Application of meshless local Petrov Galerkin method (MLPG5) for EIT forward problem

Abstract

This paper deals with the application of Meshless Local Petrov Galerkin method(MLPG) for solving the forward problem associated with Electrical Impedance Tomography(EIT). Out of six variants of MLPG, here we have used MLPG5 to solve the forward problem. Use of Heaviside function as test function eliminates the domain integral term from the weak formulation. Radial point interpolation method(RPIM) is used and the resulting shape functions possess Kronecker delta function property. Hence no modification is required in the weak form to impose essential boundary conditions. Aforementioned advantages have motivated us to apply this meshless method for EIT forward problem. Numerical accuracy of this method is determined by finding the relative error between numerical solution and exact solution. There are several factors that can affect the accuracy such as local sub domain size(αsub), influence domain size(αinf), nodal density, type and shape of radial basis functions used. A detailed parametric study is carried out in this paper. Radial basis functions(RBF) such as Multiquadrics(MQ), Inverse Multiquadrics(INVMQ) and Gaussian are used for function interpolation along with linear polynomial. Shape parameter of RBFs is also varied to study the effect on accuracy. The solution obtained from the meshless method and analytical method are compared with solutions from Finite Element Method using linear triangular elements. Relative error is found to be minimum when αsub lies in the range of 0.6–0.7 for MQ and INVMQ and αinf > 5 since the boundary voltage profile should be the same when current is injected into the circular medium from any angle under homogeneous conditions. For Gaussian function, minimum error is observed for αsub in the interval 0.5–0.6 and αinf > 5. Gaussian has shown the lowest error compared to MQ and INVMQ. Forward problem in EIT usually deals with nonhomogeneous medium and therefore we need to extend this study to nested domains. Since it is very difficult to get exact solution for nonhomogeneous medium, same conductivity value is given for both background(outer domain) and the inhomogeneity(inner domain) with some additional coupling conditions to be satisfied at the interface. Resulting boundary potentials are matched with the exact solution of homogeneous medium. MQ is used for MLPG5 implementation and error is calculated for different values of αsub and αinf. Minimum error is obtained at αsub = 0.6 for all values of αinf. It is better to select αinf > 5 to ensure the boundary profiles to overlap in all current injection cycles.