Adrien Leygue

The Proper Generalized Decomposition for Advanced Numerical Simulations

Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom.Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab software used to generate these examples.

Real Time Simulation of Biological Soft Tissues: A PGD Approach

We introduce here a novel approach for the numerical simulation of nonlinear, hyperelastic soft tissues at kilohertz feedback rates necessary for haptic rendering. This approach is based upon the use of proper generalized decomposition techniques, a generalization of PODs. Proper generalized decomposition techniques can be considered as a means of a priori model order reduction and provides a physics-based meta-model without the need for prior computer experiments. The suggested strategy is thus composed of an offline phase, in which a general meta-model is computed, and an online evaluation phase in which the results are obtained at real time. Results are provided that show the potential of the proposed technique, together with some benchmark test that shows the accuracy of the method.

Original article: Proper Generalized Decomposition based dynamic data driven inverse identification

Dynamic Data-Driven Application Systems-DDDAS-appear as a new paradigm in the field of applied sciences and engineering, and in particular in Simulation-based Engineering Sciences. By DDDAS we mean a set of techniques that allow the linkage of simulation tools with measurement devices for real-time control of systems and processes. One essential feature of DDDAS is the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically control the measurement process. DDDAS need accurate and fast simulation tools using if possible off-line computations to limit as much as possible the on-line computations. With this aim, efficient solvers can be constructed by introducing all the sources of variability as extra-coordinates in order to solve the model off-line only once. This way, its most general solution is obtained and therefore it can be then considered in on-line purposes. So to speak, we introduce a physics-based meta-modeling technique without the need for prior computer experiments. However, such models, that must be solved off-line, are defined in highly multidimensional spaces suffering the so-called curse of dimensionality. We proposed recently a technique, the Proper Generalized Decomposition-PGD-able to circumvent the redoubtable curse of dimensionality. The marriage of DDDAS concepts and tools and PGD off-line computations could open unimaginable possibilities in the field of dynamic data-driven application systems. In this work we explore some possibilities in the context of on-line parameter estimation.

The rheology of multiwalled carbon nanotube and carbon black suspensions

This paper is concerned with a direct experimental and modeling comparison of the rheology of carbon black (CB) and multiwalled carbon nanotube (CNT) suspensions within a Newtonian epoxy matrix. Experimental observations of the effect of shear on CB and CNT microstructure are reported for a range of CB and CNT suspension concentrations. Steady shear, time dependent shear behavior, and oscillatory linear viscoelasticity (LVE) of the suspensions are reported and remarkably strong similarities were observed between the CB and CNT suspension rheology, for example, 4u2009wt. % CB and 0.4u2009wt. % CNT suspensions. Optical observations showed that both the CB and CNT microstructures were shear rate sensitive and a structure-dependent hybrid Maxwell-Voigt phenomenological model with a yield stress was developed that gave a reasonable fit to the rheological data. The structure model parameters for both systems were found to be of a similar order of magnitude, although the onset of rheology development for the two systems...

Separated representations of 3D elastic solutions in shell geometries

BackgroundThe solution of 3D models in degenerated geometries in which some characteristic dimensions are much lower than the other ones -e.g. beams, plates, shells,...- is a tricky issue when using standard mesh-based discretization techniques.MethodsSeparated representations allow decoupling the meshes used for approximating the solution along each coordinate. Thus, in plate or shell geometries 3D solutions can be obtained from a sequence of 2D and 1D problems allowing fine and accurate representation of the solution evolution along the thickness coordinate while keeping the computational complexity characteristic of 2D simulations. In a former work this technique was considered for addressing the 3D solution of thermoelastic problems defined in plate geometries. In this work, the technique is extended for addressing the solution of 3D elastic problems defined in shell geometries.ResultsThe capabilities of the proposed approach are illustrated by considering some numerical examples involving different degrees of complexity, from simple shells to composite laminates involving stiffeners.ConclusionsThe analyzed examples prove the potentiality and efficiency of the proposed strategy, where the computational complexity was found evolving as reported in our former works, proving that 3D solutions can be computed at a 2D cost.

In situ synchrotron wide-angle X-ray diffraction investigation of fatigue cracks in natural rubber.

Natural rubber exhibits remarkable mechanical fatigue properties usually attributed to strain-induced crystallization. To investigate this phenomenon, an original experimental set-up that couples synchrotron radiation with a homemade fatigue machine has been developed. Diffraction-pattern recording is synchronized with cyclic loading in order to obtain spatial distributions of crystallinity in the sample at prescribed times of the mechanical cycles. Then, real-time measurement of crystallinity is permitted during uninterrupted fatigue experiments. First results demonstrate the relevance of the method: the set-up is successfully used to measure the crystallinity distribution around a fatigue crack tip in a carbon black filled natural rubber for different loading conditions.

Towards a framework for non‐linear thermal models in shell domains

Purpose – The purpose of this paper is to solve non‐linear parametric thermal models defined in degenerated geometries, such as plate and shell geometries.Design/methodology/approach – The work presented in this paper is based in a combination of the proper generalized decomposition (PGD) that proceeds to a separated representation of the involved fields and advanced non‐linear solvers. A particular emphasis is put on the asymptotic numerical method.Findings – The authors demonstrate that this approach is valid for computing the solution of challenging thermal models and parametric models.Originality/value – This is the first time that PGD is combined with advanced non‐linear solvers in the context of non‐linear transient parametric thermal models.

On the solution of the multidimensional Langer's equation using the proper generalized decomposition method for modeling phase transitions

The dynamics of phase transition in a binary mixture occurring during a quench is studied taking into account composition fluctuations by solving Langers equation in a domain composed of a certain number of micro-domains. The resulting Langers equation governing the evolution of the distribution function becomes multidimensional. Circumventing the curse of dimensionality the proper generalized decomposition is applied. The influence of the interaction parameter in the vicinity of the critical point is analyzed. First we address the case of a system composed of a single micro-domain in which phase transition occurs by a simple symmetry change. Next, we consider a system composed of two micro-domains in which phase transition occurs by phase separation, with special emphasis on the effect of the Landau free energy non-local term. Finally, some systems consisting of many micro-domains are considered.

Real-time in silico experiments on gene regulatory networks and surgery simulation on handheld devices

AbstractSimulation of all phenomena taking place in a surgical procedure is a formidable task that involves, when possible, the use of supercomputing facilities over long time periods. However, decision taking in the operating room needs for fast methods that provide an accurate response in real time. To this end, Model Order Reduction (MOR) techniques have emerged recently in the field of Computational Surgery to help alleviate this burden. In this paper, we review the basics of classical MOR and explain how a technique recently developed by the authors and coined as Proper Generalized Decomposition could make real-time feedback available with the use of simple devices like smartphones or tablets. Examples are given on the performance of the technique for problems at different scales of the surgical procedure, form gene regulatory networks to macroscopic soft tissue deformation and cutting.

Identification of non uniform thermal contact resistance in automated tape placement process

In this work focussing on the thermal modeling of the automated tape placement process applied to thermoplastic material, we study the thermal properties of the ply interfaces during in-situ consolidation. Through the comparison of experimental measurements and numerical simulations, we show that it is necessary to consider the existence of an interply thermal contact resistance (TCR). Furthermore, we show that in order to correctly predict the measured temperatures, the value of the thermal resistance has to evolve along the process although a very simple evolution law is sufficient.