TY - JOUR

T1 - Bifurcation & chaos in nonlinear structural dynamics

T2 - Novel & highly efficient optimal-feedback accelerated Picard iteration algorithms

AU - Wang, Xuechuan

AU - Pei, Weicheng

AU - Atluri, Satya N.

PY - 2018/12

Y1 - 2018/12

N2 - 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 ti≤t≤ti+1 are corrected by a weighted (with a matrix λ) integral of the error from ti 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 (ti+1−ti) 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.

AB - 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 ti≤t≤ti+1 are corrected by a weighted (with a matrix λ) integral of the error from ti 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 (ti+1−ti) 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.

KW - Collocation method

KW - Nonlinear dynamics

KW - Picard iteration method

KW - Structural vibrations

KW - Variational method

UR - http://www.scopus.com/inward/record.url?scp=85047261045&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2018.05.008

DO - 10.1016/j.cnsns.2018.05.008

M3 - Article

AN - SCOPUS:85047261045

VL - 65

SP - 54

EP - 69

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

ER -