A STOCHASTIC MULTI-STEP TRANSVERSAL LINEARIZATION METHOD (MTL) IN ENGINEERING DYNAMICS

摘要

I. INTRODUCTIONAn implicit linearization procedure, referred to as a multi-step transversal linearization (MTL), isproposed for strong numerical solutions of non-linear SDE-s acted upon by white noise excitations. Inthis method, the process of linearization of the stochastic vector field is done in such a way that thelinearized solution manifold may have repeated transversal intersections with the targeted solutionmanifold of the non-linear oscillator. The linearization of the non-linear part of the vector field isperformed conditionally via a multi-step Taylor like interpolation so as to be consistent with the typicalcharacteristics of the white noise excitation (for instance, √O h Wiener increments over a time step ofh etc.). Such an interpolating expansion of the non-linear part of the operator over a set of discretizationpoints results in a conditionally linearized and integrable set of SDE-s, whose exact solution may beexplicitly constructed in terms of the discretized (unknown) state variables. It may be mentioned here thatthe suggested functional expansion converts non-linear part of the vector field into a linearized one withan explicit dependence on time and so the linearized vector field may be considered as a modified(conditional) forcing function. However, an expansion of the non-linear terms needs the discretizedvalues of the state variables at the grid (interpolation/discretization) points. Since these discretized valuesare not known a-priori, the expansion of the non-linear vector field is only conditional, I.e., it isconditioned on the anticipated (possible) knowledge of the discretized state variables at the grid points.Such an anticipatory expansion of the non-linear part of the vector field may include as many gridpoints forward along the time axis as desired and is so performed that the linearized and non-linear vectorfields remains instantaneously identical in form at these points of discretization. Finally, based on thecondition of transversal intersections of the linearized and non-linear solution manifolds at the points ofdiscretization, a set of coupled non-linear algebraic equations in terms of the discretized state variables isestablished. A limited numerical verification of the MTL procedure is provided for a few stochasticallyexcited low-dimensional non-linear oscillators.