A Continuation method with combined restrictions for nonlinear structure analysis

In structural engineering, many problems present nonlinear behavior associated to a large number of sources, ranging from the presence of large displacements and large strains to the constitutive material behavior. The simulation of such complex structural systems with the finite element method often requires the use of analysis control schemes, called continuation methods. These schemes aim at depicting equilibrium paths, associated with system stiffness loss and to the occurrence of load and displacement limit points. Among them, the arc length and the strain control methods are very popular. However, under certain conditions, these methods either fail to provide the full system response or require a large number of iterations for convergence. In the present work, a continuation method with combined restrictions is proposed to provide the full system response in the presence of both geometric nonlinearities and material elasto-plastic softening. Two or more restrictions are combined resulting in a more robust scheme for the solution of the nonlinear system of equilibrium equations. Consistency of the extended system is reached through the least squares method. The efficiency of the combined control is compared to the application of the restrictions individually in a Newton-Raphson solution scheme. An automatic increment adaptation strategy for the control parameters is introduced which helps stabilize the incremental process with multiple restrictions. In the analysis of the highly non-linear problems studied in this paper, the proposed method with combined restrictions proves to be more robust than the standard procedure with single restrictions. In the cases where the single restrictions failed, the combined restriction was able to provide the full solution.