Discrete variational method for partial differential equations of functionally gradient beam vibration

Abstract

The field of engineering is becoming increasingly complex. In order to adapt to the numerical simulation of solving the partial differential equation of functionally graded beam vibration, a higher order stable numerical algorithm has been constructed. Differential quadrature method is used in discrete space domain. The discrete variational method is constructed in the time domain. The index differential Algebraic equation are obtained by combining the two methods. The discrete variational scheme is constructed for simulation. The results indicate that under long-term simulation, both the velocity and displacement constraints of the Runge Kutta method have defaulted. Displacement constraint values differ by 5 × 10 - 10. The velocity, displacement and acceleration constraints of the discrete variational method are stable. Compared with the Runge Kutta method, the constraint magnitude is reduced. The speed constraint is maintained at within 2.5 × 10 - 15. The displacement constraint level is maintained at within 1 × 10 - 16. This indicates that the discrete variational method has high accuracy and good stability when solving problems such as the vibration equation of functionally graded beams. When the step sizes are h= 0.1 m and h= 0.01 m, the accuracy of the discrete variational method is close. The larger the step size h, the higher the computational efficiency of the discrete variational method. The discrete variational method can maintain structural and energy conservation, making it suitable for long-term simulations. This has a good effect on solving complex problems in the field of partial differential equations.