Computational Mechanics New advances in the forced response computation of periodicstructures using the wave finite element ( WFE ) method
The wave finite element (WFE) method is investigated to describe the harmonic forced response of onedimensional periodic structures like those composed of com plex substructures and encountered in engineering applica tions. The dynamic behavior of these periodic structures is analyzed over wide frequency bands where complex spatial dynamics, inside the substructures, are likely to occu r. Within the WFE framework, the dynamic behavior of periodic structures is described in terms of numerical wave modes. Their computation follows from the consideration of the finite element model of a substructure that involves a large number of internal degrees of freedom. Some rules of thumb of the WFE method are highlighted and discussed to circumvent numerical issues like ill-conditioning and i stabilities. It is shown for instance that an exact analytic relation needs to be considered to enforce the coherence between positive-going and negative-going wave modes. Besides, a strategy is proposed to interpolate the frequency r esponse functions of periodic structures at a reduced number of discrete frequencies. This strategy is proposed to tackl e the problem of large CPU times involved when the wave modes are to be computed many times. An error indicator is formulated which provides a good estimation of the level of accuracy of the interpolated solutions at intermediate poi nts. Adaptive refinement is carried out to ensure that this error indicator remains below a certain tolerance threshold. Numerical experiments highlight the relevance of the propose d approaches.
On the direct solution of wave propagation for repetitive structures
Abstract The direct method of solving the wave constants for a repetitive structure with given frequency ω is developed in this paper. The analogy between structural mechanics and optimal control theory is applied. The dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation. The weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach. The symplectic eigenproblem is derived for skew-symmetric matrices, a cell triangular decomposition for skew-symmetric matrices is introduced to reduce the generalized eigenproblem, and the symplectic Householder transformation is introduced to cell tri-diagonalize the skew-symmetric matrix to solve the eigenequation. The problem of a wave impinging on the boundary is considered by the eigenvector expansion method, which could also be applied to a wave scattering when passing over an abnormal part. Numerical examples show that resonance will occur for the localized vibration case.
A Study of the Changes Over Time in State Anxiety in a Computer-Assisted Instructional Program
The purposes of this study were to analyze (1) the changes over time in the state anxiety level of children in a computer-assisted program of reading instruction and in classroom reading instruction, (2) the changes in state anxiety patterns as related to gender, achievement level, and ethnicity, and (3) the difference in anxiety levels of the students while in computer-assisted reading instruction and classroom reading instruction.