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The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibration and the upper bound solution of the forced vibration of solid structures, by making use of the important softening effects of the node-based smoothed finite element method (NS-FEM). Through the gradient smoothing technique, the strain-displacement matrix is obtained in the smoothing domain based on the element mesh nodes. Subsequently, the stiffness matrix is computed in a manner consistent with the standard finite element method (FEM). Here, the practical Lanczos algorithm and the modal superposition technique are employed to obtain the frequencies, modes, and transient responses of a given homogeneous structure. For three-dimensional (3D) solid structures, the automatically generated four-node tetrahedron (T4) element meshes are utilized. The results obtained from the NS-FEM are compared with the standard FEM in terms of accuracy, convergence and computational efficiency.

The Lanczos method has rapidly become the preferred method of solution for the generalized eigenvalue problems. The recent emergence of parallel computers has aroused much interest in the practical implementation of the Lanczos algorithm on these high performance computers. This paper describes an implementation of a generalized Lanczos algorithm on a distributed memory parallel computer, with specific application to structural dynamic analysis.

One major cost in the parallel implementation of the generalized Lanczos procedure is the factorization of the (shifted) stiffness matrix and the forward and backward solution of triangular systems. In this paper, we review a parallel sparse matrix factorization scheme and propose a strategy for inverting the principal block submatrix factors to facilitate the forward and backward solution of triangular systems on distributed memory parallel computers. We also discuss the different strategies in the implementation of mass-matrix-vector multiplication and how they are used in the implementation of the Lanczos procedure. The Lanczos procedure implemented includes partial and external selective reorthogonalizations. Spectral shifts are introduced when memory space is not sufficient for storing the Lanczos vectors. The tradeoffs between spectral shifts and Lanc-zos iterations are discussed. Numerical results on Intel’s parallel computers, the iPSC/860 hypercube and the Paragon machines will be presented to illustrate the effectiveness and scalability of the parallel generalized Lanczos procedure.

We review in this article a recently proposed energy-global method that is capable of calculating the entire transition amplitude matrix with a single Lanczos propagation. This method requires neither explicit computation nor storage of the eigenfunctions, rendering it extremely memory efficient. Procedures are proposed to handle situations where "spurious" eigenvalues aggregate around true eigenvalues due to round-off errors. This method is amenable to both real-symmetric and complex-symmetric Hamiltonians. Applications to molecular spectra and reactive scattering are presented. Its relationships with other methods are also discussed.

Using an efficient single Lanczos propagation method, we report the resonance emission spectra of HCN and DCN from a number of low-lying vibrational levels of the Ã-state manifold. Our calculations represent the first such undertaking in which a high-quality ab initio based potential energy surface of the excited (ù A″) state and a transition dipole surface were used. The results show a significant improvement over previous theoretical work in reproducing experimental stimulated emission pumping spectra of HCN. The improved theory-experiment agreement is attributed to the accurate Ã-state potential energy surface, while the impact of the transition dipole function was found to be relatively minor.

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.