A meshfree weak-strong (MWS) form method for time dependent problems

Abstract

A meshfree weak-strong (MWS) form method, which is based on a combination of both the strong form and the local weak form, is formulated for time dependent problems. In the MWS method, the problem domain and its boundary are represented by a set of distributed field nodes. The strong form or the collocation method is used to discretize the time-dependent governing equations for all nodes whose local quadrature domains do not intersect with natural (derivative or Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The natural boundary conditions can then be easily imposed to produce stable and accurate solutions. The moving least squares (MLS) approximation is used to construct the meshfree shape functions in this study. Numerical examples of the free vibration and dynamic analyses of two-dimensional structures as well as a typical microelectromechanical system (MEMS) device are presented to demonstrate the effectivity, stability and accuracy of the present MWS formulation.