A comparison of constitutive models for describing the flow of uncured styrene-butadiene rubber

Uncured styrene-butadiene rubber (SBR) can be modelled as a viscoelastic material with at least two different relaxation mechanisms. In this paper we compare multi-mode constitutive models combining two viscoelastic modes (linear and/or nonlinear) in three possible ways. Our particular choice of the two modes was inspired by models originally developed to describe the response of asphalt binders. We select the model that best fits the experimental data obtained from a modified stress relaxation experiment in the torsional configuration of the plate-plate rheometer. The optimisation of the five model parameters for each model is achieved by minimising the weighted least-squares distance between experimental observations and the computer model output using a tree-structured Parzen estimator algorithm to find an initial guess, followed by further optimisation using the Nelder-Mead simplex algorithm. The results show that the model combining the linear mode and the nonlinear mode is the most suitable variant to describe the observed behavior of SBR in the given regime. The predictive capabilities of the three models are further examined in changed experimental and numerical configurations. Full data and code to produce the figures in this article are included as supplementary material

Investigation of the Sharkskin melt instability using optical Fourier analysis

An optical method allowing the characterization of melt flow instabilities typically occurring during an extrusion process of polymers and polymer compounds is presented. It is based on a camera‐acquired image of the extruded compound with a reference length scale. Application of image processing and transformation of the calibrated image to the frequency domain yields the magnitude spectrum of the instability. The effectiveness of the before mentioned approach is shown on Styrene‐butadiene rubber (SBR) compounds, covering a wide range of silica filler content, extruded through a Göttfert capillary rheometer. The results of the image‐based analysis are compared with the results from the sharkskin option, a series of highly sensitive pressure transducers installed inside the rheometer. A simplified version of the code used to produce the optical analysis results is included as supplementary material

Dynamic composition of solvers for coupled problems in DOLFINx

Recent developments in DOLFINx allow for the block assembly of linear algebraic systems arising from discretisations of coupled partial differential equations. Each algebraic block represents a subproblem associated with a coupling of the unknown fields. Designing and implementing robust and scalable solution and preconditioning strategies for block-structured linear systems is an active area of research. In this contribution we show how DOLFINx can now exploit one of the most significant features of PETSc; the dynamic composition of the hierarchical solver and preconditioner options at runtime, see Brown et al [1]. The idea is inspired by the work of Kirby and Mitchell [2] that was originally implemented in the Firedrake Project. One of the most significant benefits of the approach is the possibility to construct advanced preconditioners that require structure beyond a purely algebraic problem description, eg the pressure-convection-diffusion (PCD) approximation of the Schur complement for the Navier–Stokes equations, see Silvester et al [3]. We illustrate the capabilities of our implementation on examples ranging from incompressible flow of a viscous fluid, through temperature-driven convection, to flows described by rate-type viscoelastic fluid models. References [1] J. Brown, M. G. Knepley, D. A. May, L. C. McInnes, and B. Smith, "Composable Linear Solvers for Multiphysics," in 2012 11th International Symposium on Parallel and Distributed Computing, Munich, Germany, Jun. 2012, pp. 55–62, doi: 10.1109/ISPDC.2012.16. [2] R. C. Kirby and L. Mitchell, "Solver Composition Across the PDE/Linear Algebra Barrier," SIAM J. Sci. Comput., vol. 40, no. 1, pp. C76–C98, 2017, doi: 10.1137/17M1133208. [3] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. 2014, doi: 10.1093/acprof:oso/9780199678792.001.0001. Acknowledgements The present work is supported by the National Research Fund, Luxembourg in the frame of the Industrial Fellowship project RIFLE (13754363). The experiments presented in this work were carried out using the HPC facilities of the University of Luxembourg