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

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

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

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

MODEL REDUCTION METHODS IN OPTION PRICING

In this work we introduce the Proper Orthogonal Decomposition (POD)approach to the valuation of contingent claims for one–dimensional price models.First, we present the POD in the context of an abstract Hilbert space and we givean application for the numerical pricing of Double Barrier Options. In a finitedimension setting, we show the model reduction method for Finite Differenceschemes of implicit type. In particular, we construct the reduced version of theCrank–Nicolson scheme and some numerical examples are given.Model Reduction, Proper Orthogonal Decomposition, Finite Difference Schemes, Crank–Nicolson Scheme.

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

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

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

Advanced modeling of the out-of-autoclave thermoplastics prepreg consolidation

Nowadays, composite materials are replacing metallic ones thanks to their excellent mechanical performances and reduced weight. However, many difficulties are encountered during composite forming processes. In fact, autoclave curing process is too expensive and limits the part size to the autoclave dimensions. Out-Of-Autoclave processes reduce substantially the cost of forming processes. However, the absence of autoclave pressure in out-of-autoclave manufacturing processes leads nowadays to high porosity and poor consolidation at the interface between the tows [1]. Moreover, the effect of the process parameters on the consolidation is still unknown and thus controlling the final parts quality is not obvious. Despite the high potential offered by the Out-of- Autoclave processes, only few researches has been made in the last few years, in order to quantify the consolidation of the tows while using such processes [2]. In fact, only few models addressing void dynamics in thermoplastic composites has been carried out [3, 4]. In this work, we are using a novel coupled approach involving modeling and simulation in order to quantify the consolidation in Out-of-Autoclave processes. Advanced model reduction techniques (POD, PGD ...) are employed in order to predict thermal fields during manufacturing processes and coupled to the subsequent squeeze flow

Empowering engineering with data, machine learning and artificial intelligence: a short introductive review

Simulation-based engineering has been a major protagonist of the technology of the last century. However, models based on well established physics fail sometimes to describe the observed reality. They often exhibit noticeable differences between physics-based model predictions and measurements. This difference is due to several reasons: practical (uncertainty and variability of the parameters involved in the models) and epistemic (the models themselves are in many cases a crude approximation of a rich reality). On the other side, approaching the reality from experimental data represents a valuable approach because of its generality. However, this approach embraces many difficulties: model and experimental variability; the need of a large number of measurements to accurately represent rich solutions (extremely nonlinear or fluctuating), the associate cost and technical difficulties to perform them; and finally, the difficulty to explain and certify, both constituting key aspects in most engineering applications. This work overviews some of the most remarkable progress in the field in recent years