Vibration analysis for elastic multi‐beam structures by the C⁰‐continuous time‐stepping finite element method

Abstract

SUMMARYSome C 0 -continuous time-stepping finite element method is proposed for investigating vibration analysis of elastic multi-beam structures. In the time direction, the C 0 -continuous Galerkin method is used to discretize the generalized displacement field. In the space directions, the longitudinal displacements and rotational angles on beams are discretized using conforming linear elements, while the transverse displacements on beams are discretized by the Hermite elements of third order. The error of the method in the energy norm is proved to be O(h +k 3 ), where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. The finite difference analysis in time is developed to discuss the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low-frequency regime. The method has also been extended to study some nonlinear problems. A number of numerical tests are included to illustrate the computational performance of the method.

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

“…In the temporal direction, unlike these finite difference schemes in [7,10,11], we develop a C 0 -continuous Galerkin linear finite element method to descritize the time derivatives (cf. [4,[12][13][14][15]). Since the approximate solution function is a continuous piecewise linear polynomial in the whole temporal interval, the regularity assumptions on the exact solution in the error analysis can be relaxed [4,16,17] (see Remark 1 in what follows).…”

“…We establish sharp stability estimates for the proposed numerical method (see Lemma 1). By introducing an unusual interpolation operator Ĩ satisfied by (14) and (15) (cf. [13,18]), we further derive optimal a-priori error estimates respectively in the L 2 norm and the energy norm (see Theorem 1).…”

A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes

“…It is remarked that all these methods are equal-sized in time direction. In this paper, we are devoted to proposing a fully discrete method for linear wave equations with the temporal discretization carried out by the C 0continuous time-stepping methods as used in [21,22] while the spatial discretization by the VEM in [2], respectively. We develop optimal error estimates in the H 1 seminorm and L 2 norm by using the energy method, which are equal to O(h k + τ 3 ) and O(h k+1 +τ 3 ), respectively, where h and τ denote respectively the mesh sizes in space and time, and k denotes the order of the VEM approximation.…”