An Improved Dynamic Boundary Condition in SPH Method

Dynamic boundary condition (DBC) has been widely used in SPH method. However, in certain situations, it was found that a few fluid particles could break through the boundary or were not reflected specularly. Of course, these phenomena are unphysical. To improve the performance of DBC, an improved dynamic boundary condition (IDBC) was presented in this paper. To prevent fluid particles from breaking through the boundary, the repulsive force of boundary particles was enhanced by expanding the equation of state into a higher order. To deal with the asymmetry of DBC, a rectangular support domain attached to boundary particles and a corresponding correction factor are proposed. The results of three test cases showed that the performance of IDBC was satisfied.

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>