弱连续条件下的九参三角形板元

THE 9-PARAMETERS TRIANGULAR PLATE ELEMENT IN WEAK FORM

首先对薄板弯曲平衡方程的弱形式进行了推导,导出保证单元收敛的弱协调条件,即三角形顶点函数值连续和三边的法向导数积分连续这两个条件;对比拟协调元、广义协调元和双参数法中所使用的3个积分连续条件,本条件更弱;再对这3个积分协调条件的构成方法进行了总结和分析,现有采用积分连续条件构造的有限元大都采用了这些构成方法.采用弱协调条件构造有限元,比原来的构造范围更广,井以此构造出几种单元作为算例.采用这种构成法还可构造多种单元,它们都具有采用最小势能原理法构成有限元的简便的优点,并在任意网格下收敛到真解.

The equilibrium equations for the thin plate bending problem in weak form are presented in this paper, and proposed the weak conforming conditions assured the convergence, that is, the displacement continuity at the triangle vertex and the normal derivative integral continuity along the boundary of the triangular element. The proposed weak continuity condition seems much weaker compared with the three kinds of integral continuity conditions which are used in the quasi-conforming element, the generalized element and double set parameter method. The constructed conditions of the above three integral conforming methods have been summarized and analyzed in this paper, the common used finite elements which satisfy the integral continuity conditions are mostly based on the above method. To construct finite elements by the weak conforming condition presented in this paper, we will have a more extensive range of choices, and several elements are constructed as numerical examples. More types of finite element can be constructed using this method, which possess the advantages of simplicity and convenience, just as the common used finite element based on the principle of minimum potential energy, and can converge to the analytical solution in the limit of arbitrary mesh sub-divisions.