AN ACCELERATED INCREMENTAL ALGORITHM TO TRACE THE NONLINEAR EQUILIBRIUM PATH OF STRUCTURES

Abstract

THIS PAPER DEALS WITH THE CONVERGENCE ACCELERATION OF ITERATIVE NONLINEAR METHODS. A NEW ITERATIVE ALGORITHM, NAMED THE THREE–POINT METHOD, WHICH IS BASED ON THE NEWTON–RAPHSON METHOD FOR SOLVING NONLINEAR SYSTEMS OF EQUATIONS, IS PROPOSED. IN TERMS OF COMPUTATIONAL COST, EACH ITERATION OF THE THREE–POINT METHOD REQUIRES THREE EVALUATIONS OF THE FUNCTION. THE PROPOSED METHOD HAS A CONVERGENCE ORDER OF EIGHT, AND IT IS IMPORTANT TO NOTE THAT ITS IMPLEMENTATION DOES NOT REQUIRE THE COMPUTATION OF HIGHER ORDER DERIVATIVES COMPARED TO MOST OTHER METHODS OF THE SAME ORDER. TO TRACE THE EQUILIBRIUM PATH BEYOND THE LIMIT POINT, A NORMAL FLOW ALGORITHM IS IMPLEMENTED INTO A COMPUTER PROGRAM. THE THREE–POINT METHOD IS APPLIED AS AN INNER STEP IN THE NORMAL FLOW ALGORITHM. THE PROCEDURE CAN BE USED FOR STRUCTURES WITH COMPLEX BEHAVIOR, INCLUDING: UNLOADING, SNAP–THROUGH, ELASTIC POST–BUCKLING AND INELASTIC POST–BUCKLING ANALYSES. SEVERAL NUMERICAL EXAMPLES ARE GIVEN TO ILLUSTRATE THE EFFICIENCY AND PERFORMANCE OF THE NEW METHOD. RESULTS SHOW THAT THE NEW METHOD IS COMPARABLE WITH THE WELL–KNOWN EXISTING METHODS AND GIVES BETTER RESULTS IN CONVERGENCE SPEED.