Sequential boundary heat flux estimation using the method of fundamental solutions and bayesian filters

In many thermal engineering problems the boundary conditions are not fully known, since there are technical difficulties in obtaining data. When boundary data is inaccessible, for example, it is necessary to deal with an Inverse framework to reach this unknown information. To obtain the solution of an inverse problem as an iterative process the direct problem needs to be called repeatedly which makes meshless methods implementing, as the Method of Fundamental Solutions (MFS), attractive. In the MFS procedure a linear system of fundamental solutions is solved in order to obtain the solution of the respective partial differential equation (PDE) from the studied problem. To solve the parabolic heat equation without needing to perform transformation of the Parabolic equation into Elliptic or treat the time component separately the fundamental solution of the parabolic heat equation was used. The presented case consists in a classical one-dimensional IHCP of boundary heat flux estimation using intrusive measurements. The inverse problem was solved by three different Bayesian filters schemes combined with the MFS: Sampling Importance Resampling (SIR) filter, Auxiliary Sampling Importance Resampling (ASIR) filter and Unscented Kalman Filter (UKF). The measurements from the inverse procedure were generated using Finite Difference Method (FDM) to avoid the inverse crime. The presented results shown good agreement in the three schemes even using few collocation points in the MFS procedure.