Time-discontinuous Galerkin Quadrature Element Method For Multi-degree-of-freedom Dynamics

With the development of modern engineering technology,engineering structure is becoming more and more complex.People put forward higher requirements for the dynamic analysis of engineering structures to get complex structural dynamic characteristics,thus,meet the requirements of structural seismic fortification.With the rapid development of computer technology,high-precision solutions of dynamic characteristics of complex structures can be obtained by numerical methods.Weak form quadrature element method is a kind of high-order interpolation numerical calculation method developed recently.It has been widely studied because of its high accuracy and high efficiency.At the same time,the time-discontinuous Galerkin method allows the displacement and velocity to be discontinuous in the time domain.This time-discontinuous structure helps to realize the convergence of the method and estimat errors.It has the advantages of unconditional stability and fast convergence in dealing with dynamic problems.This paper attempts to combine the time-discontinuous Galerkin method with the quadrature element method to establish the time-discontinuous Galerkin quadrature element method for solving multi-degree of-freedom problems.With the active research of scholars,time-discontinuous Galerkin method appears in many forms.In this paper,we discuss the time-discontinuous Galerkin method,the time-discontinuous Galerkin method with least square term,and the two field timediscontinuous Galerkin method with displacement and velocity independent interpolation.By combining the quadrature element method with the above methods,the corresponding time-discontinuous Galerkin quadrature element method is established.The dynamic properties of the algorithm are deeply studied,and the feasibility of the proposed method is verified by some classical examples.In this paper,three discontinuous Galerkin quadrature element methods are unconditionally stable,low dispersion and non overshoot dynamic methods.The timediscontinuous Galerkin method/least square includes a variable parameter τ,the timediscontinuous Galerkin method is a special case of τ = 0.We can control τ to filter the dynamic response of a specific frequency band.The cost of the two field method is higher than that of the single field method,and the accuracy of the two field method is one order higher than that of the single field method.Compared with the Newmark method,Wilson-θ method and other classical dynamic methods,this method has much smaller numerical dissipation and dispersion in the low frequency band.In the undamped system,"superconvergence phenomenon" appears,and the algorithm shows high convergence rate in a certain interval.The timediscontinuous Galerkin method /least square,and the two field time-discontinuous Galerkin method realizes the " asymptotic annihilation ",and filters the influence of false high-frequency modes due to artificial discretization in the multi-degree-offreedom system,but still maintains the accuracy of low-frequency mode results.Three examples are used to study the performance of each algorithm.Both damped and undamped systems are included,which proves that the physical damping of the system will affect the accuracy of the algorithm.A classical example of structural dynamics,a multi-degree-of-freedom system with rigid and flexible parts,is selected to discuss the dissipation characteristics of the algorithm at high frequency.In the study of the algorithm,"bifurcation " has caused great trouble to the work.We explain this phenomenon by analogizing it to a simple harmonic vibration system.At the end of the paper,the derivation of two field time-discontinuous Galerkin quadrature element is given.The research shows that the algorithm needs more computational cost,but can not get better algorithm accuracy,so this paper does not discuss it too much.