Lagrangian–Eulerian enforcement of non-homogeneous boundary conditions in the Particle Finite Element Method

Abstract

The Particle Finite Element Method (PFEM) is a Lagrangian finite element method with frequent remeshing, particularly suited for the simulation of fluid motions with evolving free surfaces, e.g., in the case of breaking waves or fluid–structure interactions with large displacements of the interaction surface. While the method has been successfully employed in a number of different engineering applications, there are several circumstances of practical interest where the Lagrangian nature of the method makes it difficult to enforce non-homogeneous boundary conditions. A novel mixed Lagrangian–Eulerian technique is proposed to the purpose of simplifying the imposition of this type of conditions with the PFEM. The method is simple to implement and computationally convenient, since only nodes on the boundary are considered Eulerian, while nodes inside the fluid body maintain their Lagrangian nature. A number of 2D and 3D examples, with analytical and numerical validations, confirm the excellent performance of the method.

Appendix: Finite element matrices

Let us consider the linear tetrahedron finite element of the standard 3D PFEM discretization. The global matrices involved in Eqs. (25)–(26) are obtained with standard assembly procedure of the following elemental matrices [65]:

$$\begin{aligned} \mathbf {M}_v&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \rho \mathbf {N}^{vT}\mathbf {N}^v \, {\mathrm{d}}\varOmega \\ \mathbf {M}_p&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \mathbf {N}^{pT}\mathbf {N}^p \, {\mathrm{d}}\varOmega \\ \mathbf {K}_{\mu }&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \mathbf {B}^T \mathbf {d}_\mathrm{{dev}} \mathbf {B}\, {\mathrm{d}}\varOmega \\ \mathbf {D}&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \left( \mathbf {B}^T \mathbf {m}\mathbf {N}^p \right) \, {\mathrm{d}}\varOmega \\ \mathbf {F}_{\mathrm{ext}}&= \sum _{e=1,\dots N^e} \left[ \int _{\varOmega ^t_e} \rho \mathbf {N}^{vT} \mathbf {b}\, {\mathrm{d}}\varOmega + \int _{\varGamma _{N,e}^t} \mathbf {N}^{vT} \mathbf {h}\, \mathrm{{d}}\varGamma \right] \\ \mathbf {K}_c ^v&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \mathbf {N}^{vT}\mathbf {G}^v \, {\mathrm{d}}\varOmega \\ \mathbf {K}_c^p&= \sum _{e=1,\dots N^e} \int _{\varOmega ^t_e} \mathbf {N}^{pT}\mathbf {G}^p \, {\mathrm{d}}\varOmega \end{aligned}$$

In the previous expressions, the vector $\mathbf {m}$ is defined as $\mathbf {m}=[1\ 1\ 1\ 0\ 0\ 0]^{\mathrm{T}}$, while the $\mathbf {B}$ matrix containing the shape functions derivatives can be divided in $n_n$ nodal blocks ($n_n=4$ for the tetrahedron):

$$\begin{aligned} \mathbf {B}=\left[ \mathbf {B}_1, \dots , \mathbf {B}_{n_n}\right] \end{aligned}$$

(35)

Each $\mathbf {B}_i$ nodal block is a $\left( 3 \times 6\right)$ matrix defined as:

$$\begin{aligned} \mathbf {B}_i^T = \left[ \begin{array}{ccc} \frac{\partial N^v_i}{\partial x} &{} 0 &{} 0 \\ 0 &{} \frac{\partial N^v_i}{\partial y} &{} 0 \\ 0 &{} 0 &{} \frac{\partial N^v_i}{\partial z}\\ \frac{\partial N^v_i}{\partial y} &{} \frac{\partial N^v_i}{\partial x}&{} 0 \\ 0 &{} \frac{\partial N^v_i}{\partial z} &{} \frac{\partial N^v_i}{\partial y}\\ \frac{\partial N^v_i}{\partial z} &{} 0 &{}\frac{\partial N^v_i}{\partial x} \\ \end{array} \right] \end{aligned}$$

(36)

The $\mathbf {d}_\mathrm{{dev}}$ matrix contains the material properties of the considered fluid. For a Newtonian fluid it is given by:

$$\begin{aligned} \mathbf {d}_\mathrm{{dev}}= \left[ \begin{array}{cccccc} 4/3 &{} 2/3 &{} 2/3 &{} 0 &{} 0 &{} 0 \\ 2/3 &{} 4/3 &{} 2/3 &{} 0 &{} 0 &{} 0 \\ 2/3 &{} 2/3 &{} 4/3 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{array} \right] \end{aligned}$$

(37)

The matrices $\mathbf {K}_c^v$ and $\mathbf {K}_c^p$ are expressed in terms of $\mathbf {G}^v$ and $\mathbf {G}^p$ which can be defined in terms of their nodal blocks: The matrices related to the convective terms can be defined

$$\begin{aligned} \mathbf {G}^v&=\left[ \mathbf {G}^v_1, \dots , \mathbf {G}^v_{n_n}\right] \end{aligned}$$

(38)

$$\begin{aligned} \mathbf {G}^p&=\left[ \mathbf {G}^p_1, \dots , \mathbf {G}^p_{n_n}\right] \end{aligned}$$

(39)

Each $\mathbf {G}^v_i$ block is a diagonal $\left( 3\times 3\right)$ matrix:

$$\begin{aligned} \mathbf {G}^v_i= \text {diag} \left[ \left( \mathbf {C}^T \mathbf {N}^{vT} \mathbf {L}_i \right) _1 , \left( \mathbf {C}^T \mathbf {N}^{vT} \mathbf {L}_i \right) _2 , \left( \mathbf {C}^T \mathbf {N}^{vT} \mathbf {L}_i \right) _3 \right] \end{aligned}$$

(40)

where the following matrix of shape function derivatives is defined:

$$\begin{aligned} \mathbf {L}_i = \left[ \begin{array}{ccc} \frac{\partial N^v_{i,1}}{\partial x}&{}\frac{\partial N^v_{i,2}}{\partial x}&{}\frac{\partial N^v_{i,3}}{\partial x}\\ \frac{\partial N^v_{i,1}}{\partial y}&{}\frac{\partial N^v_{i,2}}{\partial y}&{}\frac{\partial N^v_{i,3}}{\partial y}\\ \frac{\partial N^v_{i,1}}{\partial z}&{}\frac{\partial N^v_{i,2}}{\partial z}&{}\frac{\partial N^v_{i,3}}{\partial z}\\ \end{array} \right] \end{aligned}$$

(41)

Finally, each $\mathbf {G}^v_i$ block is given by a single component:

$$\begin{aligned} \mathbf {G}^p_i=\mathbf {C}^T \mathbf {N}^{vT} \mathbf {L}^p_i \end{aligned}$$

(42)

where $\mathbf {L}^p_i$ is introduced:

$$\begin{aligned} \mathbf {L}^p_i=\left[ \frac{\partial N^p_{i}}{\partial x},\frac{\partial N^p_{i}}{\partial y},\frac{\partial N^p_{i}}{\partial z} \right] ^{\mathrm{T}} \end{aligned}$$

(43)