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For solving electroencephalographic forward problem, coupled method of finite element method (FEM) and fast moving least square reproducing kernel method (FMLSRKM) which is a kind of meshfree method is proposed. Current source modeling for FEM is complicated, so source region is analyzed using meshfree method. First order of shape function is used for FEM and second order for FMLSRKM because FMLSRKM adopts point collocation scheme. Suggested method is tested using simple equation using 1-, 2-, and 3-dimensional models, and error tendency according to node distance is studied. In addition, electroencephalographic forward problem is solved using spherical head model. Proposed hybrid method can produce well-approximated solution.

Electroencephalography (EEG) is a technique to measure and analyze the scalp potential [

Solving forward problem of EEG is to obtain potential distribution of head due to neuronal electric current [

Because meshfree methods do not need finite elements, mesh generation process is not required. So, difficult situation to generate finite elements such as small objects inside a large analysis domain [

In this paper, a new hybrid method of meshfree method and FEM for forward solver is proposed. There were some tries to surmount disadvantages of FEM and meshfree method by combining two methods in previous studies [

In FMLSRKM [

After moving least square scheme is applied, approximated solution (

For the sake of an explanation, in this section, 1-dimensional shape functions are considered. Because point collocation method is adopted for FMLSRKM, quadratic shape function is used. In FEM, influence of a shape function is confined in an element (see Figure

Shape functions of (a) FEM and (b) FMLSRKM. A shape function of FEM is a partition of unity, but that of FMLSRKM is not.

So, an interfacial area between FEM domain and FMLSRKM domain has one finite element shape function and many FMLSRKM shape functions. Hence, adequate weighting functions should be considered for shape functions of the hybrid method. While linear weighting functions are applied on both FEM and meshfree in the previous study [

Combination of shape functions in an interfacial area of FEM and FMLSRKM. (a) Shape functions of FEM, (b) shape functions of FMLSRKM, and (c) shape functions of hybrid method. Shape function of FMLSRKM is fully adopted as shape function of the interfacial area (inside two vertical dotted lines).

The suggested hybrid method has some advantages. First, the system matrix of the hybrid method is sparse almost same as that of FEM, if most of analysis domain can remaine as finite element and meshfree region is well defined and specific. Second, adaptive approach is easy for meshfree region. In many cases, meshfree region may be a high-error area, because a region which is difficult to analyze is selected as meshfree region. Third, point collocation scheme is used for FMLSRKM, which makes applying boundary condition easy. Forth, the region of FMLSRKM has less error, since FMLSRKM of this paper has quadratic shape function.

Tests are performed using 1-, 2-, and 3-dimensional models with

Test problems. In these problems, numbers, variables, and solutions are unitless. (a) 1D, (b) 2D. The solid line along

In Figures

The results of the hybrid method. (a), (b), and (c) are the results of Figures

In this paper, error was evaluated using various intervals of nodes, or

The error of approximation by the hybrid method. The error of FEM is higher than that of FMLSRKM because of the orders of bases for two methods.

To investigate the error of FMLSRKM region, inhomogeneous forcing term should be applied. So,

Errors of the hybrid method along

Examples of FEM and FMLSRKM. (a) Linear approximation of FEM shows maximum difference at the middle of the interval, and (b) Quadratic approximation of FMLSRKM crosses the exact solution.

Relative error of FEM and FMLSRKM according to the interval of nodes.

Governing equation and boundary condition of EEG forward problem are

Test for EEG forward problem. (a) A dipole shown as an arrow is in the middle of FMLSRKM region, (b) result using FEM, and (c) result using the suggested hybrid method.

A hybrid method of FMLSRKM and FEM is suggested in this paper, and it shows good result of 1-, 2-, and 3-dimensional boundary value problems. In addition, error of the hybrid method is studied, and proposed method shows good convergence in 2-dimensional model. In Figure

We also tested the hybrid method for EEG forward problem and verified that the method can produce small-error solution. Actually, it is possible to reduce error of solution by

This research shows that Poisson and Laplace equations were successfully handled by the suggested hybrid method. Therefore, realistic problems can be treated such as moving objects and singular points, and it is expected that the hybrid method will take its own role for problems with special conditions as well as EEG forward problem.

This study was supported by a grant from the Korea Healthcare Technology R and D Project, Ministry for Health, Welfare and Family Affairs, Republic of Korea (A090794).