板壳问题的三维无网格伽辽金直接分析法

Direct analysis for plate and shell by three-dimensional element free Galerkin method

对于板壳问题．共有三种数值模拟方案：线性或非线性的板壳理论、退化连续体方案和直接三维连续体方案。无网格法近似函数可具有C1甚至更高的连续性，便于在Kirchhoff—Love理论中应用。但当各种无网格法用于Mindlin—Reissner板理论时．会遇到数值锁死的困扰。对比之下，三维连续体方案是最简单，最精确但并不常用的一种方案。无网格法近似函数具有高度光滑性，在板壳的厚度方向仅布置2～5层点就可以很好地捕捉此方向场的梯度，同时还可以在一定参数范围内避免剪切和体积锁死，在处理复杂本构关系、非线性板壳等问题中更是具有很大优势。本文采用无网格伽辽金法（EFG）和三维连续体方案分析了线性板壳问题，与有限单元法做了对比，并讨论了数值锁死等问题。

There are three approaches in numerical simulation of plate and shell structures： plate and shell theory approach, degenerated continuum approach, direct three-dimensional （3D） continuum approach. Because meshfree methods can easily establish approximation functions with C1 or higher order of continuity, they are widely used in Kirchhoff-Love shell theory. When applying in Mindlin- Reissner shell theory, meshfree methods fall into the dilemma of numerical locking. Among these three approaches, direct 3D continuum approach is the simplest and most accurate one in principle. However, it is the least popular one in practice because of innate drawbacks of FEM. Because of the high order of continuity of approximation functions, meshfree method could deploy only 2 layers- 5 layers of particles to capture the field gradient in the thickness direction, and at the same time, it also could alleviate locking in some ranges of factors. This approach shows great advantages when treating with materials with complicated constitutive law, and nonlinear shell. In this paper, element free Galerkin method and direct 3D continuum approach are employed to analyze the problems of linear plate and shell, and numerical results are compared with those obtained by finite element method. Finally, numerical locking is investigated.