Abstract
In this work, we implement the implicit boundary integral method for a homogeneous Hele-Shaw problem with a multi-connected domain. This method base on the solution of layer potential integral for the Laplace equation. The numerical technique is easy to implement, base on the idea of averaging the parameterization near the boundary and applying the Coarea formula. This technique changes the boundary integral into the Riemann integral that numerically easy to compute. The difficulty in the computation of hypersingular integral occurs to compute the normal velocity of free boundary. We use a collocation technique to eliminate the hypersingular part in the integral equation. Also, we show the numerical results and its computation performance due to the appearance of a non-invertible matrix.
Highlights
I T is well known for the finite difference method (FDM) and finite element method (FEM) are usually used to solve a partial differential equation numerically
The notion is based on averaging the parameterization of boundary integral equation by using delta Dirac function, use the Coarea formula, such that the boundary integral changes the form into the Riemann integral defined in Rn
This work implements the method names implicit boundary integral method to approximate the Laplacian in two dimension that included in the Hele-Shaw problem, and naively compute the normal velocity to update the free boundary
Summary
I T is well known for the finite difference method (FDM) and finite element method (FEM) are usually used to solve a partial differential equation numerically. FDM is to be implemented, but computationally it is not efficient for moving boundary, since the domain mesh grid is not fixed in time. It happens as well in FEM; this method depends on constructing the mesh grid for a given domain. The Hele-Shaw cell concept is implemented to model dendritic form in crystalization, which has been studied following the growth pattern of SafmanTaylor in Hele-Shaw cell [7] This model is used in the application of the surface evolution between viscous fluid and melting crystal, for instance, solidification or melting process. For the detail explanation how this model works for plastic molding and petroleum extraction, see [6]
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