approach has several problems, including a high computational cost caused by the
need to generate virtual particles, and leakage of particles at the angled edges of
surfaces.
Regarding polygon wall boundary models in the MPS method, Harada et al. [5]
derived the force exerted on a fluid particle from a wall from impulse-momentum
relationship at the particles near the wall. This force modeling can be classified as
repulsive-force model, so Harada’s model has the same problem of the instability
and strange behavior of fluid particles near the wall. Yamada et al. [6] focused on
the E-MPS method and they proposed another formulation. Although this is a
natural expansion of the MPS differential operator models, the model causes
excessive pressure oscillations.
The ERP model is based on the mirror particle approach and can satisfy the
boundary conditions on walls without virtual particles, and it is versatile enough to
treat arbitrarily shaped boundaries and arbitrary movements. The ERP model adds a
repulsive force adaptively only when the boundary conditions are not satisfied near
the angled edges. The ERP model has the following characteristics:
• Wall boundaries are represented explicitly.
• Generation of virtual particles and the need to make special adaptations for
angled edges are not required.
• The pressure Neumann boundary condition and the slip/no-slip condition on the
walls are satisfied.
3 Verification of the ERP Model
To verify the accuracy of the ERP model applied to the E-MPS method quantitatively, we analyzed a hydrostatic pressure problem in a rectangular vessel. The
numerical results were compared with the theoretical solution
p ¼ q g
j jh
where q is the fluid density, g is the gravitational acceleration vector, and h is the
depth of the static water surface. The initial configuration, in which the depth of
the rectangular vessel is 0.1 m and the width is 0.04 m, is illustrated in Fig. 1. In the
E-MPS computation, weak compressibility causes vertical vibrations of the fluid
surface. To reach the static state as quickly as possible, we chose a relatively high
value for the kinematic viscosity. The conditions used in the analysis are listed in
Table 1, in which l
0 is the initial particle spacing.
Figure 3 shows the pressures of fluid particles computed by (a) the ERP model,
(b) the ERP model using only the repulsive force, (c) Harada’s model, and
(d) Yamada’s model. These are the results at the 200,000th step, at which the
pressure field can be regarded to be in a steady state. In Fig. 2, snapshots obtained
A Hybrid Finite Element and Mesh-Free …
305
need to generate virtual particles, and leakage of particles at the angled edges of
surfaces.
Regarding polygon wall boundary models in the MPS method, Harada et al. [5]
derived the force exerted on a fluid particle from a wall from impulse-momentum
relationship at the particles near the wall. This force modeling can be classified as
repulsive-force model, so Harada’s model has the same problem of the instability
and strange behavior of fluid particles near the wall. Yamada et al. [6] focused on
the E-MPS method and they proposed another formulation. Although this is a
natural expansion of the MPS differential operator models, the model causes
excessive pressure oscillations.
The ERP model is based on the mirror particle approach and can satisfy the
boundary conditions on walls without virtual particles, and it is versatile enough to
treat arbitrarily shaped boundaries and arbitrary movements. The ERP model adds a
repulsive force adaptively only when the boundary conditions are not satisfied near
the angled edges. The ERP model has the following characteristics:
• Wall boundaries are represented explicitly.
• Generation of virtual particles and the need to make special adaptations for
angled edges are not required.
• The pressure Neumann boundary condition and the slip/no-slip condition on the
walls are satisfied.
3 Verification of the ERP Model
To verify the accuracy of the ERP model applied to the E-MPS method quantitatively, we analyzed a hydrostatic pressure problem in a rectangular vessel. The
numerical results were compared with the theoretical solution
p ¼ q g
j jh
where q is the fluid density, g is the gravitational acceleration vector, and h is the
depth of the static water surface. The initial configuration, in which the depth of
the rectangular vessel is 0.1 m and the width is 0.04 m, is illustrated in Fig. 1. In the
E-MPS computation, weak compressibility causes vertical vibrations of the fluid
surface. To reach the static state as quickly as possible, we chose a relatively high
value for the kinematic viscosity. The conditions used in the analysis are listed in
Table 1, in which l
0 is the initial particle spacing.
Figure 3 shows the pressures of fluid particles computed by (a) the ERP model,
(b) the ERP model using only the repulsive force, (c) Harada’s model, and
(d) Yamada’s model. These are the results at the 200,000th step, at which the
pressure field can be regarded to be in a steady state. In Fig. 2, snapshots obtained
A Hybrid Finite Element and Mesh-Free …
305