Do Wan Kim

Axial Green's function method for steady Stokes flow in geometrically complex domains

Axial Greens function method (AGM) is developed for the simulation of Stokes flow in geometrically complex solution domains. The AGM formulation systematically decomposes the multidimensional steady-state Stokes equations into 1D forms. The representation formula for the solution variables can then be derived using the 1D Greens functions only, from which a system of 1D integral equations is obtained. Furthermore, the explicit representation formula for the pressure variable enable the unique AGM approach to facilitating the stabilization issue caused by the saddle structure between velocity and pressure. The convergence of numerical solutions, the simple axial discretization of complex solution domains, and the nature of integral schemes are demonstrated through a variety of numerical examples.

Localized axial Green's function method for the convection-diffusion equations in arbitrary domains

A localized axial Greens function method (LAGM) is proposed for the convection-diffusion equation. The axial Greens function method (AGM) enables us to calculate the numerical solution of a multi-dimensional problem using only one-dimensional Greens functions for the axially split differential operators. This AGM has been developed not only for the elliptic boundary value problems but also for the steady Stokes flows, however, this paper is concerned with the localization of the AGM. This localization of the method is needed for practical purpose when computing the axial Greens function, specifically for the convection-diffusion equation on a line segment that we call the local axial line. Although our focus is mainly on the convection-dominated cases in arbitrary domains, this method can solve other cases in a unified way. Numerical results show that, despite irregular types of discretization on an arbitrary domain, we can calculate the numerical solutions using the LAGM without loss of accuracy even in cases of large convection. In particular, it is also shown that randomly distributed axial lines are available in our LAGM and complicated domains are not a burden.

Extended Meshfree Point Collocation Method for Electromagnetic Problems With Layered Singularity

A novel meshfree point collocation method for solving the electromagnetic potential problems with layered singularity is presented. The approximation based on the moving least squares technique is enriched by Heaviside step function, scissor function and wedge function. It effectively models discontinuities in electromagnetic potential and its gradient fields. Direct discretization of the partial differential equations yields the system of equations so that the computational speed is very fast compared to the Galerkin formulation. Solving the system provides the nodal solutions and jump strengths on the singular layer at the same time. Numerical results show that the developed method is highly efficient and reliable in solving electrostatic problems with charge layer or insulating layer.

A computational modeling of blood flow in asymmetrically bifurcating microvessels and its experimental validation: Modeling of blood flow in asymmetrically bifurcating microvessels

Microvascular transport is complex due to its heterogeneity. Many researchers have been developing mathematical and computational models in predicting microvascular geometries and blood transport. However, previous works were focused on developing simulation models, not on validating suggested models with microvascular geometry and blood flow in the real microvasculature. In this paper, we suggest a computational model for microvascular transport with experimental validation in its geometry and blood flow. The geometry is generated by controlling asymmetric conditions of microvascular network. Also, the blood flow in microvascular networks is predicted by considering in vivo viscosity, Poiseuille flow model, and hematocrit redistribution by plasma skimming. The suggested model is validated by the measured data in rat mesentery. Also, the microvascular transport in a case of mouse cortex is predicted and compared against experimental data to check applicability of the suggested model.

Axial Green Function Method for Axisymmetric Electromagnetic Field Computation

1D Green functions are applied to efficiently calculate the electrostatic field axisymmetric in 3D. The point of this paper is laid on the computation of the axisymmetric numerical solution which may have derivative discontinuity due to the transmission condition across the interface between different kinds of matter. A combined axial Green function method(AGM) is proposed to accurately solve this axisymmetric problem.

Axial green function method for axisymmetric electromagnetic field computation

Only with 1-D Green function for 1-D elliptic differential operator, we can solve 2-D/3-D general elliptic problems by applying the axial Green function method (AGM). An extension of AGM is proposed to enforce Neumann boundary conditions. This extension is directly available for 2-D problems with straight boundaries parallel to axes on which Neumann boundary conditions are assigned. It is thoroughly attributed to the specific axial Green functions associated with the Neumann conditions. Moreover, since this extended AGM (XAGM) in 1-D satisfies the transmission condition across an interface along which the permittivity is discontinuous, it can be applied to 2-D problems with interfaces parallel to axes without loss of accuracy. Finally, we apply the XAGM in 2-D to 3-D axisymmetric electric potential problems with variable and/or even discontinuous permittivities along interfaces. Owing to the cylindrical coordinate transform, the transformed problem is tractable to solve using this XAGM. Arbitrary distribution of axial lines is available, which must be a marked advantage of XAGM compared to other methods.