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Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.

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The states of the barrier D Ϫ center which consists of a positive ion located on the z axis at a distance from the x-y plane and two electrons in the same plane bound by the ion are investigated based on a direct diagonalization method in finite and relatively high magnetic fields. The energies of the barrier D Ϫ states, the binding energies for the barrier D Ϫ states, and the expectation values of both the distance of the electron from the origin and distance between two electrons are obtained as functions of the applied magnetic field strength ␥ perpendicular to the x-y plane and the distance between the positive ion and the x-y plane. The effects of the higher Landau levels become small as ␥ and increase. The change of symmetry of the barrier D Ϫ ground state is possible in finite magnetic fields. Both the distance of the electron from the origin and distance between two electrons vary discontinuously with the changes of symmetry of the barrier D Ϫ ground states. Our calculations indicate that the phase transitions of the barrier D Ϫ ground states can be observed in real systems.

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