Stabilization of Measurement-Integrated Simulation by Elucidation of Destabilizing Mechanism

Abstract

Measurement-integrated (MI) simulation is a numerical flow analysis method with a feedback mechanism from measurement of a real flow. It correctly reproduces a real flow under inherent ambiguity in a mathematical model or a computational condition. In this paper we theoretically investigated the destabilization phenomenon of MI simulation, in which analysis error suddenly increases at some critical feedback gain. This phenomenon has been considered as instability of a closed-loop feedback system, but present study treated it as that of a numerical scheme. First, the mechanism of the destabilization phenomenon was investigated based on the sufficient condition of the convergence of iterative calculation of existing MI simulation. It was found that the feedback signal in the source term destabilized the iterative calculation. Then, a new MI simulation scheme was derived by evaluating the feedback signal in the linear term to remove the cause of the destabilization. The validity of the present theoretical analysis was verified for examples treated in former studies of MI simulations: blood flow in an aneurismal aorta with ultrasonic measurement, blood flow in a cerebral aneurism with magnetic resonance measurement, Karman vortex street behind a square cylinder with PIV measurement, and fully developed turbulent flow in a square duct with ideal measurement. Occurrences of destabilization phenomenon in all the examples were well explained by the condition of this study, especially for cases of relatively small time steps and large feedback gains. Furthermore, the new MI simulation scheme realized the analysis without the destabilization phenomenon. The present theoretical result confirming that the destabilization phenomenon is not the instability of the feedback system but that of a numerical scheme is generally applicable to MI simulations using the velocity error for the feedback signal.