Complexity Adaptation Strategy for Order-Adaptive Elements

Abstract

Many studies have worked on reducing dynamic analysis convergence problems. Currently, there are some robust algorithms. However, they can fail when dealing with complex structures due to numerical and especially physical instabilities. Some of them can also be time-consuming procedures. On the other hand, the intrinsic truncated error in structural analysis decreases when the shape function order is raised (p-refinement). Nevertheless, this action will increase the complexity and, thus, structural convergence problems. Therefore, the solution proposed is to adapt the complexity when physical instabilities are predicted, and Hermite interpolation can be used to state well-posed results. The physical instability can be predicted by analyzing strain energy outliers and moment-curvature rule abnormalities. Moreover, to get more realistic results, nonlinear elements with plastic length were developed. Since no previous references have worked with these kinds of nonlinear high-order elements, using a set of sigmoid functions in the stiffness matrix integral was the solution to obtain generalized high-order Timoshenko beams. Another contribution of this work was establishing an appropriate manner of getting the maximum permissible error in p-adaptive methods. Finally, some examples were made to prove the formulation's robustness and show how influential the truncated error can be.