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In order to improve the performance of the membrane element with vertex rigid rotational freedom, a new method to establish the local Cartesian coordinate system and calculate the derivatives of the shape functions with respect to the local coordinates is introduced in this paper. The membrane elements with vertex rigid rotational freedom such as GQ12 and GQ12M based on this new method can achieve higher precision results than traditional methods. The numerical results demonstrate that the elements GQ12 and GQ12M with this new method can provide better membrane elements for flat shell elements. Furthermore, this new method presented in this paper offers a new approach for other membrane elements used in flat shell element to improve the computing accuracy. [ABSTRACT FROM AUTHOR]

The existence of stability switches and Hopf bifurcations for the second-order delay differential equation x′′t+ax′t-τ+bxt=0,  t>0, with complex coefficients, is studied in this paper. [ABSTRACT FROM AUTHOR]