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.
Highlights
Various algorithms are available for numerical integration of Ordinary Differential Equations (ODEs)
A highly accurate and highly efficient Differential Quadrature Time Finite Element Method (DQTFEM) was developed in the present study by using
A highly accurate and highly efficient DQTFEM was developed in the present study by using the DQ rule, the Gauss–Lobatto integration rule, and the general form of Hamilton’s variational principle for solving structural dynamic problems involving damping and external forces
Summary
Various algorithms are available for numerical integration of Ordinary Differential Equations (ODEs). The DQFEM uses Lagrange trial functions for C0 and C1 FEs and possesses high accuracy of DQM and the generality of standard FEM In this context, similar to DQFEM for spatial domain [47,48], Hamilton’s variational principle together with the DQ rule and the Gauss–Lobatto quadrature rule are used to develop a TFEM in the present work, where the initial conditions can be readily imposed as in standard FEM. Similar to DQFEM for spatial domain [47,48], Hamilton’s variational principle together with the DQ rule and the Gauss–Lobatto quadrature rule are used to develop a TFEM in the present work, where the initial conditions can be readily imposed as in standard FEM This novel method is called the DQ Time Finite Element Method (DQTFEM).
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