Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

ABSTRACT A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.

摘要 针对具有非线性特性的结构问题，求解其平衡路径或荷载-位移曲线的绝大多数数值方法均以牛顿-拉夫逊（Newton-Raphson）迭代格式为基础，并耦合路径跟踪方法。本文提出了基于Potra-Pták方法、切比雪夫（Chebyshev）方法与超哈尔利（super-Halley）方法，并结合线性弧长路径跟踪方法的新型算法。采用这些方法的核心动机在于其具备三阶收敛精度。为阐明所提方法的应用潜力，本文针对文献中已有的几何非线性空间桁架与平面桁架问题开展分析。于此研究方向中，本文将采用位置有限元法（Positional Finite Element Method），该方法将节点坐标而非位移作为非线性系统的求解变量。仿真数值结果表明，所开发的计算算法能够有效获取包含荷载与位移极限点的平衡路径。与经典的牛顿-拉夫逊法及修正牛顿-拉夫逊法相比，所实现的迭代方法在时间步数量、收敛所需累计迭代次数以及计算耗时方面均展现出更优的计算效率。