A symplectic finite element method based on Galerkin discretization for solving linear systems

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems. For the dynamic responses of continuous medium structures, the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation. Thus, a symplectic finite element method with energy conservation is constructed in this paper. A linear elastic system can be discretized into multiple elements, and a Hamiltonian system of each element can be constructed. The single element is discretized by the Galerkin method, and then the Hamiltonian system is constructed into the Birkhoffian system. Finally, all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme. Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate, it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm. The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.