Bifurcation & Chaos in Nonlinear Structural Dynamics: Novel & Highly Efficient Optimal-Feedback Accelerated Picard Iteration Algorithms

A new class of algorithms for solving nonlinear structural dynamical problems are derived in the present paper, as being based on optimal-feedback-accelerated Picard iteration, wherein the solution vectors for the displacements and velocities at any time in a finitely large time interval are corrected by a weighted (with a matrix) integral of the error. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weighting functions; thus we present 3 algorithms denoted as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3. The interval in the 3 OFAPI algorithms can be several hundred times larger than the increment required in the finite difference based implicit or explicit methods, for the same stability and accuracy. Moreover, the OFAPI algorithms-2, 3 do not require the inversion of the tangent stiffness matrix, as is required in finite difference based implicit methods. It is found that OFAPI algorithms-1, 2, 3 (especially OFAPI algorithm-2) are far more superior to the currently popular implicit and explicit finite difference methods in terms of computational speed, accuracy, and convergence.

This paper is online published on "Communications in Nonlinear Science and Numerical Simulation". It can be viewed via the following URL:

http://www.depts.ttu.edu/coe/CARES/pdf/Xuechuan_Wang_2018.pdf