The general solution of the inverse heat source calibration problem in weld modelling

In weld modelling, under limited temperature information (e.g. having just fusion boundary), heat source calibration (i.e. estimating heat input efficiency, heat source shape and heat loss) could lead to multiple apparently equivalent solutions, due to the cumulative influence of heat parameters on the induced temperature field. In this paper, an algorithm to estimate heat parameters is presented, which can recognise the existence of multiple good solutions and determine them simultaneously, given any set of temperature evidence points. The algorithm consists of two steps: (1) Reduction of the solution interval by quantifying the uncertainty induced by the limited temperature evidence, with an iterative process of linearising the temperature in respect to heat parameters, and (2) Determination of the exact, general solution of the system by utilising the particular solution of minimum objective function and the kernel of the homogeneous system of equations. For demonstration of the method’s performance on solving the inverse heat source calibration problem, a finite element solver of quasi-stationary state is used, to simulate the temperature field of the direct problem, firstly on an assumed simulated temperature field, and then on temperature data from an arc weld experiment. The results show that the proposed algorithm, within, typically, few iterations (six or less iterations for the representative cases examined in this study), is capable of determining the general solution of heat parameters, that is, not only one heat parameter solution which fulfils the evidence but all possible solutions in case of under-determined systems. This makes the proposed method superior to other optimisation methods previously suggested for heat parameter estimation of ill-posed thermal problems which were able to find just one possible solution at a time.