基于整体位移模式的平面孔洞结构数值模拟方法

针对平面孔洞问题提出了一种新的数值模拟方法。本文通过水平集方法引入孔洞边界、力边界和位移边界水平集函数，利用边界水平集函数来构造边界试探项，将试探空间表示为二元幂级数与边界试探项的线性组合；同时提出一种基于水平集方法的位移边界条件施加方法，利用位移边界水平集函数来构造满足位移边界条件的近似位移场，并给出了相应的刚度矩阵和载荷矩阵表达式。与FEM、XFEM、无网格法等方法相比，该方法无需将求解域离散，具有较低的计算成本、特性良好的刚度矩阵和较为广泛的适用性。数值算例验证了该方法的有效性。

A NUMERICAL SIMULATION METHOD BASED ON ENTIRE DISPLACEMENT MODE FOR PLAIN HOLE STRUCTURE

There are a large number of structures with discontinuities such as holes and inclusions in engineering, the existence of voids and other defects plays an important role in the integrity and stability of the structure, and affects the macro mechanical property of the material. Therefore, developing numerical simulation methods for such structure has great directive significance for engineering application. At present, the main numerical simulation methods for such structure mainly include FEM, XFEM and meshless methods, each of these methods has its advantages and disadvantages such as blending elements in XFEM and displacement boundary condition in meshless methods. This paper proposes a new kind of numerical simulation method based on entire displacement mode for plain hole structure. In the paper, we introduce level set functions for hole boundary and force boundary by level set method, construct boundary trial function with the aid of boundary level set function, and express the trial space as the linear combination of power series and boundary trial functions. Meanwhile, this paper also presents a new displacement boundary processing method based on level set method, which can be hopefully applied in other existing meshless methods. In the method, we involve the displacement boundary condition into the of approximate displacement field, the displacement boundary level set function is introduced to construct the approximate displacement field satisfying displacement boundary condition, and consequently we obtain the formula of balanced equation, stiffness matrix and load matrix. Comparing with FEM, XFEM and element free methods, this method doesn’t need to scatter the solution area, it can obtain the reversible stiffness matrix directly, avoid the ill condition problem of stiffness matrix, and decrease the difficulty of solving the linear equations, furthermore it is also suitable for hole structure with more complicated shape. Finally a numerical case is presented to testify the validity of the method.