Abstract
A novel hybrid method, which simultaneously possesses the efficiency of Fourier spectral method (FSM) and the applicability of the finite element method (FEM), is presented for the vibration analysis of structures with elastic boundary conditions. The FSM, as one type of analytical approaches with excellent convergence and accuracy, is mainly limited to problems with relatively regular geometry. The purpose of the current study is to extend the FSM to problems with irregular geometry via the FEM and attempt to take full advantage of the FSM and the conventional FEM for structural vibration problems. The computational domain of general shape is divided into several subdomains firstly, some of which are represented by the FSM while the rest by the FEM. Then, fictitious springs are introduced for connecting these subdomains. Sufficient details are given to describe the development of such a hybrid method. Numerical examples of a one-dimensional Euler-Bernoulli beam and a two-dimensional rectangular plate show that the present method has good accuracy and efficiency. Further, one irregular-shaped plate which consists of one rectangular plate and one semi-circular plate also demonstrates the capability of the present method applied to irregular structures.
Highlights
The needs for engineers to accurately predict performance of dynamic systems and to produce optimal designs as well as fast progress in hardware performance and constant decrease in the price of computers all lead to increasing popularity and application of the finite element method (FEM) in engineering
The purpose of the current study is to extend the Fourier spectral method (FSM) to problems with irregular geometry via the FEM and attempt to take full advantage of the FSM and the conventional FEM for structural vibration problems
A straight beam with the following parameters is used in the calculation: beam length L = LFEM+ LFSM = 1 m, in which LFEM and LFSM represent the length of the FEM and FSM zones on the beam structure; Young’s modulus E = 2.1 × 1011 Pa; the second moment of inertia of the cross-section I = πr4/4 with r = 0.01 m
Summary
The needs for engineers to accurately predict performance of dynamic systems and to produce optimal designs as well as fast progress in hardware performance and constant decrease in the price of computers all lead to increasing popularity and application of the FEM in engineering. Besides simplifying a model to reduce computational workload, many attempts have been made within the last decades in order to replace or improve the conventional FEM with other methods [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] Some of these are able to solve mid-frequency vibration problems with some degrees of success. As a novel Fourier expansion method, the convergence speed of the FSM is improved substantially compared with the traditional Fourier series method It is a suitable systematic method for solving a wide range of engineering and scientific problems. Compared with local methods such as the FEM, as an analytical method which uses global basis functions, the FSM shows intrinsic difficulties in satisfying boundary conditions for irregular-shaped domains like other global methods [16] such as spectral methods or spectral finite element methods [13, 14]
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