An efficient matrix tridiagonalization method for 3D finite element analysis of free vibration Abstract   In order to avoid mechanical resonance, a vibrating structure needs to be designed such that its working frequency interval is sufficiently far from its natural frequencies. Natural frequencies of a mechanical system are obtained from free vibration analysis commonly done by the finite element method. One important step in the analysis is matrix tridiagonalization commonly performed by the block Lanczos method. However, the classical block Lanczos method suffers numerical instability due to the loss of matrix orthogonality. In this paper, we demonstrate the implementation of the block Lanczos method with an orthogonality fixing scheme for 3D free vibration problems. The solution accuracy and computational time of this method are compared with those of the classical block Lanczos method and the Householder method. The results show that the block Lanczos method with the orthogonality fixing scheme employed in this work can effectively avoid the numerical instability due to the loss of matrix orthogonality. Furthermore, the block Lanczos method with the orthogonality fixing scheme provides solution accuracy as good as the Householder method while using significantly less computational time.

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application In this paper, a leastsquares spectral method and a nonconforming leastsquares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a suitable preconditioner and show that the condition number of the preconditioned system is (O((ln W)2)) (where W is degree of polynomials). Then we establish the error estimates of the proposed method. Specific numerical results based on linear elasticity problem, fourth order problem and sixth order problem are discussed to reflect the efficiency of the proposed method.