An overview of the proper generalized decomposition with applications in computational rheology

We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates

Modelling the effect of particle inertia on the orientation kinematics of fibres and spheroids immersed in a simple shear flow

Simulations of flows containing non-spherical particles (fibres or ellipsoids) rely on the knowledge of the equation governing the particle motion in the flow. Most models used nowadays are based on the pioneering work of Jeffery (1922), who obtained an equation for the motion of an ellipsoidal particle immersed in a Newtonian fluid, despite the fact that this model relies on strong assumptions: negligible inertia, unconfined flow, dilute regime, flow unperturbed by the presence of the suspended particle, etc. In this work, we propose a dumbbell-based model aimed to describe the motion of an inertial fibre or ellipsoid suspended in a Newtonian fluid. We then use this model to study the orientation kinematics of such particle in a linear shear flow and compare it to the inertialess case. In the case of fibres, we observe the appearance of periodic orbits (whereas inertialess fibres just align in the flow field). For spheroids, our model predicts an orbit drift towards the flow-gradient plane, either gradually (slight inertia) or by first rotating around a moving oblique axis (heavy particles). Multi-Particle Collision Dynamics (MPCD) simulations were carried out to assess the model predictions in the case of inertial fibres and revealed similar behaviours

A simple microstructural viscoelastic model for flowing foams

The numerical modelling of forming processes involving the flow of foams requires taking into account the different problem scales. Thus, in industrial applications a macroscopic approach is suitable, whereas the macroscopic flow parameters depend on the cellular structure: cell size, shape, orientation, etc. Moreover, the shape and orientation of the cells are induced by the flow. A fully microscopic description remains useful to understand the foam behaviour and the topological changes induced by the cell elongation or distortion, however, from an industrial point of view, microscopic simulations remain challenging to address practical applications involving flows in complex 3D geometries. In this paper, we propose a viscoelastic flow model where the foam microstructure is represented from suitable microstructure descriptors whose evolution is governed by the macroscopic flow kinematics. This is a post-peer-review, pre-copyedit version of an article published in International journal of material forming

Microscopic modelling of orientation kinematics of non-spherical particles suspended in confined flows using unilateral mechanics

The properties of reinforced polymers strongly depend on the microstructural state, that is, the orientation state of the fibres suspended in the polymeric matrix, induced by the forming process. Understanding flow-induced anisotropy is thus a key element to optimize both materials and process. Despite the important progresses accomplished in the modelling and simulation of suspensions, few works addressed the fact that usual processing flows evolve in confined configurations, where particles characteristic lengths may be greater than the thickness of the narrow gaps in which the flow takes place. In those circumstances, orientation kinematics models proposed for unconfined flows must be extended to the confined case. In this short communication, we propose an alternative modelling framework based on the use of unilateral mechanics, consequently exhibiting a clear analogy with plasticity and contact mechanics. This framework allows us to revisit the motion of confined particles in Newtonian and non-Newtonian matrices. We also prove that the confined kinematics provided by this model are identical to those derived from microstructural approache

From dilute to entangled fibre suspensions involved in the flow of reinforced polymers: A unified framework

Most suspension descriptions nowadays employed are based on Jeffery model and some of its phenomenological adaptations that do not take into account the possible existence of a relative velocity between the fibres and the suspending fluid when the fibre interactions increase. It is expected that at very low density of contacts, as predicted by standard suspension models, fibres move with the suspending fluid velocity. When the density of fibre interactions becomes extremely high and a percolated network of fibre contacts is established within the suspension, fibres cannot move anymore and then the fluid flows throughout the rigid or moderately deformable entangled fibre skeleton, like a fluid flowing through a porous medium. In between these two limit cases, one could expect that fibres move but with a velocity lower than the one of the suspending fluid. Thus, two contributions are expected, one coming from standard suspension theory in which fibres and fluid move with the same velocity, and the other resulting in a Darcy contribution consisting of the relative fibre/fluid velocity. In this paper, we elaborate a general model able to adapt continuously to all these flow regimes

A simple microstructural viscoelastic model for flowing foams

The numerical modelling of forming processes involving the flow of foams requires taking into account the different problem scales. Thus, in industrial applications a macroscopic approach is suitable, whereas the macroscopic flow parameters depend on the cellular structure: cell size, shape, orientation, etc. Moreover, the shape and orientation of the cells are induced by the flow. A fully microscopic description remains useful to understand the foam behaviour and the topological changes induced by the cell elongation or distortion, however, from an industrial point of view, microscopic simulations remain challenging to address practical applications involving flows in complex 3D geometries. In this paper, we propose a viscoelastic flow model where the foam microstructure is represented from suitable microstructure descriptors whose evolution is governed by the macroscopic flow kinematics

On the Peterlin approximation for finitely extensible dumbbells

For the simplest non-linear kinetic theory of dilute polymeric solutions (FENE dumbbells), the pre-averaging Peterlin approximation used to derive a macroscopic constitutive equation (FENE-P) is shown to have a significant impact on the statistical and rheological properties of the model. This is illustrated in simulations of transient elongational flows by means of standard and stochastic numerical techniques. (C) 1997 Elsevier Science B.V

Parallel Finite-element Algorithms Applied To Computational Rheology

We review the work of our research group over the last 4 years towards the development of efficient parallel finite element algorithms. Target applications are physical problems described by means of non-linear sets of partial differential or integro-differential equations of mixed type, and solved in complex geometries using unstructured finite element meshes. A typical example considered in this paper is the flow of viscoelastic fluids. The complexity of the governing equations is such that it prevents the use of established parallel numerical algorithms developed for elliptic problems. After a brief discussion of viscoelastic governing equations and related sequential numerical techniques, we describe a generic parallel approach to the assembly and solution of finite element equation sets. Automatic load balancing schemes and mesh partitioning methods are discussed. Finally, the proposed algorithms are evaluated in the simulation of viscoelastic Rows described by integral and differential constitutive equations. Results are reported for various distributed memory MIMD parallel computers, including the INTEL IPSC/860 hypercube, the CONVEX Meta Series, and a heterogeneous network of engineering workstations