弹性力学中的立兹法和屈列弗兹法的一般推导
The General Derivation of Ritz Method and Trefftz Method in Elastomechanics
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摘要: 本文采用一般的数学表示形式推导了线弹性力学中的立兹法和屈列弗兹法,证明了立兹法给出相应泛函极值的上限,屈列弗兹法则给出其下限.同时发现,特征值问题(例如自振频率问题)泛函变分法中的屈列弗兹法同求特征值的放松边界条件下限法是一致的.当然,此处的推导结果,也适用于一类泛函的变分法中,这类泛函的欧拉方程是线性正定的.Abstract: This paper derives the Ritz method and Trefftz method in linear elastomechanis with the help of general mathematical expressions. Thus it is proved that Ritz method gives the upper bound of the corresponding functional extremurn, while Trefftz method gives its lower bound. At the same time it has been found that the eigenvalue problem (e.g. thenatural frequency problem) concerning the functional variational method in Trefftz method is in concord with the lower bound method of the loosened boundary condition which seeks for the eigenvalue. Of course, the results of this derivation are also applicable to the sort of functional variational method of which Euler's equation is linear positive definite.
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[1] 钱伟长,《变分法和有限元》,上册,科学出版社(1980). [2] Weinstein, A. and Chien, W. Z.(钱伟长), On the vibration of a clamped plate under tension. Quarterly of Applied Mathematics,1, 1(1943), 61-68. [3] Weinstein, D. H., Modified Ritz Method. Proc. Nat. Acad. Sci., 20 (1934),529-532. [4] MacDonald, J. K. L.,on the modified variational method, Phys. Rev., 46(1934), 828-829. [5] Kohn, W., A note on Weinstein's variational method. Phys. Rev., 71 (1947), 902. -
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