膜结构极小曲面找形分析的一种线性化近似方法及其有限元线法求解

A LINEARIZED METHOD FOR MINIMAL SURFACE FORM-FINDING OF MEMBRANE STRUCTURES AND ITS FEMOL SOLUTION

膜结构的极小曲面找形分析是一个非线性问题,求解时需要进行大量非线性迭代,并需要一个合理的初始解作为收敛的保证,计算繁琐、量大且难度高。该文利用积分中值定理和归一化手段对曲面面积的表达式进行一种特殊的线性化,将原非线性问题转化为线性问题,使问题得到本质性的简化。该线性问题的解答作为原问题高质量的近似解,既可用于结构的初步设计阶段了解膜面的大概形状,亦可作为精细的找形分析中非线性迭代求解的初始解。该线性化方法的误差主要来源于映射参数分布的不均匀性,对于常见的可用平行四边形剖分的膜,其逼近精度相当高。有限元线法(FEMOL)是一种基于常微分方程(ODE)求解的半解析方法,其高度的解析性和解的光滑性特别适合于膜结构的分析。该文采用高次线法单元分析求解转化后的线性问题,只需一次求解,无需任何迭代。数值算例表明:该方法是一种简单、高效、高逼近度的膜结构找形分析方法。

The minimal surface form-finding analysis of membrane structures is a nonlinear problem. Its strong nonlinearity makes its computation to be a great challenge which usually needs a lot of iterations and a rational initial solution to guarantee the convergence of the solution process. This paper substantially simplifies this problem into a linear problem by using the integral mean-value theorem and normalization technique, the solution of which approaches the minimal surface with high accuracy. This method can be used either for an approximate form of the membrane surface in the primary design stage or for an initial solution for further computation of the original nonlinear problem. The error of the present method mainly comes from the non-uniformity of the mapping parameters, and thus this method works very well for most membranes whose shapes are close to parallelogram. As a semi-analytical method based on ordinary differential equation （ODE） techniques, the finite element method of lines （FEMOL） is very suitable for membrane problems due to its semi-analytical property and smoothness of its solutions. FEMOL is applied to the linearized problem proposed in the paper. Numerical examples given show that this linearized method is simple, efficient and reliable with highly satisfactory accuracy.