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 t in a finitely large time interval t_i≤t≤t_i+1 are corrected by a weighted (with a matrix λ) integral of the error from t_i to t. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weighting functions λ; thus we present 3 algorithms denoted as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3. The interval (t_i+1−t_i) in the 3 OFAPI algorithms can be several hundred times larger than the increment (Δt) 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) require several orders of magnitude of less computational time than the currently popular implicit and explicit finite difference methods, and provide better accuracy and convergence.