The Schrödinger Equation

The finest scale in the matter description consists of the one of quantum mechanics. In this chapter we revisit some valuable concepts of quantum mechanics, and more particularly the Schrödinger equation governing the time evolution of the so-called wavefunction, from which expectations can be easily derived. Another important output is the interatomic potentials responsible of the chemical bonds determining the structure and properties of materials. Even if the quantum framework is able to define the big picture, an important difficulty remains: the Schrödinger equation is defined in a highly multidimensional space and its solution is in most cases unattainable

Ab-Initio Calculations

Due to the difficulties found in the direct solution of the Schrödinger equation, different simplified approaches were proposed and are nowadays widely used. Among them, those most usually employed are the Hartree–Fock and the Density Functional Theory, which we revisit in the present chapter. The former makes use of nonstandard numerical approximations in order to calculate the wavefunction while circumventing the curse of dimensionality, whereas the latter involves the electronic density that is now defined in three dimensions but requires deeper analyses to retain the most relevant features present in the wavefunction description in a coarse 3D model

Kinetic Theory Models

Discrete techniques (MD or BD), despite their conceptual simplicity, are very often too expensive from the computational point of view. Kinetic theory approaches seem, in many cases, a suitable compromise between the accuracy of finer descriptions and the computational efficiency of macroscopic descriptions. In this chapter, we revisit some kinetic theory models. Even if there is a common rationale for deriving the different models, in order to emphasize their physical contents, we will follow a diversity of alternative routes to derive them

A journey around the different scales involved in the description of matter and complex systems

This book covers the main scales of description of matter, starting at its finest level, the quantum scale, moving through ab-initio, molecular dynamics, coarse grained approaches, to finish at the scale of kinetic theory models that allows a nice compromise between the rich but expensive microscopic descriptions and the computationally cheap but sometimes too coarse macroscopic descriptions. The book addresses undergraduate and graduate students, as well as beginners in multi-scale modeling of materials. (4th cover, excerpt from publisher's website

Modeling the kinematics of multi-axial composite laminates as a stacking of 2D TIF plies

Thermoplastic composites are widely considered in structural parts. In this paper attention is paid to sheet forming of continuous fiber laminates. In the case of unidirectional prepregs, the ply constitutive equation is modeled as a transversally isotropic fluid, that must satisfy both the fiber inextensibility as well as the fluid incompressibility. When the stacking sequence involves plies with different orientations the kinematics of each ply during the laminate deformation varies significantly through the composite thickness. In our former works we considered two different approaches when simulating the squeeze flow induced by the laminate compression, the first based on a penalty formulation and the second one based on the use of Lagrange multipliers. In the present work we propose an alternative approach that consists in modeling each ply involved in the laminate as a transversally isotropic fluid – TIF - that becomes 2D as soon as incompressibility constraint and plane stress assumption are taken into account. Thus, composites laminates can be analyzed as a stacking of 2D TIF models that could eventually interact by using adequate friction laws at the inter-ply interfaces.Peer ReviewedPostprint (published version

Computational vademecums for a fast and reliable simulation of RTM processes

Proper Generalized Decomposition methods allow to obtain an efficient solution for multi-parametric problems without the need for simulating numerous problems to obtain a response surface. Instead, PGD obtains a priori a reduced solution in the form of a finite sum of separable functions, easy to store in memory so as to be evaluated under real-time constraints. The present work proposes to use this tool to optimize the main RTM process parameters, the injection flow rate and the injection/mould temperature, in order to ensure the complete filling of the mould and reasonable fabrication costs (fabrication time, mould heating). To do so, the two process parameters should be introduced in the model as new coordinates, and the Proper Generalized Decomposition method used to solve the multiparametric model then obtained. By using this procedure, we could build computational vademecums, having the two parameters of interest as coordinates, allowing the fabricant to define the best compromise between injection time and process cost (mould heating) while ensuring the complete filling of the mould. In this work, after revisiting some applications of PGD in RTM processes, the separability of parametric RTM solutions will be evaluated.Peer ReviewedPostprint (published version

kPCA-Based Parametric Solutions Within the PGD Framework

Parametric solutions make possible fast and reliable real-time simulations which, in turn allow real time optimization, simulation-based control and uncertainty propagation. This opens unprecedented possibilities for robust and efficient design and real-time decision making. The construction of such parametric solutions was addressed in our former works in the context of models whose parameters were easily identified and known in advance. In this work we address more complex scenarios in which the parameters do not appear explicitly in the model—complex microstructures, for instance. In these circumstances the parametric model solution requires combining a technique to find the relevant model parameters and a solution procedure able to cope with high-dimensional models, avoiding the well-known curse of dimensionality. In this work, kPCA (kernel Principal Component Analysis) is used for extracting the hidden model parameters, whereas the PGD (Proper Generalized Decomposition) is used for calculating the resulting parametric solution

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

On the modelling of the aggregates' elasticity in a concentrated suspension of CNTs

International audienceSuspensions involving nanoparticules - in particular nano bers and nanotubes - are in much use in the development of functional materials. Thus in order to optimize the usage of these materials and their fabrication, it is essential to have a thorough knowledge of the microstructure and its evolution. In this work, the objective is to develop a two-scale kinetic theory description of concentrated suspensions including the modelling of nanotube aggregates and their evolution

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