Stochastic Differential Equations for the Deterioration Processes of Engineering Systems

Abstract

A critical part of Life-Cycle Analysis of engineering systems is the modeling of their deterioration over time. A system might be subject to different deterioration processes that might impair its ability to sustain the future levels of demand. The attainment of a given level of deterioration might also prompt maintenance operations that may disrupt its ability to provide a regular service. Recently proposed formulations model the time-varying reliability of a system by looking at the evolution of state-variables that define the characteristics of the system. These state-dependent formulations rely heavily on the chosen models for the evolution of the state-variables over time. However, most models available in literature rely on simplifying assumptions that disregard the true nature of the processes, either by discretizing the time domain or by assuming independence among different processes acting on the system at the same time. This paper proposes to use a system of Stochastic Differential Equations to model the evolution of the state variables over time. The proposed formulation captures the continuous nature of the processes and takes into account the possible interactions among them. In addition, results from stochastic calculus could be used to facilitate the simulation of the processes and to obtain closed-form solutions for the distribution of the state variables over time. Moreover, the proposed models can be calibrated based on periodical monitoring of the state variables, should that be performed via Non-Destructive Evaluation or Structural Health Monitoring. A procedure for calibration is introduced and a brief explanatory example is provided.