随机桁架结构几何非线性问题的混合摄动-伽辽金法求解

提出应用混合摄动$\!$-$\!$-$\!$伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法(generalizedpolynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动$\!$-$\!$-$\!$伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性. 

HYBRID PERTURBATION-GALERKIN METHOD FOR GEOMETRICAL NONLINEAR ANALYSIS OF TRUSS STRUCTURES WITH RANDOM PARAMETERS1)

The hybrid perturbation-Galerkin stochastic finite element method is used to solve the geometrical nonlinear truss structures with random parameters. The power polynomial expansions are adopted to express the random secant elastic modulus and random responses with respect to displacement terms, respectively. Using the high-order perturbation method, the coefficients of the power polynomial expansions can be obtained, so that the expression of the geometrical nonlinear displacement can be determined. The coefficients terms of the power polynomial expansions obtained are used as the Galerkin trial functions, and the Galerkin projection technique is employed to determine the coefficients of these trial functions. Since the trial functions come from the linear combination of the perturbation solutions, the trial functions selected are self-adaptive to the nonlinear problem. The numerical example about multi random variables with different probability distributions show that since no probability conversion is required for the proposed method so that the conversion errors in the calculation process are avoidable, which results in that the accuracy of the proposed method is higher than that of generalized polynomial chaos method (GPC method). Meanwhile, when the results are equally accurate, the nonlinear algebraic equations about the coefficients of the trial functions obtained by the proposed method is more easy to be solved than that by the GPC method, and the calculation cost of the suggested method is less than that of the GPC method. When the fluctuation of the random variable becomes large, the statistical moments of the structural response calculated by the hybrid perturbation-Galerkin method are closer to the results of the Monte Carlo simulation method than that of the high-order perturbation method, which illustrates that the hybrid perturbation-Galerkin method is effective in solving the stochastic geometric nonlinear problems.