Property of intrinsic drift coefficients in globally-evolving-based generalized density evolution equation for the first-passage reliability assessment

The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE) provides a promising tool to obtain the instantaneous probability density of any quantity of interest of stochastically excited high-dimensional systems. By introducing an absorbing boundary process (ABP) determined by failure criteria, a corresponding GE-GDEE can be constructed and solved to assess the first-passage reliability. The construction of the intrinsic drift coefficient of the GE-GDEE is the crucial step. In the present paper, the property of the intrinsic drift coefficients of the GE-GDEE for ABPs is studied. In particular, the intrinsic drift coefficients of some linear systems are exactly constructed via analytical solutions. Compared to the GE-GDEE of the original quantity of interest, the intrinsic drift coefficients of GE-GDEE of the corresponding ABP in both linear and nonlinear systems change considerably in the vicinity of the boundary, curving towards zero, but vary very little far away from the boundary. This physically means that the apparent damping is reduced, thus leading to an unconservative estimate of failure probability if the intrinsic drift coefficients of the original quantity of interest rather than those of the ABP are adopted. Interestingly, the failure probability by solving the GE-GDEE of the original quantity of interest is in the same order of magnitude as the true value and thus can be an acceptable approximate result, particularly for low failure probability estimate problems. The findings in the paper provide insightful guidance on constructing the intrinsic drift coefficients of the GE-GDEE of ABPs for the first-passage reliability evaluation of high-dimensional nonlinear systems.