Integrated Local Petrov-Galerkin Sinc Method for structural mechanics problems Wesley C.H. Slemp, * Rakesh K. Kapania † and Sameer B. Mulani ‡ Virginia Polytechnic Institute and State University, Blacksburg, VA, 24060 An integrated, local Petrov-Galerkin Sinc method is introduced and applied to static structural mechanics problems. The method approximates the highest derivative in the local weak form of the governing equation on a rectangular grid, and the lower derivatives and unknown function are found by numerical indefinite integration. We suggest that the essential boundary conditions may be applied by the traditional penalty method or by a method in which the stiffness matrix is reduced to eliminate dependent degrees of freedom. We propose and compare the performance of three basis functions in terms of their accuracy and convergence properties for two problems: a one-dimensional tapered bar and a two-dimensional plane-stress elasticity problem. Our results indicate that the present method can provide greater accuracy than the Sinc method based on Interpolation of Highest-Derivative, an integration based collocation method suggested by Li and Wu [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling, Vol. 31, No. 1, 2007, 1-9]. However, the present weak form based approach requires generally greater computational time for the same number of Sinc points. I. Introduction In recent years, Sinc approximation has been a highly studied topic in the research literature particu- larly in conjunction with methods for numerical integration and solving two-point boundary value problems (BVP). Sinc approximation has been used as the basis function in both the Sinc-collocation method and the Sinc-Galerkin method because of the ease with which it may handle the presence of singularities or un- bounded domains. 1 Further, the Sinc function is highly effective at capturing oscillating behavior in space, and is thus quite useful for solutions with such characteristics. Difficulties in the Sinc-collocation method arise in applying the method to a BVP with mixed Neumann type boundary conditions because the derivatives of the Sinc functions at the boundaries are undefined. 2 Narsimhan et al. 3 used finite difference method to calculate the derivatives of the dependent variables near the boundaries. Wu, Li, and Kong 2 addressed this issue by introducing a Sinc-collocation method with boundary treatment (SCMBT) which they showed to provide good convergence and easy treatment of the boundary conditions. A lot of attention has been given to the Sinc-Galerkin method and its efficiency has been proved for both linear and nonlinear BVP’s. 4–7 Al-Khaled 8 compared the Sinc-Galerkin method with He’s homotopy perturbation method for singular two-point BVP’s. 9, 10 El-Gamel 11 applies the Sinc-Galerkin method to a fifth-order BVP and compares the results with sixth-degree B-Spline functions. The results of El-Gamel’s study 11 indicate that the Sinc-Galerkin generally performs better than the B-spline approach. The traditional finite element method, the Sinc-Galerkin and Sinc-collocation methods, and most meshless methods approximate the primary variables through interpolation and the derivatives of the basis functions are computed. With such methods, errors in the primary variable are amplified through differentiation. * National Defense Science and Engineering Graduate (NDSEG) Research Fellow, Aerospace and Ocean Engineering, Blacks- burg, VA, Student Member, AIAA, Sensors and Structural Health Monitoring Group. † Mitchell Professor, Aerospace and Ocean Engineering, Blacksburg, VA, Associate Fellow, AIAA, Sensors and Structural Health Monitoring Group. ‡ Post Doctoral Fellow, Aerospace and Ocean Engineering, Blacksburg, VA, Member, AIAA 1 of 30 American Institute of Aeronautics and Astronautics 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th 4 - 7 May 2009, Palm Springs, California AIAA 2009-2392 Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.