Integrated Local Petrov-Galerkin Sinc Method for Structural Mechanics Problems

A novel method for solving static boundary-value problems named the integrated local Petrov-Galerkin sine method is introduced. The method uses the process of numerical indefinite integration based on the double-exponential transformation to develop basis functions for a local Petrov-Galerkin type numerical method. Because the developed basis functions do not satisfy the Kronecker delta property, essential boundary conditions are imposed using the traditional penalty method and the Lagrange multiplier method. Three basis functions are introduced, and the accuracy and efficiency of the method is examined for two problems: a one-dimensional tapered bar with vanishing tip area and a two-dimensional plane-stress elasticity problem. The numerical results indicate that the integrated local Petrov-Galerkin sine method can provide greater accuracy than the sine method based on Interpolation of highest derivative. For the two example problems studied, the method's high rate of convergence can provide greater accuracy of stresses for the same computational cost as a displacement-based C-0-continuous and a mixed finite element. However, the still in development integrated local Petrov-Galerkin sine method suffers from requiring a more fully populated stiffness matrix and relatively high computational cost of the matrix factorization.