基于DDA的弹性力学全高阶多项式位移逼近方法及其实例验证

基于维尔斯特拉斯多项式函数的逼近定理，通过DDA高阶全多项式位移函数条件下的弹性力学推导，提出了一个逼近弹性力学连续位移函数真解的全多项式位移函数逼近方法。该方法采用完整的高阶多项式位移函数，以不同阶次条件下的多项式系数为未知数，以单纯形积分为解析积分方法，通过建立和求解平衡方程，逐步逼近弹性体真解。在对单纯形积分计算过程研究的基础上，给出了三维空间单纯形计算图解法，该图解法诠释了三维空间单纯形积分公式中各变量间的逻辑关系及计算过程的图形表达。基于上述方法，编写了相应计算程序，并以一个三维简支梁受均布荷载及一个四周固定的弹性薄板受集中力作用两算例为实例，验证了所提方法的可行性。实例计算结果表明，随着逼近函数阶次的提高，数值方法获得的多项式函数计算值均单调地逐步逼近解析解。在文中所用的6阶多项式函数逼近中，简支梁实例位移计算误差小于0.2%，弹性薄板实例位移误差小于0.91%，并且，两算例与解析解位移差值都在微m级。

A DDA based complete and high order polynomial displacement approximation method in elastic mechanics and its cases verification

According to Weierstrass theorem for polynomial approximation, and based on elastic mechanics derivation where the complete and high order polynomial displacement function used in traditional DDA is employed, a DDA based complete and high order polynomial function approximation method is presented in this paper. In the method mentioned above, the complete and high order polynomial function is used as displacement function of the elastic domain, the coefficients of the basic functions of coordinate variables in polynomial are used as unknowns, and the simplex integration method is used as the analytic integration, and the simultaneous equations are established and solved in order to approach the elastic solution. Based on the study of the calculating process for simplex integration, a diagram interpretation in three-dimensional condition is presented, which is helpful to illustrate the logical relationship among the variables in the simplex integration formula and the whole calculation process for integration in a given tetrahedron. At the end of this paper, the corresponding calculation code is developed and two calculation cases, one is a three-dimensional simply supported beam with uniform loading while the other is a boundary fixed elastic plate with point loading, verify the feasibility of this method. It can be indicated by the calculation result that, with the increase of the order of polynomial function, the result of polynomial function calculated by this numerical method approximate the analytical solution monotonously. According to the polynomial approximation function with six-order in this paper, the error of displacement for simply supported beam is less than 0.2%, while the error of displacement for elastic plate is less than 0.91%. In addition, in two cases, the displacement difference, compared to the analytical solution, is within micrometer.