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 forced response of waveguides using the wave and finite element method
The forced response of waveguides subjected to time harmonic loading is treated. The approach starts with the wave and finite element (WFE) method where a segment of the waveguide is modeled using traditional finite element methods. The mass and stiffness matrices of the segment are used to formulate an eigenvalue problem whose solution yields the wave properties of the waveguide. The WFE formulation is used to obtain the response of the waveguide to a convected harmonic pressure (CHP). Since the Fourier transform of the response to a general excitation is a linear combination of the responses to CHPs, the response to a general excitation can be obtained via an inverse Fourier transform process. This is evaluated analytically using contour integration and the residue theorem. Hence, the approach presented herein enables the response of a waveguide to general loading to be found by: (a) modeling a segment of the waveguide using finite element methods and post-processing it to obtain the wave characteristics, (b) using Fourier transform and contour integration to obtain the wave amplitudes and (c) using the wave amplitudes to find the response at any point in the waveguide. Numerical examples are presented.
The Influence of the Supervision of Psychotherapy upon being Therapeutic: II: Modes of Influence of the Supervisory Relationship *
Through supervision, the student develops those innate personal characteristics which are highly important in being therapeutic. In making conscious use of “reality identification” by the student with his teacher, we must take care to avoid certain pitfalls. What is not wanted is transference identification. To help avoid this, we must select appropriate teachers as supervisors. The student should also have more than one supervisor, as this tends to protect against the development of over-idealization on the part of the student, and of omnipotence on the part of the teacher. Personal characteristics of both students and supervisors are extremely important. A good supervisory experience is the most likely vehicle for developing those qualities in the student which will, in time, lead him to function most therapeutically.