膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节，但在数学上，膜结构的极小曲面找形分析是一个高度非线性问题，一般无法求得其解析解，因此数值方法成为重要工具。近年来，基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功，基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上，该文提出一种基于EEP法的二维非线性有限元自适应方法，并成功将之应用于膜结构的找形分析。其主要思想是，通过将非线性问题用Newton法线性化，引入现有的二维线性问题的自适应求解技术，进而实现二维有限元自适应分析技术从线性到非线性的跨越，将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率，对网格自适应地进行调整，最终得到优化的网格，其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展，并给出数值算例用以表明该方法的可行性和可靠性。

A NEW ADAPTIVE FEM FOR MINIMAL SURFACES FORM-FINDING OF MEMBRANE STRUCTURES

Form-finding analysis is a key step of the design of membrane structures.The minimal-surface form-finding problem of membrane structures is a highly nonlinear problem in mathematics,and no analytic solutions are available in general.Thusly,numerical methods are an important approach.In recent years,remarkable success has been made in the adaptive analysis of both1D nonlinear finite element method(FEM)and2D linear FEM based on element energy projection(EEP)super-convergent technique.A new adaptive strategy for2D nonlinear FEM is developed and applied to the form-finding of membrane structures.In this method,by linearizing nonlinear problems into a series of linear problems via the Newton method,the existing2D linear adaptive strategy based on EEP technique can be incorporated into a nonlinear solution procedure.As a result,an adaptive mesh is automatically generated and adjusted by the algorithm to guarantee to produce a satisfactory solution with the results satisfying the user-preset error tolerance by maximum norm.Pertinent numerical examples are presented to demonstrate the feasibility and effectiveness of the newly developed method.