Improved numerical prediction and reliability-based optimization of transient heat conduction problem with interval parameters

In this paper, a high-order interval parameter perturbation method (HIPPM) and a reliability-based optimization model are proposed to solve the transient heat conduction problem with uncertainties in both the material properties and initial/boundary conditions. Interval variables are used to quantitatively describe the uncertain parameters with limited information. A modified stability theory is proposed and used to select space step and time step in the interval discrete schemes. Compared with the traditional first-order perturbation method, HIPPM can yield more accurate ranges of the uncertain temperature field by adopting the higher order terms of the Neumann series to approximate the interval matrix inverse. In the following investigated optimization model, a satisfaction degree of interval is employed to deal with the interval constraint functions. Given a reliability index representing the confidence level, uncertain constraints can be transformed into deterministic ones. The proposed HIPPM is used to predict the intervals of the constraints, and whereby eliminate the optimization nesting. A numerical example modeling a thermal protection system is presented to demonstrate the feasibility and efficiency of the proposed method.