基于有限元差分法的形态创构方法研究

Research on the morphopoiesis based on finite difference method

在形态创构方法中,较多的是建立在反映目标函数与设计变量关系的敏感度计算基础上。然而,利用解析法确定敏感度不仅需要复杂的数学推导,而且对目标函数的解析性要求较高,当形态从弹性发展到弹塑性时,难度进一步增加。首先,本文利用Taylor级数展开式推导出目标函数敏感度的中心差分格式,并分析讨论差分格式的精度与收敛性以及采用差分格式敏感度时的迭代步长确定方法。其次,以曲面结构高度参数坐标为设计变量、以结构应变能作为目标函数,建立了基于有限元差分法的形态创构方法。同时通过以往的工程实例作为算例,考察了方法的特性与力学性能变化特征。最后,利用ANSYS有限元软件对所得到的结构进行了应力分布特性以及结构稳定性与初始缺陷对极限承载力影响的分析,验证了方法的可靠性。该方法可以扩展到弹塑性问题,而且也可以克服以往敏感度分析中对目标函数解析性的苛刻要求。

Morphogenesis methods are frequently based on sensitivity calculations that involve the relationship between the objective function and the design variables.However,applying the analytical method to determining the sensitivity not only requires complicated mathematical derivation,but also requires high analytical ability of the objective function.The difficulty may increase further as the morphology proceeds from elastic state to elastoplastic state.Firstly,the central difference scheme for the sensitivity of the objective function was derived by using Taylor series expansion formula.Then,the accuracy and convergence of the difference scheme and the determination method of iterative step for the sensitivity of the difference scheme were analyzed and discussed.Secondly,a morphogenesis method based on finite element difference was established by taking the parametric coordinate of the surface structure height as the design variable and the structural strain energy as the objective function.Meanwhile,the characteristics of the method and the variation of mechanical properties were investigated by taking the previous practical engineering as an example.Finally,the finite element software ANSYS was used to analyze the stress distribution characteristics of the structure and the influence of structural stability and initial defects on the ultimate bearing capacity,and the reliability of the method were verified.This method can be extended to the elastoplastic problem and can overcome the strict requirement of analytic function in sensitivity analysis.