A new semi-analytical method with diagonal coefficient matrices for potential problems
In this paper, a new semi-analytical method is proposed for solving boundary value problems of two-dimensional (2D) potential problems. In this new method, the boundary of the problem domain is discretized by a set of special non-isoparametric elements that are introduced for the first time in this paper. In these new elements, higher-order Chebyshev mapping functions and new special shape functions are used. The shape functions are formulated to provide Kronecker Delta property for the potential function and its derivative. In addition, the first derivative of shape functions are assigned to zero at any given control point. Finally, using weighted residual method and implementing ClenshawCurtis quadrature, the coefficient matrices of equations system become diagonal, which results in a set of decoupled governing equations for the whole system. This means that the governing equation for each degree of freedom (DOF) is independent from other DOFs of the domain. Validity and accuracy of the present method are fully demonstrated through four benchmark problems.
semi-analytical method diagonal coefficient matrices potential problems non-isoparametric element clenshawcurtis quadrature chebyshev polynomials decoupled differential equations