有限元方法中的光滑积分伪弱形式

A quasi weak form of smoothed integral for finite element method

提出了一种光滑积分伪弱形式,将光滑积分拓展至被积函数非偏导项求解。结合光滑应变技术和伪弱形式,可实现有限元系统方程统一光滑积分求解,即对刚度矩阵和质量矩阵中的应变矩阵和形函数矩阵均可进行光滑积分处理,并转化为光滑子域的边界积分。光滑积分伪弱形式与光滑应变技术比较,增加了形函数矩阵不定积分处理过程,且没有降低有限元求解对形函数连续性的要求。不过,伪弱形式改变了单元积分的求解形式,连续质量矩阵求解也无需坐标映射和雅可比矩阵计算。以轴对称二维问题为研究对象,结果表明极度不规则三角形和四边形单元光滑积分伪弱形式在静态和动态有限元方程求解中也具有很好的精度。

A quasi weak form of smoothed integral is developed for the integrand that does not contain the derivative. In the formulations of finite element method, the smoothed integrals for strain matrix and shape functions can be handled respectively by smoothing strain technique and the present quasi weak form, and all the smoothed domain integrals in the stiffness matrix and consistent mass matrix can be transformed into line integral along boundary of smoothing cells. Comparing with the smoothing strain technique, an indefinite integral of shape functions is added in the quasi weak form, and the requirement for continuity of the shape functions is not decreased. However, the integral form is changed and the coordinate mapping and computing of Jacobian matrix can be avoided in the computation of consistent mass matrix. In this work, the proposed quasi weak form is extended to static and structure dynamic analyses of axisymmetric models. Numerical examples show that the proposed method has a good accuracy and convergence properties even for extremely irregular triangular and quadrilateral elements.