}_{i}^\text {s}=\varvec{F}_{\text {int}, \, i}^\text {s}+\varvec{F}_{\text {ext}, \, i}^\text {s}+\varvec{Q}_{(i)}\left( \varvec{v}^\text {f}_{i}-\varvec{v}^\text {s}_{i}\right) . \end{aligned}$$
(52)
This equation is solved for the solid acceleration at nodes by using the pressure of the fluid phase as input a datum, which is the solution of the momentum balance equation for the fluid phase in Eqs. (30) and (31). Here, the mass matrix for the solid skeleton is lumped as
$$\begin{aligned}{}[\varvec{M}^\text {s}]_{\alpha }=\sum _{p=1}^{N^{p}}m^{p}N_{\alpha }(\varvec{x}^{p}), \end{aligned}$$
(53)
where \(m^{p}\) is the mass of the solid particle computed as \(m^p=(1-n^p)\rho ^\text {s}\varOmega ^p\). Also, the nodal values of the internal and external forces are written as, respectively,
$$\begin{aligned}&[\varvec{F}_{\text {int}, \, i}^\text {s}]_{\alpha }=-\sum _{p=1}^{N^{p}}\frac{\partial N_{\alpha }(\varvec{x}^{p})}{\partial x_j}\sigma _{ij}^{\prime }\varOmega ^{p}\nonumber \\&\quad +\int _{\varOmega ^\text {f}}\left( (1-n)\frac{\partial N_{\alpha }^{e}}{\partial x_i}-N_{\alpha }^{e}\frac{\partial n}{\partial x_i}\right) p^\text {f} d\varOmega , \end{aligned}$$
(54)
$$\begin{aligned}&[\varvec{F}_{\text {ext}, \, i}^\text {s}]_{\alpha }=\sum _{p=1}^{N^{p}}N_{\alpha }(\varvec{x}^{p})m^{p}b_i^\text {s}+\int _{\partial \varOmega ^\text {s}}N_{\alpha }(\varvec{x}^\text {s})t_i(\varvec{x}^\text {s})d\Gamma , \end{aligned}$$
(55)
in which the fluid pressure must be interpolated as \( p^\text {f} = \sum _{\beta =1}^{N^{n}}{N_\beta ^e p_{\beta }^\text {f}}\).
Time integration
The semi-discrete equations derived above are discretized with a discrete set of times, and the numerical solution is obtained at a current time. In what follows, the discrete approximation at time \(t^L\) is indicated by superscript L and the time increment is defined as \(\varDelta t\equiv t^{L+1}-t^{L}\).
The time discretized forms of the momentum balance and the mass balance equations for the fluid phase, Eqs. (30) and (31), can be written as
$$\begin{aligned}&(\varvec{M}+\varvec{M}_{\text {s}})\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }+(\varvec{A}+\varvec{A}_{\text {s}})\varvec{v}_i^{\text {f}, \, L+\theta }+(\varvec{G}_{\text {s},\,i}-\varvec{G}_i)\varvec{p}^{\text {f}, \, L+\theta }\nonumber \\&\quad +(\varvec{Q}_{(i)}+\varvec{Q} _{\text {s}, {{(i)}}})(\varvec{v}_i^{\text {f}, \, L+\theta }-\varvec{v}_i^{s,L+\theta })+\varvec{S}_{\text {c},\, ij}^\text {s}\varvec{v}_{j}^{s,L+\theta }+\varvec{S}_{\text {c},\, ij}^\text {f}\varvec{v}_{j}^{\text {f}, \, L+\theta }\nonumber \\&\quad \quad =(\varvec{F}+\varvec{F}_{\text {s}})\varvec{b}_i^{\text {f}, \, L}, \nonumber \\ \end{aligned}$$
(56)
$$\begin{aligned}&\varvec{C}^\text {s}_{i}\varvec{v}_i^{s, L+\theta }+\varvec{C}^\text {f}_{i}\varvec{v}_i^{\text {f}, \, L+\theta }+\varvec{M}_{\text {p},\,i}\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }+\varvec{A}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }+\varvec{G}_{\text {p}}\varvec{p}^{\text {f}, \, L+\theta }\nonumber \\&\quad +\varvec{Q}_{\text {p},\,i}(\varvec{v}_i^{\text {f}, \, L+\theta }-\varvec{v}_i^{s, L+\theta })=\varvec{F}_{\text {p},\,i}\varvec{b}_i^{\text {f}, \, L},\nonumber \\ \end{aligned}$$
(57)
which are simultaneously solved for the fluid phase velocities \(v_i^{\text {f}, \, L+1}\) and fluid pressure \(p^{\text {f}, \, L+1}\). Here, the range of \(\theta \) is set to be \(0\le \theta \le 1\). Also, the nodal variables (time derivative, velocities, pressure) are approximated as
$$\begin{aligned}&\frac{\partial \varvec{v}_i^\text {f}}{\partial t}\bigg |^{L+\theta }\approx \frac{\varvec{v}_i^{\text {f}, \, L+1}-\varvec{v}_i^{\text {f}, \, L}}{\varDelta t}, \end{aligned}$$
(58)
$$\begin{aligned}&\varvec{v}_i^{\text {f}, \, L+\theta }\approx \varvec{v}_i^{\text {f}, \, L+\frac{1}{2}}=\frac{\varvec{v}_i^{\text {f}, \, L+1}+\varvec{v}_i^{\text {f}, \, L}}{2}, \end{aligned}$$
(59)
$$\begin{aligned}&\varvec{v}_i^{\text {s}, \, L+\theta }\approx \varvec{v}_i^{\text {s}, \, L}, \end{aligned}$$
(60)
$$\begin{aligned}&\varvec{p}^{\text {f}, \, L+\theta }\approx \varvec{p}^{\text {f}, \, L+1}, \end{aligned}$$
(61)
where \(\theta \) has been set at 1 in Eqs. (58) and (61), 1/2 in Eq. (59) and 0 in Eq. (60). To explicitly approximate the advection velocities \(\bar{v}_i^\text {f}\), we adopt the Adams-Bashforth method [12], as
$$\begin{aligned} \varvec{\bar{v}}_i^\text {f}=\frac{3}{2}\varvec{v}_i^{\text {f}, \, L}-\frac{1}{2}\varvec{v}_i^{\text {f}, \, L-1}. \end{aligned}$$
(62)
Also, the continuity term \(\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, \, L+\theta }\) is implicitly represented, while the interaction terms, \(\varvec{Q}_i\varvec{v}_i^{\text {f}, \, L+\theta }, \, \varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }\) and \(\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L+\theta }\), are evaluated explicitly as
$$\begin{aligned}&\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{C}_{i}^\text {f}\varvec{v}_i^{\text {f}, L+1}, \end{aligned}$$
(63)
$$\begin{aligned}&\varvec{Q}_{i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_i\varvec{v}_i^{\text {f}, \, L}, \end{aligned}$$
(64)
$$\begin{aligned}&\varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_{\text {s},\,i}\varvec{v}_i^{\text {f}, \, L}, \end{aligned}$$
(65)
$$\begin{aligned}&\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, L+\theta }=\varvec{Q}_{\text {p},\,i}\varvec{v}_i^{\text {f}, \, L}. \end{aligned}$$
(66)
where \(\theta \) has been set at 1 in Eq. (63) and 0 in Eqs. (64), (65) and (66). Using Eqs. (56)-(66) with some straightforward calculations, we obtain the final versions of Eqs. (56) and (57) as
$$\begin{aligned}&(\varvec{M}+\varvec{M}_{\text {s}})\frac{\varvec{v}_i^{\text {f}, \, L+1}}{\varDelta t}+\left( \varvec{A}+\varvec{A}_{\text {s}}+\varvec{S}_{\text {c},\, ij}^\text {f}\right) \frac{\varvec{v}_{j}^{\text {f}, \, L+1}}{2}+\left( \varvec{G}_{\text {s},\,i}-\varvec{G}_i\right) \varvec{p}^{\text {f}, \, L+1}\nonumber \\&\quad =(\varvec{M}+\varvec{M}_{\text {s}})\frac{\varvec{v}_i^{\text {f}, \, L}}{\varDelta t}-\left( \varvec{A}+\varvec{A}_{\text {s}}+\varvec{S}_{\text {c},\,ij}^\text {f}\right) \frac{\varvec{v}_{j}^{\text {f}, \, L}}{2}-\varvec{S}_{\text {c},\,ij}^\text {s}\varvec{v}_{j}^{\text {s}, \, L}\nonumber \\&\quad \quad +(\varvec{Q}_{(i)}+\varvec{Q}_{\text {s},\,({i})})(\varvec{v}_i^{\text {s}, \, L}-\varvec{v}_i^{\text {f}, \, L})+(\varvec{F}+\varvec{F}_{\text {s}})\varvec{b}_i^{\text {f}, \, L}, \nonumber \\ \end{aligned}$$
(67)
$$\begin{aligned}&\left( \varvec{C}^\text {f}_{i}+\frac{\varvec{M}_{\text {p},\,i}}{\varDelta t}+\frac{\varvec{A}_{\text {p},\,i}}{2}\right) \varvec{v}_i^{\text {f}, \, L+1}+\varvec{G}_{\text {p}}\varvec{p}^{\text {f}, \, L+1}=\left( \frac{\varvec{M}_{\text {p},\,i}}{\varDelta t}-\frac{\varvec{A}_{\text {p},\,i}}{2}\right) \varvec{v}_i^{\text {f}, \, L}\nonumber \\&\quad \quad +\varvec{Q}_{\text {p},\,i}(\varvec{v}_i^{\text {s}, \, L}-\varvec{v}_i^{\text {f}, \, L})-\varvec{C}^\text {s}_{i}\varvec{v}_i^{\text {s}, \, L}+\varvec{F}_{\text {p},\,i}\varvec{b}_i^{\text {f}, \, L},\nonumber \\ \end{aligned}$$
(68)
from which the fluid phase velocity and pressure are determined for the next time step.
The time-discretized form of the momentum balance equation for the solid phase (52) can be written as
$$\begin{aligned} \varvec{M}^{\text {s}, \, L+\theta }\varvec{a}_{i}^{\text {s}, \, L+\theta }=\varvec{F}_{\text {int}, \, i}^{\text {s}, \, L+\theta }+\varvec{F}_{\text {ext}, \, i}^{\text {s}, \, L+\theta }+\varvec{Q}_{(i)}\left( \varvec{v}^{\text {f}, \, L+\theta }_{i}-\varvec{v}^{\text {s}, \, L+\theta }_{i}\right) . \end{aligned}$$
(69)
Using the nodal pressure, \(\varvec{p}^{\text {f}, \, L+1}\), obtained from Eqs. (67) and (68), this is solved for \(\varvec{a}_{i}^{\text {s}, \, L}\) explicitly with \(\theta =0\). After the nodal acceleration of the solid skeleton, \(\varvec{a}_{i}^{\text {s}, \, L}\), is obtained, the nodal velocity vector in the \(x_i\)-direction is updated as
$$\begin{aligned} \varvec{v}_{i}^{\text {s}, \, L+1}=\varvec{v}_{i}^{\text {s}, \, L}+\varDelta t\varvec{a}_{i}^{\text {s}, \, L}, \end{aligned}$$
(70)
and then the velocity and position vectors of each solid material point can be updated by the following formulae:
$$\begin{aligned}&v_{i}^{p, \, L+1}=v_{i}^{p, \, L}+\varDelta t\sum _{\alpha =1}^{N^n} a_{i\alpha }^{\text {s}, \, L}N_{\alpha }(\varvec{x}^{p, \, L}), \end{aligned}$$
(71)
$$\begin{aligned}&x_{i}^{p, \, L+1}=x_{i}^{p, \, L}+\varDelta t\sum _{\alpha =1}^{N^n}v_{i\alpha }^{\text {s}, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L}). \end{aligned}$$
(72)
Prior to updating the state valuables of each solid material point, we adopt the MUSL procedure [32] to “refine” the nodal velocity vector by an additional mapping process. Then, the deformation gradient of a solid material point can be computed as
$$\begin{aligned} F_{ij}^{p, \, L+1}=\left( \delta _{ik}+\varDelta t \sum _{\alpha =1}^{N^n}\frac{\partial N_{\alpha }(\varvec{x}^{p, \, L})}{\partial x_k}v_{i \alpha }^{\text {s}, \, L+1}\right) F_{kj}^{p, \, L},\quad \end{aligned}$$
(73)
and its determinant \(J^{p, \, L+1}=\det \varvec{F}^{p, \, L+1}\) is used to update its volume and the porosity as, respectively,
$$\begin{aligned} \varOmega ^{p, \, L+1}&=J^{p, \, L+1} \varOmega ^{p, \, 0}, \end{aligned}$$
(74)
$$\begin{aligned} n^{p, \, L+1}&=1-\frac{(1-n^{p, \, 0})}{J^{p, \, L+1}}. \end{aligned}$$
(75)
According to Eq. (4), the permeability \(k^{p, \, L+1}\) can also be calculated using \(n^{p, \, L+1}\) as
$$\begin{aligned} k^{p, \, L+1}=\frac{D_{50}^2\left( n^{p, \, L+1}\right) ^3}{150\left( 1-n^{p, \, L+1}\right) ^2}. \end{aligned}$$
(76)
Here, the upper limit value of the solid porosity \(n^{p, \, L+1}\) has to be set to suppress numerical instability. In this study, it will be set at \(n^{p, \, L+1}=0.99\) in the numerical examples.
Mapping of state variables at node
Nodal values for the solid skeleton are calculated by the mapping procedures established in the previous studies [3, 38].
As is the case when using the conventional solid-fluid coupled MPM, the weighted least square approximation is adopted to evaluate the nodal velocity vector and the permeability at each node as
$$\begin{aligned}&\varvec{v}_{i}^{\text {s}, \, L+1}=\sum _{p=1}^{N^p}\frac{m^{p, \, L} \, v_i^{p, \, L+1} \, N_{\alpha }(\varvec{x}^{p, \, L})}{m^{p, \, L} \, N_{\alpha }(\varvec{x}^{p, \, L})}, \end{aligned}$$
(77)
$$\begin{aligned}&k_{\alpha }^{L+1}=\sum _{p=1}^{N^{p}}\frac{m^{p, \, L+1}k^{p, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L+1})}{m^{p, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L+1})}. \end{aligned}$$
(78)
The volume fraction of a grid node, \(\vartheta _{\alpha }^{L+1}\), is computed using \(\varOmega ^{p, \, L+1}\) as
$$\begin{aligned} \vartheta _{\alpha }^{L+1}=\frac{\sum _{p=1}^{N^{p}}\varOmega ^{p, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L+1})}{\varOmega _{\alpha }}. \end{aligned}$$
(79)
Here, \(\varOmega _{\alpha }\) is the nodal volume defined as
$$\begin{aligned} \varOmega _{\alpha }=\int _{\varOmega ^e} N_{\alpha }^e d\varOmega , \end{aligned}$$
(80)
which is carried out element-wise with the Gaussian quadrature rule [41]. Similarly, we map the porosities of solid particles onto the value at a node such that
$$\begin{aligned} n_{\alpha }^{L+1}=1-\frac{\sum _{p=1}^{N^{p}}(1-n^{p, \, L+1})\varOmega ^{p, \, L+1}N_{\alpha }(\varvec{x}^{p, \, L+1})}{\varOmega _{\alpha }}. \end{aligned}$$
(81)
to suppress numerical instability due to the interpolation of the porosity gradient [38].
Interface representation
It is assumed that the solid phase is homogeneous and that the fluid phase consists of liquid (water) and gas (air) phases, which are identified by superscripts “\(\ell \)” and “g” on their variables and symbols, respectively. Under this circumstance, the fluid phase has an interface with the solid skeleton or another fluid phase within an element, as illustrated in Fig. 2, and moves with time. The moving interface must be carefully handled to determine the domains of the phases involved so that the momentum balance and continuity equations derived above can properly be evaluated.
There are two kinds of methods to determine the geometry of a free surface that is an interface between either a liquid and a gas or a fluid and a solid. One is the interface tracking method, which uses a Lagrange description to directly express the geometry using a moving grid. The other one is the interface-capturing method, which uses an Eulerian or spatially fixed grid to track the time evolution of the interface. Since our target problems involve wave generation and propagation, and complex free surfaces, we employ the phase-field method, which is an interface-capturing method in this study. The details of this capturing method are provided in Appendix B.
Numerical algorithm
The numerical algorithm for the proposed hybrid MPM–FEM is summarized as follows:
-
(i)
Map the information carried by solid material points to the background grid. The nodal values of mass, velocity, volume fraction, porosity, and permeability
$$\begin{aligned} \left( M_{\alpha }^{\text {s}, L},\varvec{v}_{i}^{\text {s}, \, L}, \vartheta _{\alpha }^{L},n_{\alpha }^{L},k_{\alpha }^{ L}\right) \end{aligned}$$
are calculated using Eqs. (53), (77), (79), (81) and (78), respectively;
-
(ii)
Compute the volume fraction, porosity and permeability of the fluid phase \(\left( \vartheta ^{L}, \, n^{L}, \, k^{L}\right) \) with Eqs. (51);
-
(iii)
Solve Eqs. (67) and (68) simultaneously for the velocities and pressure \(\left( \varvec{v}_i^{\text {f}, L+1}, \, p^{\text {f}, L+1}\right) \) of the fluid phase for the next time step;
-
(iv)
Following the procedure provided in Appendix B, obtain the interface function \(\phi ^{L+1}\) to capture the interface between different physical domains and then update the fluid density and viscosity around the interface between the gas and liquid phases according to Eqs. (89) and (90);
-
(v)
Solve Eq. (69) for the nodal acceleration vector of the solid skeleton \(a_i^{\text {s}, \, L}\);
-
(vi)
Update the velocity and position of the solid material points using Eqs. (71) and (72), respectively;
-
(vii)
Initialize the computational grid of the solid skeleton for the next time step.
Some remarks
It is worth mentioning that both the FEM and MPM adopt weak formulations and therefore have almost the same mathematical structures. This is particularly advantageous for solid-liquid coupled problems, because the domains of solid and liquid phases overlaps with each other in most problems of interest. In fact, since the numerical algorithms of MPM are inevitably similar to that of FEM, the information exchanging between solid and liquid phases is easier than other CFD approaches such the finite difference method.
Nevertheless, surprisingly few studies have so far been made at combining FEM and MPM for solid-liquid coupled problems, for which liquid and solid phases are separately described in the Eularian and Lagrangian frames, respectively. Although there have been some studies to connect FEM and MPM with different goals in the literature [20, 21], they employ a sequential coupling method to transition between a moderately large deformation state that can be dealt with FEM and an extremely large deformation state that can more suitably be handled by MPM. Thus, the novelty of this study is obvious.
A remark is also made on the computational cost of time integration. Since the explicit time integration scheme is adopted for the solid phase, the time step size controlled by the CFL condition must be sufficiently small. Even though the implicit time integration is employed for fluid phase, the overall time step size must be controlled by that for the solid phase. Nevertheless, the size needs not be as small as that of the existing solid-liquid MPM that assumes large bulk modulus of elasticity to represent the weakly compressible liquid phase. This is another advantage over the previous explicit MPM-MPM framework for solid-liquid coupled problems.
However, the proposed method is still time-consuming and therefore not suitable for large-scale and long-duration phenomena that might include submarine landslide-induced tsunamis. This is one of the major limitations and remains to be resolved in future studies. The other major limitation is that the formulation is only applicable to either fully saturated or dry soils but not to unsaturated ones. Therefore, some important phenomena such as rainfall-induced landslides cannot be handled by the present scheme and give us a significant challenge.