A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle

Abstract

An accurate and efficient Differential Quadrature Time Finite Element Method (DQTFEM) was proposed in this paper to solve structural dynamic ordinary differential equations. This DQTFEM was developed based on the differential quadrature rule, the Gauss–Lobatto quadrature rule, and the Hamilton variational principle. The proposed DQTFEM has significant benefits including the high accuracy of differential quadrature method and the generality of standard finite element formulation, and it is also a highly accurate symplectic method. Theoretical studies demonstrate the DQTFEM has higher-order accuracy, adequate stability, and symplectic characteristics. Moreover, the initial conditions in DQTFEM can be readily imposed by a method similar to the standard finite element method. Numerical comparisons for accuracy and efficiency among the explicit Runge–Kutta method, the Newmark method, and the proposed DQTFEM show that the results from DQTFEM, even with a small number of sampling points, agree better with the exact solutions and validate the theoretical conclusions.

Keywords

• differential quadrature rule
• time finite element method
• Gauss–Lobatto quadrature rule