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

Multi-scale modeling of complex fluids and deformable fibrous media for liquid composite molding

In the last few years, the interest of the aerial and terrestrial transport industry in the fabrication of textile-reinforced composite materials has sensibly grown. This is basically due to the remarkable properties of these materials, which combine high mechanical strength with reduced weight. The manufacturing techniques that provide better control on the final quality of the components rely on autoclave curing: heat and pressure are applied on vacuum bags to achieve high volume fractions of the reinforcement and low number of defects due to the presence of voids. Nevertheless, autoclave curing implies high costs for the acquisition of the vessel and the process is energy and time consuming. To reduce the production costs, the industry has increased its interest in out-of-autoclave processing technologies, that is, liquid composite molding (LCM) techniques. In its most basic version, the technique consists in the injection of a catalyzed resin into a closed cavity, where a pre-placed fiber stack lies. When the resin has completely permeated the preform, the mold is subject to high temperatures to induce the curing of the resin to obtain the composite. The current challenge for this technology is to achieve the same quality standards for the final component as those achievable with in-autoclave processing. In LCM processes, the final quality of the component depends on several factors, such as: the structure of the textile, the arrangement of the layers, the adaption to the mold, the compaction process, the operating conditions, the geometry of the component, the configuration of the injection points for the resin, the physical and chemical interactions between the resin and the textile. All these factors affect the correct saturation of the reinforcement, and therefore process parameters must be adequately controlled in order to guarantee the required quality standards for the composite. In this sense, mold filling simulation software is a valuable tool for the process optimization; however the permeability of the reinforcement is required as an input parameter. An accurate evaluation of the permeability of the reinforcement however, represents a challenging task. Fibrous preforms for LCM generally present a hierarchical structure: the fibers are bunched in yarns, which in turn are bundled in a fabric. This structure, undergoes complex deformations during the production process: 1) during the compaction in the mold and 2) during the injection of the resin. This issue remarkably complicates an accurate evaluation of the permeability of the reinforcement and may be at the origin of the scatter observed in the experimental measurements. From a modeling point of view, the different length scales to be taken into account (typically ranging between one and three orders of magnitude) hinders a proper simulation of the deformation of the textile. The typical diameter of the fibers ranges indeed in few micrometers, while the characteristic dimension of the yarns is in the order of the millimeter. This issue represents a constraint for standard numerical approaches due to computational limits. In order to account for the effect on the permeability of the deformation of the hierarchical structure of the preform, multi-scale modeling techniques must be adopted. The objective of the thesis is the development of novel theoretical and numerical frameworks to account for the effect on the permeability of the multi-scale deformations that the textile undergoes during the two aforementioned stages of the process. The development focuses on the fiber-yarn level in 2D, where the yarn is always modeled as suspension of fibers by analogy with a complex fluid. The numerical implementations use computational fluid dynamic (CFD) tools. In order to address the problem, the permeability of a textile preform for LCM is first analyzed by experimental means. A standard CFD approach is then adopted for the simulation of a representative elementary volume of the textile; it is shown that, by means of this approach, the experimental permeability cannot be recovered over the full range of porosities. An X-ray computed microtomography of the textile is then performed. The obtained data are used for the virtual reconstruction of the exact geometry of the textile after its use for LCM. The simulations with this latter geometry provide better results; however the uncertainties on permeability still hold, and the permeability is always overestimated. These uncertainties are discussed in detail and motivate the work described hereafter. The first modeling block of the thesis concerns the analysis of the deformation that the textiles undergo during the compaction in the mold. A continuum model is first developed and validated for the squeeze flow of epoxy-based materials, the rheology of which is given by a viscoplastic constitutive law. The model is then applied to the compaction of yarns, where a viscoplastic behavior for the fiber bundle is assumed in the quasi-static regime of compression and by an analogy with flowing granular media. The rheological parameters are obtained from experimental data by a simplified analytical model for the deformation of the yarns under compaction. The commercial CFD code ANSYS Fluent is adopted for the numerical solution. The model yields information about the evolution of the fiber volume fraction during the compaction and is found to correctly recover the experimental force for high compression ratios. The second modeling block of the thesis concerns the analysis of the deformation that the textiles undergo during the injection of the resin. A numerical framework is first developed and validated for the direct numerical simulation of dilute colloidal suspensions of polymeric molecules. The numerical method consists in a coupled finite-volume/lattice-Boltzmann solution: finite volume method for hydrodynamics and lattice Boltzmann method for the sub-grid-scale physics. For computational efficiency, the lattice Boltzmann solution is accelerated on a graphic processing unit (GPGPU) with a tailored implementation and efficiently coupled with the macroscopic solver (ANSYS Fluent). The numerical method is then exploited for the solution of a mesoscopic model for the flow-induced fiber dynamics during the injection. A statistical model for the fiber dynamics is derived, based on analogy of the yarn with a non-Brownian suspension of particles with confining potentials. The fiber topology during the injection is recovered by a topological invariant and yields information about the change in permeability due to the clustering of fibers in steady-state, fully-saturated conditions. The results are presented in the form of phase diagrams, which show that in the deformable case the permeability can be up to one order of magnitude lower than in the rigid case. On the basis of the results obtained, the following main conclusions can be drawn: 1. The model developed for the compaction in the mold showed to be appropriate for a phenomenological analysis of the deformation of the yarns under compression. The model allows to analyze quantitatively the evolution of the fiber volume fraction, which yields useful information for a better understanding of the distribution of the fibers before the injection. 2. The model developed for the fiber dynamics during the injection, allows to analyze their topology induced by the fluid flow. The clustering of fibers significantly reduces the permeability at the fiber level, which could explain the overestimation obtained with simplified numerical approaches. The phase diagrams obtained for the permeability, both at the yarn and fiber level, allow to identify the best operating conditions for the infiltration of the resin. The proposed models have been developed using fluid dynamic techniques, which opens the possibility for a unified framework for the analysis, and ultimately, for a more precise estimation of the permeability. This work aims to represent a first tentative in this direction

Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics

This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micro-macro (or multiscale) models. Some elements of mathematical and numerical analysis are provided. These notes closely follow the lectures delivered by the second author at the Chinese Academy of Science during the Workshop "Stress Tensor Effects on Fluid Mechanics", in January 2010

Multiscale Simulation of Polymeric Fluids using Sparse Grids

The numerical simulation of non-Newtonian fluids is of high practical relevance since most complex fluids developed in the chemical industry are not correctly modeled by classical fluid mechanics. In this thesis, we implement a multiscale multi-bead-spring chain model into the three-dimensional Navier-Stokes solver NaSt3DGPF developed at the Institute for Numerical Simulation, University of Bonn. It is the first implementation of such a high-dimensional model for non-Newtonian fluids into a three-dimensional flow solver. Using this model, we present novel simulation results for a square-square contraction flow problem. We then compare the results of our 3D simulations with experimental measurements from the literature and obtain a very good agreement. Up to now, high-dimensional multiscale approaches are hardly used in practical applications as they lead to computing times in the order of months even on massively parallel computers. This thesis combines two approaches to reduce this enormous computational complexity. First, we use a domain decomposition with MPI to allow for massively parallel computations. Second, we employ a dimension-adaptive sparse grid variant, the combination technique, to reduce the computational complexity of the multiscale model. Here, the combination technique is used in a general formulation that balances not only different discretization errors but also considers the accuracy of the mathematical model

Brownian dynamics simulations of fine-scale molecular models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005.Includes bibliographical references (leaves 105-111).One of the biggest challenges in non-Newtonian fluid mechanics is calculating the polymer contribution to the stress tensor, which is needed to calculate velocity and pressure fields as well as other quantities of interest. In the case of a Newtonian fluid, the stress tensor is linearly proportional to the velocity gradient and is given by the Newton's law of viscosity, but no such unique constitutive equation exists for non-Newtonian fluids. In order to predict accurately a polymer's rheological properties, it is important to have a good understanding of the molecular configurations in various flow situations. To obtain this information about molecular configurations and orientations, a micromechanical representation of a polymer molecule must be proposed. A micromechanical model may be fine scale, such as the Kramers chain model, which accurately predicts a real polymer's heological properties, but at the same time possesses too many degrees of freedom to be used in complex flow simulations, or it may be a coarse-grained model, such as the Hookean or the FENE dumbbell models, which can be used in complex flow analysis, but have too few degrees of freedom to adequately describe the rheology. The Adaptive Length Scale (ALS) model proposed by Ghosh et al. is only marginally more complicated than the FENE dumbbell model, yet it is able to capture the rapid stress growth in the start-up of uniaxial elongational flow, which is not predicted correctly by the simple dumbbell models. The ALS model is optimized in order to have its simulation time as close as possible to that of the FENE dumbbell.(cont.) Subsequently, the ALS model is simulated in the start-up of the uniaxial elongational and shear flows as well as in steady extensional and shear flows, and the results are compared to those obtained with other competing rheological models such as the Kramers chain, FENE chain, and FENE dumbbell. While a 5-spring FENE chain predicts results that are in very good agreement with the Kramers chain, the required simulation time clearly makes it impossible to use this model in complex flow simulations. The ALS model agrees better with the Kramers chain than does the FENE dumbbell in the start-up of shear and elongational flows. However, the ALS model takes too long to achieve steady state, which is something that needs to be explored further before the model is used in complex flow calculations. Understanding of this phenomena may explain why the stress-birefringence hysteresis loop predicted by the ALS model is unexpectedly small. In general, if polymer stress is to be calculated using Brownian dynamics simulations, a large number of stochastic trajectories must be simulated in order to predict accurately the macroscopic quantities of interest, which makes the problem computationally expensive. However, recent technological advances as well as a new simulation algorithm called Brownian configuration fields make such problems much more tractable. The operation count in order to assess the feasibility of using the ALS model in complex flow situations yields very promising results if parallel computing is used to calculate polymer contribution to stress. In an attempt to capture polydispersity of real polymer solutions, the use of multi-mode models is explored.(cont.) The model is fit to the linear viscoelastic spectrum to obtain relaxation times and individual modes' contributions to polymer viscosity. Then, data-fitting to the dimensionless extensional viscosity in the startup of the uniaxial elongational flow is performed for the ALS and the FENE dumbbell models to obtain the molecule's contour length, bmax. It is found that the results from the single-mode and the four-mode ALS models agree much better with the experimental data than do the corresponding single-mode and four-mode FENE dumbbell models. However, all four models resulted in a poor fit to the steady shear data, which may be explained by the fact that the zero-shear-rate viscosity obtained via a fit to the dynamic data by Rothstein and McKinley and used in present simulations, tends to be somewhat lower than the steady-state shear viscosity at very low shear rates, which may have caused a mismatch between the value of ... used in the simulation and the true ... of the polymer solution. As a motivation for using the ALS model in complex flow calculations, the results by Phillips, who simulated the closed-form version of the model in the benchmark 4:1:4 contraction- expansion problem are presented and compared to the experimental results by Rothstein and McKinley [49]. While the experimental observations show that there exists a large extra pres- sure drop, which increases monotonically with increasing De above the value observed for a Newtonian fluid subjected to the same flow conditions, the simulation results with a closed-form version of the FENE dumbbell model, called FENE-CR, exhibit the opposite trend.(cont.) The ALS-C model, on the other hand, is able to predict the trend correctly. The use of the ALS-C model in another benchmark problem, namely the flow around an array of cylinders confined between two parallel plates, also shows very promising results, which are in much better agreement with experimental data by Liu as compared to the Oldroyd-B model. The simulation results for the ALS-C and the Oldroyd-B models are due to Joo, et al. [28] and Smith et al. [50], respectively. Overall, it is concluded that the ALS model is superior to the commonly used FENE dumb- bell model, although more work is needed to understand why it takes significantly longer than the FENE dumbbell to achieve steady state in uniaxial elongational flows, and why the stress birefringence hysteresis loop predicted by the ALS model is much smaller than that of the other rheological models.by Irina Burmenko.S.M

Enabling microscopic simulators to perform system-level analysis of viscoelastic flows

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 249-261).State-of-the-art methods for simulating viscoelastic flows couple the conservation equations for mass and momentum with a model from kinetic theory that describes the microstructural state of the polymer. Introduction of appropriate numerical discretization and boundary conditions for these equations leads to a hybrid simulation for studying the dynamic behavior of polymeric liquids in complex geometries. This approach represents a rare example of a successful multiscale solution of a physical problem, as it allows investigation of arbitrary models of kinetic theory. The simulations, however, are not amenable to standard numerical techniques for system-level stability, bifurcation, and control analysis as this requires closed form equations. These simulation either use stochastic descriptions for the polymer microstructure that cannot be reduced to closed form, or involve equations for the evolution of a distribution of polymer conformations, which can only be written in closed form by invoking mathematical closure approximations that can have a significant qualitative impact on the predictive ability of these simulations. The focus of this thesis was to develop a novel numerical method that can enable hybrid simulations to perform system-level analysis of polymeric flows. This numerical approach has been applied directly to kinetic theory models and hybrid simulations to obtain stationary states and associated bifurcations and stability information. The method is general in its applicability in that it treats kinetic theory models and hybrid simulations as black boxes that are then used to obtain system-level information without any modification. The methods developed here are illustrated in a variety of problems.(cont) Steady state results have been obtained for the non-interacting rigid dumbbell model in steady shear, and for the free-draining bead-spring chain model in both steady shear and uniaxial elongation that are in excellent agreement with previous studies and steady state computed from direct integration. The method is also applied to a hybrid simulation for the pressure-driven flow of non-interacting rigid dumbbells in a planar channel with a linear array of equally spaced cylinders. The computed steady state is in agreement with direct integration and qualitatively matches previous computations with closed models. Bifurcation analysis has been performed for the Doi model at equilibrium with the Onsager excluded volume potential. This analysis agrees with previous studies and accurately predicts the isotropic-nematic transition and turning point for the unstable to stable transition on the prolate solution branch. Bifurcation analysis has also been performed for the Doi model in the weak shear flow limit for the Maier-Saupe excluded volume potential. It is found that stable stationary solutions are lost at a limit point beyond which time-periodic tumbling orbits are the only stable solution. This transition occurs via an infinite period global bifurcation, while the limit point approaches a threshold value as the shear rate approaches zero. This result matches a recently published scaling analysis and demonstrates the ability of the method to provide general bifurcation analysis of kinetic theory models. Stability analysis of the fiber-spinning process for polymeric fluids has also been performed by using a hybrid simulation that couples the one-dimensional conservation equations for mass and momentum with a stochastic description for the configuration fields of the Hookean dumbbell model. The steady-state velocity profiles are in good agreement with previous studies with the Oldroyd-B model.(cont) The analysis predicts onset of the draw resonance instability via a Hopf bifurcation and subsequent stabilization via second Hopf bifurcation in draw ratio parameter space. This result is in good agreement with experimentally observed behavior during polymer fiber-spinning.by Zubair Anwar.Ph.D