Bayesian calibration of a model of polymer die swell using data from a laser–based measurement system

Our goal is to produce a predictive model linking apparent shear rate with polymer die
swell for a single extrusion system. We propose the use of a Kennedy O’Hagan-type approach for calibrating this model, where the die swell is modelled as a sum of a
deterministic simulator, a Gaussian process and an additive white noise term. For the
deterministic simulator we use a model that links the shear rate and two parameters
related to the shear–thinning and relaxation time of a polymer to the die swell. The
role of the Gaussian process is to capture the inherent structural uncertainty induced
by the missing physical processes such as wall slip and non-isothermal conditions in
the derivation of the simulator. The parameter calibration of the full model is per-
formed using a subjective Bayesian methodology where the solution is characterised
by the posterior distribution of the parameters given the observed data. We condi-
tion our model on experimental data produced from a capillary rheometer fitted with
a laser-based die swell measurement system. We implement the models using a high–
level probabilistic programming language and explore the resulting posterior using the
No–U–Turn Sampler (NUTS). Our results show that the experimental swell data leads
to a contraction in the posterior distribution with respect to the prior on the param-
eter related to the relaxation time of the polymer. In addition we demonstrate that
the Kennedy O’Hagan-type model structure leads to improved fit of the model within
the range of experimental data without sacrificing the simulator’s extrapolative power
outside.