For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary. Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case. The proof for the general boundary instead of the circular boundary is also derived. The Burton{Miller method is employed to eliminate spurious eigenvalues in the multiply connected case. Moreover, a modi ed method considering only the real-part formulation is provided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytical prediction and numerical results are found.
SUMMARYFor a plane elasticity problem, the boundary integral equation approach has been shown to yield a non-unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical di culty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not imbedded in the in uence matrix for any case.
Intact seeds from freshly harvested fruits of Myrica rubra (Sieb et Zucc.) were dormant and required 8 weeks of warm stratification followed by 12 weeks of cold stratification for germination. Exogenous application of gibberellic acid (GA(3)) to intact fresh seeds was effective in breaking dormancy, with > 70% of seeds germinating when treated with 5.2 mM GA(3) and incubated at a day/night temperature of 30/20 degrees C for 20 weeks. Removing the hard endocarp or endocarp plus seed coat of fresh seeds promoted germination, and addition of GA(3) to the embryo accelerated germination. The gibberellins GA(1) and GA(4) were more effective than GA(3) in promoting germination of seeds with the endocarp removed. Endogenous contents of GA(1), GA(3), GA(4), GA(7) and GA(20) were quantified by gas chromatography-mass spectrometry-selected ion monitoring in the endocarps, seed coats and embryos of fresh seeds treated with 5.2 mM GA(3). The content of GA(3) decreased in the endocarp during incubation, whereas GA(1) contents increased in the endocarp and seed coat. A high GA(1) content was detected in the endocarps and embryos of newly germinated seeds. We speculate that GA(3) was converted to GA(1) during incubation and that GA(1) is involved in seed germination. Endogenous abscisic acid (ABA) contents were measured in fresh seeds and in warm and cold stratified seeds. The ABA content in fresh seeds was distributed in the order endocarp > seed coat > embryo, with the content in the endocarp being about 132-fold higher than in the seed coat and embryo. Total ABA content of seeds subjected to warm or cold stratification, or both, was 8.7- to 14.0-fold lower than that of fresh seeds. Low contents of endogenous GA(1), GA(3), GA(7) and GA(20), but elevated contents of GA(4), were found in the seed coats and endocarps of warm plus cold stratified seeds and in the seed coats and embryos of newly germinated seeds. These observations, coupled with the finding that GA stimulated germination of dormant Myrica seeds, provide evidence that endogenous ABA inhibited release of dormancy and that endogenous gibberellins, especially GA(4) or GA(1), or both, are involved in germination.
Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.
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