基于欧拉插值的最小二乘混合配点法在弹性力学平面问题中的应用

APPLICATION OF THE EULER INTERPOLATION-BASED LEAST SQUARE MIXED COLLOCATION METHOD IN ELASTIC PLANE PROBLEMS

由于常规配点型无网格法存在求解不稳定、精度差和求解高阶导数等问题,提出了基于欧拉插值的最小二乘混合配点法。该方法同时以位移和应变作为未知量,通过欧拉插值将未知变量的导数表达出来,同时在插值中引入高斯权函数,并代入微分方程,从而形成以位移和应变为未知数的超定方程组,然后形成最小二乘意义下的法方程,法方程和相应的位移边界条件、应力边界条件一起形成定解体系。该方法不需要域积分,是一种真正的无网格法。一些典型的弹性力学平面问题表明本文方法具有良好的精度。

Conventional collocation meshless methods suffer from instability, poor accuracy and high-order derivation problems in the solution processes. The least square mixed collocation method based on the Euler interpolation is proposed in this study to overcome the above problems. Both the displacement and the strain are unknowns, the derivatives of which are expressed by using the Euler interpolation. The Gaussian weight function is introduced, resulting in over-determined systems. The over-determined systems are to be solved subjected to boundary conditions in terms of stress and displacement. This proposed method does not require domain integration, which is essentially truly meshless. Typical examples of plane elastic problems are used to demonstrate the high accuracy of the method.