Application of a Bayesian Filter to Estimate Unknown Heat Fluxes in a Natural Convection Problem

Abstract

Sequential Monte Carlo (SMC) or Particle Filter Methods, which have been originally introduced in the beginning of the 50’s, became very popular in the last few years in the statistical and engineering communities. Such methods have been widely used to deal with sequential Bayesian inference problems in fields like economics, signal processing, and robotics, among others. SMC Methods are an approximation of sequences of probability distributions of interest, using a large set of random samples, named particles. These particles are propagated along time with a simple Sampling Importance distribution. Two advantages of this method are: they do not require the restrictive hypotheses of the Kalman filter, and can be applied to nonlinear models with non-Gaussian errors. This papers uses a SMC filter, namely the ASIR (Auxiliary Sampling Importance Resampling Filter) to estimate a heat flux in the wall of a square cavity undergoing a natural convection. Measurements, which contain errors, taken at the boundaries of the cavity are used in the estimation process. The mathematical model, as well as the initial condition, are supposed to have some error, which are taken into account in the probabilistic evolution model used for the filter.

State Estimation Problems in Heat Transfer

“…In this section, we apply the Bayesian filters described above to state estimation problems in heat transfer that have been recently addressed by our group. These problems include: (i) the estimation of a position-dependent transient heat source in a plate [26]; (ii) the estimation of the temperature field in oil pipelines [29]; (iii) the estimation of a transient line source and the solidification front in a phase-change problem [24]; and (iv) the estimation of the transient boundary heat flux in a natural convection problem [25]. For all cases, simulated temperature measurements were used in the inverse analysis.…”

“…In this paper, we present the application of the Kalman filter and of two different algorithms of the Particle filter, namely the sampling importance resampling and auxiliary sampling importance resampling [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], to state estimation problems in heat transfer [19][20][21][22][23][24][25][26][27][28][29][30]. Before focusing on the applications of interest, the state estimation problem is defined and the Kalman and particle filters are described.…”