Some Incipient Techniques For Improving Efficiency in Computational Mechanics

This contribution presents a review of different techniques available for alleviating simulation cost in computational mechanics. The first one is based on a separated representation of the unknown fields; the second one uses a model reduction based on the Karhunen-Loève decomposition within an adaptive scheme, and the last one is a mixed technique specially adapted for reducing models involving local singularities. These techniques can be applied in a large variety of models

On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition

This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encoun- tered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations

One and two-fiber orientation kinetic theories of fiber suspensions

http://dx.doi.org/10.1016/j.jnnfm.2012.10.009The morphology influencing rheological properties of suspensions of rigid spheres constitutes the flow induced collective ordering of the spheres characterized by two or more sphere distribution functions. When the rigid spheres are replaced by rigid fibers, the collective order in the position of the spheres is replaced by the flow induced orientation of the fibers that suffices to be characterized by one-fiber orientation distribution function. A flow induced collective ordering of fibers (both in position and orientation), that can only be characterized by two or more fiber distribution functions, can still however constitute an important part of the morphology. We show that two types of interaction among fibers, one being the Onsager-type topological interaction entering the free energy and the other the hydrodynamics interaction entering the dissipative part of the time evolution, give indeed rise to a collective order in the orientation influencing the rheology of fiber suspensions

Simulating microstructure evolution during passive mixing

The prediction of microstructure evolution during passive mixing is of major interest in order to qualify and quantify mixing devices as well as to predict the final morphology of the resulting blend. Direct numerical simulation fails because of the different characteristic lengths of the microstructure and the process itself. Micro-macro approaches could be a valuable alternative but the computational cost remains tremendous. For this reason many authors proposed the introduction of some microstructural variables able to qualify and quantify the mixing process at a mesoscale level. Some proposals considered only the effects induced by the flow kinematics, other introduced also the effects of shape relaxation due to the surface tension and coalescence. The most advanced integrate also the break-up process. However, the derivation of the evolution equations governing the evolution of such microstructural variables needs the introduction of some closure relations whose impact on the computed solution should be evaluated before applying it for simulating complex mixing flows. In this work we consider the Lee and Park’s model that considers the flow kinematics, the surface tension, the coalescence and the break-up mechanisms in the evolution of the area tensor. The accuracy of both a quadratic closure and an orthotropic relations will be analyzed in the first part of this work, and then the resulting closed model by using a quadratic closure will be used for simulating complex mixing flows

Proper general decomposition (PGD) for the resolution of Navier–Stokes equations

In this work, the PGD method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions. In the first stage, the equations of Stokes and Burgers will be solved. Then, we will solve the Navier–Stokes problem in the case of the lid-driven cavity for different Reynolds numbers (Re = 100, 1000 and 10,000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.Région Poitou-Charente

Direct numerical simulation of complex viscoelastic flows via fast lattice-Boltzmann solution of the Fokker–Planck equation

Micro–macro simulations of polymeric solutions rely on the coupling between macroscopic conservation equations for the fluid flow and stochastic differential equations for kinetic viscoelastic models at the microscopic scale. In the present work we introduce a novel micro–macro numerical approach, where the macroscopic equations are solved by a finite-volume method and the microscopic equation by a lattice-Boltzmann one. The kinetic model is given by molecular analogy with a finitely extensible non-linear elastic (FENE) dumbbell and is deterministically solved through an equivalent Fokker–Planck equation. The key features of the proposed approach are: (i) a proper scaling and coupling between the micro lattice-Boltzmann solution and the macro finite-volume one; (ii) a fast microscopic solver thanks to an implementation for Graphic Processing Unit (GPU) and the local adaptivity of the lattice-Boltzmann mesh; (iii) an operator-splitting algorithm for the convection of the macroscopic viscoelastic stresses instead of the whole probability density of the dumbbell configuration. This latter feature allows the application of the proposed method to non-homogeneous flow conditions with low memory-storage requirements. The model optimization is achieved through an extensive analysis of the lattice-Boltzmann solution, which finally provides control on the numerical error and on the computational time. The resulting micro–macro model is validated against the benchmark problem of a viscoelastic flow past a confined cylinder and the results obtained confirm the validity of the approach

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

Deterministic solution of the kinetic theory model of colloidal suspensions of structureless particles

A direct modeling of colloidal suspensions consists of calculating trajectories of all suspended objects. Due to the large time computing and the large cost involved in such calculations, we consider in this paper another route. Colloidal suspensions are described on a mesoscopic level by a distribution function whose time evolution is governed by a Fokker–Plancklike equation. The difficulty encountered on this route is the high dimensionality of the space in which the distribution function is defined. A novel strategy is used to solve numerically the Fokker–Planck equation circumventing the curse of dimensionality issue. Rheological and morphological predictions of the model that includes both direct and hydrodynamic interactions are presented in different flows

Dynamique entrepreneuriale force du succes des mecanismes d'aide des petites et moyennes entreprises

L’objet de ce papier est d’interroger les relations pouvant exister entre la dynamique entrepreneuriale et le succès des petites et moyennes entreprises.Les résultats de cette étude montrent que l’expérience, la formation et la motivation sont positives et significatives suggérant ainsi leurs effets substantiels sur le succès des PME. Par contre, l’âge du dirigeant et niveau d’instruction paraissent négatifs et non significatifs.Mot Clés: PME, dynamique entrepreneuriale, petites entreprises, entrepreneur, capital humai

Simulating microstructure evolution during passive mixing

The prediction of microstructure evolution during passive mixing is of major interest in order to qualify and quantify mixing devices as well as to predict the final morphology of the resulting blend. Direct numerical simulation fails because of the different characteristic lengths of the microstructure and the process itself. Micro-macro approaches could be a valuable alternative but the computational cost remains tremendous. For this reason many authors proposed the introduction of some microstructural variables able to qualify and quantify the mixing process at a mesoscale level. Some proposals considered only the effects induced by the flow kinematics, other introduced also the effects of shape relaxation due to the surface tension and coalescence. The most advanced integrate also the break-up process. However, the derivation of the evolution equations governing the evolution of such microstructural variables needs the introduction of some closure relations whose impact on the computed solution should be evaluated before applying it for simulating complex mixing flows. In this work we consider the Lee and Park’s model that considers the flow kinematics, the surface tension, the coalescence and the break-up mechanisms in the evolution of the area tensor. The accuracy of both a quadratic closure and an orthotropic relations will be analyzed in the first part of this work, and then the resulting closed model by using a quadratic closure will be used for simulating complex mixing flows