Finite Element Methods For Vibration Analysis Of Elastic Multi-structures

Elastic multi-structures usually consist of a number of elastic substructures with thesame or different dimensions (three-dimensional bodies, plates, rods, etc.) coupled bysome proper junctions, which are widely used in engineering applications. This thesis isintended to present a number of finite element methods for vibration analysis of elastic multi-structures. Error estimates are established and some numerical examples are given to showthe efficiency of these methods.To begin with, some C⁰-continuous time stepping finite element method is proposedfor investigating vibration analysis of elastic multi-beam structures. In the time direction,the C⁰-continuous Galerkin method is used to discretize the generalized displacement field.In the space directions, the longitudinal displacements and rotational angles on beams arediscretized using conforming linear elements, while the transverse displacements on beamsare discretized by the Hermite elements of third degree. An error bound of the method inthe energy norm is obtained. The finite difference analysis in time is developed to assessthe temporal behaviour of the algorithm. The method has also been extended to study somenonlinear problems. A number of numerical tests are included to illustrate the computationalperformance of the method.In addition, the C⁰-continuous time stepping finite element method is applied to solvevibration problems of plane elasticity. In the time direction, we use the C⁰-continuousGalerkin method to discretize the displacement field, and in the space directions, we usethe usual P_r-1-conforming element with r≥2. An error estimate in the energy norm isestablished.Furthermore, the semi and fully discrete finite element methods are proposed for in-vestigating vibration analysis of elastic plate-plate structures. In the space directions, thelongitudinal displacements on plates are discretized by conforming linear elements, and thecorresponding transverse displacements are discretized by the Morley element, leading toa semi-discrete finite element method for the problem under consideration. Applying thesecond order central difference to discretize the time derivative, a fully discrete scheme is obtained, and two approaches for choosing the initial functions are also introduced. An erroranalysis in the energy norm for the semi and fully discrete methods is provided.At last, semi and fully discrete lumped mass finite element methods are proposed forinvestigating vibration analysis of elastic plate-plate structures. In the space directions, thelongitudinal displacements on plates are discretized by conforming linear elements, and thecorresponding transverse displacements are discretized by the Morley element. By meansof the technique of lumped masses, a semi-discrete lumped mass finite element method isobtained. Applying the second order central difference to discretize the time derivative, afully discrete lumped mass scheme is obtained, and two approaches for choosing the initialfunctions are also introduced. An error analysis in the energy norm for the semi and fullydiscrete lumped mass methods is presented.